TABLES
COMPLEX HYPERBOLIC AND
CIRCULAR FUNCTIONS
KENNELLY
UNIVERSITY OF CALIFORNIA
AT LOS ANGELES
6<TU£_puPuAA*--''
.^. T
ii-
TABLES
OF
COMPLEX HYPERBOLIC AND
CIRCULAR FUNCTIONS
BY
A. E. KENNELLY, Sc.D., A.M.
PROFESSOR OF ELECTRICAL ENGINEERING IN HARVARD UNIVERSITY
SECOND EDITION
REVISED AND ENLARGED
CAMBRIDGE
HARVARD UNIVERSITY PRESS
LONDON: HUMPHREY MILFORD
OXFORD UNIVERSITY PRESS
1921
First edition, March, 1914
Second edition, February, 1021
Library
PREFACE
THE tables in this book present hyperbolic and circular functions of a complex
variable, both in polar and rectangular coordinates. Such complex functions
have not hitherto been published, except over a very restricted range. They have
important applications in electrical engineering. For instance, it is possible with
their help to find in a few minutes the potential, current and power, at any point
of an alternating-current line-conductor of known constants and terminal condi-
tions; whereas the same problem, to a like degree of precision, without aid from
these functions, and by older methods, would probably occupy hours of labor and
cover several sheets of computing-paper.
Although the principal application of these functions at the present time is in
dealing with alternating-current lines, especially those of either great length or
high frequency; yet it seems likely that other uses will develop for them.
The author desires to acknowledge his indebtedness, for suggestions and help,
to a number of workers, both in mathematical and practical fields; and particu-
larly to Messrs. C. L. Bouton, W. Duddell, E. V. Huntington, F. B. Jewett, John
Perry, H. J. Ryan, and E. B. Wilson.
A. E. K.
HARVARD UNIVERSITY
January, 1914.
I
PREFACE TO THE SECOND EDITION
i IN preparing the second edition of this book, six new tables have been computed.
J These are actually extensions of the tables I to VI already incorporated. It has
been considered advisable to add the new material in new tables at the end of the
volume rather than to recast the original tables in such a manner as to include the
new matter. The new matter has been found necessary in certain departments of
electrical engineering to which complex hyperbolic functions may be advanta-
geously applied.
A. B. K.
HARVARD UNIVERSITY
June, 1920.
442282
TABLE OF CONTENTS
PAGE
I. HYPERBOLIC SINES, sinh (p /d) = r /_y . . . 5 45° to 90° ...... 2
H. HYPERBOLIC COSINES, cosh (P /5) = r fa . " " " " ...... 8
HI. HYPERBOLIC TANGENTS, tanh (P /S) = r fy_ " " " " ...... 14
IV. CORRECTING FACTOR. ^^ . " " " " 20
a
V. CORRECTING FACTOR. ^-? ....... " " " " 26
(7
VL. FUNCTIONS OF SEMI-IMAGINARIES. / (p 745°) ............. 32
VII. HYPERBOLIC SINES, sinh (x + iq) = u + w ............. 42
VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv ............ 58
IX. HYPERBOLIC TANGENTS, tanh (x + iq) = « + iv ........... 74
X. HYPERBOLIC SINES, sinh (x + iq) = r /_y_ .............. 90
XL HYPERBOLIC COSINES, cosh (x + iq) = r /_y_ ............. 106
XII. HYPERBOLIC TANGENTS, tanh (* + »</) = r /_y_ ............ 122
XIII. FUNCTIONS OF 4 + iq. f(4+iq)=u+iv ..... ..... . 138
f(4+iq) =r/7 ............ 139
XIV. SEMI-EXPONENTIALS. — and logio (~
XV. REAL HYPERBOLIC FUNCTIONS. / (x + i o) = u + i o ......... 144
XVI. SUBDIVISIONS OF A DEGREE .................... 150
EXPLANATORY TEXT ........................ 151
XVH. HYPERBOLIC SINES. sinh(p/6) = r /r 5 o° to 45° ....... 214
XVIII. HYPERBOLIC COSINES. cosh(p/5) = r/r "" " " ....... 216
XIX. HYPERBOLIC TANGENTS. tanh(p/«) = r [y_ "" " " ....... 218
XX. CORRECTING FACTOR. s- " " " " ....... 220
XXI. CORRECTING FACTOR. " " " " ....... 222
a
XXII. FUNCTIONS OF SEMI-IMAGINARIES. / (P /45°) . . . . ....... 224
XXIII. HYPERBOLIC FUNCTION FORMULAS ................. 226
TABLES OF COMPLEX HYPERBOLIC
AND CIRCULAR FUNCTIONS
TABLE I. HYPERBOLIC SINES, sinh (p /S) = r
O.I
o-3
0.4
45
46
47
48
49
O.IOOOO
0.099993
0.099987
0.099981
0.099975
45.096
46.095
47-095
48.094
49.094
O.2OOOO
0.19995
O.I9990
0.19986
0.19982
45-383
46.382
47-381
48.380
49-378
0.30001
0.29985
0.29969
0.29954
0.29939
45.860
46.858
47-856
48.854
49.852
0.40005
0.39968
0-39931
0-39893
0.39856
46-532
47-529
48.526
49.520
50.5*3
0.50016
0.49944
0.49872
0.49799
0.49727
47-391
48.388
49-385
50-378
51-368
So
Si
52
53
54
0.099970
0.099965
0.099960
0.099955
0.099950
50.094
51-093
52.092
53-091
54-090
0.19976
0.19972
0.19968
0.19963
0.19959
50-376
Si-374
52.371
0.29923
0.29907
0.29892
0.29877
0.29862
50.848
51.842
52-834
53-826
54.818
0.39820
0.39784
0.39748
0.39712
0.39676
51-506
52-497
53.486
54-472
55-458
0.49656
0.49585
0.49514
0-49444
0-49374
52-357
53-342
55-304
56.281
55
56
57
58
59
0.099944
0.099939
0.099933
0.099928
0.099922
55-089
56.088
57-086
58.085
59-083
0-19955
O.I995I
0.19946
O.I994I
0.19937
55-357
56.352
57-347
58-342
59-336
0.29847
0.29833
0.29819
0.29804
0.29790
55-809
56.799
58*773
59.760
0.39641
0.39607
0-39572
0.39538
0.39505
56.440
57-421
58.400
59-378
60.354
0.49305
0.49238
0.49172
0.49106
0.49041
57-254
58.226
59-195
60.160
61.123
6o
61
62
63
64
0.099917
0.099912
0.099907
0.099902
0.099897
60.082
61.081
62.079
63-077
64-075
0.19934
0.19929
0.19925
O.I992I
0.19918
60.331
61.324
62.317
63.309
64.301
0.29777
0.29764
0.29751
0.29738
0.29725
60.746
61.731
62.715
63.698
64.680
0-39473
0.39441
0.39409
0-39379
0-39349
61.330
62.302
63-273
64.243
65.212
0.48978
0.48916
0.48855
0.48795
0.48738
62.085
63.042
63.998
64.950
65.900
65
66
67
68
69
0.099893
0.099889
0.099885
0.099881
0.099877
65-073
66.071
67.069
68.066
69.064
0.
o.
0.
0.
0.
19914
19911
19908
19904
19901
65-293
66.284
67-275
68.265
69-255
0.29712
0.29700
0.29689
0.29678
0.29667
65.661
66.641
67.621
68.599
69-577
0.39320
o.39293
0.39266
0.39240
0.39214
66.179
67.144
68.108
69.070
70.030
0.48681
0.48627
0.48575
0.48523
0.48473
66.848
67.794
68.738
69.679
70.617
70
72
73
74
0.099873
0.099869
0.099865
0.099861
0.099858
70.062
71-059
72.057
73-054
74-051
0.19898
0.19895
0.19892
0.19889
0.19887
70.245
7I-235
72.225
73-214
74-203
0.29657
0.29647
0.29637
0.29628
0.29620
70-555
71-532
72.508
73-483
0.39190
0.39167
0.39*45
0.39123
0.39104
70.990
71.948
72.905
73-861
74.817
0.48426
0.48380
0.48338
0.48296
0.48257
71-554
72.489
73.422
74-354
7S-284
75
76
77
78
79
0.099855
0.099852
0.099850
0.099847
0.099845
75-048
76.045
77-042
78.039
79.036
0.19885
0.19883
0.19881
0.19878
0.19876
75-I92
76.180
77-168
78-156
79.144
0.29612
0.29604
0.29596
0.29590
0.28584
75-432
76.406
77-379
78.351
79-322
0.39084
0.39066
0.39050
0.39034
0.39018
75-771
76.724
77.676
78.628
79-578
0.48220
0.48184
0.48152
0.48121
0.48093
76.212
77.138
78.062
78.986
79.909
80
81
82
83
84
0.099843
0.099841
0.099839
0.099838
0.099837
80.033
81.030
82.026
83-023
84.020
0.19875
0.19873
0.19872
0.19871
0.19870
80.131
81.118
82.106
83-093
84.080
0.29578
0.29573
0.29569
0.29566
0.29563
80.294
81.266
82.238
83.209
84.180
0.39004
0-38993
0.38983
0.38973
0.38965
80.528
8i.477
82.425
83-373
84.321
0.48067
0.48044
0.48023
0.48004
0.47987
80.830
81.750
82.669
83-587
84-505
85
86
87
88
89
0.099836
0.099835
0.099834
0.099833
0.099832
85.017
86.014
87.011
88.008
89.004
0.19869
0.19868
0.19868
0.19867
0.19867
85-067
86.054
87.041
88.028
89.014
0.29561
0.29559
0.29557
0.29555
0-29553
85.150
86.120
87.090
88.060
89.030
0.38958
0.38952
0.38948
0.38946
0.38944
85.268
86.215
87.162
88.108
89.054
0.47972
0.47960
0.47952
0.47947
0-47945
85-422
86.338
87-254
88.170
89.085
90
0.099831
90.000
0.19867
90.000
0.29552
90.000
0.38942
90.000
0-47943
90.000
Note.
sinh (o 18) = o /8.
Examples.
sinh (0.3 765°) = o
.29712 7j
>5°.66i =
0.29712 /
65°.39'.4
o".
sinh
[2]
TABLE I. HYPERBOLIC SINES, sinh
= r /y. CONTINUED
0.6
0.7
0.8
0.9
i.o
45
46
47
48
49
0.60042 48.440
0.59918 49-437
0-59793 50.434
0.59667 51.426
0.59542 52.414
0.70094 49.676
0.69894 50.679
0.69695 51.676
0.69497 52.666
0.69299 53.652
0.80184 S^oS
0.79885 52-112
0.79587 53-109
0.79291 54-099
0.78996 55-o82
0.90327 52.728
0.89904 53.735
0.89482 54.734
0.89060 55.725
0.88640 56.707
1-00553 54-531
0-99975 55-540
0-99394 56.550
0.98816 57-543
0.98242 58.525
50
51
52
53
54
0.59418 53.398
0.59295 54-379
0.59174 55-355
0.59053 56-326
0.58932 57.293
0.69102 54-632
0.68907 55.606
0.68713 56.574
0.68521 57.537
0.68331 58.493
0.78703 56-058
0.78412 57-026
0.78124 57.987
0.77838 58.940
0-77555 59-886
0.88224 57.679
0.87810 58.642
0.87400 59.595
0.86993 60.538
0.86590 61.472
0.97672 59-495
0.97105 60.453
0.96543 61.399
0.95986 62.333
0-95435 63.255
55
56
57
58
59
0.58814 58.256
0.58698 59.215
0.58583 60.171
0.58469 61.122
0.58357 62.069
0.68144 59-445
0.67959 60.391
0.67776 61.331
0.67595 62.265
0.67419 63.193
0.77275 60.824
0.76999 61.755
0.76727 62.678
0.76459 63.593
0.76195 64.502
0.86192 62.396
0.85800 63.312
0.85414 64.218
0.85034 65.114
0.84660. 66.000
0.94890 64.166
0-94353 65.065
0.93825 65.952
0.93305 66.827
0.92795 67.691
6o
61
62
63
64
0.58249 63.013
0.58142 63.953
0.58037 64.889
0.57934 65.821
0.57834 66.749
0.67247 64.117
0.67078 65.036
0.66912 65.951
0.66749 66.859
0.66591 67.762
0.75938 65.405
0.75686 66.300
0-75439 67.189
0.75197 68.070
0.74962 68.944
0.84295 66.878
0.83937 67.747
0.83587 68.607
0.83244 69.458
0.82909 70.300
0.92295 68.544
0.91805 69.385
o-9I325 70-215
0.90856 71.033
0.90400 71.841
65
66
67
68
69
0-57737 67.674
0.57643 68.596
0-57553 69.515
0.57465 70.430
0-57379 7I-342
0.66437 68.661
0.66288 69.554
0.66145 70.444
0.66005 71.329
0.65870 72.209
0-74733 69.812
0.74512 70.674
0.74298 7i.529
0.74091 72-379
0.73891 73-223
0.82585 71.136
0.82270 71.962
0.81967 72.780
0.81672 73.590
0.81387 74.392
0.89957 72-637
0.89527 73-424
0.89111 74.201
0.88708 74.968
0.88320 75.723
70
72
73
74
0.57297 72.251
0.57219 73.157
0.57H5 74-061
0.57074 74-962
0.57006 75.860
0.65740 73-o85
0.65616 73.957
0.65498 74.825
0-65385 75-689
0.65278 76.550
0.73698 74-061
0.73513 74.894
0-73337 75-722
0.73169 76.544
0.73009 77-361
0.81114 75.187
0.80853 75.975
0.80602 76.756
0.80363 77.530
0.80137 78.298
0.87947 76-469
0.87589 77-207
0.87247 77.936
0.86921 78.656
0.86612 79.368
75
76
77
78
79
0.56941 76-756
0.56881 77.649
0.56824 78.540
0.56772 79.429
0.56724 80.317
0.65176 77.408
0.65081 78.263
0.64992 79.114
0.64909 79.962
0.64832 80.808
0.72858 78.174
0.72716 78.982
0.72583 79-787
0.72459 80.588
0.72345 81.385
0.79924 79.059
0.79723 79-8i5
0-79535 80.566
0-79359 81.312
0.79197 82.053
0.86320 80.072
0.86045 80.769
0.85788 81.458
0.85549 82.141
0.85328 82.818
80
81
82
83
84
0.56679 81.203
0.56638 82.087
0.56602 82.970
0.56570 83.852
0.56542 84.732
0.64761 81.652
0.64697 82.493
0.64640 83.332
0.64589 84.169
0.64545 85.005
0.72241 82.179
0.72146 82.969
0.72061 83.757
0.71985 84.543
0.71919 85.327
0.79048 82.789
0.78913 83.522
0.78792 84.251
0.78685 84.976
0.78592 85.699
0.85125 83.489
0.84940 84.156
0.84774 84-817
0.84628 85.474
0.84501 86.128
85
86
8?
88
89
0.56518 85.612
0.56498 86.490
0.56483 87.368
0.56473 88.246
0.56466 89.123
0.64507 85.839
0.64476 86.673
0.64452 87.506
0.64435 88.338
0.64425 89.169
0.71863 86.109
0.71817 86.889
0.71781 87.668
0.71755 88.446
0.71740 89.223
0.78513 86.420
0.78448 87.138
0.78397 87.855
0.78361 88.571
0.78339 89.286
0.84393 86.779
0.74305 87.426
0.84236 88.071
0.84186 88.715
0.84156 89.358
9P
0.56464 90.000
0.64422 90.000
0.71736 90.000
0.78333 90.000
0.84147 OO.OPO
Examples, sinh (i.o /go") = 0.84147 /go0.
sinh"1 (0.87947 /76°.469) = 1.0/70°.
t3]
TABLE I. HYPERBOLIC SINES, sinh (p IS) = r {y. CONTINUED
1.4
i-S
45
1.1089
56.519
1.2138
58.692
1.3205
61.034
1.4297
63-568
1-5418
66.262
46
I.IOI2
57-543
1.2037
59.726
1.3078
62.092
1.4138
64.639
1.5222
67-355
47
1-0935
58.555
I-I937
60.748
1.2951
63.128
1-3979
65.689
1.5027
68.426
48
1.0858
59-553
1.1838
6i.753
1.2824
64.142
1.3822
66.717
1.4834
69.474
49
1.0782
60.536
I-I739
62.741
1.2699
65-I37
1.3665
67.723
1.4642
70.496
50
1.0706
61.506
1.1641
63.712
1.2574
66.113
1-3509
68.707
I-445I
71.492
5i
1.0630
62.461
I-I543
64.666
1.2450
67.068
1-3355
69.668
1.4262
72.462
52
1-0556
63.401
1.1446
65.603
1.2327
68.004
1.3202
70.605
1.4075
73-405
53
1.0482
64.327
1-1350
66.523
1.2206
68.919
1-3051
7i.5i9
1.3889
74-321
54
1.0409
65-239
1.1256
67-425
1.2086
69.814
1.2902
72.409
1.3706
75.210
55
1.0336
66.136
1.1162
68.310
1.1967
70.688
1.2754
73-275
I.3525
76.072
56
1.0265
67.019
1.1070
69.178
1.1850
7I-542
1.2609
74.117
1-3347
76.907
57
1.0195
67.888
1.0979
70.028
I-I735
72.376
1.2466
74-936
1.3172
77.713
58
1.0126
68.742
1.0890
70.860
1.1622
73.188
1.2325
75-730
1.3000
78.491
59
1.0058
69.581
1.0802
7I-675
1.1511
73.980
1.2187
76.500
1.2831
79.240
6o
0.99920
70.406
1.0716
72.474
1.1403
74-752
1.2052
77.246
1.2665
79.964
61
0.99269
71.218
1.0632
73-255
1.1296
75-502
1.1919
77.967
1-2503
80.657
62
0.98633
72.016
1.0550
74-oi9
1.1192
76.232
1.1790
78.663
1-2345
81.321
63
0.98013
72.800
1.0470
74.767
1.1091
76.942
1.1664
79-335
1.2191
81.956
64
0.97409
73-570
1.0392
75-497
1.0992
77.632
1.1541
79-983
1.2040
82.562
65
0.96821
74.3?7
1.0316
76.211
1.0895
78.301
1.1421
80.607
1.1894
83.140
66
0.96251
75-07I
1.0242
76.909
1.0802
78.950
1-1305
81.207
1.1752
83.690
67
0.95698
75-8oi
1.0171
77-590
1.0712
79.580
1.1193
81.783
1.1615
84.211
68
0.95165
76-519
I.OIO2
78.257
1.0625
80.191
1.1084
82.335
1.1482
84.704
69
0.94650
77.225
1.0035
78.907
1.0540
80.783
1.0979
82.865
I.I354
85.169
70
0.94156
77-9i8
0.99712
79-543
1.0459
Si-357
1.0879
83.373
1.1231
85.607
71
0.93682
78.600
0.99099
80.164
1.0382
81.912
1.0783
83.858
1.1113
86.018
72
0.93229
79.271
0.98514
80.771
1.0308
82.450
1.0691
84.322
1. 1000
86.402
73
0.92798
79-931
0-97957
81-365
1.0237
82.971
1.0603
84-765
1.0893
86.761
74
0.92388
80.580
0.97428
81.946
1.0170
83.476
1.0520
85.188
1.0791
87.096
75
0.92001
81.220
0.96927
82.514
1.0107
83.966
1.0441
85-59I
1.0694
87.406
76
0.91637
81.850
0.96456
83.069
1.0047
84.440
1.0367
85-975
1.0603
87.693
77
0.91296
82.471
0.96015
83.614
0.99916
84.900
1.0298
86.342
1.0518
87.958
78
0.90978
83.084
0.95605
84.148
0.99397
85-346
1.0233
86.692
1.0439
88.201
79
0.90685
83.689
0.95226
84.672
0.98917
85-780
1.0173
87.026
1.0366
88.425
80
0.90416
84.286
0.94878
85-187
0.98477
86.202
1.0119
87.346
1.0299
88.630
81
0.90172
84.877
0.94562
85-693
0.98077
86.614
1.0069
87.651
1.0238
88.818
82
0-89953
85.462
0.94279
86.191
0.97715
87.016
1.0024
87.944 .
1.0183
88.991
83
0.89759
86.041
0.94029
86.683
0.97400
87.408
0.99845
88.227
1.0135
89.149
84
0.89590
86.616
0.93811
87.169
0.97124
87-793
0.99502
88.499
1.0093
89.294
85
0.89447
87.186
0.93626
87.649
0.96890
88.171
0.99210
88.762
1.0057
89.429
86
0.89330
87-753
0-93474
88.124
0.96698
88.544
0.98971
89.018
1.0027
89-554
87
0.89238
88.317
0.93356
88.596
0.96548
88.912
0.98785
89.269
1.0004
89-673
88
0.89173
88.879
0.93272
89.065
0.96441
89.277
0.98652
89-515
0.99880
89-785
89
0.89134
89.440
0.93221
89-533
0.96377
89.639
0.98572
89.758
0.99782
89.893
DO
0.89121
90.000
0.93204
90.000
0.96356
90.000
0.98545
90.000
0.99749
90.000
Examples.
sinh (i.
i/45°) =
1.1089 /j
;6°.si9 = 1.1089 /s6°.3i'.o8'.
sinh"1 (1.1084 /82°.335) =
1.4 /68°.
[4]
TABLE I. HYPERBOLIC SINES, sinh (p /5) = r /% CONTINUED
1.6
1.8
1.9
45
I-6575
69.117
1.7776
72.133
1.9029
75.292
2-0343
78.590
2.1726
82.016
46
1-6338
70.241
1-7493
73-288
1.8693
76.486
1.9947
79.829
2.1266
83-304
47
1.6103
71-339
1.7210
74.418
I-8359
77.651
1-9554
81.037
2.0809
84.560
48
1.5868
72.409
1.6929
75-5I5
1.8027
78.784
1.9165
82.2IO
2.0356
85.778
49
I-563S
73-451
1.6651
76.582
1.7697
79.883
1.8779
83.346
1.9907
86.958
5°
1.5404
74-465
I-6375
77.618
1-7370
80.946
1.8396
84.444
1.9462
88.098
51
I-5I75
75-449
1.6102
78.620
1.7046
81.974
1.8016
85.503
1.9022
89.196
52
1.4949
76.402
1-5831
79-590
1.6726
82.966
1-7643
86.524
1.8588
90-253
53
I-4725
77-325
1-5563
80.527
1.6410
83.921
1-7273
87.505
1.8160
91.267
54
1.4504
78.218
1-5299
81.429
1.6098
84.839
1.6909
88.444
1-7737
92.236
55
1.4285
79.079
1-5039
82.296
I-579I
85-718
1.6550
89.342
1.7320
93.160
56
1.4070
79.910
1.4782
83.128
1.5488
86.558
1.6196
90.197
1.6911
94.039
57
1-3859
80.709
1-4530
83-924
1.5190
87.358
1.5848
9I.OO8
1.6509
94.869
58
1-3651
Si-475
1.4282
84.683
1.4898
88.116
i.S5o6
91-773
1.6114
95-650
59
1-3447
82.209
1.4039
85.406
1.4611
88.834
1-5171
92.493
1.5726
96.382
60
I-3247
82.910
1.3800
86.091
1-4330
89.510
1.4843
93.167
1-5347
97.062
61
1-3051
83.578
1-3567
86.738
1-4055
90.142
1-4523
93.792
1.4976
97-688
62
1.2860
84.212
1-3339
87-347
1.3787
90.730
1.4210
94.368
1.4614
98.261
63
1.2674
84.813
1.3117
87.917
I.3525
91.274
•3904
94.893
1.4260
98.778
64
1.2492
85-380
1.2901
88.447
1.3270
Qi-774
.3606
95.368
i-39J5
99.238
65
1.2316
85-9I3
1.2690
88.938
1.3022
92.228
•33i6
95-792
i.358o
99.639
66
1.2145
86.413
1.2486
89.390
1.2781
92.636
•3035
96.162
I.325S
99.980
t57
1.1979
86.879
1.2288
89.802
1.2548
92.997
.2762
96.478
1.2940
100.261
68
1.1819
87.311
1.2097
90.175
1.2322
93-3*2
.2499
96.741
1.2635
100.479
69
1.1664
87.710
1.1913
90.508
1.2104
93-58o
.2244
96.949
1.2340
100.634
70
1.1516
88.076
1.1736
90.8OI
1.1895
93.802
.1999
97.102
1.2056
100.725
7i
I-I373
88.410
1.1566
9 LOSS
1.1694
93.976
.1764
97.199
1.1783
100.750
72
1.1237
88.712
1.1403
91.271
1.1502
94.104
•1539
97.241
1.1521
100.709
73
1.1107
88.982
1.1248
91.448
1.1318
94.187
• 1324
97.227
1.1271
100.602
74
1.0984
89.221
I. HOI
91.588
1.1144
94-224
1.1119
97.I58
1.1033
100.428
75
1.0867
89.431
1.0961
91.692
1.0979
94-215
1.0925
97-034
1.0807
100.187
76
1-0757
89.613
1.0829
91.760
1.0823
94.162
1-0743
96.856
1.0594
99.880
77
1.0654
89.767
1.0706
91.794
1.0677
94.068
1.0571
96.625
1-0394
99.507
78
1.0558
89.894
I.059I
9!-795
1.0541
93-933
1.0411
96.343
1.0207
99.070
79
i .0469
89.997
1.0485
9I-765
1-0415 .
93-7S8
1.0262
96.OI2
1.0033
98.571
80
1.0388
90.076
1.0388
91-705
1.0299
93-545
1.0126
95-633
0.98729
98.011
81
1.0314
90.134
1.0299
91.618
1.0194
93.298
I.OOO2
95.209
0.97271
97-394
82
1.0248
90.171
I.O2I9
9I-505
1.0099
93.019
0.98899
94-744
0.95958
96.722
83
1.0189
90.191
I.OI49
91.370
1.0015
92.711
0.97907
94.240
0.94792
96.000
84
1.0138
90.194
1.0088
91.214
0.99422
92-374
0.97043
93-702
0-93775
95-233
85
1.0095
90.184
I.OO36
91.039
0.98802
92.015
0.96309
93-133
0.92911
94.426
86
i. 0060
90.161
0.99929
90.850
0.98293
91.636
0.95706
92-539
0.92201
93-584
87
1.0032
90.129
0.99596
90.649
0.97896
91.241
0.95236
91.923
0.91646
92.713
88
I.OOI2
90.090
0-99357
90.438
0.97612
90.834
0.94899
91.291
0.91248
91.821
89
I.OOOO
90.046
0.99214
90.220
0.97442
90.419
0.94697
90.648
0.91009
90.914
90
Q-99957
90.000
0.99166
90.000
0.97385
90.000
0.94630
90.000
0.90930
90.000
Examples, sinh (1.6 /6o°) = 1.3247 /82°.9io = 1.3247 /82°.S4'.36V
sinh"1 (1.1999 /97°.iQ2) = 1.9 /7Q°.
[Si
TABLE I. HYPERBOLIC SINES, sinh (p /5) = r /jy. CONTINUED
2-3
2.4
2-5
45
2.3190
85.558
2-4745
89.205
2.6404
92.946
2.8177
96.769
3.0079
100.661
46
2.2658
86.905
2.4135
90.613
2.5707
94.419
2.7386
98.312
2.9185
102.278
47
2.2131
88.213
2-3530
91.981
2.5017
95-85i
2.6603
99.813
2.8301
103.852
48
2.1608
89.482
2.2936
93-307
2-4334
97.241
2.5829
101.271
2.7429
105-383
49
2.1090
90.711
2.2337
94.592
2.3659
98.588
2.5065
102.685
2.6568
106.869
SO
2.0577
91.898
2.1750
95.834
2.2992
99.890
2.4311
104.053
2.5720
108.309
Si
2.0071
93.042
2.1171
97.031
2.2334
101.146
2.3568
105.375
2.4885
109.701
52
I-957I
94.142
2.0600
98.181
2.1685
102.354
2.2836
106.648
2.4064
111.044
S3
1.9078
95-I97
2.0037
99.284
2.1046
103.514
2.2117
107.871
2-3257
1I2-337
54
1.8592
96.205
1.9483
100.338
2.0418
104.623
2.1410
109.042
2.2465
"3-578
55
1.8114
97-i65
1.8938
101.342
1.9801
105.680
2.0714
110.160
2.1687
114.765
56
1.7644
98.076
1.8402
102.294
I-9J95
106.683
2.0032
111.223
2.0925
115.896
57
1.7182
98.935
1.7876
103.193
1. 8600
107.629
1.9364
112.228
2.0178
116.970
58
1.6729
99.742
1.7360
104.035
1.8016
108.518
1.8709
"3-174
1.9447
117-984
59
1.6284
100.494
1.6854
104.820
1-7445
109.347
i. 8068
114.059
1-8733
118.936
60
1.5849
101.191
I-6359
105.546
1.6886
110.114
1.7441
114.880
1.8035
119.823
61
I-5424
101.830
I.5875
106.210
1.6340
110.816
1.6829
115-634
1-7353
120.642
62
1.5008
102.410
1.5402
106.811
1.5807
111.451
1.6232
116.319
1.6688
121.391
63
1.4603
102.929
1.4941
107.345
1.5287
112.016
1.5649
116.931
1.6040
122.067
64
1.4208
103.386
1.4492
107.811
1.4780
112.509
1.5081
117.467
1.5409
122.665
65
1.3824
103.777
1-4055
108.207
1.4286
112.926
1.4528
117.924
1.4794
123.182
66
J-345i .
104.101
1.3630
108.529
1.3806
113.264
I-399I
118.297
1.4196
123.613
67
1.3089
104.357
1.3218
108.775
I-334I
"3-519
1.3469
118.583
1.3616
123-954
68
1.2738
104.542
1.2819
108.943
1.2890
113.688
1.2963
118.777
1-3052
124.198
69
1.2399
104.655
1-2433
109.029
1-2453
113.767
1-2473
118.874
1.2506
124.341
70
1.2072
104.694
1.2060
109.030
1.2031
"3-752
1.1999
118.868
1.1977
124.376
7i
1.1758
104.656
1.1701
108.944
1.1624
113.638
1.1541
118.754
1.1466
124.296
72
I-I457
104.541
I-I356
108.769
1.1232
113.422
1.1099
118.526
1.0972
124.092
73
1.1168
104.348
1.1026
108.501
1.0857
113.100
1.0675
118.177
1.0497
123-755
74
1.0893
104.074
1.0711
108.138
1.0498
112.667
1.0268
117.701
1.0040
123.277
75
1.0632
103.719
1.0411
107.678
i-oi55
112.118
0.98795
117.091
0.96019
122.648
76
1-0385
103.283
1.0126
107.119
0.98292
111.449
0.95090
116.341
0.91831
121.857
77
1-0153
102.766
0.98581
106.459
0.95214
110.659
0.91576
II5-443
0.87843
120.891
78
0-99353
102.168
0.96065
105.699
0.92320
109.744
0.88261
114.390
0.84063
119.740
79
0.97332
101.491
0.93722
104.839
0.89615
108.701
0.85152
113.178
0.80503
118.392
80
0.95468
100.736
0.91557
103.879
0.87109
107-531
0.82256
111.803
0.77173
116.836
81
0-93765
99907
0-89575
102.822
0.84808
106.233
0.79589
110.261
0.74083
115.063
82
0.92229
99.008
o 87784
101.671
0.82719
104.810
0.77158
108.550
0.71251
113.065
83
0.90862
98.043
0.86187
100.432
0.80853
103.267
0.74974
106.676
0.68692
110.841
84
0.89668
97-oiS
0.84789
99.111
0.79215
101.611
0-73051
104.643
0.66422
108.390
85
0.88652
95-933
0.83596
97-7I5
0.77813
99.851
0.71396
102.462
0.64461
105.723
86
0.87817
94.806
0.82613
96.254
0.76655
98.000
0.70026
100.147
0.62827
102.855
87
0.87164
93-639
0.81845
94.740
0-75747
96.071
0.68948
97.717
0.61536
99.811
88
0.86696
92.443
0.81293
93-183
0.75095
94.082
0.68171
95-197
0.60602
96.624
89
0.86414
91.226
0.80961
91-599
0.74702
92.051
0.67703
92.615
0.60036
93-338
90
0.86321
90.000
0.80850
90.000
0-74571
90.000
0.67546
90.000
0.59847
90.000
Examples, sinh (2.5 775° = 0.96019 /i22°.648 = 0.96019 /i220.38'.53g.
sinh"1 (0.60036 /93°-338) = 2.5 789°.
[6J
TABLE I.
2.6
HYPERBOLIC SINES, sinh (p IS) = r /T. CONTINUED
2.7 2.8 2.9 3.0
o
o
o
o
0
o
4S
3.2121
104.613
3-43i8
108.614
3-6685
112.653
3-9236
116.721
4.1986
120.814
46
3-HI5
106.307
3-3*91
110.386
3-5426
114.506
3-7832
118.658
4.0426
122.832
47
3.0123
107.957
3.2079
112. Il6
3.4186
116.317
3-6453
120.551
3-8895
124.808
48
2.9144
109.564
3-0985
II3.80I
3.2966
118.084
3-5098
122.400
3-7394
126.741
49
2.8179
111.126
2.9908
115.442
3.1767
119.806
3.3768
124.205
3-5923
128.630
5°
2.7229
112.641
2.8849
117.037
3.0590
121.483
3-2465
125.965
3-4483
130.474
5i
2.6295
114.109
2.7809
118.585
2.9436
123.113
3.1189
127.679
3.3076
132.272
S2
2.5378
115.528
2.6789
120.084
2.8306
124.694
2.9941
129.345
3.1701
134.024
53
2.4478
116.897
2-5789
121.532
2.720O
126.226
2.8721
130.963
3-0359
135.729
54
2-3595
118.214
2.4809
122.929
2.6118
127.708
2.7529
132.532
2.9051
137.386
55
2.2729
119.476
2.3850
124.274
2.5060
129.138
2.6366
134-050
2.7777
138.994
56
2.1882
120.684
2.2913
125.563
2.4027
130.515
2.5232
135-517
2.6536
140.553
57
2-1053
121.834
2.1996
126.797
2.3019
131-837
2.4127
136.932
2-5330
142.062
58
2.0242
122.925
2.IIOI
127.973
2.2036
133-103
2-3051
138.293
2.4157
143-519
59
1.9450
123.954
2.0228
129.088
2.1077
134-3"
2.2004
139.598
2.3018
144.924
60
1.8677
124.918
1-9377
130.140
2.0144
135-459
2.0986
140.847
2.1912
146.276
61
1-7923
125.816
I.8547
I3I.I28
1.9236
136.544
1.9997
142.037
2.0839
147-574
62
1.7187
126.644
1-7739
132.047
1-8352
137-565
1.9036
143.167
1.9798
148.817
63
1.6470
127.399
1-6952
132.895
1-7493
138.519
1.8103
144.235
1.8789
150.004
64
1-5772
128.077
1.6187
133.669
1.6658
I39-403
1.7198
145-239
1.7812
151-133
65
1-5094
128.674
1-5442
I34-364
I-5847
140.214
1.6319
146.176
1.6865
152.204
66
1-4435
129.185
1.4719
134.978
1.5060
140.948
1-5467
147.044
1.5948
I53-2I4
67
1-3794
129.606
1.4016
I35-504
I-4295
141.599
1.4641
147.838
1.5061
154.161
68
1.3171
129.930
J-3334
135-935
1-3553
142.164
1.3840
148.555
1.4202
I55-043
69
1.2568
130.151
1.2673
136.266
1-2833
142.634
1.3063
149.190
1-3370
155-858
70
1-1983
130.262
1.2031
136.489
1.2136
I43-005
1.2310
149.738
1.2565
156.602
7i
1.1417
130.254
1.1409
136.596
1.1460
143.267
1.1581
150.191
1.1785
157.271
72
1.0870
130.117
1.0807
136.576
1.0804
143.411
1.0874
150.541
1.1030
157.860
73
1.0341
129.841
1.0226
136.415
1.0169
143.424
1.0189
150.780
1.0298
158.363
74
0.98316
129.414
0.96631
I36.IOI
0.95550
143.292
0.95251
150.894
0.95893
158.772
75
0.93420
128.822
0.91207
I35-6I7
0.89606
142.996
0.88819
150.867
0.89022
159.077
76
0.88720
128.049
0.85985
134-943
0.83862
142.516
0.82587
150.680
0.82356
159-265
77
0.84227
127.079
0.80965
134.058
0.78319
141.827
0.76551
150.309
0.75887
I59-3I9
78
0.79948
125.893
0.76157
I32-935
0.72979
140.897
0.70706
149.721
0.69605
159.216
79
0.75891
124.471
0.71568
I3I-S42
0.67847
139.686
0.65054
148.876
0.63503
158.925
80
0.72068
122.793
0.67215
129.847
0.62934
138.145
0-59597
147.721
0-57573
158.404
81
0.68499
120.837
0.63109
I27.8l2
0.58252
136.217
0.54340
146.183
0.51809
I57.592
82
0.65203
118.583
0.59275
125.396
0.53822
I33-833
0.49294
144.169
0.46213
156.404
83
0.62196
116.016
0-55739
122.554
0.49674
130.908
0.44484
141-553
0.40789
154.709
84
0.59507
113.126
0.52536
119.252
0.45849
127-351
0.39946
138.168
0.35548
152-313
85
0.57164
109.910
0.49705
II5-458
0.42399
123.066
0.35738
133-795
0.30521
148-903
86
0.55191
106.382
0.47291
III.l64
0-39394
117.968
0.31946
128.168
0.25775
143-974
87
0.53621
102.571
0-45344
106.390
0.36915
112.015
0.28691
121.005
0.21433
136.701
88
0.52479
98.524
0.43912
IOI.I94
0.35055
105.244
0.26143
112. IIO
0.17731
125.843
89
0.51783
94-307
0.43033
95-682
0-33893
97.809
0.24496
101.576
0.15092
110.091
90 0.51550 90.000 0.42738 90.000 0.33499 90.000 0.23925 90.000 0.14112 90.000
Examples, sinh (3.0 /5o°) = 3.4483 /
sinh""1 (0.15092 /iio°.Q9i)
SQ°-474 = 3-4483 /i3Q°-28/.26>.
- 3-Q /89°-
[7]
TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /_y
0-3
0.4
45
I.OOOOI
0.287
I.OOOI3
1.148
1.00067
2.578
I.O02IO
4.578
1.00519
7.141
46
0.99983
0.287
0-99943
1.146
0.99910
2-577
0-99933
4-584
1.00085
7-159
47
0.99966
0.286
0.99873
1.144
0-99753
2.576
0.99655
4-584
0.99649
7.166
48
0.99948
0.285
0.99804
1.141
0-99597
2-571
0.99378
4.578
0.99214
7-165
49
0.99931
0.283
0-99735
1.136
0.99441
2.562
0.99101
4.566
0.98780
7-155
5°
0.99914
0.282
0.99666
1.131
0.99285
2-551
0.98824
4-551
0.98347
7-137
51
0.99897
0.280
0-99597
1.123
0.99131
2-536
0.98547
4-528
0.97917
7.109
52
0.99880
0.278
0.99529
1-115
0.98977
2-519
0.98274
4.500
0.97490
7-073
S3
0.99863
0.275
0.99462
1.105
0.98825
2.498
0.98003
4.467
0.97065
7.028
54
0.99846
0.272
0-99395
1.094
0.98674
2-474
0-97734
4427
0.96644
6-973
55
0.99830
0.269
0.99329
1.081
0.98525
2.447
0.97468
4-382
0.96226
6.910
56
0.99814
0.265
0.99263
1.067
0.98377
2.417
0.97205
4-332
0.95814
6.838
57
0.99798
0.261
0.99199
1.052
0.98232
2-383
0.96945
4.276
0.95406
6.756
58
0.99782
0-257
0.99135
1.036
0.98089
2-347
0.96690
4.214
0.95005
6.666
59
0.99766
0-253
0.99073
1.018
0.97948
2.308
0.96438
4.147
0.94609
6.567
60
0.99751
0.249
0.99012
0.999
0.97810
2.267
0.96191
4-075
0.94219
6.460
61
0.99736
0.244
0.98952
0.979
0.97674
2.221
0.95948
3-998
0.93838
6-343
62
0.99721
0.238
0.98893
0-957
0.97541
2-173
0.95711
3-9I4
0-93465
6.217
63
0.99707
0.232
0.98835
0-934
0.97411
2.123
0.85478
3-826
0.93099
6.083
64
0.99693
0.226
0.98780
0.910
0.97285
2.07O
0-95251
3-733
0.92741
5-941
65
0.99679
O.22O
0.98725
0.885
0.97163
2.014
0.95031
3-635
0.92393
5-789
66
0.99666
O.2I4
0.98672
0.859
0.97044
1-955
0.94816
3-532
0-92054
5-631
67
0.99653
0.207
0.98621
0.832
0.96927
1.894
0.94608
3-423
0.91725
5-463
68
0.99641
O.2OO
0.98571
0.804
0.96814
1.830
0.94407
3-3io
0.91407
5.288
69
0.99629
0.193
0.98523
0-775
0.96706
1.764
0.94213
3-!93
0.91100
5.106
70
0.99617
0.185
0.98477
0.744
0.96601
1.696
0.94026
3.072
0.90805
4.916
7i
0.99606
0.177
0.98433
0.713
0.96500
1.626
0.93846
2.946
0.90521
4.718
72
0.99596
0.169
0.98391
0.681
0.96404
1-553
0.93674
2.816
0.90248
4-5I3
73
0.99586
0.161
0.98350
0.648
0.96313
1.479
0.93510
2.682
0.89989
4.302
74
0.99576
0.153
0.98312
0.614
0.96227
1.402
0-93354
2-544
0.89742
4.085
75
0.99567
0.144
0.98276
0.580
0.96145
1-324
0.93207
2.403
0.89508
3-86i
76
0.99559
0-135
0.98242
0-545
0.96068
1.244
0.93069
2.258
0.89288
3-631
77
0-99551
0.126
0.98210
0.509
0.95996
1.162
0.92938
2. Ill
0.89081
3-396
78
0-99544
0.117
0.98181
0.472
0.95929
1.078
0.92816
1.960
0.88889
3-155
79
0-99537
0.108
0.98154
0-435
0.95866
0-993
0.92705
1.807
0.88711
2.910
80
0.99531
0.099
0.98128
0-397
0.95808
0.908
0.92603
1.652
0.88548
2.660
81
0.99525
0.090
0.98105
0-359
0.95756
0.821
0.92509
1.494
0.88399
2.406
82
0.99520
0.080
0.98085
0.320
0.95710
0.732
0.92425
1-333
0.88266
2.149
83
0-995 1 5
0.070
0.98067
0.281
0.95670
0.643
0.62351
1.170
0.88147
1.888
84
0.99511
0.060
0.98051
0.242
0-95635
0-552
0.92287
i. 006
0.88044
1.624
85
0.99508
0.050
0.98037
0.203
0.95605
0.461
0.92233
0.841
0.87957
1-357
86
0.99506
0.040
0.98026
0.163
0.95580
0.369
0.92189
0.675
0.87886
1.087
87
0.99504
0.030
0.98018
0.123
0.95560
0.277
0.92154
0.508
0.87830
0.816
88
0.99502
0.02O
0.98012
0.082
0-95545
0.185
0.92128
0.340
0.87790
0-544
89
0.99501
O.OIO
0.98009
0.041
0-95537
0.093
0.92112
0.171
0.87766
0.272
90 0.99500 o.ooo 0.98007 o.ooo 0.95534 o.ooo 0.92106 o.ooo 0.87758 o.ooo
Note. cosh (o/5) = i.o /o^for all values of d.
Examples, cosh (0.5 /8i°) = 0.88399 /2°.4o6 = 0.88399 /2°.24/.22r.
cosh"1 (0.97810 /2°.267) = 0.3 /6o°.
[8]
TABLE II. HYPERBOLIC COSINES, cosh (p /8) = r /T. CONTINUED
0.6
0.7
0.8
0.9
45
1.01070
10.254
1.01982
13.890
1.03360
18.010
1.05333
22.567
1.08031
27.487
46
1.00449
10.291
1.01136
13.960
1.02263
18.132
1.03959
22.755
1.06358
27.762
47
0.99825
10.315
1.00289
14.013
1.01164
18.231
1.02583
22.919
1 .04680
28.011
48
0.99199
10.327
0.99441
14.050
1.00063
18.309
1.01203
23-059
1.03000
28.235
49
0-98575
10.326
0.98594
14.071
0.98963
18.368
0.99822
23.176
1.01315
28.433
50
0-97953
10.313
0.97748
14-075
0.97864
18.405
0.98443
23.267
0.99632
28.603
Si
0-97333
10.287
0.96906
14.061
0.96768
18.421
0.97064
23-332
0.97950
28.743
52
0.96716
10.248
0.96066
14.031
0.95675
18.414
0.95690
23-372
0.96270
28.854
53
0.96103
10.196
0.95232
13.982
0.94587
18.384
0.94320
23-383
0.94596
28.933
54
0-95495
10.131
0.94404 /
13.916
0.93506
18.332
0.92957
23-367
0.92928
28.981
55
0.94893
10.053
0.93582
13-831
0.92432
18.256
0.91603
23.321
0.91267
28.994
56
0.94296
9.961
0.92768
13.728
0.91367
18.156
0.90257
23.246
0.89615
28.974
57
0.93706
9-856
0.91962
13.606
0.90312
18.031
0.88922
23.140
0.87976
28.917
58
0.93125
9.738
0.91166
13-465
0.89270
17.881
0.87602
23.003
0.86350
28.823
59
0.92552
9.606
0.90382
13.306
0.88240
17-705
0.86294
22.833
0.84739
28.689
6o
0.91987
9.461
0.89607
13.127
0.87222
17.505
0.85001
22.631
0.83142
28.518
61
0.91433
9-303
0.88846
12.929
0.86221
17.278
0.83727
22.394
0.81564
28.303
62
0.90889
9-J3i
0.88100
12.713
0.85237
17.024
0.82471
22.122
0.80008
28.047
63
0.90357
8-945
0.87369
12.476
0.84271
16.743
0.81237
2I.8I4
0.78475
27.745
64
0.89838
8.747
0.86653
12. 22O
0.83324
16.435
0.80025
21.471
0.76966
27.396
65
0.89332
8.536
0.85954
"•945
0.82398
16.100
0.78837
21.091
0.75484
26.999
66
0.88838
8.312
0.85272
11.652
0.81494
15-738
0.77675
20.673
0.74029
26.555
67
0.88358
8.076
0.84609
n-339
0.80613
15-349
0.76540
2O.2I8
0.72603
26.061
68
0.87894
7.827
0.83966
11.007
0.79757
I4-93I
0-75435
19.722
0.71212
25-5I3
69
0.87445
7.565
0.83344
10.656
0.78927
14.486
0.74362
I9.I88
0.69857
24.911
70
0.87012
7.292
0.82744
10.287
0.78125
14.014
0.73322
I8.6I5
0.68539
24-254
71
0.86596
7.007
0.82166
9.900
0.77352
I3-5I5
0.72317
iS.OOI
0.67261
23-54I
72
0.86197
6.711
0.81611
9.496
0.76610
12.990
0.71347
I7.348
0.66026
22.772
73
0.85816
6.404
0.81081
9-075
0.75899
12.438
0.70416
16.656
0.64836
21.946
74
0.85454
6.086
0.80576
8.637
0.75220
11.860
0.69527
15.926
0.63692
21.061
75
0.85 in
5.758
0.80097
8.183
0.74575
11.257
0.68681
I5-I56
0.62599
20.115
76
0.84787
5.420
0.79646
7.7I3
0.73965
10.630
0.67878
14-347
0.61560
19.111
77
0.84484
5-073
0.79222
7.228
0.73392
9-979
0.67121
I3.502
0.60577
18.049
78
0.84201
4.718
0.78826
6.729
0.72856
9.306
0.66412
12.621
0.59652
16.929
79
0.83939
4-355
0.78459
6.217
0.72359
8.612
0.65753
II.7O6
0.58790
15-753
80
0.83698
3-984
0.78121
5-694
0.71901
7.897
0.65145
10-755
0.57991
14.521
81
0.83479
3.606
0.77814
5-!59
0.71484
7.164
0.64589
9.776
0.57259
I3-237
82
0.83282
3.221
0.77538
4-613
0.71108
6.413
0.64087
8.768
0.56596
11.904
83
0.83108
2.831
0.77293
4-057
0.70774
5-647
0.63641
7-733
0.56005
10.526
84
0.82957
2.436
0.77079
3-493
0.70483
4.867
0.63252
6.674
0.55487
9.105
85
0.82828
2.037
0.76898
2.922
0.70236
4-075
0.62921
5-595
0.55046
7.647
86
0.82722
1.634
0.76750
2-345
0.70033
3.272
0.62648
4.498
0.54683
6.157
87
0.82640
1.228
0.76634
1.763
0.69875
2.462
0.62435
3.386
0.54399
4.642
88
0.82581
0.820
0.76551
1.177
0.69761
1.645
0.62283
2.263
0.54195
3.106
89
0.82546
0.410
0.76501
0.589
0.69693
0.823
0.62192
I-I33
0.54072
i-SS^
9°
0.82534
o.ooo
0.76484
o.ooo
0.69671
o.ooo
0.62161
o.ooo
0.54030
o.ooo
Examples, cosh (0.7 771°) = 0.82166 /g°.goo = 0.82166 /g°
cosh"1 (0.69857 /24°.9ii) = 1.0/69°.
[9]
TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /jy. CONTINUED
1.4
45
46
47
48
49
I-H57
1.0959
1-0759
1-0559
J-0359
32.686
33-o67
33-424
33-754
34-056
I. 1608
1.1376
I. 1143
1.0910
1.0676
38.076
38.582
39.063
39-517
39-943
1.2163
1.1897
1.1630
1.1363
1.1095
43-570
44.210
44.827
45-421
45.989
1.2830
1.2529
.2227
.1926
.1624
49.084
49.864
50-625
51-365
52.083
1.3616
1-3279
1.2941
1.2604
1.2267
54-550
55-471
56.378
57-266
58.134
50
Si
52
53
54
1.0158
0.99578
0-97577
0.95580
0.93589
34.328
34-570
34-780
34-957
35-!00
1.0443
1.0209
0.99757
0.97428
0.95104
40.341
40.709
41.045
41-349
41.618
1.0828
1.0561
1.0294
1.0027
0.97613
46.529
47.041
47-525
47-978
48.400
.1322
.IO2I
.072O
.O42I
I.OI22
52-777
53-446
54.090
54.708
55-299
1.1931
1.1596
1.1262
1.0929
1.0598
58.982
59.8lO
6o.6l8
61.407
62.175
55
56
57
58
59
0.91605
0.89630
0.87666
0.85716
0.83781
35.206
35-275
35-305
35-^93
35-238
0.92788
0.90481
0.88186
0.85902
0.83633
41.850
42.045
42.201
42.314
42.383
0.94964
0.92325
0.89697
0.87082
0.84482
48.788
49.141
49.458
49.736
49-973
0.98242
0.95277
0.92327
0.89393
0.86474
55-862
56-396
56.900
57-372
57-Sii
1.0268
0.99406
0.96149
0.92912
0.89696
62.921
63.645
64-347
65.025
65.681
60
61
62
63
64
0.81862
0.79961
0.78081
0.76223
0.74390
35.I39
34.992
34-794
34-545
34.242
0.81379
0.79144
0.76929
0.74735
0.72565
42.406
42.379
42.299
42.164
41.970
0.81898
0.79331
0.76783
0.74256
0.71752
50.168
50-315
50.412
50.457
50.446
0-83573
0.80692
0.77831
0.74991
0.72173
58.214
58.578
58.902
59-185
59.421
0.86502
0.83332
0.80186
0.77065
0.73969
66.313
66.919
67.497
68.048
68.571
65
66
67
68
69
0.72584
0.70808
0.69063
0.67352
0.65677
33-881
33-459
32.974
32-423
31-804
0.70420
0.68304
0.66218
0.64164
0.62146
4i-7i3
41.389
40.993
40.520
39.966
0.69271
0.66815
0.64385
0.61985
0.59618
50-373
50.235
50.026
49-739
49.368
0-69377
0.66605
0.63858
0.61136
0.58441
59.608
59-742
59.8i7
59.828
59-769
o. 70899
0.67855
0.64836
0.61844
0.58877
69.064
69.524
69.950
70.340
70.689
70
7i
72
73
74
0.64043
0.62452
0.60907
0.59410
0.57966
31.112
30.344
29.498
28.570
27-558
0.60166
0.58227
0.56334
0.54489
0.52697
39-324
38.589
37-754
36.812
35.756
0.57283
0.54984
0.52724
0.50506
0.48335
48.905
48-342
47.668
46.870
45-937
0-55775
0.53137
0.50529
0-47955
0.45414
59-633
59.410
59.090
58-659
58-104
0-55937
0.53022
0.50134
0.47271
0-44434
70.995
7I-253
7L458
7I.6O2
71.677
75
76
77
78
79
0.56579
0-55252
0-53990
0.52796
0-51675
26.458
25.269
23.989
22.616
21.151
0.50963
0.49292
0.47688
0.46159
0.44710
34.578
33-271
31.828
30.242
28.507
0.46215
0.44151
0.42149
0.40216
0.38360
44.856
43-6io
42.180
40.548
38.692
0.42910
0.40446
0.38028
0.35658
0-33345
57405
56.542
55.487
54.206
52-659
0.41622
0.38836
0.36075
0-33339
0.30630
71.672
71-573
71.361
71.015
70.501
80
81
82
83
84
0.50631
0.49669
0.48794
0.48009
0.47320
19-594
17.946
16.210
14-391
12.495
0.43349
0.42083
0.40919
0.39867
0.38935
26.616
24-567
22.359
19-993
17-475
0.36591
0.34921
0.33362
0.31928
0.30635
36-59I
34-223
3!-569
28.608
25-333
0.31099
0.28928
0.26848
0.24877
0.23039
50.800
48.566
45-888
42.681
38.853
0.27951
0.25304
0.22691
O.2OI22
0.17608
69.774
68.774
67-4I3
65-561
63.018
85
86
87
88
89
0.46730
0.46241
0-45857
0.45581
0.45415
10.529
8.501
6.422
4-304
2-159
0.38129
0.37457
0.36927
0.36545
0.36314
14.813
I2.O22
9-I2I
6.132
3.081
0.29499
0.28538
0.27770
0.27208
0.26865
21.739
17.838
13-659
9-247
4.668
0.21368
0.19902
0.18684
0.17765
0.17193
34-303
28.949
22.746
15-729
8.051
0.15169
0.12853
O.IO7IO
0.08871
0.07570
59-474
54.420
47.029
36-059
20.198
90
0.45360
o.ooo
0.36236
O.OOO
0.26750
o.ooo
0.16997
o.ooo
0.07074
o.ooo
Examples, cosh (1.3 /73°) = 0.50506 /46°.87o = 0.50506 /46°.52'.i2"
cosh"1 (0.07074 /o°) = 1.5 /9Q°-
[10]
TABLE II. HYPERBOLIC COSINES.
1.6 1.7 i
cosh (p ^S) = r /y. CONTINUED
8 1.9 2.0
0
o
o
o
0
0
45
1-4524
59.916
I-5556
65.149
1.6714
70.229
1.7999
75.I52
I.94I3
79.922
46
.4149
60.974
1.5141
66.336
1.6257
71.536
1.7496
76.569
1.8861
81.437
47
•3774
62.021
1.4727
67.516
1.5802
72.837
1.6997
77-979
1-8313
82.947
48
.3400
63-051
1.4316
68.681
1-5350
74.125
1.6502
79-378
1.7771
84-445
49
.3028
64.066
1.3906
69-833
1.4901
75.403
i. 6012
80.768
1-7235
85-935
50
.2657
65.065
1-3499
70.973
1.4456
76.671
1-5526
82.149
1.6706
87.417
.2287
66.049
I-3095
72.IOO
1.4015
77.929
1.5046
83-522
1.6184
88.890
52
.1919
67.018
1.2693
73-217
1-3579
79.179
1-4572
84.890
1.5669
90-356
53
•1554
67.971
1.2295
74-324
I.3I47
80.422
1.4104
86.252
1.5162
91.817
54
.1191
68.909
1.1900
75-421
1.2719
81.659
1-3643
87.609
1.4664
93-275
55
1.0831
69-833
1.1509
76.508
1.2297
82.892
1.3188
88.963
1.4174
94.731
56
1-0473
70.744
I.II2I
77.589
1.1879
84.122
1.2740
90.318
1-3693
96.188
57
1.0118
71.640
1.0737
78.663
1.1467
85-350
1.2299
91.674
1.3221
97.646
58
0-97653
72.522
1-0358
79.730
1.1061
86.576
1.1865
93.032
1.2758
99.108
59
0.94160
73-392
0.99826
80.795
i. 066 1
87.808
I-I439
94-398
1-2305
100.577
60
0.90699
74-249
0.96117
81.859
1.0266
89.045
1.IO2O
95-773
1.1861
102.057
61
0.87268
75-094
0.92451
82.919
0.98772
90.287
1.0609
97.I58
1.1427
103.548
62
0.83871
75.926
0.88831
83-982
0.94946
91.540
I.O207
98.559
1.1003
105.054
63
0.80508
76.748
0.85256
85-048
0.91182
92.807
0.98119
99.978
1.0589
106.580
64
0.77177
77o6i
0.81727
86.121
0.87480
94.092
0.94254
101.420
1.0186
108.129
65
0.73879
78.365
0.78245
87.204
0.83842
95.400
0.90472
102.891
0.97928
109.707
66
0.70615
79.161
O.748IO
88.300
0.80268
96.734
0.86773
104.394
0.94100
111.318
67
0.67383
79-951
0.71420
89.415
0.76759
98.101
O.86l6o
105-937
0.90378
112.968
68
0.64184
80.736
0.68078
90.551
o.733i4
99.508
0.79631
107.527
0.86764
114.663
69
0.61019
81.519
0.64782
91.717
0.69935
100.962
0.76190
109.170
0.83259
116.410
70
0.57887
82.300
0.61533
92.917
0.66624
102.473
0.72838
110.875
0.79865
118.215
0.54786
83.084
0.58331
94.161
0.63379
104.050
0.69576
112.653
0.76583
120.087
72
0-51715
83.872
0.55175
95-458
0.60202
105.706
0.66406
114.514
0.73414
122.035
73
0.48674
84.668
0.52065
96.819
0.57095
107.453
0.63329
116.471
0.70361
124.068
74
0.45663
85-477
0.49003
98.258
0.54061
109.308
0.60349
"8-537
0.67428
126.197
75
0.42680
86.304
0.45988
99-793
0.51100
111.291
0.57470
120.726
0.64618
128.432
76
0.39724
87.156
0.43021
101.445
0.48217
"3-423
0.54696
123.058
0.61934
130.785
77
0.36794
88.041
0.40104
103.241
0.45415
"5-732
0.52030
125.550
0.59381
133.267
78
0.33888
88.972
0.37238
105.215
0.42701
118.248
0.49480
128.224
0.56965
135.890
79
0.31006
89.963
0.34428
107.409
0.40082
121.008
0.47055
131.100
0.54691
138.664
80
0.28146
91.036
0.31679
109.880
0-37567
124.055
0.44764
134.201
0.52568
141.601
81
0.25309
92.221
0.28997
112.701
0-35167
127.438
0.42617
137-553
0.50603
144.713
82
0.22494
93-56i
0.26395
"5-971
0.32897
131.210
0.40626
141.172
0.48805
148.003
83
0.19700
95.128
0.23887
119.816
0.30779
135-433
0.38806
145.082
0.47184
151.474
84
0.16926
97-033
0.21497
124.413
0.28838
140.167
0.37174
149.292
0-4575°
155-127
85
0.14177
99-473
O.I926I
129.984
0.27102
145-463
0-35749
153.806
0.44514
158.954
86
0.11467
102.829
O.I723I
136.806
0.25603
151-356
0.34550
158.614
0.43486
162.943
87
0.08809
107.920
0.15476
145.176
0.24378
157-847
0-33593
163.691
0.42674
167.072
88
0.06261
116.850
0.14093
I55-306
0.23469
164.877
0.32895
168.991
0.42088
171.315
89
0.04025
136-057
O.I3I95
167.116
0.22911
172.324
0.32472
174-453
0.41734
175-637
90
0.02920
180.000
0.12884
180.000
0.22720
180.000
0.32329
180.000
0.41615
180.000
Examples, cosh (2.0 /go0) = 0.41615 /i8o°.
cosh"1 (0.54691 /i38°.664) = 2.0 779°.
TABLE II. HYPERBOLIC COSINES, cosh (p {8) = r /_V CONTINUED
2.2
2-3
2.4
2-5
45
2.0958
84-55I
2.2636
89.050
2-4449
93.438
2.6403
97.730
2.8502
101.944
46
2.0350
86.159
2.1966
90.740
2.3711
95.202
2-5589
99.564
2.7603
103.842
47
1-9749
87-755
2.1306
92.417
2.2984
96.952
2.4788
101.380
2.672O
105.720
48
I-9I55
89-341
2-0655
94.081
2.2269
98.687
2.4001
103.179
2.5853
107.578
49
1.8569
90.917
2.0013
95-734
2.1567
100.408
2.3229
104.962
2.5004
109.417
5°
1.7992
92.484
1.9382
97.376
2.0877
102.116
2.2473
106.729
2.4173
111.238
51
1.7424
94.043
1-8763
99.009
2.O2OO
103.812
2.1732
108.482
2.3361
113.041
52
1.6865
95-594
I-8I55
100.633
1-9537
105.498
2.1008
IIO.220
2.2568
114.826
53
1.6316
97.140
1-7559
102.249
1.8889
107.173
2.0301
III.944
2-1795
"6.593
54
1-5777
98.681
1.6976
103.858
1-8255
108.838
1.9612
113.656
2.IO42
118.344
55
I-5249
IOO.22O
1.6406
105.462
1.7637
110.495
1.8940
"5.356
2.0310
120.079
56
1-4732
101.758
1.5848
107.063
1-7034
112.145
1.8286
117.045
1-9599
121.799
57
1.4226
103.296
I-5303
108.662
1.6447
113.789
1.7651
118.724
1.8909
123.504
58
i-373i
104.836
1.4772
110.261
1.5876
115-430
1-7035
120-394
1.8241
125-195
59
1.3248
106.382
1.4256
111.862
1.5322
117.069
1.6438
122.057
1-7594
126.874
60
1.2776
107.936
1-3754
113.467
1.4784
118.708
1-5859
123.715
1.6969
128.541
61
1.2317
109.501
1.3266
115.079
1.4262
120.347
1.5299
125.369
1.6366
130.198
62
1.1870
III.O79
1.2792
116.700
1.3758
121.991
1.4758
I27.O20
1.5785
131.846
63
I-I435
112.673
1.2332
118.333
1.3270
123.641
1-4237
128.671
1.5226
133.486
64
I.IOI2
114.288
1.1887
119.982
1.2799
125.300
I.3736
130.322
1.4689
135.120
65
I. O6O2
115.928
I-I457
121.650
1-2345
126.970
1-3254
I3L977
I.4I73
136.749
66
I.02O4
H7-597
1.1042
123-339
1.1908
128.654
1.2791
I33-638
1.3679
138.374
67
0.98193
119.299
1.0642
125-055
1.1488
130.355
1-2347
I35-306
1.3207
139.998
68
0.94474
I2I.O4I
1.0256
126.801
1. 1086
132.077
1.1923
136.985
1.2756
141.622
69
0.90886
122.827
0.98860
128.581
1.0700
133-823
1.1518
138.676
1.2327
143-247
70
0.87429
124.662
0.95308
130.400
1-0332
135-595
1.1132
140.382
1.1919
144.876
7i
0.84105
126.554
0.91908
132.263
0.99804
137-397
1.0765
I42.IO6
i-I532
146.511
72
0.80915
128.509
0.88663
134.174
0.96464
I39-233
1.0417
143.850
1.1166
148.153
73
0.77861
130.533
0-85573
136.137
0.93297
141.106
i. 0088
I45.6l6
1.0821
149.803
74
Q-74945
132.634
0.82638
138.158
0.90303
143.019
0.97785
147407
1.0497
151.464
75
0.72171
134.818
0.79862
140.242
0.87481
144.976
0.94877
149.225
1.0193
I53-I37
76
0.69541
137.094
0.77246
142.393
0.84833
146.980
0.92156
I5I.07I
0.99093
154-823
77
0.67060
139.468
0.74789
144.615
0.82360
149-033
0.89622
152.948
0.96459
156.524
78
0.64732
141.946
0.72497
146.911
0.80062
151-136
0.87276
I54-857
0.94026
158.241
79
0.62558
144-534
0.70372
149.285
0-77940
153-291
0.85117
156.798
0.91793
159-974
80
0.60544
147.237
0.68416
I5I-738
0.75996
155-499
0-83145
158.771
0.89757
161.723
81
0.58697
150.058
0.66632
154-271
0.74231
157.760
0.81361
160.778
0.87918
163.490
82
0.57021
152.997
0.65024
156.883
0.72646
160.074
0.79764
162.817
0.86276
165.274
83
0.55523
156.054
0.63595
159-572
0.71244
162.438
0-78354
164.888
0.84829
167.074
84
0.54209
I59-225
0.62349
162.334
0.70025
164.848
0.77131
166.987
0-83575
168.889
85
0-53085
162.503
0.61287
165.164
0.68990
167.301
0.76095
I69.II2
0.82518
170.717
86
0.52156
165.876
0.60413
168.054
0.68141
169.793
0.75247
171.260
0.81650
172-558
87
0.51428
169.329
0-59731
170.994
0.67479
172.316
0.74587
I73.427
0.80979
174.409
88
0-50905
172.849
0.59242
173-973
0.67006
174.863
0.74116
I75.6IO
0.80499
176.269
89
0.50590
176.413
0.58948
176.980
0.66722
177.427
0.73834
177.803
0.80207
178.134
90
0.50485
iSo.OOO
0.58850
180.000
0.66628
180.000
0.73740
iSo.OOO
0.80114
180.000
Examples, cosh (2.2 745°) = 2.2636 /89°.O5o = 2.2636 /89°.o3'.oo<'.
cosh"1 (1.0821 /I49°.8o3) = 2.5 773°.
[12]
TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /T. CONTINUED
2.6
2.7
2.9
45
3-°753
106.093
3-3I63
110.190
3-5741
114.248
3-8497
118.275
4-1443
122.282
46
2.9758
I08.O5I
3.2O62
112.207
3-4523
116.322
3-7I46
I 20.406
3-9945
124.469
47
2.8783
109.987
3-0984
114.199
3-3329
118.368
3-5826
122.506
3-8483
126.623
48
2.7827
III.9OI
2.9928
116.166
3.2162
' 120.387
3-4536
124-575
3-7057
128.743
49
2.6891
113-793
2.8896
118.108
3.1023
122.378
3.3278
I26.6I4
3-5667
130.828
5°
2-5977
115.663
2.7889
120.025
2.9913
124.340
3-2053
128.621
3-43J5
132.879
5i
2.5085
II7.5I2
2.6908
121.917
2.8832
126.273
3.0861
130.595
3.3001
134.894
52
2.4216
119.340
2-5952
123.784
2.7780
128.177
2.9703
132.536
3.1726 '
136.873
53
2.3369
I2I.I47
2.5023
125.626
2.6759
130.052
2.8580
134-444
3.0490
I38.8I5
54
2-2545
122.931
2.4I2O
127.442
2.5769
131.898
2.7492
136.320
2-9294
I4O.72O
55
2.I74S
124.695
2.3245
129.233
2.4809
I33.7I5
2.6439
138.162
2.8137
142.587
56
2.0970
I26.44O
2.2397
131.000
2.3880
I35-502
2.5421
139.969
2.7020
I44-4I5
57
2.0219
128.165
2.1577
132.742
2.2983
I37-259
2-4439
141.742
2-5943
146.203
58
1.9492
129.872
2.0785
134.460
2. 2Il8
138.987
2.3492
143-479
2.4905
I47-95I
59
1.8790
131.560
2.0021
136-154
2.1284
140.685
2.2580
145.180
2.3907
149.659
60
1.8113
I33-23I
1.9284
137.824
2.0481
142.354
2.1703
146.846
2.2949
I5I-32S
61
1.7460
134.885
1.8575
I39.47I
1.9709
143-993
2.0861
148.476
2.2030
152.949
62
1.6832
136.523
1.7894
141.095
1.8968
145.601
2.0053
150.071
2.1148
I54-53Q
63
1.6229
138.146
I.724I
142.696
1.8259
147.179
1.9280
151.627
2.0304
156.067
64
1.5650
139-754
I.66I5
144.276
1.7580
148.728
1.8541
I53-I46
1.9498
I57-559
65
1.5096
141.349
1. 6oi6
I45-834
1.6931
150.247
1-7835
154.628
1.8729
159.005
66
1.4566
142.932
1-5445
I47-37I
I.63II
I5L737
1.7162
156.071
1.7997
160.406
67
1.4060
144.504
1.4900
148.887
I.572I
I53.I98
1.6522
157-477
1.7301
161.759
68
1-3578
146.066
1.4382
150-384
I.5l6l
154.628
I-59I4
158.844
1.6640
163.064
69
1.3120
I47.62O
1.3889
151.863
1.4629
156.030
1-5338
I6O.I72
1.6013
164.321
70
1.2685
149.164
1.3422
I53-322
I.4I26
157.404
1-4793
161.460
1.5420
165.528
7i
1.2273
I50-705
1.2981
154.764
1.3650
158.749
1.4278
162.709
1.4861
166.685
72
1.1885
152.239
1-2565
156.189
1.3202
160.066
1-3793
163.919
1-4335
167.792
73
i-^Q
I53-770
1.2174
157-599
1.2782
I6I.355
1.3338
165.090
1.3840
168.848
74
1.1176
I55-298
1.1807
158.994
1.2388
162.617
1.2912
166.222
1-3377
169.853
75
1.0855
156.825
1.1465
160.374
I.2O2O
163.853
1.2514
167.315
1.2946
170.806
76
1.0556
I58-35I
1.1147
161.742
I.I677
165.063
1.2144
168.370
1.2545
171.708
77
1.0278
159.878
1.0852
163.096
I.I360
166.248
1. 1802
169.387
1.2174
172.560
78
I.OO22
161.406
1.0580
164.440
1.1069
167.409
1.1486
170.368
1.1832
173.362
79
0.97880
162.937
1-0331
165.773
1 .0803
168.547
I.I2OO
I7I-3I3
1.1519
174.116
80
0-95745
164.471
1.0105
167.096
1.0561
169.664
1.0938
172.225
J-^SS
174.823
81
0.93821
I66.008
0.99006
168.411
1.0342
170.760
I.O7O2
173.104
1.0979
175-485
82
0.92106
167.549
0.97189
169.718
I.OI47
171.837
I.O492
173-953
1.0751
176.104
83
0.90596
169.094
0.95592
171.018
0.99765
172.897
1.0307
174-774
1-0550
176.685
84
0.89290
170.643
0.94211
172.312
0.98287
173.942
I.OI48
175-571
1.0377
177.229
85
0.88l87
172.196
0.93045
173.601
0.97041
174-973
I.OOI4
176.344
1.0231
177.742
86
0.87286
I73.752
0.92093
174.886
0.96025
175-993
0.99042
177.099
I.OII2
178.228
87
0.86587
I75.3II
0.91354
176.168
0.95235
177.003
0.98190
177.838
I.OOI9
178.691
88
0.86089
176.873
0.90827
177-447
0.94671
178.006
0.97582
178.566
0.99528
179.137
89
0.85789
178.436
0.90512
178.724
0-94334
179.004
0.97217
I79-285
0.99130
I7.9-S7I
90
0.85689
iSo.OOO
0.96407
180.000
O.94222
iSo.OOO
0.97096
iSo.OOO
0.98999
180.000
Examples, cosh (2.8 785°) = 0,97041 /I74°.g73 = 0.97041 /i74°.g8/.23*
cosh"1 (i.5420/i65°.528) = 3.0 /7Q°.
[13]
TABLE III. HYPERBOLIC TANGENTS, tanh (p [$) = r
o-3
0.4
o-S
45
46
47
48
49
o.ioooo 44.812
o.ioooi 45.812
O.IO002 46.812
0.10003 47.812
0.10004 48.813
0.19997 44-235
0.20006 45.236
0.20015 46.237
0.20024 47-239
0.20034 48.242
0.29981 43.282
0.30012 44.281
0.30043 45.280
0.30075 46.283
0.30107 47.290
0.39921 41-954
0-39995 42.945
0.40069 43.942
0.40143 44-942
0.40217 45-947
0.49757 40.250
0.49902 41.229
0.50047 42.219
0.50192 43.213
0.50340 44.213
50
Si
52
53
54
0.10006 49.813
0.10007 50.813
0.10008 51.814
0.10009 52.816
o.iooio 53.818
0.20043 49-245
0.20053 50.251
0.20062 51.256
0.20071 52.262
0.20081 53.268
0.30138 48-297
0.30169 49.306
0.30201 50.315
0.30232 51.328
0.30263 52.344
0.40293 46.955
0.40370 47-969
0.40446 48.986
• 0.40521 50.005
0.40596 51.031
0.50490 45.220
0.50639 46.233
0.50789 47-252
0.50939 48.276
0.51089 49.308
55
56
57
58
59
o.ioon 54.820
0.10013 55.823
0.10014 56.825
0.10015 57.828
0.10016 58.830
0.20090 54.276
0.20099 55-285
0.20107 56.295
0.20115 57-306
0.20124 58.318
0.30294 53.362
0-30325 54-382
0-30355 55-404
0-30385 56.426
0,30414 57.452
0.40671 52.058
0.40746 53.089
0.40820 54.124
0.40892 55.164
0.40964 56.207
0.51239 50.344
0.51389 51.388
0-51538 52.439
0.51687 53.494
0.51835 54.556
6o
61
62
63
64
0.10017 59-833
0.10018 60.837
0.10019 61.841
O.I002O 62.845
O.IOO2O 63.849
0.20132 59.332
0.20140 60.345
0.20148 61.360
0.20156 62.375
0.20164 63.391
0.30444 58.479
0.30473 59-5Io
0.30501 60.542
0.30528 61.575
0-30555 62.610
0.41037 57-255
0.41107 58.305
0.41176 59-359
0.41244 60.417
0.41311 61.479
0-51983 55-625
0.52128 56.700
0.52271 57.781
0.52412 58.867
0-52552 59-959
65
66
67
68
69
O.IOO2I 64.853
0.10022 65.857
O.IOO23 66.862
O.IOO24 67.866
O.IOO25 68.871
0.20171 64.408
0.20179 65.425
0.20186 66.443
0.20193 67.461
0.20199 68.480
0.30580 63.647
0.30605 64.686
0.30630 65.727
0.30655 66.769
0.30678 67.813
0.41376 62.544
0.41441 63.612
0.41504 64.685
0.41565 65.760
0.41623 66.837
0.52689 61.058
0.52824 62.164
0.52957 63.276
0.53085 64.391
0.53209 65.511
70
72
73
74
O.IOO26 69.877
0.10026 70.882
0.10027 71.888
0.10028 72.893
0.10028 73.898
0.20206 69.501
O.2O2I2 7O.522
0.20217 71.544
O.2O223 72.566
O.2O228 73.589
0.30701 68.859
0.30722 69.906
0.30742 70-955
0.30762 72.005
0.30781 73-056
0.41680 67.918
0.41735 69.002
0.41788 70.089
0.41838 71,179
0.41886 72.272
0-53330 66.638
0.53446 67.771
0.53560 68.909
0-53669 70-052
0-53773 71-200
75
76
77
78
79
O.IOO29 74.904
0.10029 75-910
0.10030 76.916
0.10030 77.922
0.10031 78.928
O.2O234 74.612
0.20239 75.635
0.20243 76.659
0.20246 77.684
0.2O25O 78.709
0.30799 74.108
0.30816 75.162
0.30831 76.217
0.30846 77-273
0.30860 78.329
0.41932 73.368
0.41975 74.466
0.42016 75.566
0.42054 76.668
0.42088 77.771
0.53872 72.351
0-53965 73-507
0.54054 74-667
0.54138 75.831
0.54215 76.999
80
81
82
83
84
0.10031 79-934
0.10032 80.940
0.10032 81.946
0.10032 82.953
0.10033 83.960
0.20254 79-734
0.20257 80.759
0.20260 81.785
0.20263 82.812
0.20265 83.838
0.30872 79-386
0.30884 80.445
0.30894 81.506
0.30904 82.567
0.30912 83.628
0.42120 78.876
0.42150 79-983
0.42177 81.092
0.42201 82.203
0.42222 83.315
0.54285 78.170
0-54349 79-344
0.54407 80.520
0.54459 81.699
0.54503 82.881
85
86
87
88
89-
0.10033 84.967
0.10033 85.974
0.10033 86.981
0.10033 87.988
0.10033 88.994
0.20267 84.864
0.20268 85.891
0.20270 86.918
0.20270 87.946
0.20271 88.973
0.30920 84.689
0.30926 85.751
0.30930 86.813
0.30933 87.875
0.30934 88.937
0.42239 84.427
0.42252 85.540
0.42264 86.654
0.42274 87.768
0.42279 88.883
0.54540 84.065
0.54571 85.251
0.54596 86.438
0.54616 87.626
0.54628 88.813
90
0.10033 90.000
0.20271 90.000
0.30934 90.000
0.42280 90.000
0.54631 90.000
Note.
tanh (o /J) = o /j
Examples.
tanh (0.5 /6o°) =
0.51983 /55°.625
= 0.51983 /55°-37'.
,?o".
[14]
TABLE III. HYPERBOLIC TANGENTS, tanh (p 5) = r /j. CONTINUED
0.6
0.7
0.8
0.9
45
0.59406 38.183
0.68732 35.786
0-77577 33-098
0-85756 30.161
0.93077 27.044
46
0.59650 39.146
0.69109 36.719
0.78117 33.980
0.86480 30.980
0.93909 27.784
47
0.50898 40.119
0.69495 37.663
0.78671 34.878
0.87229 31.815
0.94950 28.539
48
0.60149 4I-°99
0.69888 38.617
0.79240 35.790
0.88001 32.666
0.95938 29.308
49
0.60403 42.088
0.70287 39.581
0.79824 36.715
0.88799 33-531
0.96966 30.092
5°
0.60660 43.085
0.70694 40-557
0.80421 37.653
0.89620 34.412
0.98032 30.892
5i
0.60920 44.092
0.71107 41-545
0.81031 38.605
0.90466 35.310
0.99136 31.710
52
0.61182 45.107
0.71527 42.543
0-81655 39-573
0.9*337 36.223
1.00282 32.545
53
0.61447 46.130
0.71952 43-555
0.82291 40.556
0.92231 37.155
1.01469 33.400
54
0.61713 47.162
0.72382 44-577
0.82940 41.554
0.93150 38.105
1.02697 34-274
55
0.61980 48.203
0.72817 45.612
0.83601 42.568
0.94094 39.075
1.03970 35.172
56
0.62248 49.254
0-73257 46.662
0.84274 43.599
0.95063 40.066
1.05287 36.091
57
0.62517 50.315
0.73700 47-725
0.84957 44-647
0.96056 41.078
1.06648 37-035
58
0.62785 51.384
0.74145 48.800
0.85649 45.712
0.97069 42.111
1.08054 38.004
59
0-63053 52-463
0.74593 49-888
0.86351 46.797
0.98106 43.167
1.09506 39.002
60
0.63322 53-552
0.75047 50.990
0.87063 47.900
0.99168 44.247
1.11009 40.026
61
0.63588 54-650
0-75499 52-107
0.87781 49.022
1.00251 45-353
1.12555 41.08,2
62
0-63852 55-758
0-75950 53-238
0.88504 50.165
1.01353 46.486
1.14144 42.168
63
0.64115 56.876
0.76400 54.383
0.89232 51.327
1.02471 47-645
I-I5777 43-289
64
0.64376 58.002
0.76848 55.542
0.89965 52.509
1.03604 48.831
I-I7454 44-445
65
0.64633 59.138
0.77294 56.716
0.90698 53.712
1-04753 50.044
1.19173 45-638
66
0.64886 60.284
0-77737 57-902
0.9U33 54.936
1.05916 51.289
1.20935 46.869
67
0-65135 61.439
0.78177 59.105
0.92166 56.180
1.07090 52.562
1.22737 48.140
68
0.65380 62.603
0.78609 60.322
0.92894 57.448
1.08268 53.868
1.24569 49-455
69
0.65618 63.777
0.79033 61.553
0.93616 58.737
1.09447 55.204
1.26429 50.812
70
0.65850 64.959
0.79450 62.798
0.94332 60.047
1.10627 56.572
1.28316 52.215
?i
0.66075 66.150
0.79858 64.057
0.95037 61.379
1.11803 57.974
1.30221 53.666
72
0.66294 67.350
0.80256 65.329
0.95727 62.732
1.12972 59.408
1.32140 55.164
73
0.66505 68.559
0.80641 66.614
0.96402 64.106
1.14126 60.874
1.34063 56.710
74
0.66708 69.775
0.81014 67.913
0.97060 65.501
1.15260 62.372
I-35986 58-307
75
0.66902 70.998
0.81371 69.225
0.97697 66.917
1.16370 63.903
1.37894 59-957
76
0.67086 72.228
0.81713 70.550
0.98311 68.352
1.17450 65.468
1-39775 61.658
77
0.67261 73.466
0.82038 71.886
0.98898 69.808
1.18493 67.064
1.41620 63.409
78
0.67425 74-7"
0-82345 73.233
0-99455 71-282
1.19495 68.691
1.43412 65.212
79
0.67577 75.962
0.82632 74.591
0.99981 72.773
1.20447 70.347
1.45141 67.065
80
0.67718 77.219
0.82899 75.958
1.00473 74.282
1.21344 72.034
1.46790 68.968
81
0.67847 78.481
0.83144 77.334
1.00926 75.805
1.22179 73-746
1.48345 70-919
82
0.67964 79.749
0.83366 78.719
I-OI339 77-344
1.22946 75.483
1.49790 72.913
83
0.68068 81.021
0.83564 80.112
1.01710 78.896
1.23640 77-243
1.51110 74-948
84
0.68159 82.296
0.83738 81.512
1.02036 80.460
1.24253 79-025
1.52289 77-023
85
0.68236 83.575
0.83887 82.917
1.02316 82.034
1.24781 80.825
I-533H 79.I32
86
0.68299 84.856
0.84009 84.328
1.02548 83.617
1.25219 82.640
1.54170 81.269
87
0.68349 86.140
0.84105 85.743
1.02730 85.206
1.25566 84.468
1.54848 83.429
88
0.68385 87.426
0.84173 87.161
1.02860 86.801
1.25814 86.308
1-55339 85.609
89
0.68406 88.713
0.84214 88.580
1.02937 88.400
1.25963 88.153
I-55637 87.802
90
0.68413 90.000
0.84229 90.000
1.02960 90.000
1.26015 90.000
1-55740 90-000
Eaxmples.
tanh (0.9 /77°)
= 1.18493 /67°.o64
= 1.18493 /67°-03'.
50*.
(0.66708
= Q.6 /74°.
[15]
TABLE III. HYPERBOLIC TANGENTS, tanh (p IS) = r £y. CONTINUED
i.i
1.2
1.4
45
0.99389
23-833
1.0457
20.616
1.0857
17.464
1. 1143
14.484
1-1323
11.712
46
1.0049
24.476
1.0582
21.144
1.0993
17.882
1.1284
14-775
1.1464
11.884
47
1.0164
25-131
1.0713
21.685
1.1136
18.301
1.1433
15.064
1.1613
12.048
48
1.0283
25-799
1.0851
22.236
1.1286
18.721
1.1590
15.352
1.1770
12.208
49
1.0408
26.480
1.0995
22.798
I-I445
19.148
1.1756
15.640
1.1936
12.362
5°
1-0539
27.178
1.1147
23-371
1.1613
19.584
1.1932
I5-930
1.2112
12.510
1.0676
27.891
1.1307
23-957
1.1789
20.027
1.2118
16.222
1.2299
12.652
52
1.0818
28.621
1.1474
24-558
1.1976
20.479
1-2315
16.515
1.2498
12.787
S3
1.0967
29.370
1.1650
25-I74
1.2173
20.941
1.2524
16.811
1.2709
12.914
54
1.1122
30.139
1.1835
25.807
1.2381
21.414
1.2746
17.110
1-2933
13-035
55
I.I284
30.930
1.2030
26.460
1.2602
21.900
1.2982
I7-4I3
I.3I72
13-151
56
I-I453
3J-744
1.2235
27-I33
1.2836
22.401
1.3234
17.721
L3427
13.262
57
1.1629
32-583
1.2450
27.827
1.3084
22.918
1.3502
18.036
1.3700
13.366
58
1.1813
33-449
1.2677
28.546
I.3346
23-452
1.3787
18.358
1.3992
13.466
59
1.2005
34-343
1.2916
29.292
1.3626
24.007
1.4093
18.689
I-430S
13-559
6o
1.2206
35-267
.3168
30.068
I.3923
24.584
1.4421
19.032
1.4642
13-651
<6i
1.2415
36.226
•3434
30.876
1-4239
25.187
1.4772
19.389
I-5005
13-738
62
1.2632
37.222
•3714
31.720
I.4576
25.820
1.5148
19.761
I-5396
13.824
63
1.2858
38-255
.4009
32.603
1-4935
26.485
1-5553
20.150
1.5819
13.908
64
1.3094
39.328
.4321
33-527
I-53I9
27.186
1.5990
20.562
1.6277
i3-99i
65
1-3339
40.446
1.4649
34.498
I-5729
27.928
1.6463
20.999
1.6776
14.076
66
1-3593
41.612
1-4995
35-520
1.6167
28.715
1.6974
21.465
I-73I9
14.166
67
1.3856
42.827
1-5359
36.597
1.6637
29-554
1.7528
21.966
I.79H
14.261
68
1.4129
44.096
1-5743
37-737
1.7140
30.452
1.8131
22.507
1.8566
14.364
69
1.4411
45.421
1.6148
38-941
1.7680
3I-4I5
1.8788
23.096
1.9285
14.480
70
1.4702
46.806
1.6573
40.219
1.8260
32-452
1.9506
23.740
2.0078
14.612
71
1.5001
48.256
1.7020
41-575
1.8882
33-570
2.0293
24.448
2.0959
14-765
72
1-5307
49-773
1.7488
43-017
I-955I
34-782
2.1158
25.232
2.1942
14.944
73
1.5620
51.361
.1-7977
44-553
2.0269
36.101
2.2III
26.106
2.3043
.15-159
74
I-5938
53-022
1.8488
46.190
2.1041
37-539
2.3164
27.084
2.4285
15-419
75
1.6261
54.762
1.9019
47-936
2.1869
39.110
2-4332
28.186
2.5694
15-734
76
1-6585
56.581
1.9568
49.798
2.2757
40.830
2.5631
29-433
2.7303
16.120
77
1.6910
58.482
2.0134
51-786
2.3706
42.720
2.7078
30-855
2-9*57
16.597
78
1.7232
60.468
2.0713
53.906
2.4716
44-798
2.8697
32.486
3-1312
17.186
79
I-755°
62.538
2.1299
56.165
2.5786
47.088
3-0509
34-367
3-3843
17.924
80
1.7858
64.692
2.1887
58.571
2.6912
49.611
3.2536
36-546
3-6845
18.856
81
1-8155
66.931
2.2471
61.126
2.8085
52-391
3.4807
39.085
4.0459
20.044
82
1.8435
69.252
2.3040
63.832
2.9290
55-447
3-7337
42.056
4.4876
21.578
83
1.8696
71.650
2-3585
66.690
3.0506
58.800
4-0135
45-546
5-0365
23-588
84
1-8933
74.121
2.4094
69.694
3-I704
62.460
4.3188
49.646
5-73"
26.276
85
1.9142
76.657
2-4555
72.836
3-2845
66.432
4.6429
54-459
6.6297
29-955
86
1.9318
79.252
2-4955
76.102
3-3884
70.706
4-9729
60.069
7.8015
35-134
87
1.9460
81.895
2.5281
79-475
3-4767
75-253
5.2871
66.523
9-34I3
42.644
88
1.9564
84-575
2.5523
82.933
3-5446
80.030
5-5531
73-786
11.259
53-726
89
1.9627
87.281
2.5671
86.452
3-5875
84.971
5-7333
81.707
13.182
69-695
90
1.9648
90.000
2.5721
90.000
3.6021
90.000
5.9978
90.000
14.101
90.000
Examples.
tanh (i
4 /64°)
= 1-5990
/20°.562
= 1.5990 /20°.33'
43'-
tanh-1 (1.7550 /62°.538 = i.i /79°.
[16]
TABLE III. HYPERBOLIC TANGENTS, tanh (p /8) = r £/. CONTINUED
1.6
1.8
1.9
45
1.1413
9.201
1.1428
6.984
1-1385
5-063
.1302
3-438
1.1191
2.094
46
1.1548
9.267
I.I553
6.952
1.1499
4-950
.1401
3.260
1.1275
1.867
47
1.1691
9.318
1.1686
6.902
1.1618
4.814
•1505
3-058
1-1363
1.613
48
1.1842
9-358
1.1826
6.834
I-I743
4-659
.1614
2.832
I-I455
1-333
49
I.2OO2
9-38S
1.1974
6-749
1.1876
4.480
.1728
2.578
I-I550
1-023
So
.2171
9.400
1.2131
6.645
1.2016
4-275
.1848
2.295
1.1650
0.681
•2351
9.400
1.2296
6.520
1.2163
4-045
•1975
1.981
I-I754
0.306
52
•2541
9-384
1.2472
6-373
1.2318
3-787
.2107
1.634
1.1863
0.103
53
.2744
9-354
1.2659
6.203
1.2483
3-499
•2247
1-253
1.1977
0.550
54
.2960
9-309
1.2857
6.008
1.2657
3.180
•2394
0-835
1.2096
1.039
55
.3190
9.246
1.3067
5.788
1.2842
2.826
•2549
0-379
I.222O
I-57I
56
•3435
9.166
1-3292
5-539
1-3038
2.436
.2713
O.I2I
1.2350
2.149
57
-3698
9.069
1-3532
5.261
1-3247
2.008
1.2886
0.666
1.2487
2.777
58
•3979
8-953
1.3788
4-953
1.3469
1.540
1.3069
1.259
1.2630
3.458
59
.4281
8.817
1.4063
4.611
1.3706
1.026
1.3263
1.905
1.2781
4-195
6o
1.4606
8.661
I-4358
4.232
1-3959
0.465
1-3470
2.606
1.2939
4-995
61
1.4956
8.484
I-4675
3.819
1-4230
0.145
1.3689
3.366
I.3I06
5.860
62
1-5334
8.286
1.5017
3.365
1.4521
0.810
1.3922
4.191
I.328I
6-793
63
1-5743
8.065
1.5386
2.869
1-4833
1-533
1.4170
5-085
1.3466
7.802
64
1.6187
7.819
I-5785
2.326
1.5169
2.318
1-4435
6.052
1.3662
8.891
65
1.6670
7-548
1.6218
1-734
I-553I
3.172
1.4718
7.099
1.3868
10.068
66
1.7198
7-252
1.6690
1.090
i-59~23
4.098
1.5022
8.232
1.4086
11-338
67
1.7777
6.928
1.7206
0.387
1-6347
5.104
1-5347
9-459
I-43I7
12.707
68
1.8413
6-575
1.7770
0.376
1.6807
6.196
1.5696
10.786
1.4562
14.184
69
1.9116
6.191
1-8389
1.209
1.7308
7.382
1.6070
12.221
1.4821
I5.776
70
1.9893
5-776
1.9072
2.116
1-7854
8.671
1.6474
13-773
I-5095
17.490
2.0759
5.326
1.9828
3.106
1.8451
10.074
1.6908
15-454
I-5385
19-337
72
2.1728
4.840
2.0667
4.187
1.9105
11.602
1.7376
17-273
I-5693
21.326
73
2.2819
4-3U
2.1603
5.371
1.9823
13.266
1.7881
19.244
I.60I9
23.466
74
2.4053
3-744
2.2652
6.670
2.0613
15.084
1.8425
21.379
1.6363
25.769
75
2.5461
3.127
2.3834
8.101
2.1484
17.076
1.9011
23.692
1.6725
28.245
76
2.7079
2-457
2.5172
9-685
2.2446
19.261
1.9641
26.202
I.7I05
30-905
77
2.8957
1.726
2.6696
11.447
2.3510
21.664
2.0317
28.925
I-7503
33-76o
78
3-IJ57
0.922
2.8442
13-420
2.4685
24-3I5
2.1040
31.881
I.79I7
36.820
79
3.3766
0.034
3-0455
15.644
2.5984
27.250
2.1809
35.088
I-8344
40.093
80
3.6909
0.960
3.2790
18.175
2.7416
30.510
2.2621
38.568
. 1.8781
43-590
81
4-0755
2.087
3-55I9
21.083
2.8987
34.140
2.3469
42.344
1.9222
47-319
82
4-5558
3-390
24.466
3.0700
38.191
2-4344
46.428
1.9661
51.281
83
5.1722
4-937
4.2487
28.446
3-2540
42.722
2.5230
50.842
2.OO9O
55-474
84
5.9896
6.839
4.6927
33-199
3-4477
47.793
2.6104
55.590
2.0497
59.894
85
7.1203
9.289
5.2102
38.945
3-6459
53.448
2.6940
60.673
2.0872
64.528
86
8.7720
12.668
5-7994
45-956
3-8392
59.720
2.7701
66.075
2.I2O2
69-359
87
11.388
17.791
6.4356
54-527
4-oi57
66.606
2.8350
71.768
2.1476
74-359
88
I5-990
26.760
7-0503
64.868
4.1592
74-043
2.8849
77.700
2.1680
79-494
89
24.844
46.011
7-5I93
76.896
4-253I
81.905
2.9163
83.805
2.1807
84.723
90 34.232 90.000 7.6968 90.000 4.2863 90.000 2.9271 90.000 2.1850 90.000
Note. Negative quantities are in heavy type.
Examples, tanh (1.6/54°) = 1.2960 /9°.309 = 1.2960 /9°.i8'.32*.
tanh (2.0/64^) = 1.3662 \8°.89i = 1.3662 \8°.53'.28'.
tr1 (1.4718 \7°.o99) = 1.9/65°.
TABLE III. HYPERBOLIC TANGENTS, tanh (p [S) = r /jy. CONTINUED
2.3
2.4
2.5
45
1.1065
1.007
i
.0932
0-155
1.0799
0.492
1.0672
0.961
1-0553
1.283
46
1.1134
0.746
i
.0987
0.127
1.0842
0.783
1.0702
1.252
1-0573
1.564
47
1. 1206
0.458
i
.1044
0.436
1.0884
I.IOI
1.0732
1.567
1.0592
1.868
48
1.1280
0.141
i
.1102
0-774
1.0927
1.446
1.0762
1.908
i. 0610
2.195
49
I-I357
0.206
i
.1161
1.142
1.0970
1.820
1.0790
2.277
1.0626
2.548
5°
I-I437
0.586
i
.1222
1.542
1.1013
2.226
1.0818
2.676
i .0640
2.929
5i
1.1519
I.OOI
i
.1284
1.978
1.1056
2.666
1.0845
3.107
1-0653
3-340
52
1.1604
1.452
i
•1347
2.452
1.1099
3-144
1.0870
3.572
1.0663
3-782
S3
1.1693
1-943
i
.1411
2.965
1.1142
3-659
1.0894
4.073
1.0671
4-256
54
1.1784
2.476
i
•1477
3-520
1.1185
4.215
1.0917
4.614
1.0676
4.766
55
1.1878
3-055
i
-1544
4.120
1.1227
4.815
1.0937
5.196
1.0678
5-314
56
1.1976
3.682
i
.l6l2
4.769
1.1268
5-462
1-0955
5.822
1.0677
5-903
57
1.2078
4.361
i
.1681
5-469
1.1309
6.160
1.0971
6.496
1.0672
6-534
58
1.2183
5-094
i
-I75I
6.226
1.1348
6.912
1.0984
7.220
1.0662
7.211
59
1.2292
5.888
i
.1822
7.042
1.1386
7.722
1.0993
7.998
1.0647
7.938
60
1.2405
6-745
i
.1894
7.921
1.1422
8-594
1.0999
8.835
1.0628
8.718
61
1.2522
7.671
i
.1967
8.869
1-1457
9-531
I.IOOI
9-735
i .0603
9.556
62
1.2644
8.669
i
.2041
9.889
1.1490
10.540
1.0999
10.701
1.0572
10.455
63
1.2771
9-744
i
.2Il6
10.988
1.1520
11.625
1.0992
11.740
1-0535
11.419
64
1.2903
10.902
i
.2192
12.171
1.1548
12.791
1 .0980
12.855
i .0490
12.455
65
I-3039
12.151
i
.2268
13-443
1-1573
14.044
1.0962
14-053
1.0438
13-567
66
1.3181
13-496
i
-2344
14.810
1-1595
15-390
1.0939
15-341
1.0378
14-761
67
I-3329
14.942
i
.2421
16.280
1.1613
16.836
1.0909
16.723
1.0310
16.044
68
1-3483
16.499
i
.2499
17-858
1.1628
18.389
1.0872
18.208
1.0232
17.424
69
1.3642
18.172
i
.2576
I9-552
1.1639
20.056
1.0829
19.802
1.0145
18.906
70
1.3808
19.968
i
.2654
21.370
1.1645
21.843
1.0779
21.514
1.0049
20.500
7i
1.3980
21.898
i
.2732
23-319
1.1647
23-759
1.0721
23-352
0.99427
22.215
72
I-4I59
23.968
i
.2809
25-405
1.1644
25.811
1-0655
25-324
0.98265
24.061
73
1-4344
26.185
i
-2885
27.636
i-.i637
28.006
1.0582
27-439
0.97006
26.048
74
1-4535
28.560
i
.2961
30.020
1.1625
30.352
1.0501
29.706
0.95649
28.187
75
1-4732
31.099
i
.3036
32.564
1. 1608
32.858
1.0413
32.134
0.94202
30.489
76
1-4934
33-8II
i
.3109
35-274
1.1587
35-531
1.0318
34-730
0.92671
32-966
77
1.5140
36.702
i
.3l8l
38.156
1.1561
38.374
1.0218
37-505
0.91068
35-633
78
I-5348
39-778
i
•3251
41.212
I-I53I
41.392
1.0113
40.467
0.89405
38.501
79
1-5559
43-043
1-3318
44-446
1.1498
44.590
1.0004
43.620
0.87701
41-582
80
1.5768
46.501
i.
.3383
47-859
1.1462
47.968
0.98931
46.968
0.85979
44-887
81
1-5974
50.151
i
•3444
51-449
1.1425
5I.527
0.97822
50-517
0.84264
48.427
82
1.6174
53.989
i
•35oo
55-212
1.1387
55-264
0.96733
54-267
0.82584
52.209
83
1.6364
58.011
i,
•3552
59.140
I-I349
59.I7I
0.95686
58.212
0.80977
56.233
84
1.6541
62.210
i,
•3599
63.223
1.1312
63-237
0.94710
62.344
0.79476
60.499
85
1.6700
66.570
i,
,3640
67.449
1.1279
67-450
0.93826
66.650
0.78120
64.994
86
1.6837
71.070
i,
3674
71.800
1.1250
71-793
0.93061
71.113
0.76946
69.703
87
1.6949
75.690
i.
37°2
76.254
1.1225
76.245
0.92439
75-710
0.75991
74.598
88
1.7031
80.406
i,
3722
80.790
1.1207
80.781
0.91979
80.413
0.75283
79-645
89
1.7081
85.187
i.
3734
85.381
1.1196
85.376
0.91696
85.188
0.74849
84.796
90
1.7099
90.000
I-3738
90.000
1.1192
90.000
0.91601
90.000
0.74702
90.000
Note.
Negative quantities are in heavy type.
Examples.
tanh (2.1 748°)
= 1.1280 /o°.i4i
= 1.1280
/o°.o8'.28*.
tanh (2.1 /49°) = 1.1357 \o°.2o6 = 1.1357 \o° i2'.22*.
(1.0318 \34°-73o) = 2.4 776°.
[18]
TABLE III. HYPERBOLIC TANGENTS, tanh
2.6 2.7 2.8
(p /$) = r /y. CONTINUED
2.9 3.0
0
o
o
o
0
0
45
1.0445
1.480
1.0348
1-576
1.0264
1-595
1.0192
1-554
1.0131
1.468
46
1.0456
1.744
I.0352
I.82I
1.0262
1.816
1.0185
1.748
I.OI2O
1.637
47
1.0465
2.030
1-0354
2.083
1.0257
2.051
1.0175
1-955
I.OI07
1.815
48
1-0473
2.337
1-0353
2.365
1.0250
2.303
1.0163
2.175
I.OOgi
2.OO2
49
1.0479
2.667
1.0350
2.666
1.0240
2.572
1.0148
2.409
I.OO72
2.198
50
1.0483
3.022
1.0344
2.988
1.0227
2.857
1.0129
2.656
I.OO49
2.405
51
1.0483
3.403
1-0335
3-332
I.02IO
3.160
1.0106
2.916
1.0023
2.622
52
1.0480
3.812
1.0323
3.700
I.OlSg
3-483
1.0080
3-191
0.99922
2.849
53
1-0475
4.250
1.0307
4.094
1.0164
3-826
1.0049
3.481
0.99572
3.086
54
1.0466
4.717
1.0286
4-513
i-OI35
4.190
1.0013
3.788
0.99174
3-334
55
1.0452
5.219
I.O26O
4-959
I.OIOI
4-577
0.99722
4.112
0.98722
3-593
56
1.0434
5.756
1.0230
5-437
1. 006 1
4.987
0.99254
4-452
0.98211
3.862
57
1.0412
6.331
I.OI94
5-945
1.0015
5-422
0.98724
4.810
0.97637
4.141
58
1-0385
6.947
I.OI52
6.487
0.99625
5-884
0.98126
5-186
0.96995
4-432
59
1-0351
7.606
I.OIO4
7.066
0.99027
6.374
0-97454
5-582
0.96279
4-735
60
1.0311
8.313
I.OO48
7.684
0-98353
6.895
0.96701
5-999
0.95481
5-049
61
1.0265
9.069
0.99847
8-343
0-97597
7-449
0.95862
6-439
0-94597
5-375
62
i. 02 1 1
9.879
0.99129
9.048
0.96751
8.036
0.94930
6.904
0.93619
5-713
63
1.0149
10.747
0.98322
9.801
0.95807
8.660
0.93897
7-392
0.92540
6.063
64
1.0079
11.677
O.9742O
10.607
0.94760
9.325
0.92756
7.907
0.91352
6.426
65
0.99990
12.675
0.96414
11.470
0.93603
10.033
0.91500
8.452
0.90047
6.801
66
0.99099
13.747
0.95300
12.393
0.92328
10.789
0.90121
9.027
0.88618
7.192
67
0.98106
14.898
0.94070
13-383
0.90927
H.599
0.88612
9.639
0.87055
7.598
68
0.97006
16.136
0.92720
14.449
0.89396
12.464
0.86964
10.289
0.85350
8.021
69
0-95794
17.469
0.91243
15-597
0.87726
13-396
0.85169
10.982
0.83495
8.463
70
0.94467
18.902
0.89635
16.833
0.85912
14-399
0.83221
11.722
0.81482
8.926
71
0.93022
20.451
0.87893
18.168
0.83950
15-482
0.81113
12.518
0.79302
9.414
72
0.91458
22.122
O.860I3
19-613
0.81834
16.655
0.78839
I3.378
0.76946
9.932
73
0.89775
23.929
0.83994
21.184
0.79562
I7.93I
0.76394
14-310
0.74408
10.485
74
0.87976
25.884
0.8l839
22.893
0.77134
19-325
0-73774
15-328
0.71683
11.081
75
0.86066
28.003
0-79552
24-757
o.7455i
20.857
0.70977
16.448
0.68765
11.729
76
0.84053
30.302
0.77140
26.799
0.71815
22.547
0.68005
17.690
0.65650
12.443
77
0.81949
32-799
O.746I2
29.038
0.68938
24:421
0.64861
19.078
0.62338
13-241
78
0.79769
35.5I3
0.71985
31-505
0.65930
26.512
0.61551
20.647
0.58829
14.146
79
0-77535
38.466
0.69278
34-231
0.62807
28.861
0.58087
22.437
0-55129
15.191
80
0.75271
41.678
O.6652O
37-249
0.59596
3L5I9
0.54486
24.504
0.51245
16.419
81
0.73010
45-171
0.63743
40.599
0.56326
34-543
0.50773
26.921
0.47190
17-893
82
0.70790
48.966
0.60990
44-322
0.53039
38.004
0.46981
29.784
0.42986
19.700
83
0.68652
53.078
0.58311
48.464
0.49791
41.989
0.43156
33-221
0.38661
21.976
84
0.66645
57.5I7
0.55765
53.060
0.46648
46.591
0-39363
37.403
0.34257
24.916
85
0.64820
62.286
0.53421
58.143
0.43691
5L907
0.35689
42.549
0.29833
28.839
86
0.63230
67.370
0.51353
63.722
0.41024
58.025
0-32255
48.931
0.25491
34-254
87
0.61928
72.740
0.49636
69.778
0.38762
64.988
0.29220
56.833
0.21393
41.990
88
0.60959
78.349
0.48347
76.253
0.37028
72.762
0.26791
66.456
0.17815
53-294
89
0.60361
84.129
0-47544
83.042
0-35929
81.195
0.25198
77.709
0.15225
69.480
90
0.60160
9O.OOO
0.47273
90.000
0-35553
90.000
0.24641
90.000
0.14255
90.000
Note. Negative quantites are in heavy type.
Examples tanh (2.6 765°) = 0.99990 /i2°.675 = 0.99990 /i2°.40/.3O*r.
tanh"1 (0.88618 \7°.i92) = 3.0/66°.
[19]
sinh 0
1ABLE,
1 V . ^UJX-IVC.^ .
L ±r\ \j r /\v^ i u
e ''
'1
O.I
0.2
o-3
0.4
0-5
0
45
46
47
48
49
o
i.ooooo 0.096
0-99993 o-°95
0.99987 0.095
0.99981 0.094
0.99975 0.094
o
i.ooooo 0.383
0.99975 0.382
0.99950 0.381
0.99930 0.380
0.99910 0.378
o
i.ooooo 0.860
0.99950 0.858
0.99897 0.856
0.99847 0.854
0.99797 0.852
o
1.00013 1.532
0.99920 1.529
0.99828 1.526
0-99733 1-520
0.99640 1.513
o
1.00032 2.391
0.99888 2.388
0.99744 2.385
0.99598 2.378
0.99454 2.368
So
Si
52
53
54
0.99970 0.094
0.99965 0.093
0.99960 0.092
0.99955 0.091
0.99950 0.090
0.99880 0.376
0.99860 0.374
0.99840 0.371
0.99815 0.367
0.99795 0.362
0.99743 0.848
0.99690 0.842
0.99640 0.834
0.99590 0.826
0.99540 0.818
0.99550 1.506
0.99460 1.497
0.99370 1.486
0.99280 1.472
0.99190 1.458
0.99312 2.357
0.99170 2.342
0.99028 2.325
0.98888 2.304
0.98748 2.281
55
56
57
58
59
0.99944 0.089
0.99939 0.088
0-99933 0.086
0.99928 0.085
0.99922 0.083
0-99775 0.357
0.99755 0.352
0.99730 0.347
0.99705 0.342
0.99685 0.336
0.99490 0.809
0.99443 0.799
0.99397 0.787
0.99347 0.773
0.99300 0.760
0.99103 1.440
0.99018 1.421
0.98930 1.400
0.98845 1.378
0.98763 1.354
0.98610 2.254
0.98476 2.226
0.98344 2.195
0.98212 2.l6o
0.98082 2.123
60
61
62
63
64
0.99917 0.082
0.99912 0.081
0.99907 0.079
0.99902 0.077
0.99897 0.075
0.99670 0.331
0.99645 0.324
0.99625 0.317
0.99605 0.309
0.99585 0.301
0.99257 0.746
0.99213 0.731
0.99170 0.715
0.99127 0.698
0.99083 0.680
0.98685 1.330
0.98603 1.302
0.98523 1.273
0.98448 1.243
0-98373 I-212
0.97956 2.085
0.97832 2.042
0.97710 1.998
0.97590 1.950
0.97476 1.900
65
66
67
68
69
0.99893 0.073
0.99889 0.071
0.99885 0.069
0.99881 0.066
0.99877 0.064
0.99570 0.293
0-99555 0.284
0.99540 0.275
0.99520 0.265
0.99505 0.255
0.99040 0.661
0.99000 0.641
0.98963 0.621
0.98927 0.599
0.98891 0.577
0.98300 1.179
0.98232 1.144
0.98165 1.108
0.98100 1.070
0.98035 1.030
0.97364 1.848
0.97254 1.794
0.97150 1.738
0.97046 1.679
0.96946 1.617
7o
72
73
74
0.99873 0.062
0.99869 0.059
0.99865 0.057
0.99861 0.054
0.99858 0.051
0.99490 0.245
0.99475 0.235
0.99460 0.225
0.99445 0.214
0.99435 0.203
0.98857 0.555
0.98823 0.532
0.98790 0.508
0.98760 0.483
0.98733 0.458
0-97975 0.990
0.97918 0.948
0.97863 0.905
0.97808 0.861
0.97758 0.817
0.96852 1.554
0.96760 1.489
0.96676 1.422
0.96592 1.354
0.96514 1.284
75
76
77
78
79
0.99855 0.048
0.99852 0.045
0.99850 0.042
0.99847 0.039
0.99845 0.036
0.99425 0.192
0.99415 0.180
0.99405 0.168
0-99395 0.156
0.99385 0.144
0.98707 0.432
0.98680 0.406
0.98653 0.379
0.98633 0.351
0.98613 0.322
0.97710 0.771
0.97665 0.724
0.97625 0.676
0.97585 0.628
0-97545 0-578
0.96440 1. 212
0.96368 1.138
0.96304 I.O62
0.96246 0.986
0.96190 0.909
80
81
82
83
84
0.99843 0.033
0.99841 0.030
0.99839 0.026
0.99838 0.023
0.99837 O.O2O
0-99375 0.131
0.99365 0.118
0.99360 o.i 06
0.99355 0.093
0.99350 0.080
0.98593 0.294
0.98577 0.266
0.98563 0.238
0-98553 0.209
0.98543 0.180
0.97512 0.528
0.97483 0.477
0.97458 0.425
0-97433 0.373
0.97413 0.321
0.96134 0.830
0.96088 0.750
0.96046 0.669
0.96008 0.587
0.95974 0.505
85
86
87
88
89
0.99836 O.OI7
0.99835 0.014
0.99834 o.on
0.99833 0.008
0.99832 0.004
0-99345 0.067
0.99340 0.054
0-99335 0.041
0-99335 0.028
0-99335 0.014
0.98537 0.150
0.98530 O.I2O
0,98523 0.090
0.98517 O.66o
0.98510 O.O3O
0-97395 0.268
0.97380 0.215
0.97370 0.162
0.97365 0.108
0.97360 0.054
0.95944 0.422
0.95920 0.338
0.95904 0.254
0.95894 0.170
0.95890 0.085
90
0.99831 o.ooo
0-99335 o.ooo
0.98507 o.ooo
0-97355 o.ooo
0.95886 o.ooo
Note.
sinh 0 h
en 0 = 0/5..
— i.o wn
6
,
Example.
sinh (0.3 769°)
0.98891 /o°.577
— o 98891 /o°.34',
,37'.
Q-3 /69°
[20]
1AULK J.V. ^
UXS-IVC^ i 11\ \J J
6
JNTINUED
0
45
46
47
48
49
0.6
0
1.00070 3.440
0.99863 3.437
0.99655 3-434
0.99445 3426
0.99237 3-4I4
0.7
o
1.00134 4.676
0.99849 4.679
0.99564 4.676
0.99281 4 666
0.98999 4.652
0.8
o
1.00230 6.108
0.99856 6.112
0.99484 6.109
0.99114 6.099
0.98745 6.082
0.9
o
1.00363 7.728
0.99893 7.735
0.99425 7.734
0.98955 7.725
0.98488 7.707
1.0
o
1.00553 9-531
0.99975 9.546
0.90394 9-550
0.98816 9.543
0.98242 9.525
50
52
53
54
0.99030 3.398
0.98825 3.379
0.98623 3.355
0.98421 3.326
0.98220 3.293
0.98717 4.632
0.98439 4.606
0.98161 4-574
0.97887 4-537
0.97616 4.493
0.98379 6.058
0.98015 6.026
0-97655 5-987
0.97298 5.940
0.96944 5.886
0.98026 7.679
0.97567 7-642
0.97111 7.595
0.96659 7.538
0.96211 7.472
0.97672 9-495
0.97105 9.453
0.96543 9.399
0.95986 9.333
0-95435 9-255
55
56
57
58
59
0.98023 3.256
0.97830 3.215
0.97638 3.171
0.97448 3.122
0.97262 3.069
0.97349 4.445
0.97084 4.391
0.96823 4.331
0.96564 4.265
0.96313 4.193
0.96594 5.824
0.96249 5.755
0-95909 5-678
0-95574 5-593
0.95244 5.502
0.95769 7-396
0-95333 7-312
0.94904 7.218
0.94482 7.114
0.94067 7.000
0.94890 9.166
0-94353 9-065
0.93825 8.952
0.93305 8.827
0.92795 8.691
6o
61
62
63
64
0.97081 3.013
0.96903 2.953
0.96728 2.889
0.96557 2.821
0.96390 2.749
0.96067 4.117
0.95826 4.036
0.95589 3-951
0.95356 3-859
o-gs^o 3-762
0.94923 5.405
0.94608 5.300
0.94299 5.189
0.93996 5.070
0.93703 4-944
0.93661 6.878
0.93263 6.747
0.92874 6.607
0.92493 6.458
0.92121 6.300
0.92295 8.544
0.91805 8.385
0.91325 8.215
0.90856 8.033
0.90400 7.841
65
66
67
68
69
0.96228 2.674
0.96072 2.596
0.95922 2.515
0.95775 2.430
0.95632 2.342
0.94911 3.661
0.94697 3.554
0.94493 3.444
0.94293 3.329
0.94100 3.209
0.93416 4.812
0.93140 4.674
0.92873 4.529
0.92614 4.379
0.92364 4.223
0.91761- 6.136
0.91411 5.962
0.91074 5.780
0.90747 5.590
0.90430 5.392
0.89957 7.637
0.89527 7.424
0.89111 7-2OI
0.88708 6.968
0.88320 6.723
70
72
73
74
0.95495 2.251
0.95365 2.157
0.95242 2.061
0.95123 1.962
0.95010 1.860
0.93914 3.085
0.93737 2.957
0.93569 2.825
0.93407 2.689
0.93254 2.550
0.92123 4.061
0.91891 3.894
0.91671 3.722
0.91461 3.544
0.91261 3.361
0.90127 5.187
0.89837 4.975
0.89558 4.756
0.89292 4.530
0.89041 4.298
0.87947 6.469
0.87589 6.207
0.87247 5-936
0.86921 5.656
0.86612 5.368
75
76
77
78
79
0.94902 1.756
0.94802 1.649
0.94707 1.540
0.94620 1.429
0.94540 1.317
0.93109 2.408
0.92973 2.263
0.92846 2.114
0.92727 1.962
0.92617 i. 808
0.91073 3.174
0.90895 2.982
0.90729 2.787
0.90574 2.588
0.90431 2.385
0.88804 4.059
0.88581 3.815
0.88372 3.566
0.88177 3.312
0-87997 3.053
0.86320 5.072
0.86045 4.769
0.85788 4.458
0.85549 4-I4I
0.85328 3:818
80
81
82
83
84
0.94465 1.203
0.94397 1.087
0.94337 0.970
0.94283 0.852
0.94237 0.732
0.92516 1.652
0.92424 1.493
0.92343 1.332
0.92270 1.169
0.92207 1.005
0.90301 2.179
0.90183 1.969
0.90076 1.757
0.89981 1.543
0.89899 1.327
0.87831 2.789
0.87681 2.522
0.87547 2.251
0.87428 1.976
0.87324 1.699
0.85125 3.489
0.84940 3.156
0.84774 2.817
0.84628 2.474
0.84501 2.128
85
86
87
88
89
0.94197 0.612
0.94163 0.490
0.94138 0.368
0.94122 0.246
0.94110 0.123
0.92153 0.839
0.92109 0.673
0.92074 0.506
0.92050 0.338
0.92036 0.169
0.89829 1.109
0.89771 0.889
0.89726 0.668
0.89694 0.446
0.89675 0.223
0.87237 1.420
0.87164 1.138
0.87108 0.855
0.87068 0.571
0.87043 0.286
0.84393 1-779
0.84305 1.426
0.84236 1.071
0.84186 0.715
0.84156 0.358
90
0.94107 o.ooo
0.92031 o.ooo
0.89670 o.ooo
0.87037 o.ooo
0.84147 o.ooo
• E^p,,
sinh (1.0/85°)
0.84393 /i°.779
-0 84101/1° 46'
.44".
1.0/85°
[21]
sinh d
1ABI
I.I
,E IV.
L,UKK£,t
1.2
_HJNLr ? AC 1 UK. - T /y. L.ON
i-3 i-4
TINUED
1-5
0
0
0
0
O
o
45
1.0081
11.519
I.OII5
13.692
1.0158
16.034
1. 02 1 2
18.568
1.0279
21.262
46
I.OOII
"•543
I.003I
13.726
i. 0060
16.092
I.OO99
18.639
1.0148
21-355
47
0.99409
"•555
0-99475
I3.748
0-99623
16.128
0.99850
18.689
I.OOlS
21.426
48
0.98709
"•553
0.98650
13-753
0.98654
16.142
0.98729
18.717
0.98893
21-474
49
0.98018
0.97825
I3.74I
0.97685
16.137
0.97607
18.723
0.97613
2 1 .496
50
0.97327
11.506
0.97008
13.712
0.96722
16.113
0.96493
18.707
0.96340
21.492
51
0.96636
11.461
0.96192
13.666
0.95770
16.068
0-95393
18.668
0.95080
21.462
52
0.95964
11.401
0.95383
13.603
0.94822
16.004
0.94300
18.605
0.93833
21.405
53
0.95291
11.327
0-94583
I3.523
0.93892
I5-9I9
O.9322I
18.519
0.92593
21.321
54
0.94627
11.239
0.93800
I3.42S
0.92969
15.814
0.92157
18.409
0.91373
2I.2IO
55
0.93963
11.136
0.93017
I3.3IO
0.92054
15.688
O.9IIOO
18.275
0.90167
2I.O72
56
0.93318
11.019
0.92250
13.178
0.91154
I5.542
0.90064
18.117
0.88980
20.907
57
0.92682
10.888
0.91492
13.028
0.90269
I5-376
0.89043
17.936
0.87813
20.713
58
0.92054
10.742
0.90750
12.860
0.89400
15.188
0.88036
17-730
0.86667
2O.49I
59
0.91436
10.581
0.90017
12.675
0.88546
14.980
0.87050
17.500
0.85540
2O.240
6o
0.90836
10.406
0.89300
12.474
0.87715
14-752-
0.86086
17-246
0-84433
19.964
61
0.90244
10.218
0.88600
12.255
0.86892
14.502
0.85136
16.967
0-83353
I9-657
62
0.89666
10.016
0.87918
12.019
0.86092
14.232
0.84214
!6.663
0.82300
19.321
63
0.89103
9.800
0.87250
11.767
0.85315
13-942
0.83314
16.335
0.81273
18.956
64
0.88554
9-570
0.86600
11.497
0.84554
13.632
0.82436
15-983
0.80267
18.562
65
0.88019
9-327
0.85967
II. 211
0.83808
I3.30I
0.81579
15.607
0.79293
18.140
66
0.87501
9.071
0.85350
10.909
0.83094
12.950
0.80750
15.207
0.78347
17.690
67
0.86998
8.801
0.84758
10.594
0.82400
12.580
0.79950
14-783
0-77433
17.211
68
0.86514
8.519
0.84183
10.257
0.81731
12.191
0.79171
14-335
0.86547
16.704
69
0.86045
8.225
0.83625
9.907
0.81077
11.783
0.78421
13-865
0.75693
16.169
70
0.85596
7.918
0.83093
9-543
0.80454
"•357
0.77707
13-373
0.74873
15.607
71
0.85165
7.600
0.82583
9.164
0.79862
10.912
O.77O2I
12.858
0.74087
I5.0I8
72
0.84754
7.271
0.82095
8-771
0.79292
10.450
0.76364
12.322
0-73333
14.402
73
0.84362
6.931
0.81631
8.365
0.78746
9.971
0.75736
11.765
0.72620
I3.76I
74
0.83989
6.580
0.81190
7.946
0.78231
9-476
11.188
0.71940
13.096
75
0.83637
6. 2 2O
0.80773
7.514
0.77746
8.966
0-74579
10.591
0.71293
12.406
76
0.83306
5.850
0.80380
7.069
0.77287
8.440
0.74050
9-975
0.70687
11.693
77
0.82996
5-471
0.80013
6.614
0.76858
7.900
0-73557
9-342
0.70120
10.958
78
0.82707
5.084
0.79671
6.148
0.76459
7-346
0.73093
8.692
0-69593
10.201
79
0.82441
4.689
0-79355
5.672
0.76090
6.780
0.72664
8.026
0.69107
9-425
80
0.82196
4.286
0.79065
5-187
0-75751
6.202
0.72279
7-346
0.68660
8.630
81
0.81975
3.877
0.78802
4-693
0-75444
5.614
0.71921
6.651
0.68253
7.8l8
82
0.81775
3.462
0.78566
4.191
0-75165
5.016
0.71600
5-944
0.67887
6.991
83
0.81599
3.041
0.78358
3-683
0.74923
4.408
0.71319
5.227
0.67567
6.149
84
0.81445
2.616
0.78175
3.169
0-747"
3-793
0.71073
4-499
0.67287
5-294
85
0.81315
2.186
0.78022
2.649
0-74531
3-i7i
0.70864
3-762
0.67047
4.429
86
0.81209
1-753
0.77895
2.124
0.74383
2-544
0.70694
3.018
0.66847
3-554
87
0.81125
1.317
0-77797
1.596
0.74268
1.912
0.70561
2.269
0.66693
2.673
88
0.81066
0.879
0.77727
1.065
0.74185
1.277
0.70466
1.515
0.66587
1-785
89
0.81031
0.440
0.77684
0-533
0.74136
0.639
0.70409
0.758
0.66521
0.893
90
0.81019
o.ooo
0.77670
o.ooo
0.74120
o.ooo
0.70389
o.ooo
0.66499
o.ooo
E:
camole.
sinh (1.5
765°)
n ^rnn? /
- o *7n/?n-5
/TR° <•>«':•:
/65°
[22]
1ABLK IV. l^UJK.JK.Jl,V^lliN»j
1.6 1.7
r-rt^lUJ*.. - m ' IJk CONTINUED
1.8 1.9 2.O
o
o
b
0
0
o
45
1-0359
24.117
1.0456
27.I33
1.0572
30.292
1.0707
33.590
1.0863
37.016
46
1. 02 1 1
24.241
1.0290
27.288
1.0385
30.486
1.0499
33.829
1-0633
37.304
47
I.O064
24-339
1.0124
27.418
1.0199
30.651
1.0292
34.037
1.0405
37.560
48
0.99175
24.409
0.99582
27.5I5
1.0015
30.784
1.0087
34-2io
1.0178
37.778
49
0.97719
24451
0.97947
27.582
0.98317
30.883
0.98837
34-346
0-99535
37.958
50
0.96275
24-465
0.96323
27.618
0.96500
30.946
0.96821
34-444
0.97310
38.098
Si
0.94844
24-449
0.94718
27.620
0.94700
30.974
0.94821
34-503
0.95110
38.196
52
0.93431
24.402
0.93123
27.590
0.92922
30.966
0.92858
34-524
0.92940
38.253
53
0.92031
24-325
0.91547
27.527
0.91167
30.921
0.90911
34.505
0.90800
38.267
54
0.90650
24.218
0.89994
27.429
0.89433
30-839
0.88995
34-444
0.88685
38.236
55
0.89281
24.079
0.88465
27.296
0.87728
30.718
0.87106
34-342
0.86600
38.160
56
0.87938
23.910
0.86953
27.128
0.86044
30.558
0.85242
34-197
0.84555
38.039
57
0.86619
23.709
0.85471
26.924
0.84389
30.358
0.83411
34.008
0.82545
37.869
58
0.85319
23-475
0.84012
26.683
0.82767
30.116
0.81611
33-773
0.80570
37-650
59
0.84044
23.209
0.82582
26.406
0.81172
29-834
0.79847
33-493
0.78630
37.382
60
0.82794
22.910
0.81176
26.091
0.79611
29.510
0.78121
33-I67
0.76735
37.062
61
0.81569
22.578
0.79806
25.738
0.78083
29.142
0.76437
32.792
0.74880
36.688
62
0.80375
22.212
0.78464
25-347
0.76594
28.730
0-74795
32-368
0.73070
36.261
63
0.79213
2I.8I3
0.77159
24.917
0.75139
28.274
0.73179
31-803
0.71300
35.778
64
0.78075
21.380
0.75888
24.447
0.73722
27.774
0.71611
3I-368
0-69575
35.238
65
0.76975
20.913
0.74647
23.938
0.72344
27.228
0.70084
30.792
0.67900
34.639
66
0.75906
20.413
0.73447
23.390
0.71006
26.636
0.68605
30.162
0.66275
33.980
67
0.74869
19.879
0.72282
22.802
0.69711
25-997
0.67168
29.478
0.64700
33-261
68
0.73869
19.311
0.71159
22.175
0.68455
25-3"
0.65779
28.741
0.63175
32.479
69
0.72900
I8.7IO
0.70076
21.508
0.67244
24.580
0.64421
27.949
0.61700
31-634
70
0.71975
18.076
0.69035
20.801
0.66083
23.802
0.63153
27.102
0.60280
30.725
0.71081
17.410
0.68035
20.055
0.64967
22.976
0.61916
26.199
0.58915
29-750
72
O.7023I
16.712
0.67076
19.271
0.63900
22.104
0.60731
25.241
0.57605
28.709
73
0.69419
15.982
0.66165
18.448
0.62878
21.187
0.59600
24.227
0-56355
27.602
74
0.68650
15.221
0.65300
17.588
0.61911
20.224
0.58526
23-158
o-55l6S
26.428
75
0.67919
I4.43I
0.64476
16.692
0.60994
19.215
0.57500
22.034
0-54035
25-187
76
0.67231
I3.6I3
0.63700
15.760
0.60128
18.162
0.56542
20.856
0.52970
23.880
77
0.66588
12.767
0.62976
14.794
0.59317
17.068
0.55637
19.625
0.51970
22.507
78
0.65988
11.894
0.62300
13-795
0.58561
15-933
0-54795
18-343
0-51035
21.070
79
0.65431
10.997
0.61676
12.765
0.57861
I4.758
0.54011
17.012
0.50165
19-571
80
0.64925
IO.O76
0.61106
11.705
0.57217
13-545
0.53295
I5-633
0.49364
18.01 1
81
0.64463
9-J34
0.60582
10.618
0.56633
12.298
0.52642
14.209
0.48636
16.391
82
0.64050
8.171
0.60112
9-505
0.56106
11.019
0.52052
12-744
0.47979
14.722
83
0.63681
7.191
o.597oo
8.370
0.55639
9.711
0-51530
11.240
0.47396
13.000
84
0.63363
6.194
0-59341
7.214
0.55234
8-374
0.51075
9.702
0.46888
11.233
85
0.63094
5.184
0.59035
6.039
0.54890
7-oi5
0.50689
8-133
0.46455
9.426
86
0.62875
4.161
0.58782
4.850
0.54607
5-636
0.50372
6-539
0.46101
7.584
87
O.627OO
3.129
0.58586
3-649
0.54387
4.241
0.50124
4-923
0.45823
5-713
88
0.62575
2.090
0.58445
2.438
0.54229
2-834
0.49947
3.291
0.45624
3-821
89
O.6250O
1.046
0.58361
1.230
0.54135
1.419
0.49841
1.648
0.45505
1.914
90
0.62473
o.ooo
0.58333
o.ooo
0.54103
o.ooo
0.49805
o.ooo
0.45465
o.ooo
Example.
sinh (i
•8/77°)
O CO? I 7 ,
^I7°.o68
= O.CQ?!? /I7°.4'
.05'.
1.8/77°
<~'oyoi / i
[23]
sinh 6
1ABLE IV.
2.1
UUiVKJl,^ JL 1IN (j
2.2
rA^iUK.. - - r 77. CON
2.3 2.4
TINUED
2-5
o
o
0
o
o
O
45
1.1043
40.558
I.I248
44.205
1.1480
47.946
1.1740
5L769
1.2032
55-661
46
1.0789
40.905
1.0970
44.613
1.1177
48.419
1.1411
52.312
1.1674
56.278
47
1.0538
41.213
1.0695
44.981
1.0877
48.851
1.1085
52.813
1.1320
56.852
48
1.0289
41.482
1.0423
45-307
1.0580
49.241
1.0762
53.27I
1.0972
57.383
49
1.0043
41.711
I.OI53
45-592
1.0286
49.588
1.0444
53-685
1.0627
57.869
50
0.97986
41.898
0.98864
45-834
0.99965
49.890
1.0130
54-053
J.0288
58.309
Si
0.95576
42.042
0.96232
46.031
0.97104
50.146
0.98200
54-375
0.99540
58.701
52
0.93195
42.142
0.93636
46.181
0.94283
50.354
0.95150
54-648
0.96256
59.044
53
0.90848
42.197
0.91077
46.284
0.91504
50.5H
0.92154
54-871
0.93028
59-337
54
0.88533
42.205
0.88559
46.338
0.88774
50.623
0.89208
55-042
0.89860
59-578
55
0.86257
42.165
0.86082
46.342
0.86091
50.680
0.86308
55.i6o
0.86748
59-765
56
0.84019
42.076
0.83646
46.294
0-83457
50.683
0.83467
55-223
0.83700
59.896
57
0.81819
41.935
0.81255
46.193
0.80869
50.629
0.80683
55-228
0.80712
59-970
58
0.79662
41.742
0.78909
46.035
0.78331
50.518
0-77954
55-174
0.77788
59.984
59
0-77543
41.494
0.76609
45.820
0.75848
50.347
0.75283
55-059
0.74932
59-936
60
0.75471
41.191
0-74359
45-546
o.734i7
50.114
0.72671
54.880
0.72140
59-823
61
0-73447
40.830
0.72159
45-2io
0.71043
49.816
0.70121
54.634
0.69412
59.642
62
0.71467
40.410
O.7OOO9
44.811
0.68725
49-451
0.67633
54-3I9
0.66752
59-391
63
0.69538
39.929
0.67914
44-345
0.66465
49.016
0.65204
53-931
0.64160
59.067
64
0.67657
39-386
0.65873
43.811
0.64261
48.509
0.62838
53-467
0.61636
58.665
65
0.65829
38.777
0.63886
43.207
0.62113
47.926
0.60533
52.924
0.59176
58.182
66
0.64053
38.101
0.61955
42.529
0.60026
47.264
0.58296
52-297
0.56784
57.613
67
0.62329
37-357
O.6OO82
41-775
0.58004
46.519
0.56121
51-583
0.54464
56.954
68
0.60657
36-542
0.58268
40.943
0.56043
45-688
0.54013
50.777
0.52208
56.198
69
0.59043
35-655
0.56514
40.029
0.54143
44.767
0.51971
49.874
0.50024
55-341
70
0.57487
34-694
0.54818
39.030
0.52309
43-752
0.49996
48.868
0.47908
54-376
71
0.55990
33-656
0.53186
37-944
0-50539
42.638
0.48088
47-754
0.45864
72
0-54557
32.541
0.51618
36.769
0.48835
41.422
0.46246
46.526
0.43888
52.092
73
0.53181
3L348
0.50118
35-501
0.47204
40.100
0.44479
45-177
0.41988
50.755
74
0.51871
30.074
0.48682
34.I38
0.45644
38-667
0.42783
43-701
0.40160
49-277
75
0.50629
28.719
'0.47323
32-678
0.44152
37.118
0.41165
42.091
0.38408
47.648
76
0-49453
27.283
0.46027
31.119
0.42736
35-449
0.39621
40.341
0.36732
45-857
77
0.48347
25.766
0.44810
29-459
0.41397
33-659
0.38157
38.443
0.35137
43-891
78
0.47311
24.168
0.43666
27.699
0.40135
31-744
0.36775
36.390
0.33625
41.740
79
0.46349
22.491
0.42601
25-839
0.38963
29.701
0.35480
34-I78
0.32201
39-392
80
0.45461
20.736
0.41617
23.879
0.37873
27.531
0.34273
31-803
0.30869
36.836
81
0.44650
18.907
0.40716
21.822
0.36873
25-233
0.33162
29.261
0.29633
34-063
82
0.43919
17.008
0.39902
19.671
0-35965
22.810
0.32149
26.550
0.28500
31-065
83
0.43268
15-043
0.39176
I7-432
0.35153
20.267
0.31239
23.676
0.27477
27.841
84
0.42699
13-015
0.38540
15.111
0.34441
17.611
0.30438
20.643
0.26569
24.390
85
0.42215
10.933
0.37998
12.715
0.33831
14.851
0.29748
17.462
0.25784
20.723
86
0.41818
8.806
0-37551
10.254
0.33328
I2.OOO
0.29178
14.147
0.25131
16.855
87
0.41507
6.639
O.372O2
7.740
0.32933
Q.07I
0.28728
10.717
0.24614
12.811
88
0.41284
4-443
0.36951
5-183
0.32650
6.082
0.28405
7.197
0.24241
8.624
89
0.41150
2.226
0.36800
2-599
0.32479
3-05I
0.28210
3-615
0.24014
4.338
90
0.41105
o.ooo
0.36750
o.ooo
0.32422
O.OOO
0.28144
o.ooo
0.23939
0.000
Example.
sinh (2.
3/84!)
= n 7/1/1/1 T ,
f T *7^ fkT T
/I7° 36'.
ff
2.3/84° -«™->-i -"
'
/ * / *ov
[24]
1ABLE J.V.
2.6
^UJS.K.H,^ 1 1IN Vj
2-7
r A^ J. U.K..
0
2.8
- r 77. CONTINUED
2.9 3.0
0
o
e
o
o
o
45
1.2354 59-6I3
1.2710 63.614
I.3I02 67.653
1-3530 7L72I
1-3995 75.8I4
46
1.1967 60.307
1.2293 64.386
1.2652 68.506
1.3046 72.658
1-3475 76-832
47
1.1586 60.957
1.1881 65.116
I.22O9 69.317
1.2570 73-551
1.2965 77.808
48
I.I209 61.564
1.1476 65.801
I.I774 70.084
1.2103 74-400
1.2465 78.741
49
1.0838 62.126
1.1077 66.442
I.I345 70.806
1.1644 75.205
1.1974 79-630
50
1.0473 62.641
1.0685 67.037
1.0925 71.483
1.1195 75.965
1.1494 80.474
Si
I.OII3 63.109
1.0300 67.585
1.0513 72.113
1-0755 76.679
1.1025 81.272
52
0.97608 63.528
0.99219 68.084
I.OIO9 72.694
1.0324 77-345
1.0567 82.024
53
0.94145 63.897
0.95514 68.532
0.97143 73-226
0.99038 77-963
I.OI2O 82.729
54
0.90750 64.214
0.91881 68.929
0.93279 73-708
0.94928 78.532
0.96837 83.386
55
0.87419 64.476
0.88333 69.274
0.89500 74-138
0.90917 79.050
0.92590 83.994
56
0.84162 64.684
0.84863 69.563
0.85811 74.515
0.87007 79-5I7
0.88453 84.553
57
0.80973 64.834
0.81467 69.797
0.822II 74.837
0.83197 79-932
0.84433 85.062
58
0.77854 64.925
0.78152 69.973
0.78700 75.103
0.79487 80.293
0.80523 85.510
59
0.74808 64.954
0.74919 70.088
0.75275 75-3H
0.75876 80.598
0.76727 85.924
Oo
0.71835 64.918
0.71767 70.140
0.71943 75-459
0.72366 80.847
0.73040 86.276
61
0.68935 64.816
0.68692 70.128
0.68700 75.544
0.68955 81.037
0.69463 86.574
62
0.66104 64.644
0.65700 70.047
0.65543 75-565
0.65641 81.167
0.65993 86.8l7
63
0.63346 64.399
0.62786 69.895
0.62475 75.5I9
0.62424 81.235
0.62630 87.004
64
0.60662 64.077
0.59952 69.669
0-59493 75-403
0.59303 81-239
0-59373 87.133
65
0.58054 63.674
0.57193 69.364
0.56596 75.214
0.56272 81.176
0.56217 87.204
66
0.55519 63.185
0.54514 68.978
0.53785 74-948
0-53334 81.044
0.53160 87.214
67
0.53054 62.606
0.51911 68.504
0.51054 74-599
0.50486 80.838
O.5O2O3 87.161
68
0.50658 61.930
0.49386 67.935
0.48404 74.164
0.47724 80.555
0.47340 87.043
69
0.48338 61.151
0.46937 67.266
0.45832 73.634
0.45045 80.190
0.44567 86.858
70
0.46089 60.262
0.44559 66.489
0-43343 73-ooS
0.42448 79.738
0.41883 86.602
0.43911 59.254
0.42256 65.596
0.40928 72.267
0-39934 79-191
0.39283 86.271
72
0.41808 58.117
0.40026 64.576
0.38586 71.411
0-37497 78.541
0.36767 85.860
73
0-39773 56-841
0.37872 63.415
0.36321 70.424
0.35135 77.780
0.34327 85.363
74
0-37814 55-4I4
0.35789 62.101
0.34125 69.292
0.32845 76.894
0.31964 84.772
75
0.35930 53-822
0.33780 60.617
0.32002 67.996
0.30627 75.867
0.29674 84.077
76
0-34123 52.049
0.31846 58.943
0.29951 66.516
0.28478 74.680
0.27452 83.265
77
0.32395 50-079
0.29987 57.058
0.27971 64.827
0.26397 73.309
0.25296 82.319
78
0.30749 47-893
0.28206 54.935
0.26064 62.897
0.24379 71.721
0.23202 81.216
79
0.29189 45.471
0.26507 52.542
0.24231 60.686
0.22432 69.876
0.21168 79.925
80
0.27718 42-793
0.24894 49-847
0.22476 58.145
0.20551 67.721
0.19191 78.404
81
0.26346 39.837
0.23374 46.812
0.20804 55-217
0.18738 65.183
0.17270 76.592
82
0.25078 36.583
0.21954 43-396
0.19222 51.833
0.16998 62.169
0.15404 74-404
83
0.23922 33.016
0.20644 39.554
0.17741 47.908
0.15343 58.553
0.13596 71-709
84
0.22887 29.126
0.19458 35.252
0.16375 43-351
0.13774 54.i68
0.11849 68.313
85
0.21986 24.910
0.18409 30.458
0.15142 38.066
0.12323 48.795
0.10174 63.903
86
0.21227 20.382
0.17515 25.164
0.14069 31.968
0.11016 42.168
0.085917 57-974
87
0.20623 15.571
0.16794 19.390
0.13184 25.015
0.098934 34-005
0.071443 49-701
88
0.20184 10.524
0.16264 13.194
0.12520 17.244
0.090148 24.110
0.059103 37.843
89
0.19917 5.307
0.15938 6.682
0.12105 8.809
0.084469 12.576
0.050307 21.091
90
0.19827 o.ooo
0.15829 o.ooo
0.11964 o.ooo
0.082845 o.ooo
0.047040 o.ooo
Example.
sinh (3.0 /86°)
0.085917 /57°-974
J'.26*.
3-Q /86°
[25]
TABLE
V. CORRECTING FACTOF
tanh
ti.
0
O.I
0.2
0.3
0.4
o.
5
0
45
46
47
48
49
I.OOOOO
I.OOOIO
I.OO02I
1.00033
1.00044
o
o.i 88
0.188
0.188
0.188
0.187
0.99985
I.O003O
I.OO075
I.OOI2O
I.OOI7O
0
0.765
0.764
0.763
0.76l
0.758
0.99937
1.00040
1.00143
1.00250
1.00357
0
I.7I8
I.7I9
1.720
I.7I7
I.7IO
0.99803
0.99988
1.00173
1.00358
1.00543
o
3.046
3.055
3.058
3.058
3.053
0.99514
0.99804
1.00094
1.00384
1 .00680
o
4.750
4.771
4.781
4.787
4.787
So
52
53
54
1.00056
1.00068
1.00080
1.00092
1.00103
0.187
0.187
0.186
0.184
0.182
I.OO2I5
I.OO265
I.O03IO
1-00355
I.OO4O5
0-755
0.749
0.744
0.738
0.732
1.00460
1.00563
1.00670
1.00773
1.00877
1-703
1.694
1.685
1.672
1.656
1.00733
1.00925
1.01115
1.01303
1.01490
3.045
3.031
3.014
2.995
2.969
1 .00980
1.01278
1.01578
1.01878
1.02178
4.780
4.767
4-747
4.724
4.692
55
56
57
58
59
1.00114
1.00125
1.00135
1.00146
1.00156
0.180
0.177
o.i75
0.172
0.170
1.00450
1.00495
1-00535
1-00575
I.OO62O
0.724
0.715
0.705
0.694
0.682
1.00980
1.01083
1.01183
1.01283
1.01380
1.638
1.618
1.596
1-574
1.548
1.01675
1.01865
1.02050
1.02230
1.02410
2.942
2.911
2.876
2.836
2.793
1.02478
1.02778
1.03076
1-03374
1.03670
4-656
4.612
4.561
4-506
4-444
6o
61
62
63
64
1.0017
1.0018
1.0019
I.OO20
I.OO2O
0.167
0.163
0.159
0.155
0.151
1.0067
1.0070
1.0074
1.0078
I.OO82
0.668
0.655
0.640
0.625
0.609
1.0148
1.0158
1.0167
1.0176
1.0185
1.521
1.490
1.458
1-425
1.390
1.0259
1.0278
1.0294
1.0311
1.0328
2.745
2.695
2.641
2.583
2.521
1.0397
1.0425
1.0454
1.0482
1.0510
4-375
4-300
4.220
4-133
4.041
65
66
67
68
69
I.OO2I
I.OO22
1.0023
1.0024
I.OO25
0.147
0.143
0.138
0.134
0.129
I. OO86
I.OOgO
1.0093
I.OO97
I.OIOO
0.592
0-575
0-557
0-539
0.520
1.0193
I.02O2
I.O2IO
1.0218
I.O226
1-353
I.3I4
1-273
1.231
1.187
1.0344
1.0360
1.0376
1.0391
1.0406
2.456
2.388
2.315
2.240
2.163
1-0538
1.0564
1.0591
1.0617
1.0642
3-942
3-836
3-724
3-609
3-489
70
72
73
74
I.O026
1.0026
1.0027
I.OO28
I.OO28
0.123
0.118
O.II2
O.I07
O.IO2
1.0103
1.0106
1.0109
I.OII2
1.0114
0-499
0.478
0.456
0.434
0.411
1.0234
I.O24I
1.0247
1.0254
I.O260
1.141
1.094
1.045
0-995
0.944
1.0420
1.0434
1.0447
1.0460
1.0472
2.082
1.998
1.911
1.821
1.728
1. 0666
1.0689
1.0712
1-0734
1-0755
3-362
3.229
3.091
2.948
2.800
75
76
77
78
79
I.OO29
I.OO29
1.0030
I.OO3O
I.003I
O.O96
O.O9O
0.084
0.078
O.072
1.0117
1. 01 2O
I.OI22
I.OI23
I.OI25
0.388
0.365
0.341
0.316
0.291
1.0266
I.O272
1.0277
I.O282
I.O287
0.892
0.838
0.783
0.727
0.671
1.0483
1.0494
1.0504
1-0513
1.0522
1.632
1-534
1-434
1-332
1.229
1.0774
1.0793
1.081 1
1.0828
1.0843
2.649
2-493
2-333
2.169
2.OOI
80
81
82
83
84
I.003I
1.0032
1.0032
1.0032
1.0033
O.066
O.O6O
0.054
0.047
O.O4O
I.OI27
I.OI29
I.OI30
I.OI32
I-OI33
0.266
0.241
0.215
o.i 88
0.162
I.O29I
1.0295
1.0298
I.030I
1.0304
0.614
0-555
0-494
0-433
0.372
1-0530
1-0538
1.0544
1-0550
1-0555
1.124
1.017
0.908
0.797
0.685
1.0857
1.0870
i. 088 1
1.0892
1.0901
.830
.656
.480
.301
.119
85
86
87
88
89
1-0033
1.0033
1-0033
1.0033
1-0033
0.033
O.O26
0.019
O.OI2
O.OO6
I.OI34
I.OI34
I-OI35
I-OI35
I-OI35
0.136
0.109
0.082
0.054
0.027
1.0307
1.0309
I.03IO
I.03II
I.03II
0.311
0.249
0.187
0.125
0.063
1.0560
1-0563
1.0566
1.0568
1.0570
0-573
0.460
0.346
0.232
0.117
i .0908
1.0914
1.0919
1.0923
1.0926
0-935
0-749
0.562
0-374
0.187
90
1.0033
O.OOO
I.OI36
O.OOO
I-03II
O.OOO
1.0570
O.OOO
1.0926
0.000
Note.
Example.
Negative quantities are ii
tanh (0.4 /74°)
i heavy
\i°.728
type.
/ 0
= 1.0472
- 1.0472 \i-.43
41*.
[26]
TABLE V. CORRECTING
0.6
0.7
tanhfl
0
• /y. CONTINUED
0.8
0.9
45
46
47
0.99010
0.99417
0.99830
6.817
6.854
6.881
0.98189
0.98727
0.99279
9.214
9.281
9-337
0.96971
0.97646
0-98339
11.902
I2.O2O
12.122
0.95284
0.96089
0.96921
14.839
15.020
15-185
0.93077
0.93999
0.94950
17.956
18.216
18.461
48
1.00248
6.901
0.99840
9-383
0.99050
12.210
0.97779
15-334
0.95938
18.692
49
1.00672
6.912
I
00410
9.419
0.99780
12.285
0.98665
15.469
0.96966
18.908
50
I.OIIOO
6.915
I
00991
9-443
1.00526
12.347
0.99578
15-588
0.98032
19.108
1-01533
6.908
I
01581
9-455
1.01289
12.395
1.00518
15.690
0.99136
19.290
52
1.01970
6.893
I
02181
9-457
1.02069
12.427
1.01486
15-777
1.00282
19-455
53
1.02412
6.870
I
02789
9-445
1.02864
12.444
1.02479
15-845
1.01469
19.600
54
1.02855
6.838
i
03403
9-423
1.03675
12.446
1.03500
15.895
1.02697
19.726
55
1.03300
6.797
I
04024
9-388
1.04501
12.432
1.04549
15-925
1.03970
19.828
56
1.03747
6.746
i
04653
9-338
1-05343
I2.4OI
1.05626
15-934
1.05287
19.909
57
1.04195
6.685
1.05286
9-275
1.06196
12.353
1.06729
15-922
1.06648
19.965
58
1.04642
6.616
I
05921
9.200
1.07061
12.288
1.07854
15-889
1.08054
19.996
59
1.05089
6-537
I
06561
9.112
1.07939
12.203
1.09007
15.833
1.09506
19.998
60
1-05537
6.448
i
07210
9.010
1.08829
I2.IOO
1.10188
15-753
1.11009
19-973
61
1.05980
6.350
i
07856
8.893
1.09726
11.978
1.11390
15-647
1-12555
19.918
62
1.06420
6.242
1.08500
8.762
1.10630
11.835
1.12614
15-514
1.14144
19-832
63
1.06858
6.124
I
09H3
8.617
1.11540
11.673
1.13856
15-355
I-I5777
19.711
64
1.07293
5-998
I
09783
8.458
1.12456
11.491
1.15116
15-169
I.I7454
19.555
65
1.07722
5.862
I
10420
8.285
I.I3373
11.288
1.16392
14.956
1.19173
19-362
66
1.08143
5-7i6
I
"053
8.098
1.14291
11.064
1.17684
14.711
1.20935
19.131
67
1.08560
5-561
I
11683
7.895
1.15208
IO.82O
1.18989
14.438
1.22737
18.860
68
1.08967
5-397
i
12299
7.678
1. 16118
10.552
1.20298
14-132
1.24569
18.545
69
1.09363
5-223
I
12904
7-447
1.17020
10.263
1.21608
13-796
1.26429
18.188
70
1.09750
5.041
I
13500
7.202
I.I79I5
9-953
1.22919
13-428
1.28316
17-785
1.10125
4-850
I
14083
6-943
1.18796
9.621
1.24226
13.026
1.30221
17-334
72
1.10490
4-650
I
14651
6.671
1.19659
9.268
1-25524
12.592
1.32140
16.836
73
1.10842
4.441
i
15201
6.386
1.20503
8.894
1.26807
12.126
1.34063
16.290
74
1.11180
4-225
I
15734
6.087
1.21325
8-499
1.28067
11.628
1.35986
I5-693
75
1.11503
4.002
1.16244
5-775
I.22I2I
8.083
1.29300
11.097
1.37894
15-043
76
1. 11810
3-772
1-16733
5-450
1.22889
7.648
1.30500
10.532
1-39775
14.342
77
I.I2I02
3-534
I
17197
5-II4
1.23623
7.192
1-31659
9-936
1.41620
13-591
I-I2375
3.289
1.17636
4.767
I.243I9
6.718
1.32772
9-309
1.43412
12.788
79*
I.I2628
3-038
1.18046
4.409
1.24976
6.227
1.33830
8-653
I-45I4I
11.935
80
1.12863
2.781
I
18427
4.042
I.2559I
5.718
1.34827
7.966
1.46790
11.032
81
1.13078
2.519
1.18777
3.666
1.26158
5-195
1-35754
7-254
I-48345
10.081
82
I.I3273
2.251
I
19094
3-281
1.26674
4-656
1.36607
6.517
1.49790
9.087
83
I-I3447
1.979
I
19377
2.888
1.27138
4.104
1.37378
5-757
1.51110
8.052
84
I.I3598
1.704
1.19626
2.488
1-27545
3-540
1.38059
4-975
1.52289
6-977
85
.13727
1.425
I
19839
2.083
1.27895
2.966
1.38646
4-175
I-533I4
5-868
86
.13832
1.144
I
20013
1.672
I.28l85
2.383
1.39132
3-360
1.54170
4-731
87
.13915
0.860
I
20150
1.257
1.28413
1.794
1.39518
2.532
1.54848
3-571
88
•13975
0-574
I
.20247
0.839
I-28575
1.199
1-39793
1.692
1-55339
2.391
89
.I40IO
0.287
I
.20306
0.420
I.2867I
0.600
1-39959
0.847
I-55637
1.198
90 1.14022 o.ooo 1.20327 o.ooo 1.28700 o.ooo 1.40017 o.ooo 1.55740 o.ooo
Note.
Negative quantities are in heavy type.
-49'
[27]
tanh 6
1ABLE V.
I.I
^UK.IVJl,l^ 1 UN U
1.2
0
— r /j. i_or
1.4
JTINUEU
1-5
0
o
o
o
0
o
45
0.90354 2I.I67
0.87142 24.384
0.83515 27.536
0.79593 30.516
0.75487 33.288
46
0.91359 21.524
0.88183 24.856
0.84561 28.118
0.80600 31.225
0.76427 34.116
47
0.92400 21.869
0.89275 25.315
0.85661 28.699
0.81664 31.936
0.77420 34-952
48
0.93482 22.2OI
0.90425 25.764
0.86815 29.279
0.82786 32.648
0.78467 35-792
49
0.94618 22.52O
0.91625 26.2O2
0.88038 29.852
0.83971 33.360
0-79573 36.638
50
0.95809 22.822
0.92892 26.629
0.89331 30.416
0.85228 34.070
0.80747 37-490
0.97055 23.109
0.94225 27.043
0.90685 30.973
0.86557 34-778
0.81993 38.348
52
0.98345 23.379
0.95617 27.442
0.92123 31.521
0.87964 35.485
0.83320 39-213
53
0.99700 23.630
0.97083 27.826
0.93638 32.059
0.89457 36.189
0.84727 40.086
54
i .01 1 r 23.861
0.98625 28.193
0.95238 32.586
0.91043 36.890
0.86220 40.965
55
1.0258 24.O7O
1.0025 28.540
0.96938 33.100
0.92729 37.587
0.87813 41.849
56
.O4I2 24.256
1.0196 28.867
0.98738 33-599
0.94529 38.279
0.89513 42.738
57
.0572 24.417
1-0375 29.173
1.0065 34-082
0.96443 38.964
0.91333 43.634
58
.0739 24.551
1.0564 29.454
1.0266 34.548
0.98479 39.642
0.93280 44-534
59
.0914 24.657
1.0763 29.708
1.0482 34-993
1. 0066 40.311
0-95367 45-441
6o
.1096 24.733
1.0973 29.932
1.0710 35.416
1.0301 40.968
0.97613 46.349
61
.1286 24.774
I.II95 30.124
1-0953 35-8x3
1.0551 41.611
1.0003 47-262
62
.1484 24.778
1.1428 30.280
I.I2I2 36.180
1.0820 42.239
1.0264 48.176
63
1.1689 24.745
1.1674 30-397
I.I488 36.515
.1109 42.850
1.0546 49.092
64
I.I9O4 24.672
I.I934 30.473
1.1784 36.814
.1421 43.438
1.0851 50.009
65
I.2I26 24.554
1. 2208 30.502
1.2099 37.072
.1759 44.001
1.1184 50.924
66
1.2357 24.388
1.2496 30.480
1.2436 37.285
.2124 44.535
1.1546 51-834
67
1.2596 24.173
1.2799 30.403
1.2798 37.446
.2520 45.034
1.1943 52.739
68
1.2845 23.904
1.3119 30.263
i-3l85 37-548
.2951 45-493
1-2377 53.636
69
I.3IOI 23.579
1-3457 30-059
1.3600 37-585
•3420 45-904
1.2857 54.520
70
1.3365 23.194
1.3811 29.781
1.4046 37-548
•3933 46.260
I-3385 55.388
1.3637 22.744
1.4183 29.425
1-4525 37-430
.4495 46.552
1-3973 56.235
72
I.39I5 22.227
1.4573 28.983
1-5039 37-218
.5113 46.768
1.4628 57.056
73
1.4200 21.639
1.4981 28.447
1.5592 36.899
1.5794 46.894
1-5362 57-841
74
1.4489 20.978
1.5407 27.810
1.6186 36.461
1.6546 46.916
1.6190 58.581
75
1.4783 20.238
1.5849 27.064
1.6822 35.890
1.7380 46.814
1.7129 59.266
76
1.5077 19.419
1.6307 26.202
1-7505 35.170
1.8308 46.567
1.8202 59.880
77
1.5373 I8.5I8
1.6778 25.214
1.8235 34-280
1.9341 46.145
1.9438 60.403
78
1.5665 17.532
1.7261 24.094
1.9012 33.202
2.0498 45.514
2.0875 60.814
79
1.5955 16.462
1.7749 22.835
1.9835 31.912
2.1792 44-633
2.2562 61.076
80
1.6235 15.308
1.8239 21.429
2.0702 30.389
2.3240 43-454
2.4563 61.144
81
1.6505 14.069
1.8726 19.874
2.1604 28.609
2.4862 41-915
2.6973 60.956
82
1.6759 12.748
1.9200 18.168
2.2531 26.553
2.6669 39.944
2.9917 60.422
83
1.6996 11.350
1.9654 16.310
2.3466 24.200
2.8668 37-454
3-3577 59-412
84
I.72I2 9.879
2.0078 14.306
2.4388 21.540
3.0848 34-354
3.8207 57.724
85
1.7402 8.343
2.0463 12.164
2.5267 18.568
3.3164 30.541
4.4198 55-045
86
1.7562 6.748
2.0796 9.898
2.6065 I5-294
3.5521 25.931
5.2010 50.866
87
1.7691 5.105
2.1068 7.525
2.6744 11.747
3.7765 20.477
6.2275 44-356
88
1.7785 3.425
2.1269 5-067
2.7266 7.970
3.9665 14.214
7.5060 34-274
89
1.7843 I.7I9
2-1393 2.548
2.7596 4.029
4.0952 7.293
8.7880 19.305
90
1.7862 o.ooo
2.1434 o.ooo
2.7709 o.ooo
4.1413 o.ooo
9.4007 o.ooo
Note.
Negative quantities are in heavy type.
tanh (1.3 /45°)
Example.
i-3 /45°
0.83515 \27°.536 =
- 0.83515 \i7°.32'
.10".
[28]
TABLE V.
CORRECTING
^.™^ tanhfl
INTJED
1-AC ^ - r£y. CONT
o
45
46
47
48
49
1.6
o
0.71331 35-799
0.72175 36.733
0.73069 37.682
0.74013 38.642
0.75013 39-6i5
o
0.67224 38.016
0.67959 39-048
0.68741 40.098
0.69565 41.166
0.70435 42.251
1.8
0
0.63250 39-937
0.63883 41.050
0.64544 42.186
0.65239 43-341
0.65978 44.520
1.9
o
0.59484 41.562
0.60005 42.740
0.60553 43.942
0.61126 45.168
0.61726 46.422
2.0
o
0-55955 42-906
0.56375 44-133
0.56815 45-387
0.57275 46.667
0.57750 47-977
50
52
53
54
0.76069 40.600
0.77194 41.600
0.78381 42.616
0.79650 43.646
0.81000 44.691
0.71359 43-355
0.72329 44.480
o.73365 45-627
0.74465 46.797
0.75629 47-992
0.66756 45.725
0.67572 46.955
0.68433 48.213
0.69350 49.501
0.70317 50.820
0.62358 47.705
0.63026 49.019
0.63721 50.366
0.64458 51.747
0.65232 53-165
0.58250 49-319
0.58770 50.694
o-593I5 52-103
0.59885 53-550
0.60480 55-039
55
56
57
58
59
0.82438 45-754
0.83969 46.834
0.85613 47-931
0.87369 49.047
0.89256 50.183
0.76865 49.212
0.78188 50.461
0.79600 51.739
0.81106 53.047
0.82724 54.389
0.71344 52.174
0.72433 53-564
0-73594 54-992
0.74828 56.460
0.76144 57-974
0.66047 54.621
0.66911 56.121
0.67821 57.666
0.68784 59.259
0.69805 60.905
o.6noo 56.571
0.61750 58.149
0.62435 59-777
0.63150 61.458
0.63905 63-195
60
61
62
63
64
0.91288 51.339
0-93475 52.516
0.95834 53.714
0.98394 54-935
1.0117 56.181
0.84459 55-768
0.86324 57.181
0.88335 58.635
0.90506 60.131
0.92853 61.674
0-77550 59-535
0.79056 61.145
0.80672 62.810
0.82406 64.533
0.84272 66.318
0.70895 62.606
0.72047 64.366
0.73274 66.191
0.74579 68.085
0.75974 70.052
0.64695 64.995
0.65530 66.860
0.66405 68.793
0.67330 70.802
0.68310 72.891
65
66
67
68
69
1.0419 57-452
.0749 58.748
.mi 60.072
.1508 61.425
.1948 62.809
0.95400 63.266
0.98176 64.910
I.OI2I 66.6l3
1.0453 68.376
I.o8l7 7O.2O9
0.86283 68.172
0.88461 70.098
0.90817 72.104
0.93372 74-I96
0.96156 76.382
0.77463 72.099
0.79063 74.232
0.80774 76.459
0.82611 78.786
0.84579 8I.22I
0.69340 75.068
0.70430 77-338
0.71585 79-707
0.72810 82.184
0.74105 84.776
70
72
73
74
.2433 64.224
.2974 65.674
.3580 67.160
.4262 68.686
•5033 70.256
I.I2I9 72.II6
1.1663 74.IO6
I.2I57 76.187
1.2708 78.371
1.3325 80.670
0.99189 78.671
.0251 81.074
.0614 83.602
.1013 86.266
.1452 89.084
0.86705 83.773
0.88989 86.454
0.91453 89.273
O.94III 92.244
0.96974 95-379
0-75475 87.490
0.76925 90.337
0.78465 93-326
0.80095 96.466
0.81815 99.769
75
76
77
78
79
.5913 71-873
.6924 73-543
.8098 75-274
J-9473 77-078
2.1104 78.966
1.4020 83.IOI
1.4807 85.685
1.5704 88.447
1.6731 91.420
I.79I5 94-644
.1936 92.076
.2470 95.261
1.3061 98.664
1.3714 102.315
1.4436 106.250
1.0005 98.692
1.0337 IO2.2O2
1.0693 105.925
I.I074 IO9.88I
I.I478 114.088
0.83625 103.245
0.85525 106.905
0.87515 110.760
0.89585 114.820
0.91720 119.093
80
81
82
83
84
2.3068 80.960
2.5472 83.087
2.8474 85.390
3.2326 87.937
3-7435 90.839
1.9288 98.175
2.0894 IO2.O83
2.2775 106.466
2.4992 111.446
2.7604 II7.I99
1.5231 110.510
1.6104 115.140
1.7056 120.191
1.8078 125.722
1.9155 131-793
I.I906 II8.568
1.2352 123.344
I.28l3 128.428
I-3279 I33.842
1-3739 I39-590
0.93905 123.590
0.96110 128.319
0.98305 133-281
1.0045 138.474
1.0249 143.894
85
86
87
88
89
4.4502 94.289
5.4825 98.668
7.1175 104.791
9.9938 114.760
15.528 135.011
3.0648 123.945
34II4 I3L956
3.7856 I4L527
4.1472 152.868
4.4231 165.896
2.0255 138.448
2.1329 145-720
2.2309 153.606
2.3107 162.043
2.3628 170.905
1.4179 145-673
1.4579 152-075
1.4921 158.768
1.5184 165.700
1.5349 172.805
1.0436 149-528
1.0601 155-359
1.0738 161.359
1.0840 167.494
1.0904 173-723
90
21.395 180.000
4.5275 iSo.OOO
2.3813 180.000
1.5406 180.000
1.0925 180.000
Note.
Example.
Negative quantiti
tanh (2.0 /8o°)
es are in heavy type.
0.93905 \i23°.590
= 0.93905 \ 1 23°.35'
24 •
2.0 /8o°
[29]
TABLE V.
2.1
CORRECTING
2.2
FACTOI
2-.
> tanh0 _/-
CONTINUED
I 2.5
6
j
- r/j.
2.i
0
o
o
0
o
0
45
0.52690
43.993
0.49691
44.845
0.46952
45.492
0.44467
45.961
O.422I2
46.283
46
0.53019
45.254
0.49941
46.127
0.47139
46.783
0.44592
47.252
0.42292
47.564
47
0.53362
46.542
O.5O2OO
47.436
0.47322
48.101
0.44717
48.567
0.42368
48.868
48
0.53714
47.859
0.50465
48.774
0.47509
49-446
0.44842
49.908
0.42440
50.195
49
0.54081
49.206
0.50732
50.142
0.47696
50.820
0.44958
51.277
0.42504
5L548
5°
0.54462
50.586
O.5IOO9
51.542
0.47883
52.226
0.45075
52.676
0.42560
52.929
51
0.54852
52.001
0.5I29I
52.978
0.48070
53-666
0.45188
54.107
0.42612
54-340
52
0.55257
53.452
0.51578
54.452
0.48256
55-144
0.45292
55.572
0.42652
55.782
53
0.55681
54-943
0.51868
55.965
0.48443
56-659
0.45392
57.073
0.42684
57.256
54
0.56114
56.476
0.52168
57.520
0.48630
58.215
0.45488
58.614
0.42704
58.766
55
0.56562
58.055
0.52473
59-120
0.48813
59.8i5
0.45571
60.196
0.42712
60.314
56
0.57029
59.682
0.52782
60.769
0.48991
61.462
0.45646
61.822
0.42708
61.903
57
0.57514
61.361
0.53095
62.469
0.49170
63.160
0.45713
63.496
0.42688
63-534
58
0.58014
63.094
0.53414
64.226
0-49339
64.912
0.45767
65.220
0.42648
65.211
59
0.58533
64.888
0.53736
66.042
0.49504
66.722
0.45804
66.998
0.42588
66.938
60
0.59071
66.745
0.54064
67.921
0.49661
68.594
0.45829
68.835
0.42512
68.718
61
0.59629
68.671
0-54395
69.869
0.49813
70.531
0.45838
70.735
0.42412
70.556
62
0.60210
70.669
0.54732
71.889
0-49957
72.540
0.45829
72.701
0.42288
72.455
63
0.60814
72.744
0.55073
73-988
0.50087
74.625
0.45800
74.740
O.42I4O
74.419
64
0.61443
74.902
0.55418
76.171
O.5O2O9
76.791
0.45750
76.855
0.41960
76-455
65
0.62090
77.151
0.55764
78.443
0.50317
79-044
0.45675
79.053
0.41752
78.567
66
0.62767
79.496
0.56109
80.810
0.50413
81.390
0-45579
81.341
O.4I5I2
80.761
67
0.63471
81.942
0.56459
83.280
0.50491
83.836
0-45454
83.723
O.4I24O
83.044
68
0.64205
84.499
0.56814
85.858
0.50557
86.389
0.45300
86.208
0.40928
85.424
69
0.64962
87.172
0.57164
88.552
0.50604
89.056
0.45I2I
88.802
0.40580
87.906
70
0.65752
89.968
0.57518
91-370
0.50630
91.843
0.44913
91-514
0.40196
90.5OO
0.66571
92.898
0.57873
94-319
0.50639
94-759
0.44671
94-352
0.39771
93-215
72
0.67424
95-968
0.58223
97.405
0.50626
97.811
0.44396
97-324
0.39306
96.061
73
0.68305
99.185
0.58568
100.636
0.50596
101.006
0.44092
100.439
0.38802
99.048
74
0.69214
102.560
0.58914
104.020
0.50543
104.352
0-43754
103.706
0.38260
IO2.I87
75
0.70152
106.099
0.59255
107.564
0.50470
107.858
0.43388
107.134
0.37681
105.489
76
0.71114
109.811
0.59586
111.274
0.50378
111.531
0.42992
110.730
0.37068
108.966
77
0.72096
113.702
0.59914
115-156
0.50265
"5-374
0.42575
114-505
0.36427
112.633
78
0.73086
117.778
0.60232
119.212
0.50135
II9-392
0.42138
118.467
0.35762
II6.5OI
79
0.74090
122.043
0.60536
123.446
0.49991
123.590
0.41683
122.620
0.35080
120.582
80
0.75086
126.501
0.60832
127.859
0.49835
127.968
O.4I22I
126.968
0.34392
124.887
81
0.76067
131-151
O.6lIO9
132.449
0.49674
132.527
0.40759
131-517
0.33706
129.427
82
0.77019
135-989
0.61364
137-212
0.49509
137-264
0.40305
136.267
0.33033
I34.209
83
0.77924
141.011
O.6l6oO
142.140
0-49343
142.171
0.39869
141.212
0.32391
I39.233
84
0.78767
146.210
0.6l8l4
147-223
0.49183
I47.237
0.39463
146.344
0.31790
144.499
85
0.79524
I5L570
O.62OOO
152.449
0.49039
152.450
0.39094
151.650
0.31248
149.994
86
0.80176
157.070
0.62155
157.800
0.48913
157-793
0.38775
157.113
0.30778
I55-703
87
0.80710
162.690
O.62282
163.254
0.48804
163.245
0.38516
162.710
0.30396
161.598
88
0.81100
168.406
0.62374
168.790
0.48726
168.781
0.38325
168.413
0.30II3
167.645
89
0.81338
174.187
O.62427
174-381
0.48678
I74.376
0.38207
174.188
0.29940
173.796
90
0.81424
180.000
0.62445
180.000
0.48661
180.000
0.38167
180.000
0.29880
iSo.OOO
Note
Negative quantities are in heavy type.
0
1 - Atllll^l C •
2-3 /9Q°
"• \I50
= 0.48661
Zi_^
[30]
1
TABLE V.
CORRECTING
FACTOR tanh 6 - I- ^~
NUED
JT/V^ J. VJJS-. -
0
r /~y. \^<JJN 11
0
45
46
47
48
49
2.6
o
0.40173 46.480
0.40215 47-744
0.40250 49.030
0.40281 50.337
0.40304 51.667
. 2.7
o
0.38326 46.576
0.38341 47.821
0.38348 49.083
0.38344 50.365
0-38333 51-666.
2.8
o
0.36657 46.595
0.36650 47.816
0.36632 49-051
0.36607 50.303
0.36571 51-572
2.9
o
0.35145 46.554
0.35121 47-748
0.35086 48.955
0.35045 50-175
0-34997 5I-409
3-0
0
0.33770 46.468
0-33733 47-637
0.33690 48.815
0.33637 50.002
0-33573 51-198
50
5i
52
53
54
0.40319 53.022
0.40319 54-403
0.40308 55.812
0.40288 57.250
0.40254 58-717
0.38311 52.988
0.38278 54-332
0.38233 55.700
0.38174 57-094
0.38096 58.513
0.36525 52.857
0.36464 54.160
0.36389 55.483
0.36300 56.826
0.36196 58.190
0.34928 52.656
0.34848 53-9I6
0-34759 55-I9I
0.34652 56.481
0.34528 57.788
0-33497 52-405
0.33410 53.622
0-33307 54-849
o-33 J91 56-086
0-33058 57-334
55
56
57
58
59
0.40200 60.219
0.40131 61.756
0.40046 63.331
0.39942 64.947
0.39812 66.606
0.38000 59-959
0.37889 61.437
0.37756 62.945
0.37600 64.487
0.37422 66.066
0.36075 59-577
0.35932 60.987
0.35768 62.422
o.3558o 63.884
0.35367 65.374
0.34386 59-"2
0.34226 60.452
0.34043 61.810
0.33837 63.186
0.33602 64.582
0.32907 58.593
0.32737 59-862
0.32546 61.141
0.32332 62.432
0.32093 63.735
6o
61
62
63
64
0.39658 68.313
0.39481 70.069
0.39273 7I-879
0.39035 73-747
0.38765 75-677
0.37215 67-684
0.36980 69.343
0.36716 71.048
0.36416 72.801
0.36081 74.607
0.35126 66.895
0.34856 68.449
0.34554 70.036
0.34217 71.660
0-33843 73.325
0-33345 65-999
0.33056 67-439
0.32734 68.904
0.32378 70.392
0.31985 71-907
0.31827 65.049
0.31532 66.375
0.31206 67.713
0.30847 69.063
0.30451 70.426
65
66
67
68
69
0.38458 77-675
0.38115 79-747
0-37733 81.898
0.37310 84.136
0.36844 86.469
0.35709 76.470
0.35296 78.393
0.34841 80.383
0.34341 82.449
0-33794 84.597
0.33430 75-033
0.32974 76.789
0.32474 78.599
0.31927 80.464
0.31331 82.396
0.31552 73.452
0.31076 75-027
0-30556 76-639
0.29988 78.289
0.29369 79.982
0.30016 71.801
0.29539 73.I92
0.29018 74.598
0.28450 76.021
0.27832 77-463
70
7i
72
73
74
0.36333 88.902
0-35777 9I-45I
0.35176 94-122
0.34529 96.929
0-33837 99-884
0.33198 86.833
0.32553 89.168
0-31857 91.613
0.31109 94.184
0.30311 96.893
0.30683 84.399
0.29982 86.482
0.29226 88.655
0.28415 90.931
0.27548 93-325
0.28697 81.722
0.27970 83.518
0.27186 85.378
0.26343 87.310
0.25439 89.328
0.27161 78.926
0.26434 80.414
0.25649 81.932
0.24803 83.485
0.23894 85.081
75
76
77
78
79
0.33102 103.003
0.32328 106.302
0.31519 109.799
0.30680 113.513
0.29821 117.466
0.29464 99-757
0.28570 102.799
0.27634 106.038
0.26661 109.505
0.25659 113.231
0.26625 95.857
0.25648 98.547
0.24621 101.421
0.23546 104.512
0.22431 107.861
0.24475 91.448
0.23450 93.690
0.22366 96.078
0.21224 98.647
0.20030 101.437
0.22922 86.729
0.21883 88.443
0.20779 90.241
0.19610 92.146
0.18376 94.191
80
81
82
83
84
0.28950 121.678
0.28081 126.171
0.27227 130.966
0.26405 136.078
0.25633 141-517
0.24637 117.249
0.23609 121.599
0.22589 126.322
0.21597 131.464
0.20654 137.060
0.21284 111.519
0.20116 115.543
0.18943 120.004
0.17783 124.989
0.16660 130.591
0.18788 104.504
0.17507 107.921
0.16200 111.784
0.14880 116.221
0-13573 121.403
0.17082 96.419
0.15730 98.893
0.14329 101.700
0.12887 104.976
0.11419 108.916
85
86
87
88
89
0.24931 147.286
0.24319 153-370
0.23818 159.740
0.23446 166.349
0.23216 173.129
0.19786 143-143
0.19020 149.722
0.18384 156.778
0.17906 164.253
0.17609 172.042
0.15604 136.907
0.14651 144.025
0.13844 151.988
0.13224 160.762
0.12832 170.195
0.12307 127.549
O.III22 134.931
0.10074 I43-833
0.092383 I54-456
0.086890 166.709
0.099443 113-839
0.084970 120.254
0.071310 128.990
0-059383 141.294
0.050750 158.480
90
0.23138 180.000
0.17509 180.000
0.12698 180.000
0.084969 I8O.OOO
0.047517 180.000
Note.
Example.
Negative quantities are in heavy type,
tanh (2.9 /85°) , „ t a .,
2-9/85!
= 0.12307 \i27 .549
- 0.12307 \i27--32
.56'.
[31]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. /(p/4£) = r fa
p Sinh Cosh Tanh
0 /
0 /
o /
o
o.
45-00
I.
0.
o.
45-00
O.I
O.IOOOO
45.06
I.OOOOI
0.17
0.09999
44.49
O.2
O.2OOOO
45-23
1.00013
1.09
0.19997
44.14
o-3
0.30001
45-52
1.0007
2-35
0.29981
43-17
0.4
0.40005
46.32
1. 002 1
4-35
0.39921
41-57
o-S
0.50016
47-23
1.0052
7.08
0-49757
40.15
0.6
0.60042
48.27
I.OIO7
10.15
0.59406
38.11
0.7
0.70094
49.40
I.OigS
13-53
0.68732
35-47
0.8
0.80184
51.06
1.0336
18.00
0-77577
33-o6
0.9
0.90327
52.44
1-0533
22.34
0.85756
30.10
I.O
1-0055
54-32
1.0803
27.29
0.93077
27.03
i.i
1.1089
56.31
i."57
32-41
0.99389
23-50
1.2
1.2138
58.41
1. 1608
38-05
1-0457
20.36
i-3
1.3205
61.02
1.2163
43-35
1.0857
17.27
1.4
1.4297
63-34
1.2830
49-05
1. 1143
14.29
i-5
1.5418
66.15
1.3616
54-33
1-1323
11.42
1.6
1-6575
69.07
1-4524
59-55
1.1413
9.12
i-7
1.7776
72.08
I.5556
65.09
1.1428
6-59
1.8
1.9029
75.18
1.6714
70.14
1-1385
5-04
1.9
2.0343
78.36
1.7999
75-io
1.1302
3-26
2.O
2.1726
82.01
1-9413
79.56
1.1191
2.06
2.1
2.3190
85-34
2.0958
84-33
1.1065
I.OI
2.2
2-4745
89.12
2.2636
89.03
1.0932
0.09
2-3
2.6404
92.57
2.4449
93.26
1.0799
0.29
2.4
2.8177
96.46
2.6403
97-44
1.0672
0.58
2-5
3.0079
100.39
2.8502
101.56
1-0553
1.17
2.6
3.2121
104.36
3-0753
106.05
1.0445
1.29
2-7
3.43i8
108.36
3-3!63
IIO.IO
1.0348
1-34
2.8
3-6685
112.39
3-5741
114.15
1.0264
1.36
2.9
3.9236
116.43
3-8497
118.16
1.0192
1.33
3-o
4.1986
1 20.48
4-1443
122.16
1.0131
1.28
3-i
4.4948
124.56
4.4589
126.15
1.0080
1.19
3-2
4-8154
129.02
4-7955
!30.i5
1.0041
1.13
3-3
5.1586
133.09
5.I54I
I34-I3
1.0008
1.04
3-4
5-5306
137.17
5-5393
138-13
0.9984
0.56
3-5
5-9305
141.24
5.9356
142.12
0.9967
0.48
3-6
6.3603
145-31
6.3900
146.11
0.9954
0.40
3-7
6.8244
149.38
6.8606
150.10
0.9947
0.32
3-8
7-3228
153.44
7.3646
154.09
0-9943
0.25
3-9
7.8590
157.50
7.9047
158.10
0.9942
O.2O
4.0
8.4351
161.57
8.4831
162.11
0.9943
O.I4
4-i
9-0535
166.02
9.1024
166.12
0.9946
O.IO
4.2
9.7198
170.07
9.7704
170.13
0.9948
O.O6
4-3
10.434
174.11
10.481
174-15
0-9955
0.04
4.4
II. 201
178.16
11.246
178.16
0.9960
O.OO
Note. Negative quantities are in heavy type.
Examples, sinh (1.7 /45°) = 1.7776 /72°.o8'.
tanh (2.4 /45°) = 1.0672 \o°.58'.
[32]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. /(p/4^) = r /y. CONTINUED
Cosech
Sech
Coth
0.
00
45-00
i.
o.
oo
45-oo
O.I
IO.OOOO
45-06
0.99999
0.17
IO.OOOO
44-49
0.2
5.0000
45-23
0.99987
1.09
5.0008
44.14
o-3
3-3333
45-52
0-9993
2-35
3-3355
43-17
0.4
2-4997
46.32
0.9979
4-35
2.5050
41.57
0-5
1.9984
47-23
0.9948
7.08
2.0096
40.15
0.6
1.6654
48.27
0.9894
10.15
1.6830
38.11
0.7
1.4268
49.40
0.9806
13-53
1-4549
35-47
0.8
1.2471
51.06
0.9675
18.00
1.2890
33-o6
0.9.
1.1070
52.44
0.9494
22.34
1. 1660
30.10
I.O
0.9945
54-32
0.9256
27.29
1.0746
27.03
i.i
0.9018
56.31
0.8963
32.41
i. 006 1
23-50
1.2
0.8238
58.41
0.8614
38.05
0.9564
20.36
i-3
0-7573
61.02
0.8222
43-35
0.9211
17.27
1.4
0.6995
63-34
0.7793
49-05
0.8996
14.29
•5
0.6486
66.15
0-7344
54-33
0.8831
11.42
.6
0.6033
69.07
0.6885
59-55
0.8763
9.12
•7
0.5625
72.08
0.6429
65.09
0.8751
6-59
.8
0.5256
75-18
0.5981
70.14
0.8788
5-04
•9
0.4916
78.36
0.55,56
75-10
0.8848
3-26
2.0
0.4603
82.01
0-5151
79.56
0.8936
2.06
2.1
0.4312
85-34
0.4772
84.33
0.9038
I.OI
2.2
0.4041
89.12
0.4418
89.03
0.9147
0.09
2-3
0.3788
92-57
0.4090
93-26
0.9260
0.29
2.4
0-3549
96.46
0.3788
97-44
0.9370
0.58
2-5
0-3325
100.39
0-3509
101.56
0.9476
1.17
2.6
0.3114
104.36
0.3252
106.05
0-9574
1.29
2.7
0.2914
108.36
0.3016
IIO. 10
0.9663
1-34
2.8
0.2726
112.39
0.2798
114.15
0-9743
.36
2-9
0.2549
116.43
0.2598
118.16
0.9812
•33
3-o
0.2382
120.48
0.2413
122.16
0.9871
.28
3-i
0.2225
124.56
0.2243
126.15
0.9920
.19
3-2
0.2077
129.02
0.2085
130.15
0.9959
•13
3-3
0.1939
133-09
0.1940
134-13
0.9992
.04
3-4
o.i 808
137.17
0.1805
138.13
1.0016
0.56
3-5
o.i 686
141.24
0.1681
142.12
•0033
0.48
3-6
0.1572
I45-3I
0.1565
146.11
.0047
0.40
3-7
0.1465
149.38
0.1458
150.10
•0053
0.32
3-8
0.1366
153-44
0.1358
154.09
.0057
0.25
3-9
0.1272
157-50
0.1265
158.10
.0058
O.2O
4.0
0.1186
161.57
01179
162.11
.0057
O.I4
4.1
0.1105
166.02
0.1099
166.12
.0054
O.IO
4-2
0.1029
170.07
0.1024
170.13
1.0052
O.O6
4-3
0.09584
174.11
0.09541
174-15
1.0045
O.O4
4-4
0.08927
178.16
0.08892
178.16
1.0040
o.oo
Note. Negative quantities are in heavy type.
Examples, cosech (2.0 745°) = 0.4603 \82°.oi'.
coth (2.5 /45JO = 0.9476/1°. 1 7'-
[33]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. / (p /45°) = r /y. CONTINUED
Sinh
Cosh
4-5
12.026
182.19
4.6
12.909
186.23
4-7
13-858
190.27
4.8
14.876
194.30
4-9
15-968
198-33
S-o
17.140
202.36
5-i
18.397
206.39
5-2
19-747
210.42
5-3
21.195
214-45
5-4
22.750
218.48
5-5
24.418
222.50
5-6
26.219
226.53
5-7
28.141
230.56
5-8
30.192
234-59
5-9
32-405
239.02
6.0
34-784
243-05
P
Cosech
0 /
4-5
0.08316
182.19
4-6
0.07746
186.23
4-7
0.07216
190.27
4.8
0.06722
194.30
4-9
0.06263
198.33
S-o
0.05834
202.36
5-i
0.05436
206.39
5-2
0.05064
210.42
5-3
0.04718
214.45
5-4
0.04396
218.48
5-5
0.04095
222.50
5-6
0.03814
226.53
5-7
0.03554
230.56
5-8
0.03312
234.59
5-9
0.03086
239.02
6.0
0.02875
243.05
12.067
12.948
13.894
14.909
15-999
17.169
18.425
19.772
21.219
22.772
24.439
26.238
28.159
30.209
32.421
182.19
186.21
190.23
194.26
198.29
202.32
206.35
210.38
214.41
218.44
222.47
226.51
230.54
234.57
239.00
34.798 243.04
Sech
0.08288
0.07723
0.07197
0.06707
0.06250
0.05824
0.05428
0.05058
0.04713
0.04391
0.04092
0.03811
0-03551
0.03310
0.03085
182.19
186.21
190.23
194.26
198.29
202.32
206.35
210.38
214.41
218.44
222.47
226.51
230.54
234-57
239.00
Tanh
O I
0.9966 o.oo
0.9970 O.02
0.9974 O.O4
0.9978 O.O4
0.9980 O.04
0.9983
0.9985
O.O4
O.04
0.9087 O.O4
0.9989 O.O4
0.9990 O.O4
0.9992
0-9993
0-9994
0.9995
0.03
O.O2
O.O2
O.02
0.9996 O.O2
0.9996 o.oi
Coth
1.0034
o.oo
1.0030
1.0026
O.O2
O.O4
I.OO22
O.O4
I.OO2O
O.O4
I.OOI7
O.O4
I.OOI5
O.O4
I.OOI3
O.O4
I.OOII
O.O4
I.OOIO
O.04
1.0008
0.03
1.0007
1. 0006
O.O2
O.O2
1.0005
O.02
1.0004
O.O2
0.02874 243.04
1.0004 o.oi
Note. Negative quantities are in heavy type.
Examples, tanh (6.0 745°) = 0.9996 /o°.oi'.
seen (5.0 /45°) = 0.05824 \2O2°.32'.
[34]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. /(p/4s°) = r /y. CONTINUED
Sinh and cosh
Tanh and coth
Sech and cosech
6.05
36.047
245-06
I.OOO
o.oo
2.774)
<IO~2
245.06
6.io
37-349
247.08
I.OOO
o.oo
2.678
u
247.08
6.iS
38-693
249.09
I.OOO
o.oo
2.583
•
249.09
6.20
40.084
251.11
I.OOO
o.oo
2-495
"
251.11
6.25
41-524
253-12
I.OOO
o.oo
2.408
u
253.12
6.30
43-020
255-I4
I.OOO
o.oo
2.325
a
255.14
6-35
44-563
257.15
I.OOO
o.oo
2.244
u
257.15
6.40
46.171
259-I7
I.OOO
o.oo
2.166
u
259.17
6-45
47-832
261.18
I.OOO
o.oo
2.091
u
261.18
6.50
49-553
263.20
I.OOO
o.oo
2.018
a
263.20
6.55
5I-336
265.22
I.OOO
o.oo
1.948
u
265.22
6.6o
53-i83
267.24
I.OOO
o.oo
1. 880
a
267.24
6.65
55-no
269.25
I.OOO
o.oo
1.815
u
269.25
6.70
57-058
271.27
I.OOO
o.oo
1.752
u
271.27
6-75
59-I36
273.28
I.OOO
o.oo
1.691
u
273.28
6.80
61.259
275-30
I.OOO
o.oo
1.632
u
275.30
6.85
63-463
277-31
I.OOO
o.oo
1.576
u
277.31
6.90
65-746
279-33
I.OOO
o.oo
1.521
u
279.33
6-95
68.119
281.34
I.OOO
o.oo
1.468
u
281.34
7.00
70.570
283-36
I.OOO
o.oo
1.4*7
a
283.36
7-05
73.109
285-37
I.OOO
o.oo
1.368
u
285.37
7.10
75-739
287.39
I.OOO
o.oo
1.312
u
287.39
7-iS
78.473
289.40
I.OOO
o.oo
1.274
a
289.40
7.20
81.296
291.42
I.OOO
o.oo
1.230
a
291.42
7-25
84.215
293-43
I.OOO
o.oo
1.187
u
293.43
7-30
87.250
295-45
I.OOO
o.oo
1.146
a
295.45
7-35
90.386
297.46
I.OOO
o.oo
1.016
u
297.46
7.40
93-083
299.48
I.OOO
o.oo
1.074
u
299.48
745
97.009
301.49
I.OOO
o.oo
1.031
a
301.49
7-5°
100.50
303-51
I.OOO
o.oo
<IO~*
303.51
7-55
104.12
305-52
I.OOO
o.oo
9.605
u
305-52
7.60
107.86
307-54
I.OOO
o.oo
9.271
u
307.54
7-65
111.74
309-56
I.OOO
o.oo
8.949
u
309.56
7.70
115.67
3H-57
I.OOO
o.oo
8.638
u
311.57
7-75
119.94
3*3.59
I.OOO
o.oo
8-337
u
313.59
7.80
124.26
316.00
I.OOO
o.oo
8.048
u
316.00
7.85
128.71
318.02
I.OOO
o.oo
7.769
a
318.02
7.90
133-35
320.03
I.OOO
o.oo
7-499
a
320.03
7-95
138.16
322.05
I.OOO
o.oo
7.238
a
322.05
8.00
143.12
324-06
I.OOO
o.oo
6.987
u
324.06
8.05
148.28
326.07
I.OOO
o.oo
6.744
1C
326.07
8.10
153.61
328.09
I.OOO
o.oo
6.510
u
328.09
8.15
I59-I4
330.11
I.OOO
o.oo
6.284
a
330.11
8.20
164.87
332-12
I.OOO
o.oo
6.066
u
332.12
8.25
170.80
334-14
I.OOO
o.oo
5-855
a
334.14
Note. Negative quantities are in heavy type.
Examples, sinh (7.55 /45°) = cosh (7.55 /45°) = 104.12
sech (7-50/45!) = cosech (7.50/45!) = 9-95Q X io-»\303°.si'
[35]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. f (p [*£) = r
CONTINUED
Sinh and cosh
Tanh and coth
Sech and cosech
8.30
176.95
336.15
I.OOO
o.oo
5.651X10-3
336.15
8-35
183.31
338.17
I.OOO
o.oo
5-455
•
338.17
8.40
189.91
340.18
I.OOO
o.oo
5.266
«
340.18
8.45
196.75
342.20
I.OOO
o.oo
5.083
a
342.20
8.50
203.83
344-22
I.OOO
o.oo
4.906
((
344-22
8-55
211. l6
346.24
I.OOO
o.oo
4.736
«
346.24
8.60
218.76
348.25
I.OOO
o.oo
4.571
«
348.25
8.65
226.63
350.27
I.OOO
o.oo
4.413
«
350.27
8.70
234-79
352.28
I.OOO
o.oo
4.259
H
352.28
8-75
243.23
354-30
I.OOO
o.oo
4.111
a
354.30
8.80
251-99
356.31
I.OOO
o.oo
3.968
«
356.31
8.85
261.06
358.33
I.OOO
0.00
3-830
It
358.33
8.90
270.46
360.34
I.OOO
o.oo
3.698
u
360.34
8-95
280.19
362.36
I.OOO
o.oo
3.569
u
362.36
9.00
290.28
364-38
I.OOO
o.oo
3-445
"
364.38
9-05
300.73
366.39
I.OOO
o.oo
3.3253
•
366.39
9.10
3H.54
368.41
I.OOO
o.oo
3.2099
u
368.41
9-i5
322.75
370.42
I.OOO
o.oo
3.0983
u
370.42
9.20
334-37
372-44
I.OOO
o.oo
2.9908
u
372.44
9-25
346.39
37446
I.OOO
o.oo
2.8869
u
374.46
9-3°
358.85
376.47
I.OOO
o.oo
2.7867
u
376.47
9-35
371.81
378.48
I.OOO
o.oo
2.6895
a
378.48
9.40
385-15
380.50
I.OOO
o.oo
2.5964
u
380.50
9-45
399.04
382.51
I.OOO
o.oo
2.5060
u
382.51
9-5°
413-38
384.53
I.OOO
o.oo
2.4191
"
384.53
9-55
428.26
386.55
I.OOO
o.oo
2-3350
u
386.55
9.60
443.67
388.56
I.OOO
o.oo
2.2540
u
388.56
9-65
446.93
390-57
I.OOO
o.oo
2.2263
u
390.57
9.70
476.18
392.59
I.OOO
o.oo
2.IOOI
u
392.59
9-75
493-31
395-01
I.OOO
o.oo
2.0271
u
395-01
9.80
511.07
397.02
I.OOO
o.oo
1.9567
u
397.02
9-85
529.46
399-03
I.OOO
o.oo
1.8887
u
399.03
9.90
548.52
401.05
I.OOO
o.oo
1.8231
u
401.05
9-95
568.25
403.07
I.OOO
o.oo
1.7598
u
403.07
10.00
588.69
405.08
I.OOO
0.00
r.6987
u
405.08
10.05
609.89
407.09
I.OOO
o.oo
1.6397
u
407.09
IO.IO
631.84
409.11
I.OOO
0.00
1.5827
u
409.11
10.15
654-58
411.13
I.OOO
o.oo
•5277
u
411.13
IO.2O
678.14
4I3-H
I.OOO
o.oo
•4746
u
413-14
10.25
702.53
415-15
I.OOO
o.oo
•4234
a
415.15
10.30
727.81
417.17
I.OOO
o.oo
•3740
u
417.17
10-35
754-01
419.19
I.OOO
o.oo
.3262
u
419.19
10.40
781.14
421.21
I.OOO
o.oo
.2802
u
421.21
10.45
809.26
423-23
I.OOO
o.oo
•2357
a
423-23
10.50
838.38
425-24
I.OOO
o.oo
1.1928
u
425.24
Note. Negative quantities are in heavy type.
Examples.
sinh (10.0/45^) = cosh (10.0/45^) = 588.69 /405°.o8'.
sech (IO.Q /45°) = cosech (10.0/45^) = 1.6987 X io~3 \4os°.o8' = 1.6987 X io~3 \45°.o8'
[36]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. f(pl*£) = r fa. CONTINUED ;
Sinh and cosh
Tanh and coth
Sech and cosech
10.55
868.56
427.26
I.OOO
o.oo
1. 1513X10-'
427.26
io.6o
899.81
429.27
I.OOO
o.oo
1.1113
•
429.27
10.65
932.18
43L29
I.OOO
o.oo
1.0728
•
431.29
10.70
965.74
433-30
I.OOO
o.oo
1.0555
*
433.30
10.75
1,000.5
435-32
I.OOO
o.oo
do"4
435.32
10.80
1,036.5
437-33
I.OOO
o.oo
9.6478
u
437-33
10.85
1,073.8
439-35
I.OOO
o.oo
9.3128
u
439-35
10.90
1,112.4
441-36
I.OOO
o.oo
8.9892
u
441.36
10.95
1,152.5
443-38
I.OOO
o.oo
8.6770
"
443.38
II.OO
1,194.0
445-39
I.OOO
o.oo
8.3750
"
445.39
11.05
1,237.0
447.41
I.OOO
o.oo
8.0845
u
447.41
II. IO
1,281.5
449.42
I.OOO
o.oo
7.8037
u
449.42
11.15
1,327.5
451-44
I.OOO
o.oo
7-5327
a
451.44
II. 20
i,375-3
45346
I.OOO
o.oo
7.2711
u
453.46
11.25
1,424.8
455-47
I.OOO
o.oo
7.0184
u
455-47
11.30
1,476.1
457-48
I.OOO
o.oo
6.7747
u
457.48
"•35
1,529-2
459-50
I.OOO
o.oo
6.5393
a
459.50
11.40
1,584-3
561-52
.000
o.oo
6.3120
u
461.52
"•45
1,641.4
463-53
.000
o.oo
6.0929
a
463.53
11.50
1,700.3
465-54
.000
o.oo
5.8811
a
465.54
"•55
1,761.5
467.56
.000
o.oo
5.6769
u
467.56
1 1. 60
1,824.9
469-57
.000
o.oo
5-4797
a
469.57
11.65
1,890.6
471-59
I.OOO
o.oo
5-2893
u
471.59
11.70
1,958.6
474.01
I.OOO
o.oo
5-1056
u
474.01
"•75
2,029.1
476.03
I.OOO
o.oo
4.9282
u
476.03
n.8o
2,102.1
478.04
I.OOO
o.oo
4.7571
a
478.04
11.85
2,177.8
480.05
I.OOO
o.oo
4.5910
u
480.05
11.90
2,256.1
482.07
I.OOO
o.oo
4.4323
u
482.07
"•95
2,337-3
484.09
I.OOO
o.oo
4.2784
u
484.09
I2.OO
2,421.5
486.10
I.OOO
o.oo
4.1297
u
486.10
12.05
2,508.6
488.12
I.OOO
o.oo
3.9862
u
488.12
12.10
2,598.9
490.14
I.OOO
o.oo
3-8478
a
490.14
12.15
2,692.6
492.15
I.OOO
o.oo
3-7141
u
492.15
12. 2O
2,789.0
494.17
I.OOO
o.oo
.3-5856
u
494.17
12.25
2,889.7
496.18
I.OOO
o.oo
3-4605
a
496.18
12.30
2,993-7
498.20
I.OOO
o.oo
3-3403
u
498.20
12-35
3,101.4
500.21
I.OOO
o.oo
3-2243
u
500.21
12.40
3,213-1
502-23
I.OOO
o.oo
3-0143
u
502.23
12-45
3,328.3
504.24
I.OOO
o.oo
3.0042
u
504.24
12.50
3,448.5
506.26
I.OOO
o.oo
2.8998
a
506.26
12.55
3,572.6
508.27
I.OOO
o.oo
2.7991
a
508.27
12.60
3,7oi.i
510.29
I.OOO
o.oo
2.7019
u
510.29
12.65
3,834-3
5I2-3i
I.OOO
o.oo
2.6080
a
512.31
12.70
3,972.6
514-32
I.OOO
o.oo
2.5172
u
314.32
12.75
4,"5-3
516.33
I.OOO
o.oo
2.4300
a
516.33
Note. Negative quantities are in heavy type.
Examples, sinh (12.0 /45°) = cosh (12.0/45°) = 2421.5
86°. 10' = 2421.5
sech (12.75 /45°) = cosech (12.75 /45°) = 2.43 X io-3\si6°.33'.
[37]
442282
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. / (p /45!) = r /y. CONTINUED
Sinh and cosh
Tanh and coth
Sech and cosech
12.80
4,263.4
518.35
I.OOO
o.oo
2-3455 X
I0~4
518.35
12.85
4,416.8
520.37
I.OOO
o.oo
2.2641
•
520.37
12.90
4,575-7
522.38
I.OOO
o.oo
2.1854
"
522.38
12.95
4,740.5
524.39
I.OOO
o.oo
2.1095
"
524.39
13.00
4,911.0
526.41
I.OOO
o.oo
2.0362
"
526.41
13-05
5,087.8
528.43
I.OOO
o.oo
1.9655
u
528.43
13.10
5,270.9
530.44
I.OOO
o.oo
1.8972
u
430.44
13-15
5,460.6
532-45
I.OOO
o.oo
1.8313
u
532.45
13.20
5,657.0
534-47
I.OOO
o.oo
1.7677
u
534-47
13.25
5,858.5
536.49
I.OOO
o.oo
1.7061
u
536.49
13-30
6,071.6
538.50
I.OOO
o.oo
1.6470
u
538.50
13-35
6,290.1
540.51
I.OOO
o.oo
1.5898
u
540.51
13.40
6,516.5
542.53
.000
o.oo
1.5346
u
542.53
13-45
6,751.0
544-55
.000 '
o.oo
1.4813
u
544-55
13-50
6,993-9
546.57
.000
o.oo
1.4298
u
546.57
13-55
7,245-5
548.58
.000
o.oo
1.3801
u
548.58
13.60
551-00
.000
o.oo
1.3322
u
551.00
13-65
7,776.4
553-01
I.OOO
o.oo
1.2859
u
553-01
13.70
8,056.4
555-03
I.OOO
o.oo
1.2412
u
555.03
13-75
8,346.2
557-05
I.OOO
o.oo
1.1982
u
557.05
13.80
8,646.7
559-o6
I.OOO
o.oo
1.1565
u
559.06
13-85
8,957.8
561.07
I.OOO
o.oo
1.1164
u
561.07
13.90
9,280.3
563-09
I.OOO
o.oo
1.0776
a
563.09
13-95
9,614.1
565-11
I.OOO
o.oo
1.0165
u
565.11
14.00
9,960.2
567.12
I.OOO
o.oo
1.0040
u
567.12
14.05
10,318
569.14
I.OOO
o.oo
9.69i4X
I0~5
569.14
14.10
10,690
571.15
I.OOO
o.oo
9-3547
u
571.15
14-15
11,075
573-i6
I.OOO
o.oo
9.0296
u
573-16
14.20
n,473
575-iS
I.OOO
o.oo
8.7160
u
575.18
14-25
11,886
577-20
I.OOO
o.oo
8.4132
u
577.20
14.30
12,314
579-21
I.OOO
o.oo
8.I2IO
u
579-21
14-35
12,757
581.22
I.OOO
o.oo
7.8388
u
581.22
14.40
13,216
583-24
I.OOO
o.oo
7.5666
u
583.24
14-45
13,692
585-26
.000
o.oo
7.3037
u
585.26
14.50
14,184
587-27
.000
o.oo
7.0500
u
587.27
14-55
14,695
589.29
.000
o.oo
6.8050
u
289.29
14.60
15,224
591.30
.000
o.oo
6.5687
u
591.30
14.65
15,772
593-32
.000
o.oo
6.3405
u
593.32
14.70
i6,339
595-34
.000
o.oo
6.1203
u
595-34
14-75
16,927
597-35
.000
o.oo
5.9077
u
597-35
14.80
17,536
599-37
.000
o.oo
5-7024
a
599-37
14.85
18,167
601.39
.000
o.oo
5-5044
u
601.39
14.90
18,822
603.40
.000
o.oo
5-3130
u
603.40
14-9$
19,498
605.41
I.OOO
o.oo
5.1286
u
605.41
15.00
20,200
607.43
I.OOO
o.oo
4.9504
u
607.43
Note. Negative quantities are in heavy type.
Examples, sinh (14.0 /45°) = cosh (14.0 745°) = 9960.2 A67
sech (14.0 /45°)
cosech (14.0 /45°)
[38]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. /(p/45JO = r/jy. CONTINUED
Sinh and cosh
Tanh and coth
Sech and cosech
15-05
20,927
609.44
I.OOO
o.oo
4-7785 Xio~6 609.44
15.10
21,680
611.46
1. 000
o.oo
4.6120
u
611.46
15-15
22.460
613.48
I.OOO
o.oo
4-4523
u
613.48
15.20
23,269
615.49
I.OOO
o.oo
4.2980
u
615.49
15-25
24,106
617-50
I.OOO
o.oo
4.1482
u
617.50
15-30
24,973
619.52
I.OOO
o.oo
4.0040
u
619.52
15-35
25,873
621.54
I.OOO
o.oo
3.8651
u
621.54
15.40
26,802
623.55
I.OOO
o.oo
3-73Jo
u
623.55
15-45
27,768
625-57
I.OOO
o.oo
3.6012
u
625.57
15-50
28,765
627-59
I.OOO
o.oo
3.4760
u
627.59
15-55
29,803
630.00
I.OOO
o.oo
3-3554
u
630.00
15.60
30,872
632.02
I.OOO
o.oo
3-2390
u
632.02
15-65
31,987
634.04
I.OOO
o.oo
3.1263
a
634.04
15-70
33,i4o
636.05
I.OOO
o.oo
3.0170
u
636.05
15-75
34,33i
638.06
I.OOO
0.00
2.9129
u
638.06
15-80
35,569
640.08
I.OOO
o.oo
2.8110
u
640.08
15.85
36,846
642.10
I.OOO
o.oo
2.7140
a
642.10
15.90
38,i74
644.11
I.OOO
o.oo
2.6200
u
644.11
15-95
39,546
646.12
I.OOO
o.oo
2.5287
u
646.12
16.00
40,970
648.14
I.OOO
o.oo
2.4410
u
648.14
16.05
42,443
650.16
I.OOO
0.00
2.3561
a
650.16
16.10
43,97i
652.17
I.OOO
o.oo
2.2740
a
652.17
16.15
45,553
654.18
I.OOO
o.oo
2.1952
u
654.18
16.20
47,192
656.20
I.OOO
o.oo
2.1190
u
656.20
16.25
48,890
658.22
I.OOO
o.oo
2.0454
a
658.22
16.30
50,649
660.23
I.OOO
o.oo
1.9740
a
660.23
16.35
52,473
662.24
I.OOO
. o.oo
1-9055
u
662.24
16.40
54,359
664.26
I.OOO
o.oo
1.8400
u
664.26
16.45
56,316
666.28
I.OOO
o.oo
J-7757
a
666.28
16.50
58,475
668.29
I.OOO
o.oo
1.7100
u
668.29
16.55
60,444
670.31
I.OOO
o.oo
1.6544
u
670.31
16.60
62,619
672.32
I.OOO
o.oo
1.5969
u
672.32
16.65
64,872
674-34
I.OOO
o.oo
1-5415
u
674.34
16.70
57,208
676-35
I.OOO
o.oo
1.4879
u
676.35
16.75
69,626
678.36
I.OOO
o.oo
1.4362
u
678.36
16.80
72,132
680.38
I.OOO
o.oo
1.3863
u
680.38
16.85
74,727
682.40
I.OOO
o.oo
1.3382
u
682.40
16.90
77,4i8
684.41
I.OOO
o.oo
1.2917
u
684.41
16.95
80,203
686.43
I.OOO
o.oo
1.2468
u
686.43
17.00
83,088
688.45
I.OOO
o.oo
1-2035
u
688.45
17-05
86,080
690.47
I.OOO
o.oo
1.1617
u
690.47
17.10
89,176
692.48
I.OOO
o.oo
1.1214
u
692.48
I7-I5
92,387
694.49
I.OOO
o.oo
1.0824
u
694.49
17.20
95, 711
696.51
I.OOO
o.oo
1.0448
u
696.51
17-25
99,149
698-53
I.OOO
o.oo
1.0086
u
698-53
Note. Negative quantities are in heavy type.
Examples, sinh (17.0 /45°) = cosh (17.0 /45°) = 83,088 /688
= 83,088 /328C
sech (17.0 /45°) = cosech (17.0 /45°) = 1-2035 X io~5 \688°.45'-
[39]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES.
p Sinh and cosh Tanh and coth
= r/jy. CONTINUED
Sech and cosech
17.30
102,720
700.54
I.OOO
o.oo
9.7349X10-'
' 700.54
17-35
106,420
702.55
I.OOO
o.oo
9.3968
u
702.55
17.40
110,250
704.57
I.OOO
o.oo
9.0703
u
704.57
17.45
114,220
706.59
I.OOO
o.oo
8.7551
u
706.59
17-50
"8,330
709.00
I.OOO
o.oo
8.4510
u
709.00
17-55
122,590
711.01
I.OOO
o.oo
8.1576
u
711.01
17.60
127,000
7I3-03
I.OOO
o.oo
7.8741
u
713.03
17.65
131,570
715-05
I.OOO
o.oo
7.6006
u
715.05
17.70
136,300
717.06
I.OOO
o.oo
7.3365
u
717.06
17.75
141,210
719.07
I.OOO
o.oo
7.0817
u
719.07
17.80
146,290
721.09
I.OOO
o.oo
6.8356
u
721.09
17-85
i5i,55o
723.11
I.OOO
o.oo
6.5983
u
723.11
17.90
157,000
725.12
I.OOO
o.oo
6.3710
u
725.12
17-95
162,660
727.13
I.OOO
o.oo
6.1478
u
727.13
18.00
168,520
729-15
I.OOO
o.oo
5.9383
u
729.15
18.05
174,580
73I-I7
I.OOO
o.oo
5.7281
u
731-17
18.10
180,860
733-iS
I.OOO
o.oo
5.5292
u
733.18
18.15
183,530
735-20
I.OOO
o.oo
5.4488
u
735-20
18.20
194,110
737-21
I.OOO
o.oo
5.1517
u
737.21
18.25
201,100
739-23
I.OOO
o.oo
4.9727
u
739.23
18.30
208,330
741.24
I.OOO
o.oo
4.8000
u
741.24
18.35
215,830
743.26
I.OOO
o.oo
4.6332
u
743.26
18.40
223,600
745-27
I.OOO
o.oo
4-4723
u
745.27
18.45
231,650
747.29
I.OOO
o.oo
4.3168
u
747.29
18.50
239,980
749-31
I.OOO
o.oo
4.1671
u
749.31
18.55
248,620
75I-32
I.OOO
o.oo
4.0222
It
751.32
18.60
257,570
753-34
I.OOO
o.oo
3.8825
u
753.34
18.65
266,840
755-35
I.OOO
o.oo
3.7476
u
755-35
18.70
. 276,440
757-37
I.OOO
o.oo
3.6174
u
757-37
18-75
286,390
759-38
I.OOO
o.oo
3.4918
u
759-38
18.80
296,690
761.40
I.OOO
o.oo
3.3628
u
761.40
18.85
307,380
763-41
I.OOO
o.oo
3.2533
u
763-41
18.90
318,570
765-43
I.OOO
o.oo
3.1404
u
765-43
18.95
329,890
767.44
I.OOO
o.oo
3.0313
u
767.44
19.00
34i,77o
769.46
I.OOO
o.oo
2.9260
u
769.46
19.05
354,060
771-47
I.OOO
o.oo
2.8244
u
771.47
19.10
366,810
773-49
I.OOO
o.oo
2.7262
u
773-49
i9-!5
380,010
775-50
I.OOO
o.oo
2.6315
a
775-50
19.20
393,690
777-52
I.OOO
o.oo
2.5401
u
777-52
19-25
407,850
779-53
I.OOO
o.oo
2.4519
u
779-53
19.30
422,530
78i.55
I.OOO
o.oo
2.3667
u
781.55
19-35
437,730
783-57
I.OOO
o.oo
2.2845
u
783.57
19.40
453,490
785-59
I.OOO
o.oo
2.2051
u
785.59
19-45
469,810
788.00
I.OOO
o.oo
2.1285
u
788.00
19.50
486,720
790.02
I.OOO
o.oo
2.0546
u
790.02
Note. Negative qyantities are in heavy type.
Examples, sinh (19.05 /45°) = cosh (19.05 /45°) = 354,o6o /77i°-
sech (19.30 /45°) = cosech (19.3 745°) = 2.3667 X io~«'
[40]
TABLE VI
FUNCTIONS OF SEMI-IMAGINARIES. /(p/45°) = r /y. CONTINUED
Sinh and cosh
Tanh and coth
Sech and cosech
19-55
504,230
792.03
I.OOO
o.oo
1. 983 2 >
Cio~*
1 792.03
19.60
522,380
794-05
I.OOO
o.oo
1-9*53
u
794.05
19.65
541,220
796.06
I.OOO
o.oo
1.8478
"
796.06
19.70
560,650
798.08
I.OOO
o.oo
1.7837
"
798.08
19-75
599,830
800.09
I.OOO
o.oo
1.6671
"
800.09
19.80
601,730
802. ii
I.OOO
o.oo
1.6619
«
802. II
19.85
623,390
804.12
I.OOO
o.oo
1.6041
•
804.12
19.90
645,820
806.14
I.OOO
o.oo
1.5484
u
806.14
19-95
669,070
808.15
I.OOO
o.oo
1.4946
u
808.15
20.00
693,1 50
810.17
I.OOO
o.oo
1.4426
u
810.17
20.05
718,090
812.18
I.OOO
o.oo
1.3926
u
812.18
20. 10
743,930
814.20
I.OOO
o.oo
1.3442
u
814.20
20.15
770,710
816.21
I.OOO
o.oo
1.2975
u
816.21
20.20
798,440
818.23
I.OOO
o.oo
1-2525
u
818.23
20.25
827,160
820.24
I.OOO
o.oo
1.2090
u
820.24
20.30
856,940
822.26
I.OOO
o.oo
1.1669
u
822.26
20.35
887,770
824.27
I.OOO
o.oo
1.1264
a
824.27
20.40
9*9,730
826.29
I.OOO
o.oo
1.0873
u
826.29
20.45
952,820
828.30
I.OOO
o.oo
1.0496
u
828.30
20.50
987,120
830.32
I.OOO
o.oo
1.0130
u
830.32
Note. Negative quantities are in heavy type.
Example, sinh (20.0 /45°) = cosh (20.0 /45°) = 693,i5o/8io°.i7/ = 693, 150 ,
[41]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv
x = 0.05
x = 0.15
0.0
0.05
O.I
0.15
O.2
o.oo
o.oo
o.oo
o.oo
o.oo
o.oo
0.07846
0.15643
0-23345
0.30902
0.05002
0.04987
0.04945
0.04864
0.04757
o.oo
0.07856
0.15663
0.23374
0.30940
0.10017
0.09986
0.09893
0.09740
0.09526
o.oo
0.07885
0.15722
0.23461
0.31056
0.15056
0.15010
0.14871
0.14640
0.14319
o.oo
0.07934
0.15820
0.23608
0.31250
0.20134
0.20072
0.19886
0.19577
0.19148
o.oo
0.08003
0.15957
0.23813
0.31522
0.25
0-3
0-35
0.4
0-45
o.oo
o.oo
o.oo
o.oo
o.oo
0.38268
Q-45399
0.52250
0.58778
0.64944
0.04621
0.04457
0.04265
0.04047
0.03804
0.38316
0-45454
0.52313
0.58850
0.65023
0.09254
0.08925
0.08541
0.08104
0.07617
0.38460
0.45626
0.52511
0.59073
0.65270
0.13910
0.13415
0.12838
0.12181
0.11449
0.38700
0.45911
0.52839
0.59441
0.65677
0.18601
0.17939
0.17167
0.16288
0.15310
0.39036
0.46310
0.53298
0.59958
0.66248
0-5
0.55
0.6
0.65
0.7
o.oo
o.oo
o.oo
o.oo
o.oo
0.70711
0.76041
0.80902
0.85264
0.89101
0.03537
0.03249
0.02940
0.02614
0.02271
0.70796
0.76133
0.81000
0.85367
0.89208
0.07083
0.06505
0.05888
0.05234
0.04547
0.71065
0.76421
0.81307
0.85691
0.89547
0.10646
0.09778
0.08850
0.07867
0.06835
0.71508
0.76898
0.81814
0.86225
0.90105
0.14237
0.13076
0.11834
0.10520
0.09141
0.72130
0.77567
0.82525
0.86975
0.90889
0-75
0.8
0.85
0.9
0-95
o.oo
o.oo
o.oo
o.oo
o.oo
0.92388
0.95106
0.97237
0.98769
0.99692
0.01914
0.01546
0.01168
0.00783
0.00392
0.92503
0.95225
0-97359
0.98892
0.99816
0.03833
0.03095
0.02338
0.01567
0.00786
0.92850
0.95582
0.97724
0.99263
1.00191
0.05762
0.04653
0.03515
0.02355
0.01181
0.93429
0.96178
0.98333
0.99882
1.00815
0.07705
0.06222
0.04700
0.03150
0.01580
0.94242
0.97014
0.99188
1.00751
1.01692
I.O
1.05
i.i
I-I5
1.2
o.oo
o.oo
o.oo
o.oo
o.oo
I.OOOOO
0.99692
0.98769
0.97237
0.95106
o.oo
0.00392
0.00783
0.01168
0.01546
1.00125
0.99816
0.98892
0-97359
0.95225
o.oo
0.00786
0.01567
0.02338
0.03095
1.00500
1.00191
0.99263
0.97724
0.95582
o.oo
0.01181
0.02355
0.03515
0.04653
1.01127
1.00815
0.99882
0.98333
0.96178
o.oo
0.01580
0.03150
0.04700
0.06222
1.02007
1.01692
1.00751
0.99188
0.97014
I.2S
i-3
1-35
1.4
1-45
o.oo
o.oo
o.oo
o.oo
o.oo
0.92388
0.89101
0.85264
0.80902
0.76041
0.01914
0.02271
0.02614
0.02940
0.03249
0.92503
0.89208
0.85367
0.81000
0.76133
0.03833
0.04547
0.05234
0.05888
0.06505
0.92850
0.89547
0.85691
0.81307
0.76421
0.05762
0.06835
0.07867
0.08850
0.09778
0.93429
0.90105
0.86225
0.81814
0.76898
0.07705
0.09141
0.10520
0.11834
0.13076
0.94242
0.90889
0.86975
0.82525
0.77567
i-5
i-55
1.6
1.65
i-7
o.oo
o.oo
o.oo
o.oo
o.oo
0.70711
0.64944
0.58778
0.52250
0-45399
0.03537
0.03804
0.04047
0.04265
0.04457
0.70796
0-65023
0.58850
0.52313
0-45454
0.07083
0.07617
0.08104
0.08541
0.08925
0.71065
0.65270
0-59073
0-52511
0.45626
0.10646
0.11449
o. 12181
0.12838
0.13415
0.71508
0.65677
0.59441
0.52839
0.45911
0.14237
0.15310
0.16288
0.17167
0.17939
0.72130
0.66248
0.59958
0.53298
0.46310
1-75
i. 80
1.85
1.9
1-95
o.oo
o.oo
o.oo
o.oo
o.oo
0.38268
0.30902
0.23345
0.15643
0.07845
0.04621
0.04757
0.04864
0.04945
0.04987
0.38316
0.30940
0-23374
0.15663
0.07856
0.09254
0.09526
0.09740
0.09893
0.09986
0.38460
0.31056
0.23461
0.15722
0.07885
0.13910
0.14319
0.14640
0.14871
0.15010
0.38700
0.31250
0.23608
0.15820
0.07934
o. 18601
0.19148
0.19577
0.19886
0.20072
0.39036
0.31522
0.23813
0.15957
0.08003
2.0
o.oo
o.oo
0.05002
o.oo
0.10017
o.oo
0.15056
o.oo
0.20134
o.oo
Note.
Negative quantities are in heavy type.
Examples.
sinh (o.i
+ i 0.5) = 0.07083
+ * 0.71065.
sinh (o.i + i 1.2) = —0.03095 + i 0.95582.
[42]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 0.25
x = 0.3
= 0.35
x = 0.4
x = 0.45
o
0.25261
o.oo
0.30452
o.oo
0.35719
o.oo
0.41075
o.oo
0.46534
o.oo
0.05
0.25183
0.08092
0.30358
0.08202
0.35609
0.08331
0.40949
0.08482
0.46391
0.08654
O.I
0.24950
0.16135
0.30077
0.16353
0.35279
0.16611
0.40570
0.16912
0.45961
0.17254
0.15
0.24563
0.24078
0.29611
0.24403
0-34732
0.24789
0-39940
0.25237
0.45249
0.25748
O.2
0.24025
0.31872
0.28962
0.32303
0-33971
0.32814
0.39065
0.33407
0.44257
0.34084
0.25
0.23338
0.39471
0.28134
0.40003
0.33000
0.40636
0-37949
0.41371
0.42992
0.42209
0-3
0.22508
0.46825
0.27133
0-47457
0.31826
0.48208
0.36598
0.49080
0.41462
0.50074
o-35
0.21539
0.53891
0.25965
0.54619
0.30455
0.55483
0.35022
0.56486
0.39677
0.57630
0.4
0.20437
0.60625
0.24636
0.61444
0.28897
0.62416
0.33231
0.63544
0.37647
0.64831
0-45
0.19208
0.66985
0.23156
0.67889
0.27161
0.68964
0.31234
0.70210
0.35385
0.71632
o-5
0.17862
0.72932
0.21533
0.73917
0.25257
0.75086
0.29045
0.76443
0.32905
0.77992
Q-55
o. 1 6406
0.78429
0.19777
0.79488
0.23198
0.80746
0.26676
0.82206
0.30222
0.83871
0.6
0.14848
0.83443
0.17899
0.84570
0.20995
0.85908 •
0.24143
0.87461
0.27352
0.89232
0.65
0.13199
0.87942
0.15911
0.89130
0.18663
0.90540
0.21462
0.92177
0.24314
0.94044
0.7
0.11468
0.91900
0.13825
0.93140
0.16216
0.94614
0.18648
0.96324
0.21126
0.98275
0-75
0.09667
0.95290
0.11654
0.96577
0.13669
0.98105
0.15719
0.99878
0.17808
1.01901
0.8
0.07801
0.98093
0.09410
0.99418
0.11038
.00991
0.12693
1.02816
0.14380
1.04899
0.85
0.05897
1.00292
0.07109
1.01646
0.08338
•03254
0.09589
1.05120
0.10863
1.07250
0.9
0.03952
1.01871
0.04764
1.03247
0.05588
.04880
0.06426
1.06776
0.07280
1.08939
o-95
0.01982
1.02823
0.02389
1.04212
0.02803
.05860
0.03223
1.07774*
0.03651
1.09957
I.O
o.oo
1.03141
o.oo
1-04534
0.00
.06188
o.oo
1.08107
o.oo
1.10297
1.05
0.01982
1.02823
0.02389
1.04212
0.02803
.05860
0.03223
1.07774
0.03651
1.09957
i.i
0.03952
1.01871
0.04764
1.03247
0.05588
.04880
0.06426
1.06776
0.07280
1.08939
i-i5
0.05897
1.00292
0.07109
1.01646
0.08338
•03254
0.09589
1.05120
0.10863
1.07250
1.2
0.07801
0.98093
0.09410
0.99418
0.11038
.00991
0.12693
1.02816
0.14380
1.04899
1-25
0.09667
0.95290
0.11654
0.96577
0.13669
0.98105
0.15719
0.99878
0.17808
I.OIOOI
i-3
0.11468
0.91900
0.13825
0.93140
0.16216
0.94614
0.18648
0.96324
0.21126
0.98275
i-3S
0.13199
0.87942
0.15911
0.89130
0.18663
0.90540
0.21462
0.92177
0.24314
0.94044
i-4
0.14848
0.83443
0.17899
0.84570
0.20995
0.85908
0.24143
0.87461
0.27352
0.89232
1-45
0.16406
0.78429
0.19777
0.79488
0.23198
0.80746
0.26676
0.82206
0.30222
0.83871
i-5
0.17862
0.72932
0.21533
0.73917
0.25257
0.75086
0.29045
0.76443
0.32905
0.77992
1-55
0.19208
0.66985
0.23156
0.67889
0.27161
0.68964
0.31234
0.70210
0.35385
0.71632
1.6
0.20437
0.60625
0.24636
0.61444
0.28897
0.62416
0.33231
0.63544
0.37647
0.64831
1-65
0.21539
0.53891
0.25965
0.54619
0.30455
0.55483
0.35022
0.56486
0.39677
0.57630
i-7
0.22508
0.46825
0.27133
0-47457
0.31826
0.48208
0.36598
0.49080
0.41462
0.50074
i-75
0.23338
0.39471
0.28134
0.40003
0.33000
0.40636
0.37949
0.41371
0.42992
0.42209
1.8
0.24025
0.31872
0.28962
0.32303
0.33071
0.32814
0.39065
0.33407
0.44257
0.34084
1-85
0.24563
0.24078
0.29611
0.24403
0.34732
0.24789
0.39940
0.25237
0.45249
0.25748
1.9
0.24950
0.16135
0.30077
0.16353
0.35279
0.16611
0.40570
0.16912
0.45961
0.17254
i-95
0.25183
0.08092
0.30358
0.08202
0.35609
0.08331
0.40949
0.08482
0.46391
0.08654
2.0
0.25261
o.oo
0.30452
o.oo
0.35719
o.oo
0.41075
o.oo
0.46534
o.oo
Note.
Negative quantities are in heavy type.
Examples.
sinh (0.4 + i o) =
0.41075 +
to.
sinh (0.4 + * i) =
o. + i 1.08107.
[43]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 0.5
= 0.55
x = 0.6
x = 0.65
x = 0.7
o
0.05
O.I
0.15
O.2
0.52110
0.51949
0.51468
0.50670
0-49559
o.oo
0.08847 •
0.17640
0.26324
0.34846
0.57815 o.oo
0.57637 0.09063
0.57103 0.18070
0.56218 0.26965
0.54986 0.35695
0.63665
0.63469
0.62882
0.61906
0.60549
o.oo
0.09301
0.18545
0.27674
0.36633
0.69675
0.69460
0.68817
0.67750
0.66265
o.oo
0.09563
0.19066
0.28452
0.37663
0.75858
0.75625
0.74925
0.73763
0.72146
o.oo
0.09848
0.19635
0.29301
0.38787
O.25
0-3
0-35
0.40
0-45
0.48143
0.46430
0.44431
0.42158
0.39624
0.43152
0.51193
0.58918
0.66280
0.73233
0.53414
0.51514
0.49296
0.46773
0.43963
0.44204
0.52441
0.60354
0.67895
0.75018
0.58819
0.56726
0.54284
0.51506
0.48412
0.45366
0.53819
0.61940
0.69680
0.76990
0.64371
0.62081
0.59408
0.56368
0.52981
0.46641
0.55332
0.63682
0.71639
0.79154
0.70084
0.67590
0.64680
0.61371
0.57683
0.48033
0.56984
0.65582
0.73777
0.81517
0-5
0-55
0.6
0.65
0.7
0.36847
0.33842
0.30629
0.27227
0.23657
0-79735
0.85745
0.91227
0.96146
1.00472
0.40882
0.37548
0.33983
0.30208
0.26248
0.81678
0.87835
0.93450
0.98489
1.02920
0.45018
0.41347
0.37422
0.33265
0.28904
0.83825
0.90144
0.95906
1.01078
1.05626
0.49268
0.45250
0.40954
0.36405
0.31632
0.86182
0.92678
0.98602
1.03919
1.08595
0.53640
0.49266
0.44589
0.39636
0.34439
0.88754
0.95444
1.01545
1.07021
1.11836
0-75
0.8
0.85
0.9
0-95
0.19942
0.16103
0.12165
0.08152
0.04088
1.04179
1.07244
1.09647
I-II374
1.12415
0.22125
0.17866
0.13497
0.09044
0.04536
1.06717
1.09857
1.12319
1.14088
1.15154
0.24364
0.19674
0.14862
0.09959
0.04995
1.09523
1.12744
1.15271
1.17087
1. 18181
0.26663
0.21531
0.16265
0.10900
0.05467
1.12602
1.15912
1.18512
1.20379
1.21504
0.29030
0.23442
0.17709
0.11867
0.05952
1.15962
1.19374
1.22049
1.23972
1.25130
.0
•05
.1
•15
.2
o.oo
0.04088
0.08152
0.12165
0.16103
1.12763
1.12415
I-II374
1.09647
1.07244
o.oo
0.04536
0.09044
0.13497
0.17866
1.15510
1.15154
1.14088
1.12319
1.09857
0.00
0.04995
0.09959
0.14862
0.19674
1.18547
1. 18181
1.17087
1.15271
1.12744
o.oo
0.05467
0.10900
0.16265
0.21531
1.21879
1.21504
1.20379
1.18512
1.15912
o.oo
0.05952
0.11867
0.17709
0.23442
1.25517
1.25130
1.23972
1.22049
1-19374
1-25
i-35
1.4
i -45
0.19942
0.23657
0.27227
0.30629
0.33842
1.04179
1.00472
0.96146
0.91227
0.85745
0.22125
0.26248
0.30208
0.33983
0.37548
1.06717
1.02920
0.98489
0.93450
0.87835
0.24364
0.28904
0.33265
0.37422
0.41347
1.09523
1.05626
1.01078
0.95906
0.90144
0.26663
0.31632
0.36405
0.40954
0.45250
1.12602
1.08595
1.03919
0.98602
0.92678
0.29030
0.34439
0.39636
0.44589
0.49266
1.15962
1.11836
1.07021
1-01545
0-95444
i-SS
1.6
1-65
0.36847
0.39624
0.42158
0.44431
0.46430
0-79735
0.73233
0.66280
0.58918
0.5H93
0.40882
0.43963
0.46773
0.49296
0.51514
0.81678
0.75018
0.67895
0.60354
0.52441
0.45018
0.48412
0.51506
0.54284
0.56726
0.83825
0.76990
0.69680
0.61940
0.53819
0.49268
0.52981
0.56368
0.59408
0.62081
0.86182
0.79154
0.71639
0.63682
0.55332
0.53640
0.57683
0.61371
0.64680
0.67590
0.88754
0.81517
0-73777
0.65582
0.56984
1-75
1.8
1.85
1.9
1-95
0.48143
0.49559
0.50670
0.51468
0.51949
0.43152
0.34846
0.26324
o. 1 7640
6.08847
0.53414
0.54986
0.56218
0.57103
0.57637
0.44204
0.35695
0.26965
0.18070
0.09063
0.58819
0.60549
0.61906
0.62882
0.63469
0.45366
0.36633
0.27674
0.18545
0.09301
0.64371
0.66265
0.67750
0.68817
0.69460
0.46641
0.37663
0.28452
0.19066
0.09563
0.70084
0.72146
0.73763
0.74925
0.75625
0.48033
0.38787
0.29301
0.19635
0.09848
2.0
0.52110
0.00
0.57815
0.00
0.63665
o.oo
0.69675
0.00
0.75858
o.oo
Note.
Negative quantities are in heavy type.
Examples.
sinh (0.65
sinh (0.55 •
f i 0-75)
f i 1-40)
= 0.26663 + * 1.12602.
= -0.33983 + * 0.93450.
[44]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 0.75
x = 0.85
x = o.g
x = 0.95
o
0.05
O.I
0-15
O.2
0.82232
0.81978
0.8I2I9
0.79960
0.78207
o.oo
0.10158
0.20253
0.30224
0.40079
0.88811
0.88537
0.87717
0.86357
0.84464
o.oo
0.10493
0.20922
0.31222
0.41329
0.95612
0.95317
0-94435
0.92970
0.90932
o.oo
0.10855
0.21643
0.32298
0.42753
1.02652 .
1.02335
1.01388
0.99816
0.97628
o.oo
0.11244
0.22418
0.33455
0.44285
1 .09948
1.09610
1.08595
1.06911
1.04567
o.oo
o. 11661
0.23250
0.34695
0.45927
O.25
o-3
0.35
0.4
0.45
0.75972
0.73269
O./OII4
0.66527
0.62529
0.49545
0.58777
0.67647
0.76100
0.84083
0.82050
0.79131
0.75724
0.71849
0.67532
0.51182
0.60718
0.69881
0.78613
0.86859
0.88334
0.85191
0.81522
o.7735i
0.72704
0.52945
0.62811
0.72289
0.81322
0.89853
0.94838
0.91463
0.87525
0.83047
0.78057
0.54842
0.65061
0.74879
0.84235
0.93071
1.01579
0.97965
0.93747
0.88950
0.83605
0.56875
0.67473
0.77655
0.87358
0.96523
°-5
0-55
0.6
0.65
0.7
0.58146
0.53405
0.48335
0.42966
0-37332
0.91548
0.98449
1.04742
1.10390
I-iS3S6
0.62799
0.57678
0.52202
0.46403
0.40319
0.94571
1.01700
1.08201
1-14035
1.19166
0.67608
0.62095
0.56199
0-49957
0.43407
0.97830
1.05205
1.11930
1.17965
1.23274
0.72586
0.66667
0.60337
0.53635
0.46603
1-01334
1.08973
i-!5939
1.22191
1.27689
0.77745
0.71406
0.64621
0.57448
0.49916
1.05092
1.13013
1.20238
1.26722
1-32425
0.75
0.8
0.85
0.9
o-95
0.31469
0.254II
0.19197
0.12864
0.06452
1.19613
1.23132
1.25891
1.27874
1.29069
0.33986
0.27444
0.20732
0.13893
0.06968
1-23563
1.27198
1.30048
1.32097
J-3333i
0.36589
0.29546
0.22320
0.14957
0.07502
1.27822
1.31582
1.34530
1.36650
1.37927
0.39283
0.31721
0.23964
0.16058
0.08054
1.32400
1.36294
1-39349
1.41544
1.42867
0.42076
0.33976
0.25667
0.17200
0.08627
1-37309
1.41348
1.44516
1.46793
1.48164
I.O
1.05
i.i
1-15
1.2
o.oo
0.06452
0.12864
0.19197
0.25411
1.29468
1.29069
1.27874
1.25891
1.23132
o.oo
0.06968
0.13893
0.20732
0.27444
1-33743
•33331
•32097
.30048
.27198
o.oo
0.07502
0.14957
0.22320
0.29546
1.38353
•37927
•36650
•34530
.31582
o.oo
0.08054
0.16058
0.23964
0.31721
I.43309
1.42867
1.41544
1-39349
1.36294
o.oo
0.08627
0.17200
0.25667
0.33976
1.48623
1.48164
1.46793
1.44516
1.41348
1-25
i-3
i-3S
1.4
1-45
0.31469
0.37332
0.42966
0.48335
0.53405
1.19613
I-I5356
1.10390
1.04742
0.98449
0.33986
0.40319
0.46403
0.52202
0.57678
•23563
.19166
•14035
.08201
.01700
0.36589
0.43407
0.49957
0.56199
0.62095
.27822
.23274
•17965
.11930
.05205
0.39283
0.46603
0.53635
0.60337
0.66667
1.32400
1.27689
1.22191
I-I5939
1.08973
0.42076
0.49916
0.57448
0.64621
0.71406
1.37309
1.32425
1.26722
1.20238
1.13013
i-5
i-SS
1.6
1.65
i-7
0.58146
0.62529
0.66527
0.70114
0.73269
0.91548
0.84083
0.76100
0.67647
0.58777
0.62799
0.67532
0.71849
0.75724
0.79131
0.94571
0.86859
0.78613
0.69881
0.60718
0.67608
0.72704
0.77351
0.81522
0.85191
0.97830
0.89853
0.81322
0.72289
0.62811
0.72586
0.78057
0.83047
0.87525
0.91463
i-oi334
0.93071
0.84235
0.74879
0.65061
0.77745
0.83605
0.88950
0-93747
0.97965
1.05092
0.96523
0.87358
0.77655
0.67473
i-75
1.8
1.85
1.9
i-95
0.75972
0.78207
0.79960
0.81219
0.81978
0-49545
0.40079
0.30224
0.20253
0.10158
0.82050
0.84464
0.86357
0.87717
0.88537
0.51182
0.41329
0.31222
0.20922
0.10493
0.88334
0.90932
0.92970
0.94435
0.95317
0.52945
0.42753
0.32298
0.21643
0.10855
0.94838
0.97628
0.99816
1.01388
1-02335
0.54842
0.44285
0-33455
0.22418
0.11244
1.01579
1.04567
1.06911
1.08595
1.09610
0.56875
0.45927
0-34695
0.23250
0.11661
2.0
0.82232
o.oo
0.88811
o.oo
0.95612
o.oo
1.02652
o.oo
1.09948
o.oo
Note.
Negative quantities are in heavy type.
Examples.
sinh (0.8 + * 0.7)
= 0.40319 -f- * 1.19166.
sinh (0.8 + i 1.7) = —0.79131 + 10.60718.
[45]
TABLE VII. HYPERBOLIC SINES, sinh (* + iq) = u + iv. CONTINUED
= 1.05
= 1.15
0
0.05
O.I
0.15
0.2
1.17520 o.oo
1.17158 O.I2I07
I.I6073 0.24139
1.14273 0.36023
I.II768 0.47684
1.25386 o.oo
1.24999 0.12583
1.23842 0.25089
1.21921 0.37440
1.19249 0.49560
1-33565 o.oo
1-33153 0.13091
1.31920 0.26101
1.29875 0.38951
1.27028 0.51560
1.42078 o.oo
1.41640 0.13632
1.40329 0.27179
1.38152 0.40559
1.35124 0.53689
1.50946 o.oo
1.50481 0.14206
1.49088 0.28325
1.46776 0.42269
1.43558 0.55952
0.25
o-3
0-35
0.4
0-45
1.08574 0.59051
I.047II 0.70055
I.OO2O2 0.80626
0.95076 0.90700
0.89363 I.002I5
1.15841 0.61375
1.11719 0.72811
1.06909 0.83798
1.01439 0.94269
0.95344 1.04158
1.23398 0.63852
1.19007 0.75749
1.13883 0.87180
1.08056 0.98073
1.01564 1.08362
1.31263 0.66488
1.26592 0.78877
1.21141 0.90780
1.14943 1.02123
1.08037 1.12836
1.39456 0.69291
1.34494 0.82202
1.28703 0.94607
1. 22118 1.06428
1.14781 1.71593
o-5
0-55
0.6
0.65
0.7
0.83099 I.O9II2
0.76323 I.I7337*
0.69077 1.24838
0.61404 1.31569
0-53353 1-3749°
0.88661 1.13405
0.81432 1.21953
0.73700 1.29750
0.65514 1.36746
0.56924 1.42899
0.94445 1.17982
0.86743 1.26875
0.78508 1.34986
0.69787 1.42265
0.60637 1.48666
1.00464 1.22854
0.92272 1.32114
0.83511 1.40560
0.74235 1.48139
0.64502 1.54805
1.06735 1-28033
0.98032 1.37684
0.88724 1.46485
0.78869 1.54384
0.68528 1.61331
0-75
0.8
0.85
0.9
0-95
0.44973 1.42562
0.36316 1.46756
0.27435 I-50045
0.18384 1.52408
0.09221 1.53832
0.47983 1.48171
0.38746 1.52530
0.29271 1.55948
0.10615 1.58405
0.09838 1.59885
0.51113 1.54151
0.41274 1.58685
0.31180 1.62242
0.20894 1.64798
0.10479 1.66337
0.54371 1.60517
0.43904 1.65238
0.33167 1.68941
0.22226 1.71602
0.11147 1.73206
0.57765 1.67283
0.46645 1.72204
0.35238 1.76063
0.23613 1.78836
0.11843 1.80507
I.O
1-05
i.i
i.iS
1.2
o.oo 1.54308
0.09221 1.53832
0.18384 1.52408
0.27435 1.50045
0.36316 1.46756
o.oo 1.60379
0.09838 1.59885
0.19615 1.58405
0.29271 1.55948
0.38746 1.52530
o.oo 1.66852
0.10479 1.66337
0.20894 1.64798
0.31180 1.62242
0.41274 1.58685
0.00 I.7374I
0.11147 1.73206
0.22226 1.71602
0.33167 1.68941
0.43904 1.65238
o.oo 1.81066
0.11843 1.80507
0.23613 1.78836
0.35238 1.76063
0.46645 1.72204
1.25
i-3
i-35
1.4
i-45
0.44973 1.42562
0.53353 1.37490
0.61404 1.31569
0.69077 1.24838
0.76323 1.17337
0.47983 1.48171
0.56924 1.42899
0.65514 1.36746
0.73700 1.29750
0.81432 1.21953
0.51113 1.54151
0.60637 1.48666
0.69787 1.42265
0.78508 1.34986
0.86743 1.26875
0.54371 1.60517
0.64502 1.54805
0.74235 1.48139
0.83511 .40560
0.92272 .32114
0.57765 1.67283
0.68528 1.61331
0.78869 1.54384
0.88724 1.46485
0.98032 1.37684
i-5
i-SS
1.6
1.65
i-7
0.83099 1.09112
0.89363 1.00215
0.95076 0.90700
I.OO2O2 0.80626
I.047II 0.70055
0.88661 1.13405
0.95344 1.04158
.01439 0.94269
.06909 0.83798
.11719 0.72811
0.94445 1.17982
1.01564 1.08362
1.08056 0.98073
1.13883 0.87180
1.19007 0.75749
1.00464 .22854
1.08037 .12836
1.14943 .02123
1.21141 0.90780
1.26592 0.78877
1.06735 1.28033
1.14781 1.17593
1.22118 1.06428
1.28703 0.94607
1.34494 0.82202
i-7S
1.8
1.85
1.9
i-9S
1.08574 0.59051
I.II768 0.47684
I.I4273 0.36023
I.I6073 0.24139
I.I7I58 O.I2IO7
.15841 0.61375
.19249 0.49560
.21921 0.37440
.23842 0.25089
.24999 0.12583
1.23398 0.63852
1.27028 0.51560
1.29875 0.38951
1.31920 0.26101
1.33153 0.13091
1.31263 0.66488
1.35124 0.53689
1.38152 0.40559
1.40329 0.27179
1.41640 0.13632
1.39456 0.69291
1-43558 0.55952
1.46776 0.42269
1.49088 0.28325
1.50481 0.14206
2.O
1.17520 o.oo
1.25386 o.oo
1.33565 o.oo
1.42078 o.oo
1.50946 o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (i.o + * i.o) = o + i 1.54308.
sinh (i.o + i 1.5) = —0.83099 + i 1.09112.
[46]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
= 1.25
x = 1.3
* = i-35
* = 1.4
= 1-45
o
1.60192 o.oo
1.69838 o.oo
1.79909 o.oo
1.90430 o.oo
2.01427 o.oo
0.05
1.59698 0.14816
1.69315 0.15464
1.79354 0.16150
1.89843 0.16876'
2.00806 0.17644
O.I
1.58220 0.29541
1.67747 0.30832
1.77694 0.32199
1.88086 0.33647
1.98947 0.35180
0.15
1.55766 0.44084
1.65146 0.46010
1.74938 0.48051
1.85169 0.50212
1.95862 0.52498
O.2
1.52352 0.58355
1.61526 0.60905
1.71104 0.63606
1.81110 0.66466
1.91569 0.69493
0.25
1.47998 0.72267
1.56910 0.75424
1.66215 0.78769
1.75934 0.82311
1.86094 0.86060
o-3
1.42732 0.85733
1.51327 0.89478
1.60300 0.93446
1.69675 0.97649
1.79473 1.02095
o-35
1.36586 0.98670
1.44811 1.02980
1.53398 1.07548
1.62369 1.12384
1.71745 1.17502
0.4
1.29598 1.10999
1.37402 1.15848
1.45550 1.20986
1.54061 1.26427
1.62958 1.32184
0-45
1. 21811 1.22643
1.29146 1.28001
1.36804 1.33678
1.44804 1.39690
1.53166 1.46051
°-5
1.13273 1.33532
1.20094 1.39365
1.27215 1.45546
1.34655 1.52092
1.42431 1.59017
Q-55
1.04036 1.43597
1.10301 1.49870
1.16842 1.56517
1.23674 1.63556
1.30817 1.71007
0.6
0.94158 1.52777
0.99829 1.59451
1.05748 1.66523
1.11932 1.74012
1.18396 1.81935
0.65
0.83700 1.61014
0.88740 1.68048
0.94002 1.75502
0.99500 1.83394
1.05245 1.91745
0.7
0.72726 1.68260
0.77105 1.75610
0.81677 1.83399
0.86454 1.91646
0.91446 2.00373
0-75
0.61303 1.74467
0.64994 1.82089
0.68848 1.90165
0.72875 1.98717
0.77083 2.07766
0.8
0.49502 1.79600
0.52483 1.87445
0-55595 1-95759
0.58846 2.04562
0.62244 2.13878
0.85
0.37396 1.83624
0.39648 1.91646
0.41999 2.00146
0.44455 2.09147
0.47022 2.18671
0.9
0.25060 1.86517
0.26569 1.94665
0.28144 2.03299
0.29790 2.12442
0.31510 2.22115
o-9S
0.12569 1.88260
0.13325 1.96484
0.14116 2.05199
0.14427 2.14427
0.15804 2.24191
I.O
o.oo 1.88842
o.oo 1.97091
o.oo 2.05833
o.oo 2.15090
o.oo 2.24884
1.05
0.12569 1.88260
0.13325 1.96484
0.14116 2.05199
0.14427 2.14427
0.15804 2.24191
i.i
0.25060 1.86517
0.26569 1.94665
0.28144 2.03299
0.29790 2.12442
0.31510 2.22115
i-J5
0.37396 1.83624
0.39648 1.91646
0.41999 2.00146
0.44455 2.09147
0.47022 2.18671
1.2
0.49502 1.79600
0.52483 1.87445
0-55595 1-95759
0.58846 2.04562
0.62244 2.13878
I-25
0.61303 1.74467
0.64994 1-82089
0.68848 1.90165
0.72875 1-98717
0.77083 2.07766
i-3
0.72726 1.68260
0.77105 1.75610
0.81677 1-83399
0.86454 1.91646
0.91446 2.00373
i-35
0.83700 1.61014
0.88740 1.68048
0.94002 1.75502
0.99500 1.83394
1.05245 1.91745
1.4
0.94158 1.52777
0.99829 1.59451
1.05748 1.66523
1.11932 1.74012
1.18396 1.81935
i-45
1.04036 1.43597
1.10301 1.49870
1.16842 1.56517
1.23674 1.63556
1.30817 1.71007
i-5
1.13273 1.33532
.20094 1.39365
1.27215 1.45546
1-34655 1.52092
1.42431 1.59017
i-55
1. 21811 1.22643
.29146 1.28001
1.36804 1.33678
1.44804 1.39690
1.53166 1.46051
1.6
1.29598 1.10999
.37402 1.15848
I-45550 1.20986
1.54061 1.26427
1.62958 1.32184
1.65
1.36586 0.98670
.44811 1.02980
I-53398 1.07548
1.62369 1.12384
1.71745 1.17502
i-7
1.42732 0.85733
.51327 0.89478
1.60300 0.93446
1.69675 0.97649
1-79473 1-02095
i-75
1.47998 0.72267
.56910 0.75424
1.66215 0.78769
1-75934 0.82311
1.86094 0.86060
1.8
1.52352 0.58355
.61526 0.60905
1.71104 0.63606
1.81110 0.66466
1.91569 0.69493
1.85
1-55766 0.44084
.65146 0.46010
1.74938 0.48051
1.85169 0.50212
1.95862 0.52498
1.9
1.58220 0.29541
.67747 0.30832
1.77694 0.32199
1.88086 0.33647
1.98947 0.35180
i-95
1.59698 0.14816
.69315 0.15464
1-79354 0.16150
1.89843 0.16876
2.00806 0.17644
2.O I.6OI92 O.OO
1.69838 o.oo
1.79909 o.oo
1.90430 o.oo
2.01427 o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (1.35 + to) = 1.70909 + io.
sinh (1.4 + i 1.15) = -0.44455 + i 2.09147.
[47]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
* = 1.5
= i-SS
1.6
* = 1.65
x = 1.7
0
0.05
O.I
0.15
0.2
2.12928 o.oo
2.12272 0.18457
2.10307 0.36800
2.07045 0.54916
2.02507 0.72693
2.24961 o.oo
2.24268 0.19316
2.22191 0.38512
2.18745 0.57471
2.13951 0.76076
2-37557 0.00
2.36824 0.20223
2.34632 0.40320
2.30993 0.60170
2.25930 0.79648
2.50746 o.oo
2.49973 0.21180
2.47659 0.42230
2.43818 0.63019
2.38474 0.83420
2.64563 o.oo
2.63747 0.22191
2.61306 0.44245
2.57253 0.66026
2.51614 0.87400
0.25
o-3
o-3S
0.4
o-4S
1.96720 0.90023
1.89720 1.06797
1.81551 1.22913
1.72263 1.38271
1.61912 1.52777
2.07837 0.94211
2.00442 1.11766
1.91811 1.28632
1.81997 1.44704
1.71062 1.59885
2.19473 0.98636
2.11664 1.17015
2.02550 1.34672
1.92187 1.51500
1.80640 1.67393
2.31660 1.03306
2.23417 1.22556
2.13797 1.41050
2.02858 1.58674
1.90669 1.75320
2.44424 1.08235
2.35727 1.28403
2.25577 1-47779
2.14036 1.66244
2.01175 1-83684
o-S
0-55
0.6
0.65
0.7
1 -5°563 1-66341
1.38286 1.78879
1.25156 1.90314
1.11255 2.00576
0.96667 2.09601
1.59071 1.74080
1.46101 1.87201
1.32229 1.99169
1.17542 2.09908
1.02130 2.19353
1.67978 1.82254
1.54281 1.95992
1.39632 2.08522
1.24123 2.19765
1.07848 2.29654
1-77305 1.90885
1.62847 2.05274
1.47385 2.18396
1.31015 2.30173
1.13837 2.40529
1.87074 1.99992
1.71820 2.15067
1.55506 2.28816
1.38234 2.41154
1.20109 2.52005
0-75
0.8
0.85
0.9
o.QS
0.81484 2.17334
0.65798 2.23727
0.49707 2.28742
0.33309 2.32345
0.16706 2.34516
0.86089 2.27446
0.69517 2.34137
0.52516 2.39384
0.35192 2.43155
0.17650 2.45427
0.90909 2.38127
0.73409 2.45131
0.55456 2.50625
0.37163 2.54573
0.18639 2.56952
0.95957 2.49404
0.77485 2.56740
0.58536 2.62494
0.39225 2.66629
0.19673 2.69121
1.01244 2.61302
0.81754 2.68989
0.61761 2.75017
0.41387 2.79350
0.20757 2.81960
I.O
1.05
i.i
i-iS
1.2
o.oo 2.35241
0.16706 2.34516
0.33309 2.32345
0.49707 2.28742
0.65798 2.23727
o.oo 2.46186
0.17650 2.45427
0.35192 2.43155
0.52516 2.39384
0.69517 2.34137
o.oo 2.57746
0.18639 2.56952
0.37163 2.54573
0.55456 2.50625
0.73409 2.45131
o.oo 2.69951
0.19673 2.69121
0.39225 2.66629
0.58536 2.62494
0.77485 2.56740
o.oo 2.82832
0.20757 2.81960
0.41387 2.79350
0.61761 2.75017
0.81754 2.68989
1.25
i-3
i-35
1.4
1-45
0.81484 2.17334
0.96667 2.09601
1. 11255 2.00576
1.25156 1.90314
1.38286 1.78879
0.86089 2.27446
1.02130 2.19353
1.17542 2.09908
1.32229 1.99169
1.46101 1.87201
0.90909 2.38127
1.07848 2.29654
1.24123 2.19765
1.39632 2.08522
1.54281 1.95992
0.95957 2.49404
1.13837 2.40529
1.31015 2.30173
1.47385 2.18396
1.62847 2.05274
1.01244 2.61302
1.20109 2.52005
1.38234 2.41154
1.55506 2.28816
1.71820 2.15067
i-5
i-5S
1.6
1.65
i-7
1.50563 1.66341
1.61912 1.52777
1.72263 1.38271
1.81551 1.22913
1.89720 1.06797
1.59071 1.74080
1.71062 1.59885
1.81997 1.44704
1.91811 1.28632
2.00442 1.11766
1.67978 1.82254
1.80640 1.67393
1.92187 1.51500
2.02550 1.34672
2.11664 1.17015
1.77305 1.90885
1.90669 1.75320
2.02858 1.58674
2.13797 1.41050
2.23417 1.22556
1.87074 1.99992
2.01175 1.83684
2.14036 1.66244
2.25577 1-47779
2.35727 1-28403
1-75
1.8
i.8S
1.9
i-95
1.96720 0.90023
2.02507 0.72693
2.07045 0.54916
2.10307 0.36800
2.12272 0.18457
2.07837 0.94211
2.13951 0.76076
2.18745 0.57471
2.22191 0.38512
2.24268 0.19316
2.19473 0.98636
2.25930 0.79648
2.30993 0.60170
2.34632 0.40320
2.36824 0.20223
2.31660 1.03306
2.38474 0.83420
2.43818 0.63019
2.47659 0.42230
2.49973 0.21 180
2.44424 1.08235
2.51614 0.87400
2.57253 0.66026
2.61306 0.44245
2.63747 0.22191
2.0
2.12928 o.oo
2.24961 o.oo
2-37557 o.oo
2.50746 o.oo
2.64563 o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (1.7 -f 1*0.7) = 1.20109 4
• / 2.52005.
sinh (1.7 + 11-2) = -2.35727 + * 1.28403.
[48]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv, CONTINUED
= i-7S
x = 1.8
= 1.85
* = 1.9
x = 1.95
o
2.79041
o.oo
2.94217
o.oo
3.10129
o.oo
3.26816
o.oo
3.44321
o.oo
0.05
2.78181
0.23257
2.93310
0.24381
3-09173
0.25566
3.25809
0.26815
3.43259
0.28131
O.I
2.75606
0.46370
2-90595
0.48612
3.06311
0.50975
3.22793
0.53465
3.40081
0.56089
0.15
2-71331
0.69198
2.86088
0.72542
3.01560
0.76069
3.17787
0.79785
3.34807
0.83701
O.2
2.65384
0.91599
2.79817
0.96026
2.94950
1.00694
3.10821
1.05614
3.27468
1.10797
0.25
2.57800
1-13435
2.71821
1.18918
2.86522
1.24698
3-01939
1.30791
3.18111
1.37210
0-3
2.48627
1-34571
2.62149
1.41076
2.76327
1-47934
2.91196
1.55162
3.06792
1.62777
°-3S
2.37922
1.54878
2.50862
1.62365
2.64429
1.70258
2.78657
1.78576
2.93582
1.87341
0.4
2.25749
1.74231
2.38027
1.82653
2.50900
I-9I53I
2.64400
2.00889
2.78561
2.10749
0-45
2.12185
1.92509
2.23725
2.01814
2-35824
2.11624
2.48513
2.21964
2.61823
2.32858
o-5
1.97312
2.09600
2.08043
2.19731
2.19294
2.30413
2.31094
2.41670
2.43471
2.53532
o-5S
1.81223
2.25399
1.91079
2.36294
2.01413
2.47780
2.12250
2.59887
2.23618
2.72642
0.6
1.64016
2.39808
1.72937
2.51400
1.82289
2.63620
1.92098
2.76501
2.02387
2.90071
0.65
1-45799
2-52739
1-53728
2.64956
1.62042
2.77835
1.70761
2.91409
1.79907
3-05713
0.7
1.26682
2.64111
1-33572
2.76878
1.40796
2.90337
1.48372
3.04522
1.56318
3.19469
0-75
1.06784
2.73855
1.12592
2.87093
1. 18681
3.01049
1.25067
3.15757
1.31766
3.31255
0.8
0.86229
2.81911
0.90918
2.95538
0-95835
3.09904
1.00992
3-25045
1.06401
3.41000
0.85
0.65141
2.88229
0.68684
3.02161
0.72398
3.16850
0.76294
3.32330
0.80380
3.48641
o.g
0.43652
2.92769
0.46026
3.06921
0.48627
3.21841
0.51125
3.37565
0.53864
3-54134
o-95
0.21893
2-955°5
0.23084
3.09789
0.24332
3.24848
0.25642
3.40719
0.27015
3-57443
I.O
o.oo
2.96419
o.oo
3.10747
o.oo
3-25853
o.oo
3.41773
o.oo
3-58548
1.05
0.21893
2.95505
0.23084
3.09789
0.24332
3.24848
0.25642
3.40719
0.27015
3-57443
i.i
0.43652
2.92769
0.46026
3.06921
0.48627
3.21841
0.51125
3.37565
0.53864
3-54I34
i-i5
0.65141
2.88229
0.68684
3.02161
0.72398
3.16850
0.76294
3-32330
0.80380
3-48641
1.2
0.86229
2.81911
0.90918
2.95538
0.95835
3.09904
1.00992
3-25045
1.06401
3.41000
1-25
1.06784
2-73855
1.12592
2.87093
1. 18681
3.01049
1.25067
3.15757
1.31766
3-3i255
i-3
1.26682
2.64111
1-33572
2.76878
1.40796
2.90337
1.48372
3.04522
1.56318
3.19469
i-35
1-45799
2.52739
1.53728
2.64956
1.62042
2.77835
1.70761
2.91409
1.79907
3-05713
1.4
1.64016
2.39808
1.72937
2.51400
1.82289
2.63620
1.92098
2.76501
2.02387
2.90071
1-45
1.81223
2-25399
1.91079
2.36294
2.01413
2.47780
2.12250
2.59887
2.23618
2.72642
i-5
1.97312
2.09600
2.08043
2.19731
2.19294
2.30413
2.31094
2.41670
2.43471
2.53532
i-SS
2.12185
1.92509
2.23725
2.01814
2.35824
2.11624
2.48513
2.21964
2.61823
2.32858
1.6
2.25749
1-74231
2.38027
1.82653
2.50900
I-9I53I
2.64400
2.00889
2.78561
2.10749
1.65
2.37922
1.54878
2.50862
1.62365
2.64429
1.70258
2.78657
1.78576
2.93582
1.87341
i-7
2.48627"
I-3457I
2.62149
1.41076
2.76327
1-47934
2.91196
1.55162
3.06792
1.62777
i-TS
2.57800
I-I3435
2.71821
1.18918
2.86522
1.24698
3.01939
1.30791
3.18111
1.37210
1.8
2.65384
0.91599
2.79817
0.96026
2.94950
i .00694
3.10821
1.05614
3.27468
1.10797
1.85
2-71331
0.69198
2.86088
0.72542
3.01560
0.76069
3.17787
0.79785
3.34807
0.83701
1.9
2.75606
0.46370
2.90595
0.48612
3.06311
0.50975
3-22793
0.53465
3.40081
0.56089
i-95
2.78181
0.23257
2.93310
0.24381
3-09173
0.25566
3.25809
0.26815
3.43259
0.28131
2.O
2.79041
o.oo
2.94217
o.oo
3.10129
o.oo
3.26816
o.oo
3.44321
0.00
Note.
Negative quantities are in heavy type.
Examples.
sinh (1.85 + *' Q-75)
s= I. l868l
+ » 3.01049.
sinh (1.85 + i 1.35) = -1.62042 + * 2.77835.
[49]
TABLE VII. HYPERBOLIC SINES, sinh (* + ig) = u + iv. CONTINUED
x = 2.05
x = 2.15
o
0.05
O.I
0.15
0.2
3.62686 o.oo
3.61568 0.29518
3.58221 0.58854
3.52666 0.87827
3.44935 1.16258
3.81958 o.oo
3.80781 0.30978
3.77256 0.61765
3.71404 0.92172
3.63264 I.220IO
4.02186 o.oo
4.00946 0.32516
3.97235 0.64831
3.91074 0.96747
3.82501 1.28066
4.23419 o.oo
4.22113 0.34135
4.18206 0.68060
4.11720 1.01564
4.02695 1.34443
4.45711 o.oo
4-44337 0.35839
4.40223 0.71458
4-33396 1.06636
4.23896 1.41156
0.25
0-3
o-3S
0.4
0-45
3.35078 1.43973
3.23156 1.70800
3.09241 1.96574
2.93420 2.21136
2.75789 2.44335
3.52883 1.51096
3.40327 1.79250
3.23673 2.06299
3.09OII 2.32076
2.90443 2.56423
3.71571 1.58596
3.58351 1.88148
3.42920 2.16540
3.25376 2.43597
3.05825 2.69152
3.91188 1.66493
3.77269 1.97516
3.61024 2.27322
3.42553 2.55726
3.21970 2.82553
4.11783 1.74806
3.97131 2.07379
3.80031 2.38672
3.60588 2.68495
3.38921 2.96661
o-S
o-SS
0.6
0.65
0.7
2.56458 2.66027
2.35546 2.86080
2.13182 3.04368
1-89503 3-20780
1.64656 3-35214
2.70085 2.79188
2.48062 3.00232
2.24509 3.19426
1-99573 3-36649
I.73405 3-5I798
2.84389 2.93048
2.61199 3-15137
2.36399 3-35283
2.10142 3.53361
1.82589 3.69261
2.99402 3-07639
2.74988 3.30828
2.48879 3-51977
2.21236 3.70956
1.92228 3.87647
3.15165 3.23000
2.89466 3-47347
2.61982 3.69552
2.32883 3-89478
2.02349 4.07003
0-75
0.8
0.85
0.9
0-95
1-38794 3.4758i
1.12076 3.57806
0.84667 3.65825
0.56737 3-7I587
0.28456 3-75059
1.46169 3.64777
1.18032 3.75507
0.89166 3.83923
0.59751 3-8997I
0.29968 3.93615
1.53910 3.82885
1.24282 3.94148
0.93888 4.02981
0.62916 4.09329
0.31555 4-13154
1.62035 4-01950
1.30844 4-13773
0.98845 4.23046
0.66237 4-29711
0.33221 4.33726
1.70566 4.22019
1-37732 4-34433
1.04049 4.44170
0.69724 4.51167
0.34970 4-55382
I.O
1.05
i.i
i.iS
1.2
o.oo 3.76220
0.28456 3.75059
0.56737 3-71587
0.84667 3.65825
1.12076 3.57806
o.oo 3-94832
0.29968 3.93615
0-59751 3-8997I
0.89166 3.83923
1.18032 3.75507
o.oo 4.14431
0.31555 4-13154
0.62916 4.09329
0.93888 4.02981
1.24282 3.94148
o.oo 4-35067
0.33221 4.33726
0.66237 4-29711
0.98845 4.23046
1.30844 4.13773
o.oo 4.56791
0.34970 4.55382
0.69724 4.51167
1.04049 4.44170
1-37732 4-34433
I-2S
i-3
i -35
1.4
1-45
1.38794 3-47581
1.64656 3.35214
1.89503 3.20780
2.13182 3.04368
2.35546 2.86080
1.46169 3.64777
1-73405 3-5I798
1-99573 3-36649
2.24509 3.19426
2.48062 3.00232
1.53910 3.82885
1.82589 3.69261
2.10142 3.53361
2.36399 3-35283
2.61199 3.15137
1.62035 4.01950
1.92228 3.87647
2.21236 3-70956
2.48879 3-51977
2.74988 3.30828
1.70566 4.22019
2.02349 4-07003
2.32883 3-89478
2.61982 3.69552
2.89466 3-47347
i-5
i-SS
1.6
i.6S
i-7
2.56458 2.66027
2.75789 2.44335
2.93420 2.21136
3.09241 1.96574
3.23156 1.70800
2.70085 2.79188
2.90443 2.56423
3.09011 2.32076
3-23673 2.06299
3.40327 1.79250
2.84389 2.93048
3.05825 2.69152
3.25376 2.43597
3.42920 2.16540
3.58351 1.88148
2.99402 3.07639
3.21970 2.82553
3-42553 2.55726
3.61024 2.27322
3.77269 1.97516
3.15165 3.23000
3.38921 2.96661
3.60588 2.68495
3.80031 2.38672
3.97131 2.07379
1-75
1.8
1.85
1.9
i-9S
3.35078 1.43973
3-44935 1.16258
3.52666 0.87827
3.58221 0.58854
3.61568 0.29518
3.52883 1.51096
3.63264 I.22OIO
3.71404 0.92172
3.77256 0.61765
3.80781 0.30978
3.71571 1.58596
3.82501 1.28066
3.91074 0.96747
3.97235 0.64831
4.00946 0.32516
3.91188 1.66493
4.02695 1.34443
4.11720 1.01564
4.18206 0.68060
4.22113 0.34135
4.11783 1.74806
4.23896 1.41156
4.33396 1.06636
4.40223 0.71458
4-44337 0.35839
2.0
3.62686 o.oo
3.81958 o.oo
4.02186 o.oo
4.23419 o.oo
4.45711 o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (2.2 +
*' i-o) =0 + * 4.56791.
sinh (2.2 + i 1.5) = -3.15165 + i 3.23000.
[So]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
q
X = 2.2$
x = 2.3
a: = 2.35
x = 24
x = 245
0
0.05
O.I
0.15
O.2
4.69117 o.oo
4.67671 0.37633
4.63341 0.75035
4.56155 I.II974
4.46157 1.48222
4.93696 o.oo
4.92174 0.39522
4.87618 0.78799
4.80056 1.17592
4.69533 1.55659
5.19510 o.oo
5.17909 0.41509
5.13114 0.82761
5.05156 1.23504
4.94083 1.63485
5.46623 o.oo
5.44938 0.43599
5.39893 0.86930
5.31521 1.29724
5.19869 1.71719
5.75103 o.oo
5.73330 0.45799
5.68022 0.91316
5.59213 1.36269
5.46955 1.80383
0.25
0.03
o-3S
0.4
o-45
4.33407 1.83557
4.17986 2.I776O
3.99987 2.50620
3.79523 2.81935
3.56719 3.II5I2
4.56116 1.92766
4.39887 2.28685
4.20946 2.63194
3.99409 2.96081
3.75410 3-27141
4.79965 2.02457
4.62887 2.40182
4.42955 2.76426
4.20292 3.10966
3-95038 343588
5.05014 2.12655
4.87045 2.52280
4.66073 2.90349
4.42228 3.26629
4.15656 3.60894
5.31325 2.23385
5.12420 2.65009
4-90356 3-04999
4.65268 3-43I09
4-373" 3-79I04
o-5
0-55
0.6
0.65
0.7
3.31716 3.39168
3.04677 3.64734
2-7574° 3.88050
2.45II3 4.08975
2.12975 4.27377
3.49096 3-56185
3.20630 3.83034
2.90188 4.07520
2.57956 4-29494
2.24134 4.48820
3.6735I 3-74093
3-37395 4-02290
3.05360 4.28008
2.71443 4.51087
2.35853 4-71384
3.86521 3.92935
3.55003 4.22554
3.21297 4-49567
2.85610 4.73808
2.48162 4.95127
4.06659 4.12761
3-73499 4-43873
3.38036 4-72249
3.00490 4.97713
2.61091 5.20109
0-75
0.8
0.85
0.9
0-9S
1.79523 4.43145
1.44965 4.56181
1.09513 4.66404
0.73386 4-73751
0.36807 4.78178
1.88930 4-65378
1.52560 4-79058
1.15251 4.89805
0.77231 4-97521
0.38735 5-02169
1.98808 4.88776
1-60537 5-03I53
1.21277 5.14429
0.81269 5.22533
0.40760 5.27416
2.09184 5-13394
1.68916 5.28496
1.27607 5.40341
0.85511 5.48853
0.42888 5-53981
2.20082 5.39298
i.777i6 5-55162
I-34255 5-67603
0.89966 5.76545
0.45122 5-8I933
I.O
1.05
i.i
i.iS
1.2
o.oo 4.79657
0.36807 4.78178
0.73386 4.73751
1.09513 4.66404
1.44965 4.56181
0.00 5-03722
0.38735 5-02169
0.77231 4.97521
1.15251 4.89805
1.52560 4-79058
o.oo 5.29047
0.40760 5.27416
0.81269 5.22533
1.21277 5-14429
1-60537 5.03153
o.oo 5-55695
0.42888 5.53981
0.85511 5.48853
1.27607 5.40341
1.68916 5.28496
o.oo 5-83732
0.45122 5.81933
0.89966 5.76545
I.34255 5-67603
1.77716 5.55162
1.25
i-3
i-3S
1.4
1-45
1.79523 443145
2.12975 4.27377
2.45113 4.08975
2-75740 3-88050
3.04677 3.64734
1.88930 4-65378
2.24134 4.48820
2.57956 4-29494
2.90188 4.07520
3.20630 3.83034
1.98808 4.88776
2.35853 4-71384
2.71443 4.51087
3.05360 4.28008
3-37395 4-02290
2.09184 5.13394
2.48162 4,95127
2.85610 4.73808
3.21297 4.49567
3.55003 4.22554
2.20082 5.39298
2.61091 5.20109
3.00490 4-977I3
3.38036 4.72249
3-73499 443873
i-5
i-SS
1.6
1.65
i-7
3.31716 3.39168
3-56719 3-I][5i2
3.79523 2.81935
3.99987 2.50620
4.17986 2.17760
3.49096 3.56185
3.75410 3.27141
3.99409 2.96081
4.20946 2.63194
4.39887 2.28685
3-6735I 3.74093
3.95038 3.43588
4.20292 3.10966
4-42955 2.76426
4.62887 2.40182
3.86521 3.92935
4.15656 3-60894
4.42228 3.26629
4.66073 2.90349
4.87045 2.52280
4x16659 4.12761
4-373II 3.79104
4.65268 3.43109
4.90356 3-04999
5.12420 2.65009
1-75
1.8
1.85
1.9
1-95
4.33407 1.83557
4.46157 1.48222
4-56i55 I-II974
4.63341 0.75035
4.67671 0.37633
4.56116 1.92766
4-69533 1-55659
4.80056 1.17592
4.87618 0.78799
4.92174 0.39522
4.79965 2.02457
4.94083 1.63485
5.05156 1.23504
5.13114 0.82761
5.17909 0.41509
5.05014 2.12655
5.19869 1.71719
5.31521 1.29724
5.39893 0.86930
5.44938 0.43599
5.31325 2.23385
5.46955 1-80383
5.59213 1.36269
5.68022 0.91316
5-73330 0.45799
2.0
4.69117 o.oo
4.93696 o.oo
5.19510 o.oo
5.46623 o.oo
5.75103 o.oo
Note.
Negative quantities are in heavy type.
Examples, sinh (2.4 + i 0.05) = 5.4493s + i 0.43599.
sinh (2.4 + i 1.95) = -5.44938 + * 0.43599.
CsxJ
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 2.5
= 2.55
x = 2.6
x = 2.65
= 2.7
o
0.05
O.I
0.15
O.2
6.05020 o.oo
6.03155 0.48113
5.97572 0.95930
5.88304 1.43155
5.75408 1.89498
6.36451 o.oo
6.34489 0.50548
6.28615 1.00784
6.18866 1.50399
6.05301 1.99087
6.69473 o.oo
6.67409 0.53109
6.61231 1.05891
6.50976 1.58019
6.36706 2.09174
7.04169 o.oo
7.01998 0.55803
6.95500 1.11262
6.84713 1.66034
6.69705 2.19784
7.40626 o.oo
7-38343 0.58636
7.31508 1.16911
7.20163 1.74465
7.04377 2.30943
0.25
o-3
o-35
0.4
o-45
5.58966 2.34673
5.39077 2.78401
5-15865 3.20411
4.89472 3.60448
4.60062 3.98260
5.88004 2.46547
5.67082 2.92488
5.42664 3.36624
5.14900 3.78686
4.83961 4.18413
6.18512 2.59039
5-96505 3-07306
5.70819 3.53680
5.41615 3-97872
5.09071 4-39612
6.50567 2.72178
6.27419 3.22894
6.00403 3.71619
5.69685 4-18053
5-35454 4-61910
6.84249 2.85997
6.59903 3-39288
6.31488 3.90488
5.99179 4-39279
5.63177 4.85363
o-S
o-55
0.6
0.65
0.7
4.27814 4-33619
3.92929 4.66304
3.55622 4-96113
3.16122 5.22864
2.74674 5.46392
4.50039 4.55560
4.13342 4.89898
3.74097 5.21217
3-32545 5-49321
2.88943 5-74039
4-73389 4-78641
4.34788 5-14719
3-93506 5.47624
3-49799 5-77I53
3.03934 6.03123
4.97923 5.02919
4.57321 5.40827
4.13900 5-75401
3.67927 6.06427
3.19686 6.33714
5.23702 5-28454
4.80998 5.68287
4-35329 6.04606
3.86976 6.37218
3.36237 6.65891
0-75
0.8
0.85
0.9
o-9S
2.31531 5.66550
1.86961 5.83215
1.41239 5.96287
0.94646 6.05680
0.47469 6.11339
2-43559 5-952i8
1.96674 6.12727
1.48577 6.26458
0.99563 6.36327
0.49935 6.42273
2.56196 6.25374
2.06878 6.43770
1.56285 6.58199
1.04729 6.68567
0.52526 6.74814
2.69474 6.57095
2.17600 6.76424
1.64385 6.91583
1.10156 7.02478
0.55249 7.09042
2.83425 6.90458
2.28866 7.10769
1.72896 7.26697
1-15859 7.38146
0.58109 7-45043
I.O
1.05
.1
•15
.2
o.oo 6.13229
0.47469 6.11339
0.94646 6.05680
1.41239 5.96287
1.86961 5.83215
o.oo 6.44259
0.49035 6.42273
0.99563 6.36327
1.48577 6.26458
1.96674 6.12727
o.oo 6.76901
0.52526 6.74814
1.04729 6.68567
1.56285 6.58199
2.06878 6.43770
o.oo 7.11234
0.55249 7-09042
1.10156 7.02478
1.64385 6.91583
2.17600 6.76424
o.oo 7-47347
0.58109 7.45043
1-15859 7-38146
1.72896 7.26697
2.28866 7.10769
•25
•3
•35
•4
•45
2.31531 5-66550
2.74674 5.46392
3.16122 5.22864
3.55622 4.96113
3.92929 4.66304
2.43559 5-95218
2.88943 5.74039
3.32545 5-49321
3.74097 5-21217
4.13342 4.89898
2.56196 6.25374
3.03934 6.03123
3.49799 5-77153
3.93506 5-47624
4.34788 5.14719
2.69474 6.57095
3.19686 6.33714
3.67927 6.06427
4.13900 5.75401
4.57321 5-40827
2.83425 6.90458
3.36237 6.65891
3.86976 6.37218
4-35329 6.04606
4.80998 5.68287
i-5
i-SS
1.6
1.65
i-7
4.27814 4-33619
4.60062 3.98260
4.89472 3.60448
5-15865 3-20411
5.39077 2.78401
4-50039 4.55560
4.83961 4-18413
5.14900 3.78686
5.42664 3.36624
5.67082 2.92488
'4-73389 4-78641
5.09071 4-39612
5.41615 3.97872
5.70819 3.53680
5-96505 3-07306
4.97923 5-02919
5-35454 4-61910
5.69685 4-18053
6.00403 3.71619
6.27419 3.22894
5.23702 5.28454
5.63177 4.85363
5.99179 4-39279
6.31488 3.90488
6.59903 3-39288
i-75
1.8
1.85
1.9
1-95
5.58966 2.34673
5.75408 1.89498
5.88304 1.43155
5.97572 0.95930
6.03155 0.48113
5.88004 2.46547
6.05301 1.99087
6.18866 J. 50399
6.28615 1.00784
6.34489 0.50548
6.18512 2.59039
6.36706 2.09174
6.50976 1.58019
6.61231 1.05891
6.67409 0.53109
6.50567 2.72178
6.69705 2.19784
6.84713 1.66034
6.95500 1.11262
7.01998 0.55803
6.84249 2.85997
7.04377 2.30943
7.20163 1.74465
7.31508 1.16911
7-38343 0.58636
2.O
6.05020 o.oo
6.36451 o.oo
6.69473 o.oo
7.04169 o.oo
7.40626 o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (2.7 + *Q-7) = 3.36237 +
i 6.65891.
sinh (2.5 + i 1.25) = —2.31531 + i 5.66550.
[52]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 2.75
x = 2.85
x — 2.9
x = 2.95
o
0.05
O.I
0.15
0.2
7-78935 o.oo
7.76534 0.61616
7.69345 1.22852
7.57413 1.83331
7.40811 2.42680
8.19192 o.oo
8.16666 0.64750
8.09106 1.29101
7.96557 1.92656
7.79098 2.55023
8.61497 o.oo
8.58841 0.68046
8.50891 1.35673
8.37694 2.02463
8.19332 2.68005
9.05956 o.oo
9.03163 0.71512
8.94802 1.42583
8.80924 2.12776
8.61616 2.81656
9.52681 o.oo
9.49744 0.75157
9.40952 1.49851
9.26358 2.23621
9.06053 2.96012
0.25
o-3
0.35
0.4
0-45
7.19642 3.00532
6.94036 3-56531
6.64151 4-10333
6.30172 4.61604
5.92307 5.10030
7.56834 3.15818
7.29905 3.74666
6.98476 4.31204
6.62740 4.85083
6.22918 5.35972
7-959I9 3-31894
7.67599 3-93738
7-34547 4-53I53
6.96966 5.09775
6.55088 5-63254
8.36994 3.48800
8.07213 4-13793
7.72455 4-76236
7.32934 5-35742
6.88894 6.91945
8.80162 3.66578
8.48845 4-34884
8.12294 5.00509
7-70735 5.63049
7.24424 6.22116
o-S
0-55
0.6
0.65
o-7
5.50790 5-553"
5.05878 5.97168
4.57847 6.35344
4.06993 6.69602
3.53629 6.99732
5.79256 5-83556
5.32022 6.27542
4.81509 6.67660
4.28026 7.03661
3.71905 7.35323
6.09170 6.13261
5.59498 6.59486
5-06375 7.01646
4.50131 7-39479
3.91112 7.72754
6.40608 6.44498
5.88371 6.93078
5.32508 7-37385
4.73361 7.77146
4.11295 8.12115
6.73647 6.77348
6.18717 7.28404
5.63048 7.74969
4-97774 8.16757
4.32508 8.53508
0.75
0.8
0.85
0.9
o-95
2.98086 7.25548
2.40704 7.46891
1.81839 7.63629
1.21852 7-75659
0.61115 7.82907
3.13491 7-62452
2.53144 7.84881
1.91237 8.02470
1.28150 8.15112
0.64273 8.22728
3.29681 8.01263
2.66217 8.24834
2.OIII2 8.43318
1.34768 8.56604
0.67592 8.64608
3.46694 8.42078
2.79956 8.66850
2.11491 8.86275
1.41723 9.00237
0.71081 9.08649
3.64575 8.84998
2-94395 9.11031
2.22399 9-3I447
1.49032 9.46121
0-74747 9-54962
I.O
1.05
i.i
1.2
o.oo 7.85328
0.61115 7.82907
1.21852 7.75659
1.81839 7.63629
2.40704 7.46891
o.oo 8.25273
0.64273 8.22728
1.28150 8.15112
1.91237 8.02470
2.53144 7.84881
o.oo 8.67281
O.67592 8.64608
1.34768 8.56604
2.0III2 8.43318
2.66217 8.24834
o.oo 9.11458
0.71081 9.08649
1.41723 9.00237
2.11491 8.86275
2.79956 8.66850
0.00 9-579I5
0.74747 9-54962
1.49032 9.46121
2.22399 9-3I447
2.94395 9.11031
1-25
1-35
i-4
i-45
2.98086 7-25548
3.53629 6.99732
4.06993 6.69602
4.57847 6.35344
5.05878 5.97168
3.13491 7-62452
3-71905 7-35323
4.28026 7.03661
4.81509 6.67660
5.32022 6.27542
3.29681 8.01263
3.9III2 7.72754
4.50131 7-39479
5-06375 7.01646
5.59498 6.59486
3.46694 8.42078
4.11295 8.12115
4.73361 7.77146
5.32508 7-37385
5.88371 6.93078
3.64575 8.84998
4.32508 8.53508
4-97774 8.16757
5.63048 7.74969
6.18717 7.28404
i-55
1.6
1.65
5-50790 5-553"
5.92307 5.10030
6.30172 4.61604
6.64151 4-10333
6.94036 3.56531
5.79256 5-83556
6.22918 5-35972
6.62740 4.85083
6.98476 4.31204
7.29905 3.74666
6.09170 6.13261
6.55088 5.63254
6.96966 5.09775
7-34547 4-53I53
7.67599 3-93738
6.40608 6.44498
6.88894 5.91945
7.32934 5-35742
7.72455 4.76236
8.07213 4.13793
6.73647 6.77348
7.24424 6.22116
7-70735 5-63049
8.12294 5.00509
8.48845 4.34884
1-75
1.8
1.85
1.9
1-95
7.19642 3.00532
7.40811 2.42680
7.57413 1.83331
7-69345 1.22852
7-76534 0.61616
7.56834 3-15818
7.79098 2.55023
7-96557 1.92656
8.09106 1.29101
8.16666 0.64750
7.959I9 3-31894
8.19332 2.68005
8.37694 2.02463
8.50891 1.35673
8.58841 0.68046
8.36994 3.48800
8.61616 2.81656
8.80924 2.12776
8.94802 1.42583
9.03163 0.71512
8.80162 3.66578
9.06053 2.96012
9.26358 2.23621
9.40952 1.49851
9.49744 0.75157
2.0
7.78935 o.oo
8.19192 o.oo
8.61497 o.oo
9.05956 o.oo
9.52681 o.oo
Note.
Negative quantities are in heavy type.
Examples, sinh (2.9 + i Q-9) = 1-41723 + i 9.00237.
sinh (2.8 + i 1.4) = —4.81509 + * 6.67660.
[53]
TABLE VII. HYPERBOLIC SINES, sinh (* + iq) = u + iv. CONTINUED
x = 3.0
x - 3.05
x = 3.10
= 3-iS
x = 3.20
o.o
10.01787
o.oo
10.53399
o.oo
11.07645
o.oo
11.64661
o.oo
12.24588
o.oo
0.05
9.98699
0.78990
10.50150
0.83020
11.04230
0.87258
11.61070
0.91714
12.20810
0.96399
O.I
9.89454
1-57493
10.40430
1.65529
10.94010
1-73979
11.50320
1.82863
12.09510
1.92205
0.15
9.74108
2.35025
10.24290
2.47017
10.77040
2.59626
11.32480
2.72885
11.90750
2.86826
0.2
9-52757
3.11108
10.01840
3.26982
10.53430
3.43673
11.07660
3.61224
11.64650
3.79678
0.25
9-2553I
3-85273
9.73216
4.04931
10.23330
4.25602
10.76010
4-47336
11.31370
4.70190
o-3
8.92599
4-57062
9.38586
4.80383
9.86919
5.04906
10.37720
5.30690
10.91120
5-57802
0-35
8.54164
5.26034
8.98171
5-52874
9-44423
5.81097
9.93036
6.10773
10.44130
6.41976
0.4
8.10463
5.91762
8.52218
6.21956
8.96104
6-537Q6
9.42230
6.87089
9.90712
7.22191
0-45
7.61765
6.53842
8.01011
6.87204
8.42260
7.22284
8.85615
7.59170
9.31184
7-97954
o-5
7.08371
7.11891
7.44866
7.48215
7.83223
7.86409
8.23539
8.26570
8.65915
8.68797
o.SS
6.50609
7-6555I
6.84128
8.04612
7-19358
8.45686
7-56387
8.88873
7-95306
9.34284
0.6
5.88836
8.14491
6.19173
8.56049
6.51058
8.99749
6.84570
9-45697
7-19795
9.94011
0.65
5-23433
8.58409
5.50400
9.02209
5.78743
9.48264
6.08533
9.96690
6.39846
10.47610
0.7
4.54806
8.97035
4-78233
9.42805
5.02860
9-90933
5-28745
10.41540
5-55952
10.94750
0-75
3-83368
9.30131
4.03119
9-77589
4.23878
10.27490
4.45696
10.79970
4.68630
11.35140
0.8
3-09569
9-57492
3-255i8
10.06350
3.42281
10.57720
3-59900
11.11730
3.78419
11.68530
0.85
2-33863
9.78949
2.45911
10.28900
2-58575
10.81420
2.71885
11.36650
2.85874
11.94720
o.g
1.56714
9-94371
1.64788
10.45110
1.73274
10.98460
1.82193
11-54550
1.91568
12.13540
o-95
0.78599
10.03660
0.82649
10.54870
0.86905
11.08720
0.91378
11.65340
0.96080
12.24880
I.O
o.oo
10.06766
o.oo
10.58135
o.oo
11.12150
0.00
11.68946
o.oo
12.28665
1.05
0.78599
10.03660
0.82649
10.54870
0.86905
11.08720
0.91378
11.65340
, 0.96080
12.24880
i.i
1.56714
9-94371
1.64788
10.45110
1.73274
10.98460
1.82193
11-54550
1.91568
12.13540
•IS
2.33863
9.78949
2.45911
10.28900
2.58575
10.81420
2.71885
11.36650
2.85874
11.94720
.2
3-09569
9-57492
3.25518
10.06350
3.42281
10.57720
3-59900
11.11730
3.78419
11.68530
•25
3.83368
9.30131
4.03119
9.77589
4.23878
10.27490
4.45696
10.79970
4.68630
11.35140
•3
4.54806
8.97035
4.78233
9.42805
5.02860
9-90933
5.28745
10.41540
5.55952
10.94750
•35
5-23433
8.58409
5.50400
9.02209
5.78743
9.48264
6.08533
9.96690
6.39846
10.47610
•4
5.88836
8.14491
6.19173
8.56049
6.51058
8.99749
6.84570
9-45697
7.19795
9.94011
•45
6.50609
7-6555I
6.84128
8.04612
7.19358
8.45686
7-56387
8.88873
7.95306
9.34284
•5
7.08371
7.11891
7.44866
7.48215
7.83223
7.86409
8.23539
8.26570
8.65915
8.68797
•55
7.61765
6.53842
8.OIOH
6.87204
8.42260
7.22284
8.85615
7.59170
9.31184
7-97954
.6
8.10463
5.91762
8.52218
6.21956
8.96104
6.53705
9.42230
6.87089
9.90712
7.22191
.65
8.54164
5.26034
8.98171
5-52874
9.44423
5.81097
9.93036
6.10773
10.44130
6.41976
•7
8.92599
4-57062
9.38586
4.80383
9.86919
5.04906
10.37720
5.30690
10.91120
5.57802
i-75
9-25531
3-85273
9-73216
4.04931
10.23330
4.25602
10.76010
4-47336
11.31370
4.70190
1.8
9.52757
3.11108
10.01840
3.26982
10.53430
3-43673
11.07660
3.61224
11.64650
3.79678
i.8S
9.74108
2.35025
10.24290
2.47017
10.77040
2.59626
11.32480
2.72885
11.90750
2.86826
1.9
9.89454
1 -5 7493
10.40430
1.65529
10.94010
1-73979
11.50320
1.82863
12.09510
1.92205
i-95
9.98699
0.78990
10.50150
0.83020
11.04230
0.87258
11.61070
0.91714
12.20810
0.96399
2.0
10.01787
o.oo
10.53399
o.oo
11.07645
o.oo
11.64661
o.oo
12.24588
0.00
Note.
Negative quantities are in heavy type.
Examples.
sinh (3.0 + i 0.95) =
0.78599 + i 10.03660.
sinh (3.0 + * 1.05) = — 0.78599 + i 10.03660.
[54]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 3-25
x = 3.30
= 3-35
x = 3-4°
x = 3-45
o.o
12.87578
O.OO
13-53788
o.oo
14.23382
o.oo
14.96536
o.oo
I5-73432
o.oo
0.05
12.83610
1.01326
13.49615
1.06507
14.18950
1-11953
14.91923
1.17679
15.68581
1.23699
O.I
12.71726
2.02028
13.37120
2.12356
14.05851
2.23215
14.78111
2.34632
15.54061
2.46636
0.15
12.52002
3.01484
13.16388
3.16897
13.84054
3-33101
14.55188
3-50139
I5-29959
3.68053
O.2
12.24560
3.99082
12.87530
4.19483
I3-537I7
4-40933
14.23291
4.63487
14.96423
4.87198
0.25
11.89566
4.94218
12.50736
5-19485
i3-I5Q33
5-46075
13.82620
5-73977
14.53662
6.03341
o-3
11.47240
5.86309
12.06234
6.16381
12.68244
6-47795
13-33423
6.80928
14.01940
7-15765
0-35
10.97841
6.74784
11.54294
7.09279
12.13633
7-45549
12.76006
7.83682
I3-4I572
8.23775
0.4
10.41673
7-59099
10.95235
7-97905
H-5I54I
8-38705
12.10723
8.81604
12.72933
9.26706
0-45
9.79082
8.38733
10.29428
8.81610
10.82349
9.26691
n-37975
9.74090
11.96447
10.23924
o-5
9-10455
9.13197
9-57273
9.59881
10.06483
10.08964
10.58212
10.60570
11.12585
11.14830
0-55
8.36217
9.82031
8.79215
10.32233
9.24413
10.85016
9.71923
11.40512
10.21863
11.98861
0.6
7.56820
10.44810
7-95737
10.98222
8.36643
n-54379
8.79642
12.13423
9.24840
12.75501
0.65
6.72758
11.01147
7-07352
11.57440
7-437I5
12.16623
7-81938
12.78852
8.22116
13.44278
0.7
S-84548
11.50695
6.14670
12.09520
6.46202
12.71369
6.79414
I3-36397
7.14324
14.04764
0-75
4-92735
11.93150
5.18072
12.54145
5-44705
13.18275
5.72700
13.85702
6.02127
14.56595
0.8
3-97883
12.28247
4-18343
12.91035
4-39850
I3-57054
4-62455
14.26465
4.86217
14.99442
0.85
3-00579
12.55773
3.16036
13.19970
3.32282
13.87465
3-49359
14.58432
3.67311
15-33045
0.9
2.01422
12.75556
2.11780
13.40764
'2.22666
14.09323
2.34110
14.81410
2.46139
15-57196
o-95
I.OIO22
12.87474
1.06217
13-5329°
1.11677
14.22498
1.17417
14.95250
1.23450
15.71746
I.O
o.oo
12.91456
o.oo
I3-57476
0.00
14.26891
o.oo
14.99874
0.00
15.76607
1.05
I.OIO22
12.87474
1.06217
13-53290
1.11677
14.22498
1.17417
14.95250
1.23450
15.71746
i.i
2.01422
12.75556
2.11780
13.40764
2.22666
14.09323
2.34110
14.81410
2.46139
15-57196
i-iS
3-00579
12.55773
3.16036
13.19970
3-32282
13.87465
3-49359
14.58432
3.67311
15-33045
1.2
3-97883
12.28247
4.18343
12.91035
4-39850
I3-57054
4-62455
14.26465
4.86217
14.99442
1-25
4-92735
11.93150
5.18072
12.54145
5.44705
13.18275
5.72700
13.85702
6.02127
14.56595
!-3
5.84548
11.50695
6.14670
12.09520
6.46202
12.71369
6.79414
I3-36397
7.14324
14.04764
i-3S
6.72758
11.01147
7.07352
11.57440
7-43715
12.16623
7.81938
12.78852
8.22116
13.44278
1.4
7.56820
10.44810
7-95737
10.98222
8.36643
n-54379
8.79642
12.13423
9.24840
12.75501
i -45
8.36217
9.82031
8.79215
10.32233
9.24413
10.85016
9.71923
11.40512
10.21863
11.98861
i-S
9-10455
9.13197
9-57273
9.59881
10.06483
10.08964
10.58212
10.60570
11.12585
11.14830
i-SS
9.79082
8.38733
10.29428
8.81610
10.82349
9.26691
n-37975
9.74090
11.96447
10.23924
1.6
10.41673
7-59099
10.95235
7-97905
11-51541
8.38705
12.10723
8.81604
12.72933
9.26706
1.65
10.9784!
6.74784
11.54294
7.09279
12.13633
7-45549
12.76006
7.83682
13.41572
8.23775
i-7
11.47240
5.86309
12.06234
6.16381
12.68244
6-47795
I3-33423
6.80928
14.01940
7-15765
i-75
11.89566
4.94218
12.50736
5-I9485
13-15033
5-46075
13.82620
5-73977
14.53662
6.03341
1.8
I2.2456O
3.99082
12.87530
4.19483
13-53717
4.40933
14.23291
4-63487
14.96423
4.87198
1.85
I2.52O02
3.01484
13.16388
3.16897
13.84054
3-33101
14.55188
3-50I39
15.29959
3-68053
1.9
12.71726
2.02028
13.37120
2.12356
14.05851
2.23215
14.78111
2.34632
15.54061
2.46636
i-95
12.83610
1.01326
13.49615
1.06507
14.18950
I.II953
14.91923
1.17679
15.68581
1.23699
2.O 12.87578 O.OO
13.53788 o.oo
14.23382 o.oo 14.96536 o.oo 15.73432 o.oo
Note.
Examples, sinh (3.40 + * o) = 14.96536 + i o.
sinh (3.45 + i 1.45) = —10.21863 + i 11.98861.
-/
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
x = 3-50
= 3-55
x = 3.60
= 3-65
x = 3.70
0.0
16.54263
o.oo
17.39230
o.oo
18.28546
o.oo
19.22434
0.00
20.21129
o.oo
0.05
16.49163
1.30029
17-33870
1.36684
18.22900
1.43680
19.16506
1.51036
20.14900
1.58770
O.I
16.33896
2.59256
17.17817
2.72525
18.06033
2.86475
18.98765
3.01142
19.96246
3.16561
0.15
16.08555
3-86885
16.91175
4.06686
17.78022
4-27503
18.69316
4.49390
19.65290
4.72401
0.2
15-733°°
5.12129
16.54105
5-38339
17.39050
5.65896
18.28342
5.94868
19.22208
6.25327
0.2S
15.28340
6.34215
16.06840
6.66674
16.89356
7.00800
17.76096
7.36678
18.67280
7-74399
o-3
14.73960
7-5239I
15.49665
7.90898
16.29246
8.31383
17.12900
8.73946
18.00840
9.18696
t>-35
14.10490
8.65928
14.82933
9.10246
15.59090
9.56840
16.39143
10.05827
17.23295
10.57330
0.4
13-3833°
9.74126
14.07066
10.23982
14.79326
10.76400
15-55281
11.31505
16.35127
11.89444
o-4S
12.57910
10.76320
13.22520
11.31405
13.90436
11.89320
14.61830
12.50208
15.36878
13.14223
o-5
11.69740
11.71820
12.29820
12.31852
12.92978
12.94910
13-59365
13.61203
14.29155
14.30902
0-55
JO-74357
12.60210
11.29540
13.24705
11-87545
13-92515
12.48520
14.63806
13.12620
15.38760
0.6
9.72361
13.40770
10.22293
14.09390
10.74789
14.81535
11.29972
!5-57383
11.87990
16.37127
0.65
8.64360
14.13065
9.08745
14.85386
9-55412
15.61421
10.04469
16.41360
10.56037
17.25404
0.7
7.51020
14.76650
7-89594
15.52225
8.30143
16.31680
8.72766
17.15216
9-17573
18.03040
0-75
6.33°59
i5-3"3o
6.65574
16.09492
6.99754
16.91880
7.35683
17.78497
7-73453
18.69565
0.8
S-n^S
15.76170
5-37451
16.56840
5-65052
17.41650
5.94065
18.30814
6.24563
19.24560
0.85
3.86180
16.11491
4.06015
16.93970
4.26865
17.80680
4.48783
18.71843
4.71823
19.67690
0.9
2.58783
16.36878
2.72075
17.20653
2.86047
18.08732
3-00735
19.01331
3.16174
19.98688
o.QS
1.29792
16.52173
1.36458
17.36731
1.43466
18.25632
1.50832
19.19100
I-58576
20.17362
I.O
0.00
16.57282
0.00
17.42102
o.oo
18.31278
o.oo
19-25033
o.oo
20.23601
1.05
1.29792
16.52173
1.36458
17.36731
1.43466
18.25632
1.50832
19.19100
1.58576
20.17362
i.i
2.58783
16.36878
2.72075
17.20653
2.86047
18.08732
3-00735
19.01331
3.16174
19.98688
i-iS
3.86180
16.11491
4.06015
16.93970
4.26865
17.80680
4.48783
18.71843
4.71823
19.67690
1.2
5.11195
15.76170
5.37451
16.56840
5.65052
17.41650
5.94065
18.30814
6.24563
19.24560
1-25
6.33059
15-3113°
6.65574
16.09492
6.99754
16.91880
7.35683
17.78497
7-73453
18.69565
i-3
7.51020
14.76650
7.89594
15.52225
8.30143
16.31680
8.72766
17.15216
9-17573
18.03040
i-3S
8.64360
14.13065
9.08745
14.85386
9-55412
15.61421
10.04469
16.41360
10.56037
17.25404
1.4
9.72361
13.40770
10.22293
14.09390
10.74789
14.81535
11.29972
I5-57383
11.87990
16.37127
i. 45
10.74357
12.60210
11.29540
13.24705
11.87545
13.92515
12.48520
14.63806
13.12620
15.38760
i-5
11.69740
11.71820
12.29820
12.31852
12.92978
12.94910
13.59365
13.61203
14-29155
14.30902
i-55
12.57910
10.76320
13.22520
11.31405
13-90436
11.89320
14.61830
12.50208
15.36878
13.14223
1.6
13-38330
9.74126
14.07066
10.23982
14.79326
10.76400
15.55281
11-31505
16.35127
11.89444
i.6S
14.10490
8.65928
14.82933
9.10246
15-59090
9.56840
16.39143
•10.05827
17.23295 ^10.57330
i-7
14.73960
7-52391
15-49665
7.90898
16.29246
8.31383
17.12900
8.73946
18.00840
9.18696
i-75
15.28340
6-34215
16.06840
6.66674
16.89356
7.00800
17.76096
7.36678
18.67280
7-74399
1.8
15.73300
5.12129
16.54105
5-38339
17.39050
5.65896
18.28342
5.94868
19.22208
6.25327
1.85
16.08555
3-86885
16.91175
4.06686
17.78022
4.27503
18.69316
4.49390
19.65290
4.72401
1.9
16.33896
2.59256
17.17817
2.72525
18.06033
2.86475
18.98765
3.01142
19.96246
3.16561
i-95
16.49163
1.30029
17.33870
1.36684
18.22900
1.43680
19.16506
1.51036
20.14900
1.58770
2.O
16.54263
0.00
17.39230
o.oo
18.28546
0.00
19.22434
o.oo
20.21129
0.00
Note. Negative quantities are in heavy type.
Examples, sinh (3.70 + i 0.5) = 14.29155 -4-
i 14.30902
.
-
sinh (3.70 + i 1.5) = —14.29155 + * 14.30902.
[56]
TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED
= 3-75
x = 3.80
* = 3.85
x = 3.90
= 3-95
o.o
21.24878
o.oo
22.33941
o.oo
23.48589
o.oo
24.69110
o.oo
25.95806
o.oo
0.05
21.18327
1.66900
22.27052
1.75448
23.41348
1.84435
24.61500
1.93883
25.87805
2.03816
O.I
20.98716
3.32772
22.06437
3-49815
23-19673
3.67733
24.38710
3-86571
25.63849
4-06375
0.15
20.66167
4.96592
21.72216
5.22025
22.83696
5.48764
24.00888
5-76875
25.24084
6.06429
O.2
20.20879
6.57350
21.24603
6.91017
22.33640
7.26411
23.48262
7.63623
24.68760
8.02744
0.25
19.63131
8.14055
20.63891
8.55748
21.69813
8.99580
22.81160
9.45662
23.98212
9.94109
o-3
18.93280
9.65741
19-90455
10.15203
20.92608
10.67203
21-99993
11.21871
23.12881
11.79366
°-3S
18.11756
11.11473
19.04746
11.68400
20.02501
12.28246
21.05272
12.91164
22.13290
13-57315
0.4
17.19062
12.50353
18.07296
13-14392
19.00048
13.81716
I9-97556
14.52497
21.00052
15.26910
o.4S
16.15770
13.81524
16.98701
14.52281
17.85880
15.26668
18.77526
16.04874
19.73867
16.87098
o-5
15.02516
15.04177
15-79634
15.81216
16.60702
16.62208
17.45924
17-47355
18.35512
18.36873
0-55
13.79998
16.17556
14.50828
17.00402
15.25286
17.87500
16.03558
18.79065
16.85842
19.75331
0.6
12.48971
17.20963
13.13076
18.09105
13.80465
19.01770
14-51307
19.99190
15.25776
21.01610
0.65
11.10246
18.13760
11.67230
19.06655
12.27134
20.04315
12.90106
21.06988
13.56305
22.14931
0.7
9.64674
18.95373
10.14188
19.92448
10.66233
20.94503
11.20952
22.01797
11.78474
23.14597
0.75
8.13156
19.65301
8.54892
20.65958
8.98766
21.71778
9.44887
22.83030
9-93373
23.99991
0.8
6.56624
20.23113
6.90325
21.26731
7-25754
22.35664
7.62997
23.50188
8.02149
24.70590
0.85
4.96043
20.68452
5-21503
21.74391
5.48267
22.85766
.5.76402
24.02856
6.05979
25.25957
0.9
3-32404
21.01038
3-49465
22.08646
3.67400
23.21775
3.86254
24.40710
4.06074
25.65749
o-9S
1.66716
21.20670
1-75273
22.29283
1.84268
23.43470
1.93724
24.63516
2.03665
25.89724
I.O
o.oo
21.27230
o.oo
22.36178
o.oo
23.50717
o.oo
24-71135
o.oo
25-97731
1.05
1.66716
21.20670
1.75273
22.29283
1.84268
23.43470
1.93724
24.63516
2.03665
25.89724
i.i
3-32404
21.01038
3.49465
22.08646
3.67400
23.21775
3.86254
24.40710
4.06074
25-65749
i-iS
4.96043
20.68452
5-21503
21.74391
5.48267
22.85766
5.76402
24.02856
6.05979
25.25957
1.2
6.56624
20.23113
6.90325
21.26731
7.25754
22.35664
7.62997
23.50188
8.02149
24.70590
1-25
8.13156
19.65301
8.54892
20.65958
8.98766
21.71778
9.44887
22.83030
9-93373
23.99991
i-3
9.64674
18.95373
10.14188
19.92448
10.66233
20.94503
11.20952
22.01797
11.78474
23.14597
1-35
11.10246
18.13760
11.67230
19.06655
12.27134
20.04315
12.90106
21.06988
13-56305
22.14931
1.4
12.48971
17.20963
13.13076
18.09105
13.80465
19.01770
14.51307
19.99190
15-25776
21.01610
1-45
13.79998
16.17556
14.50828
17.00402
15.25286
17.87500
16.03558
18.79065
16.85842
19.75331
i-5
15.02516
15.04177
15-79634
15.81216
16.60702
16.62208
17.45924
17-47355
18.35512
18.36873
1-55
16.15770
13.81524
16.98701
14.52281
17.85880
15.26668
18.77526
16.04874
19.73867
16.87098
1.6
17.19062
12.50353
18.07296
13-14392
19.00048
13.81716
19.97556
14.52497
21.00052
15.26910
1-65
18.11756
11.11473
19.04746
11.68400
20.02501
12.28246
21.05272
12.91164
22.13290
13.57315
i-7
18.93280
9.65741
19.90455
10.15203
20.92608
10.67203
21.99993
11.21871
23.12881
11.79366
1-75.
19.63131
8.14055
20.63891
8.55748
21.69813
8.99580
22.81160
9.45662
23.98212
9.94109
1.8
20.20879
6.57350
21.24603
6.91017
22.33640
7.26411
23.48262
7.63623
24.68760
8.02744
1.85
20.66167
4.96592
21.72216
5.22025
22.83696
5.48764
24.00888
5.76875
25.24084
6.06429
1.9
20.98716
3.32772
22.06437
3.49815
23-19673
3.67733
24.38710
3-86571
25.63849
4.06375
i -95
21.18327
1.66900
22.27052
1.75448
23.41348
1.84435
24.61500
1.93883
25.87805
2.03816
2.0
21.24878
o.oo
22.33941
o.oo
23.48589
0.00
24.69110
o.oo
25.95806
o.oo
Note. Negative quantities are in heavy type.
Examples, sinh (3.90 + fr'o-75) =
9.44887 +
i 22.83030.
sinh (3.95 + i 1.95) = -25.87805 + i 2.03816.
[57]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv
x = 0.05
x = 0.15
0
I.OOOO O.OO
1.00125 o.oo
1.00500 o.oo
1.01127 o.oo
1.02007 o.oo
0.05
0.99692 o.oo
0.99816 0.00392
1.00191 0.00786
1.00815 0.01181
1.01692 0.01580
O.I
0.98769 o.oo
0.98892 0.00783
0.99263 0.01567
0.99882 0.02355
1.00751 0.03150
0.15
0.97237 o.oo
0.97385 0.01168
0.97724 0.02338
0.98333 0.03515
0.99188 0.04700
O.2
0.95106 o.oo
0.95225 0.01546
0.95582 0.03095
0.96178 0.04653
0.97014 0.06222
0.2S
0.92388 o.oo
0.92503 0.01914
0.92850 0.03833
0.93429 0.05762
0.94242 0.07705
0-3
0.89101 o.oo
0.89208 0.02271
0.89547 0.04547
0.90105 0.06835
0.90889 0.09141
o-3S
0.85264 o.oo
0.85367 0.02614
0.85691 0.05234
0.86225 0.07867
0.86975 0.10520
0.4
0.80902 o.oo
0.81000 0.02940
0.81307 0.05888
0.81814 0.08850
0.82525 0.11834
0-45
0.76041 o.oo
0.76133 0.03249
0.76421 0.06505
0.76898 0.09778
0.77567 0.13076
o.S
0.70711 o.oo
0.70796 0.03537
0.71065 0.07083
0.71508 0.10646
0.72130 0.14237
o-SS
0.64945 o.oo
0.65023 0.03804
0.65270 0.07617
0.65677 0.11449
0.66248 0.15310
0.6
0.58779 o.oo
0.58850 0.04047
0.59027 0.08104
0.59441 0.12181
0.59958 0.16288
0.65
0.52250 o.oo
0.52313 0.04265
0.52511 0.08541
0.52839 0.12838
0.53298 0.17167
0.7
0.45399 o.oo
0.45439 0.04457
0.45626 0.08925
0.45911 0.13415
0.46310 0.17939
o.7S
0.38268 o.oo
0.38316 0.04621
0.38460 0.09254
0.38700 0.13910
0.39036 o.i 8601
0.8
0.30902 o.oo
0.30940 0.04757
0.31056 0.09526
0.31250 0.14319
0.31522 0.19148
0.85
0.23345 o.oo
0.23374 0.04864
0.23461 0.09740
0.23608 0.14640
0.23813 0.19577
0.9
0.15643 o.oo
0.15663 0.04941
0.15722 0.09893
0.15820 0.14871
0.15957 0.19886
o-9S
0.07846 o.oo
0.07856 0.04987
0.07885 0.09986
0.07934 0.15010
0.08003 0.20072
I.O
o.oo o.oo
o.oo 0.05002
o.oo 0.10017
o.oo 0.15056
o.oo 0.20134
1-05
0.07846 o.oo
0.07856 0.04987
0.07885 0.09986
0.07934 0.15010
0.08003 0.20072
i.i
0.15643 o.oo
0.15663 0.04941
0.15722 0.09893
0.15820 0.14871
0.15957 0.19886
i-iS
0.23345 o.oo
0.23374 0.04864
0.23461 0.09740
0.23608 0.14640
0.23813 0.19577
1.2
0.30902 o.oo
0.30940 0.04757
0.31056 0.09526
0.31250 0.14319
0.31522 0.19148
1-25
0.38268 o.oo
0.38316 0.04621
0.38460 0.09254
0.38700 0.13910
0.39036 0.18601
i-3
0.45399 o.oo
0.45439 0.04457
0.45626 0.08925
0.45911 0.13415
0.46310 0.17939
•35
0.52250 o.oo
0.52313 0.04265
0.52511 0.08541
0.52839 0.12838
0.53298 0.17167
•4
0.58779 o.oo
0.58850 0.04047
0.59027 0.08104
0.59441 0.12181
0.59958 0.16288
•45
0.64945 o.oo
0.65023 0.03804
0.65270 0.07617
0.65677 0.11449
0.66248 0.15310
•5
0.70711 o.oo
0.70796 0.03537
0.71065 0.07083
0.71508 0.10646
0.72130 0.14237
•55
0.76041 o.oo
0.76133 0.03249
0.76421 0.06505
0.76898 0.09778
0.77567 0.13076
.6
0.80902 o.oo
0.81000 0.02940
0.81307 0.05888
0.81814 0.08850
0.82525 0.11834
•65
0.85264 o.oo
0.85367 0.02614
0.85691 0.05234
0.86225 0.07867
0.86975 0.10520
1-7
0.89101 o.oo
0.89208 0.02271
0.89547 O-O4547
0.90105 0.06835
0.90889 0.09141
1-75
0.92388 o.oo
0.92503 0.01914
0.92850 0.03833
0.93429 0.05762
0.94242 0.07705
1.8
0.95106 o.oo
0.95225 0.01546
0.95582 0.03095
0.96178 0.04653
0.97014 0.06222
1.85
0.97237 o.oo
0.97385 0.01168
0.97724 0.02338
0.98333 0.03515
0.99188 0.04700
1.9
0.98769 o.oo
0.98892 0.00783
0.99263 0.01567
0.99882 0.02355
1.00751 0.03150
1-95
0.99692 o.oo
0.99816 0.00392
1.00191 0.00786
1.00815 0.01181
1.01692 0.01580
I.OOOO O.OO I.OOI25 O.OO
1.00500 o.oo
1.01127 o.oo
1.02007 o.oo
Note.
Negative quantities are in heavy type.
Examples, cosh (o + i 0.75) = 0.38268 + « o.
cosh (0.2 + i 1.5) = —0.72130 + * 0.14237
[58]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
x = 0.25
x = 0.3
x = 0.35
x = 0.4
x = 0.45
0
o.os
O.I
0.15
0.2
1.03141 o.oo
1.02823 0.01982
1.01871 0.03952
1.00292 0.05897
0.98093 0.07806
1.04534 o.oo
1.04212 0.02389
1.03247 0.04764
1.01646 0.07109
0.99418 0.09410
i. 06 1 88 o.oo
1.05860 0.02803
1.04880 0.05588
1.03254 0.08338
1.00991 0.11038
1.08107 o.oo
1.07774 0.03223
1.06776 0.06426
1.05120 0.09589
1.02816 0.12693
1.10297 o.oo
1.09957 0.03651
1.08939 0.07280
1.07250 0.10863
1.04899 0.14380
0.25
0-3
0-35
0.4
0-4S
0.95290 0.09667
0.91900 0.11468
0.87942 0.13199
0.83443 0.14848
0.78429 0.16406
0.96577 0.11654
0.93140 0.13825
0.89130 0.15911
0.84570 0.17899
0.79488 0.19777
0.98105 0.13669
0.94614 0.16216
0.90540 0.18663
0.85908 0.20995
0.80746 0.23198
0.99878 0.15719
0.96324 0.18648
0.92177 0.21462
0.87461 0.24143
0.82206 0.26676
1.01901 0.17808
0.98275 0.21126
0.94044 0.24314
0.89232 0.27352
0.83871 0.30222
0-5
0-55
0.6
0.65
0.7
0.72932 0.17862
0.66985 0.19208
0.60625 °. 2043 7
0.53891 0.21539
0.46825 0.22508
0.73917 0.21533
0.67889 0.23156
0.61444 0.24636
0.54619 0.25965
0.47457 0.27133
0.75086 0.25257
0.68964 0.27161
0.62416 0.28897
0.55483 0.30455
0.48208 0.31826
0.76443 0.29045
0.70210 0.31234
0.63544 0.33231
0.56486 0.35022
0.49080 0.36598
0.77992 0.32905
0.71632 0.35385
0.64831 0.37647
0.57630 0.39677
0.50074 0.41462
o.7S
0.8
0.85
0.9
o-9S
0.39471 0.23338
0.31872 0.24025
0.24078 0.24563
0.16135 0.24950
0.08092 0.25183
0.40003 0.28134
0.32303 0.28962
0.24403 0.29611
0.16353 0.30077
0.08202 0.30358
0.40636 0.33000
0.32814 0.33971
0.24789 0.34732
0.16611 0.35279
0.08331 0.35609
0.41371 0.37949
0.33407 0.39065
0.25237 0.39940
0.16912 0.40570
0.08482 0.40949
0.42209 0.42992
0.34084 0.44257
0.25748 0.45249
0.17254 0.45961
0.08654 0.46391
I.O
1.05
i.i
1.2
o.oo 0.25261
0.08092 0.25183
0.16135 0.24950
0.24078 0.24563
0.31872 0.24025
o.oo 0.30452
0.08202 0.30358
0.16353 0.30077
0.24403 0.29611
0.32303 0.28962
o.oo 0.35719
0.08331 0.35609
o. 16611 0.35279
0.24789 0.34732
0.32814 0.33971
o.oo 0.41075
0.08482 0.40949
0.16912 0.40570
0.25237 0.39940
0.33407 0.39065
o.oo 0.46534
0.08654 0.46391
0.17254 0.45961
0.25748 0.45249
0.34084 0.44257
1-25
i-35
1.4
1-45
0.39471 0.23338
0.46825 0.22508
0.53891 0.21539
0.60625 0.20437
0.66985 0.19208
0.40003 0.28134
0.47457 0.27133
0.54619 0.25965
0.61444 0.24636
0.67889 0.23156
0.40636 0.33000
0.48208 0.31826
0.55483 0.30455
0.62416 0.28897
0.68964 0.27161
0.41371 0.37949
0.49080 0.36598
0.56486 0.35022
0.63544 0.33231
0.70210 0.31234
0.42209 0.42992
0.50074 0.41462
0.57630 0.39677
0.64831 0.37647
0.71632 0.35385
i-SS
1.6
1.65
0.72932 0.17862
0.78429 0.16406
0.83443 0.14848
0.87942 0.13199
0.91900 0.11468
0.73917 0.21533
0.79488 0.19777
0.84570 0.17899
0.89130 0.15911
0.93140 0.13825
0.75086 0.25257
0.80746 0.23198
0.85908 0.20995
0.90540 0.18663
0.94614 0.16216
0.76443 0.20945
0.82206 0.26676
0.87461 0.24143
0.92177 0.21462
0.96324 0.18648
0.77992 0.32905
0.83871 0.30222
0.89232 0.27352
0.94044 0.24314
0.98275 0.21126
i-75
1.8
1.85
1.9
I.9S
0.95290 0.09667
0.98093 0.07806
1.00292 0.05897
1.01871 0.03952
1.02823 0.01982
0.96577 0.11654
0.99418 0.09410
1.01646 0.07109
1.03247 0.04764
1.04212 0.02389
0.98105 0.13669
1.00991 0.11038
1.03254 0.08338
1.04880 0.05588
1.05860 0.02803
0.99878 0.15719
1.02816 0.12693
1.05120 0.09589
1.06776 0.06426
1.07774 0.03223
1.01901 0.17808
1.04899 0.14380
1.07250 0.10863
1.08939 0.07280
1.09957 0.03651
2.0
1.03141 o.oo
1.04534 o.oo
i. 06 1 88 o.oo
1.08107 o.oo
1.10297 o/oo
Note.
Negative quantities are in heavy type.
Examples, cosh (0.3 + i 0.9) = 0.16353 +
i 0.30077.
cosh (0.45 + i 1.7) = —0.98275 + i 0.21126.
[59]
TABLE VTII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
0.5
x = 0.55
x = 0.6
x = 0.65
x — 0.7
o
0.05
O.I
0.15
0.2
1.12763 o.oo
1.12415 0.04088
1.11374 0.08152
1.09647 0.12165
1.07244 0.16103
1.15510 o.oo
1.15154 0.04536
1.14088 0.09044
1.12319 0.13497
1.09857 0.17866
1.18547 o.oo
1. 18181 0.04995
1.17087 0.09959
1.15271 0.14862
1.12744 0.19674
1.21879 o.oo
1.21504 0.05467
1.20379 0.10900
1.18512 0.16265
1.15912 0.21531
1.25517 o.oo
1.25130 0.05952
1.23972 0.11867
1.22049 0.17709
1.19374 0.23442
0.25
0-3
0-35
0.4
0.45
1.04179 0.19942
1.00472 0.23657
0.96146 0.27227
0.91227 0.30629
0.85745 0.33842
1.06717 0.22125
1.02920 0.26248
0.98489 0.30208
0.93450 0.33983
0.87835 0.37548
1.09523 0.24364
1.05626 0.28904
1.01078 0.33265
0.95906 0.37422
0.90144 0.41347
1.12602 0.26663
1.08595 0.31632
1.03919 0.36405
0.98602 0.40954
0.92678 0.45250
1.15962 0.29030
1.11836 0.34439
1.07021 0.39636
1.01545 0.44589
0.95444 0.49266
0-5
0-55
0.6
0.65
0.7
0.79735 0.36847
0.73233 0.39624
0.66280 0.42158
0.58918 0.44431
0.51193 0.46430
0.81678 0.40882
0.75018 0.43963
0.67895 0.46773
0.60354 0.49296
0.52441 0.51514
0.83825 0.45018
0.76990 0.48412
0.69680 0.51506
0.61940 0.54284
0.53819 0.56726
0.86182 0.49268
0.79154 0.52981
0.71639 0.56368
0.63682 0.59408
0.55332 0.62081
0.88754 0.53640
0.81517 0.57683
0.73777 0.61371
0.65582 0.64680
0.56984 0.67590
0-75
0.8
0.85
0.9
0-9.5
0.43152 0.48143
0.34846 0.49559
0.26324 0.50670
0.17640 0.51468
0.08847 0.51949
0.44204 0.53414
0.35695 0.54986
0.26965 0.56218
0.18070 0.57103
0.09063 0.57637
0.45366 0.58819
0.36633 0.60549
0.27674 0.61906
0.18545 0.62882
0.09301 0.63469
0.46641 0.64371
0.37663 0.66265
0.28452 0.67750
0.19066 0.68817
0.09563 0.69460
0.48033 0.70084
0.38787 0.72146
0.29301 0.73763
0.19635 0.74925
0.09848 0.75625
1.0
1.05
i.i
1.2
o.oo 0.52110
0.08847 0.51949
0.17640 0.51468
0.26324 0.50670
0.34846 0.49559
o.oo 0.57815
0.09063 0.57637
0.18070 0.57103
0.26965 0.56218
0.35695 0.54986
o.oo 0.63665
0.09301 0.63469
0.18545 0.62882
0.27674 0.61906
0.36633 0.60549
o.oo 0.69675
0.09563 0.69460
0.19066 0.68817
0.28452 0.67750
0.37663 0.66265
o.oo 0.75858
0.09848 0.75625
0.19635 0.74925
0.29301 0.73763
0.38787 0.72146
1.25
1.3
1-35
1.4
i-45
0.43152 0.48143
0.51193 0.46430
0.58918 0.44431
0.66280 0.42158
0.73233 0.39624
0.44204 0.53414
0.52441 0.51514
0.60354 0.49296
0.67895 0.46773
0.75018 0.43963
0.45366 0.58819
0.53819 0.56726
0.61940 0.54284
0.69680 0.51506
0.76990 0.48412
0.46641 0.64371
0.55332 0.62081
0.63682 0.59408
0.71639 0.56368
0.79154 0.52981
0.48033 0.70084
0.56984 0.67590
0.65582 0.64680
0.73777 0.61371
0.81517 0.57683
i-SS
1.6
1.65
1-7
0.79735 0.36847
0.85745 0.33842
0.91227 0.30629
0.96146 0.27227
1.00472 0.23657
0.81678 0.40882
0.87835 0.37548
0.93450 0.33983
0.98489 0.30208
1.02920 0.26248
0.83825 0.45618
0.90144 0.41347
0.95906 0.37422
1.01078 0.33265
1.05626 0.28904
0.86182 0.49268
0.92678 0.45250
0.98602 0.40954
1.03919 0.36405
1.08595 0.31632
0.88754 0.53640
0.95444 0.49266
1.01545 0.44589
1.07021 0.39636
1.11836 0.34439
i!s
1.85
1.9
1-95
1.04179 0.19942
1.07244 0.16103
1.09647 0.12165
1.11374 0.08152
1.12415 0.04088
1.06717 0.22125
1.09857 0.17866
1.12319 0.13497
1.14088 0.09044
1.15154 0.04536
1.09523 0.24364
1.12744 0.19674
1.15271 0.14862
1.17087 0.09959
1. 18181 0.04995
1.12602 0.26663
1.15912 0.21531
1.18512 0.16265
1.20379 0.10900
1.21504 0.05467
1.15962 0.29030
1.19374 0.23442
1.22049 0.17709
1.23972 0.11867
1.25130 0.05952
2.0
• 1.12763 o.oo
1.15510 o.oo
1.18547 o.oo
1.21879 o.oo
1.25517 o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (0.6 + i 0.95) = 0.09301 + i 0.63469.
cosh (0.6 + i 1.05) = —0.09301 + i 0.63469.
[60]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
0.75
0.8
x = 0.85
x = o.g
x = 0.95
o
0.05
O.I
0.15
O.2
1.29468 o.oo
1.29069 0.06452
1.27874 0.12864
1.25891 0.19197
1.23132 0.25411
1.33743 o.oo
I-3333I 0.06968
1.32097 0.13893
1.30048 0.20732
1.27198 0.27444
1.38353 o.oo
1.37927 0.07502
1.36650 0.14957
1.34530 0.22320
1.31582 0.29546
1.43309 o.oo
1.42867 0.08054
1.41544 0.16058
1.39349 0.23964
1.36294 0.31721
1.48623 o.oo
1.48164 0.08627
1.46793 0.17200
1.44516 0/25667
1.41348 0.33976
0.25
°-3
0-35
0.4
0-45
1.19613 0.31469
1.15356 0.37332
1.10390 0.42966
1.04742 0.48335
0.98449 0.53405
1.23563 0.33986
1.19166 0.40319
1.14035 0.46403
1.08201 0.52202
1.01700 0.57678
1.27822 0.36589
1.23274 0.43407
1.17965 0.49957
1.11930 0.56199
1.05205 0.62095
1.32400 0.39283
1.27689 0.46603
1.22194 0.53635
1.15939 0.60337
1.08973 0.66667
1.37309 0.42076
1.32425 0.49916
1.26722 0.57448
1.20238 0.64626
1.13013 0.71406
o-5
0-55
0.6
0.65
0.7
0.91548 0.58147
0.84083 0.62529
0.76100 0.66527
0.67647 0.70114
0.58777 0.73269
0.94571 0.62799
0.86859 0.67532
0.78613 0.71849
0.69881 0.75724
0.60718 0.79131
0.97830 0.67608
0.89853 0.72704
0.81322 0.77351
0.72290 0.81522
0.62811 0.85191
1.01334 0.72586
0.93071 0.78057
0.84235 0.83047
0.74879 0.87525
0.65061 0.91463
1.05092 0.77745
0.96523 0.83605
0.87358 0.88950
0.77655 0.93747
0.67473 0.97965
0.75
0.8
0.85
0.9
o.95
0.49545 0.75972
0.40008 0.78207
0.30224 0.79960
0.20253 0.81219
0.10158 0.81978
0.51182 0.82050
0.41329 0.84464
0.31222 0.86357
0.20922 0.87717
0.10493 0.88537
0.52945 0.88334
0.42753 0.90932
0.32298 0.92970
0.21643 0.94435
0.10855 0.95317
0.54842 0.94838
0.44285 0.97628
0-33455 0.99816
0.22418 1.01388
0.11244 1-02335
0.56875 1.01579
0.45927 1.04567
0.34695 1.06911
0.23250 1.08595
0.11661 1.09610
.0
•05
.1
•IS
.2
o.oo 0.82232
0.10158 0.81978
0.20253 0.81219
0.30224 0.79960
0.40008 0.78207
o.oo 0.88811
0.10493 0.88537
0.20922 0.87717
0.31222 0.86357
0.41329 0.84464
o.oo 0.95612
0.10855 0.95317
0.21643 0.94435
0.32298 0.92970
0.42753 0.90932
o.oo 1.02652
0.11244 1-02335
0.22418 1.01388
0.33455 0.99816
0.44285 0.97628
o.oo 1.09948
o. 11661 1.09610
0.23250 1.08595
0.34695 1.06911
0.45927 1.04567
•25
•3
i-35
1.4
1-45
0.49545 0.75972
0.58777 0.73269
0.67647 0.70114
0.76100 0.66527
0.84083 0.62529
0.51182 0.82050
0.60718 0.79131
0.69881 0.75724
0.78613 0.71849
0.86859 0.67532
0.52945 0.88334
0.62811 0.85191
0.72290 0.81522
0.81322 0.77351
0.89853 0.72704
0.54842 0.94838
0.65061 0.91463
0.74879 0.87525
0.84235 0.83047
0.93071 0.78057
0.56875 1.01579
0.67473 0.97965
0.77655 0.93747
0.87358 0.88950
0.96523 0.83605
i-5
1-55
1.6
1.65
i-7
0.91548 0.58147
0.98449 0.53405
1.04742 0.48335
1.10390 0.42966
I.I5356 0.37332
0.94571 0.62799
1.01700 0.57678
1.08201 0.52202
1.14035 0.46403
1.19166 0.40319
0.97830 0.67608
1.05205 0.62095
1.11930 0.56199
1.17965 0.49957
1.23274 0.43407
1.01334 0.72586
1.08973 0.66667
I-I5939 0.60337
1.22194 0.53635
1.27689 0.46603
1.05092 0.77745
1.13013 0.71406
1.20238 0.64626
1.26722 0.57448
1.32425 0.49916
1-75
1.8
1.85
1.9
1-95
1.19613 0.31469
1.23132 0.25411
1.25891 0.19197
1.27874 0.12864
1.29069 0.06452
1.23563 0.33986
1.27198 0.27444
1.30048 0.20732
1.32097 0.13893
1.33331 0.06968
1.27822 0.36589
1.31582 0.29546
1.34530 0.22320
1.36650 0.14957
1.37927 0.07502
1.32400 0.39283
1.36294 0.31721
1-39349 0.23964
1.41544 0.16058
1.42867 0.08054
1.37309 0.42076
1.41348 0.33976
1.44516 0.25667
1.46793 0.17200
1.48164 0.08627
2.0
1.29468 o.oo
1.33743 0.00
1.38353 o.oo
1.43309 o.oo
1.48623 o.oo
Note.
Negative quantities are in heavy type.
Examples, cosh (0.9 +
i i.o) = o + * 1.02652.
cosh (0.9 + i i.io) = —0.22418 + i 1.01388.
[61]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
x = 1.05
0
0.05
O.I
0.15
O.2
1.54308 o.oo
1.53832 0.09221
1.52408 0.18384
1.50045 0.27435
1.46756 0.36316
1.60379 o.oo
1.59885 0.09838
1.58405 0.19615
1.55948 0.29271
1.52530 0.38746
1.66852 o.oo
1.66337 0.10479
1.64798 0.20894
1.62242 0.31180
1.58685 0.41274
1.73741 o.oo
1.73206 0.11147
1.71602 0.22226
1.68941 0.33167
1.65238 0.43904
i. 81066 o.oo
1.80507 0.11843
1.78836 0.23613
1.76063 0.35238
1.72204 0.46645
0.25
0.3
o-3S
0.4
o-4S
1.42562 0.44973
1.37490 0.53353
1.31569 0.61404
1.24838 0.69077
1.17337 0.76323
1.48171 0.47983
1.42899 0.56924
1.36746 0.65514
1.29750 0.73700
1.21953 0.81432
1.54151 0.51113
1.48666 0.60637
1.42265 0.69787
1.34986 0.78508
1.26875 0.86743
1.60517 0.54371
1.54805 0.64502
1.48139 0.74235
1.40560 0.83511
1.32114 0.92272
1.67283 0.57765
1.61331 0.68528
1.54384 0.78869
1.46485 0.88724
1.37684 0.98032
o-S
o.SS
0.6
0.65
0.7
1.09112 0.83099
1.00215 0.89363
0.90700 0.95076
0.80626 I.OO2O2
0.70055 1-04711
1.13405 0.88661
1.04158 0.95344
0.94269 1.01439
0.83798 1.06909
0.72811 1.11719
1.17982 0.94445
1.08362 1.01564
0.98073 1.08056
0.87180 1.13883
0.75749 1.19007
1.22854 1.00464
1.12836 1.08037
1.02123 I-I4943
0.90780 1.21141
0.78877 1.26592
1.28033 1-06735
1.17593 1.14781
1.06428 I. 22118
0.94607 1.28703
0.82202 1.34494
0-75
0.8
0.85
0.9
o-9S
0.59051 1.08574
0.47684 1.11768
0.36023 1.14273
0.24139 1.16073
0.12107 1.17158
0.61375 1.15841
0.49560 1.19249
0.37440 1.21921
0.25089 1.23842
0.12583 1.24999
0.63852 1.23398
0.51560 1.27028
0.38951 1.29875
0.26101 1.31920
0.13091 1.33153
0.66488 1.31263
0.53689 1.35124
0.40559 1.38152
0.27179 1.40329
0.13632 1.41640
0.69291 1.39456
0.55952 1.43558
0.42269 1.46776
0.28325 1.49088
0.14206 1.50481
I.O
1.05
i.i
i-iS
1.2
o.oo 1.17520
0.12107 1.17158
0.24139 1.16073
0.36023 1.14273
0.47684 1.11768
o.oo 1.25386
0.12583 1.24999
0.25089 1.23842
0.37440 1.21921
0.49560 1.19249
o.oo 1.33565
0.13091 1.33153
0.26101 1.31920
0.38951 1.29875
0.51560 1.27028
o.oo 1.42078
0.13632 1.41640
0.27179 1.40329
0.40559 1.38152
0.53689 1.35124
o.oo 1.50946
0.14206 1.50481
0.28325 1.49088
0.42269 1.46776
0.55952 1.43558
1.25
i-3
1-35
1.4
i.4S
0.59051 1.08574
0.70055 1.04711
O.8O626 1.00202
O.9O7OO 0.95076
I.002I5 0.89363
0.61375 1.15841
0.72811 1.11719
0.83798 1.06909
0.94269 1.01439
1.04158 0.95344
0.63852 1.23398
0.75749 1.19007
0.87180 1.13883
0.98073 1.08056
1.08362 1.01564
0.66488 1.31263
0.78877 1.26592
0.90780 1.21141
I.02I23 I.I4943
1.12836 1.08037
0.69291 1.39456
0.82202 1.34494
0.94607 1.28703
1.06428 I. 22118
1.17593 1.14781
i.S
i-SS
1.6
1.65
1.7
.09112 0.83099
.17337 0.76323
.24838 0.69077
.31569 0.61404
.37490 0.53353
1.13405 0.88661
1.21953 0.81432
1.29750 0.73700
1.36746 0.65514
1.42899 0.56924
1.17982 0.94445
1.26875 0.86743
1.34986 0.78508
1.42265 0.69787
1.48666 0.60637
1.22854 1.00464
I.32II4 0.92272
1.40560 0.835II
I.48I39 0.74235
1.54805 0.64502
1.28033 1.06735
1.37684 0.98032
1.46485 0.88724
1.54384 0.78869
1.61331 0.68528
1-75
1.8
1.85
1.9
1.95
1.42562 0.44973
X.46756 0.36316
1.50045 0.27435
1.52408 0.18384
1.53832 O.O922I
1.48171 0.47983
1.52530 0.38746
1.55948 0.29271
1.58405 0.19615
1.59885 0.09838
1.54151 0.51113
1.58685 0.41274
1.62242 0.31180
1.64798 0.20894
1.66337 0.10479
I.605I7 0.54371
1.65238 0.43904
I.6894I 0.33167
I.7I6O2 O.22226
I.732O6 O.III47
1.67283 0.57765
1.72204 0.46645
1.76063 0.35238
1.78836 0.23613
1.80507 0.11843
2.O
1.54308 o.oo
1.60379 o.oo
1.66852 o.oo
I«7374i o.oo
i. 81066 o.oo
Note.
Negative qua
n tities are in heavy
type.
Examples, cosh (1.2 4- i o) = i. 81066 + i o.
cosh (E.I + » i.i) = —0*26101 + i 1.31920.
[62]
TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED
9
x = 1.25
*= 1.3
x = i-35
x = 1.4
x = 1.45
o
0.05
O.I
o-iS
O.2
1.88842 o.oo
1.88260 0.12569
1.86517 0.25060
.83624 0.37396
.79600 0.49502
1.97091 o.oo
1.96484 1.13325
1.94665 0.26569
1.91646 0.39648
1.87445 0.52483
2.05833 o.oo
2.05200 0.14116
2.03299 0.28144
2.00146 0.41999
1-95759 0.55595
2.15090 o.oo
2.14427 0.14941
2.12442 0.29790
2.09147 0.44455
2.04562 0.58846
2.24884 o.oo
2.24191 0.15804
2.22115 0.31510
2.18671 0.47022
2.13878 0.62244
0.25
0-3
0-35
0.4
0-45
.74467 0.61303
.68260 0.72726
.61014 0.83700
1.52777 0.94158
1.43597 1.04036
1.82089 0.64994
1.75610 0.77105
1.68048 0.88740
1.59451 0.99829
1.49870 1.10301
1.90165 0.68848
1.83399 0.81677
1.75502 0.94002
1.66523 1.05748
1.56517 1.16842
1.98717 0.72875
1.91646 0.86454
1.83394 0.99500
1.74012 1.11932
1.63556 1.23674
2.07766 0.77083
2.00373 0.91446
1.91745 1.05245
1.81935 1.18396
1.71007 1.30817
o-5
o.SS
0.6
0.65
0.7
1.33532 1.13273
1.22643 1.2x811
1.10999 1.29598
0.98670 1.36586
0.85733 1-42732
1.39365 1.20094
1.28001 1.29146
1.15848 1.37402
1.02980 1.44811
0.89478 1.51327
1.45546 1.27215
1.33678 1.36804
1.20986 1.45550
1.07548 1.53398
0.93446 1.60300
1.52092 1.34655
1.39690 1.44804
1.26427 1.54061
1.12384 1.62369
0.97649 1.69675
1.59017 1.42431
1.46051 1.53166
1.32184 1.62958
1.17502 1.71745
1.02095 1-79473
0.75
0.8
0.85
0.9
o-95
0.72267 1.47998
0.58355 1-52352
0.44084 1.55766
0.29541 1.58220
0.14816 1.59698
0.75424 1.56910
0.60905 1.61526
0.46010 1.65146
0.30832 1.67747
0.15464 1.69315
0.78769 1.66215
0.63606 1.71104
0.48051 1.74938
0.32199 1.77694
0.16150 1.79354
0.82311 1.75934
0.66466 1.81110
0.50212 1.85169
0.33647 1.88086
0.16876 1.89843
0.86060 1.86094
0.69493 1.91569
0.52498 1.95862
0.35180 1.98947
0.17644 2.00806
I.O
1.05
i.i
i-iS
1.2
o.oo 1.60192
0.14816 1.59698
0.29541 1.58220
0.44084 1.55766
0.58355 1-52352
o.oo 1.69838
0.15464 1.69315
0.30832 1.67747
0.46010 1.65146
0.60905 1.61526
o.oo 1.79909
0.16150 1.79354
0.32199 1.77694
0.48051 1.74938
0.63606 1.71104
o.oo 1.90430
0.16876 1.89843
0.33647 1.88086
0.50212 1.85169
0.66466 1.81110
o.oo 2.01427
0.17644 2.00806
0.35180 1.98947
0.52498 1.95862
0.69493 1.91569
I.2S
i-3
i-35
1.4
i-4S
0.72267 1.47998
0.85733 1.42732
0.98670 1.36586
.10999 1-29598
.22643 1. 21811
0.75424 1.56910
0.89478 1.51327
1.02980 1.44811
1.15848 1.37402
1.28001 1.29146
0.78769 1.66215
0.93446 1.60300
1.07548 1.53398
1.20986 1.45550
1.33678 1.36804
0.82311 1.75934
0.97649 1.69675
1.12384 1.62369
1.26427 1.54061
1.39690 1.44804
0.86060 1.86094
1.02095 1.79473
1.17502 1.71745
1.32184 1.62958
1.46051 1.53166
i-5
•55
.6
.65
•7
.33532 1.13273
•43597 1.04036
•52777 0.94158
.61014 0.83700
.68260 0.72726
1-39365 1.20094
1.49870 1.10301
1.59451 0.99829
1.68048 0.88740
1.75610 0.77105
1.45546 1.27215
1.56517 1.16842
1.66523 1.05748
1.75502 0.94002
1.83399 0.81677
1.52092 1.34655
1.63556 1.23674
1.74012 1.11932
1.83394 0.99500
1.91646 0.86454
1.59017 1.42431
1.71007 1.30817
1.81935 1.18396
1.91745 1.05245
2.00373 0.91446
•75
.8
•85
1.9
1-95
.74467 0.61303
.79600 0.49502
.83624 0.37396
.86517 0.25060
.88260 0.12569
1.82089 0.64994
1.87445 0.52483
1.91646 0.39648
1.94665 0.26569
1.06484 0.13325
1.90165 0.68848
X-95759 0.55595
2.00146 0.41999
2.03299 0.28144
2.05200 0.14116
1.98717 0.72875
2.04562 0.58846
2.09147 0.44455
2.12442 0.29790
2.14427 0.14941
2.07766 0.77083
2.13878 0.62244
2.18671 0.47022
2.22115 0.31510
2.24191 0.15804
2.O 1.88842 O.OO
1.97091 o.oo
2.05833 o.oo
2.15090 o.oo
2.24884 o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (1.4 + i 1.9) = —2.12442 + * 0.29790.
cosh (1.4 + i 1.4) = —1.26427 + * 1.54061.
[63]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
*= 1.5
= i-55
x = 1.6
x = 1.65
x = 1.7
o
2.35241 o.oo
2.46186 o.oo
2.57746 o.oo
2.69952 o.oo
2.82832 o.oo
0.05
2.34516 0.16706
2.45427 0.17650
2.56952 0.18639
2.69121 0.19673
2.81960 0.20757
O.I
2.32345 0.33309
2.43155 0.35192
2-54573 0.37162
2.66629 °-39225
2.79350 0.41387
0.15
2.28742 0.49707
2.39384 0.52516
2.50625 0.55456
2.62494 0.58536
2.75017 0.61761
O.2
2.23727 0.65798
2.34137 0.69517
2.45131 0.73409
2.56740 0.77485
2.68989 0.81754
0.25
2.17334 0.81484
2.27446 0.86089
2.38127 0.90909
2.49404 0.95957
2.61302 1.01244
°-3
2.09601 0.96667
2.19353 1.02130
2.29654 1.07848
2.40529 1.13837
2.52005 1.20109
0.35
2.00576 1.11255
2.09908 1.17542
2.19765 1.24123
2.30173 1.31015
2.41154 1.38234
0.4
1.90314 1.25156
1.99169 1.32229
2.08522 1.39632
2.18396 1.47385
2.28816 1.55506
o-4S
1.78879 1.38286
1.87201 1.46101
1.95992 1.54281
2.05274 1.62847
2.15067 1.71820
o-S
1.66341 1.50563
1.74080 1.59071
1.82254 1.67978
1.00885 1.77305
1.99992 1.87074
0-55
1.52777 1.61912
1.59885 1.71062
1.67393 1.80640
1.75320 1.90669
1.83684 2.01175
0.6
1.38271 1.72263
1.44704 1.81997
1.51500 1.92187
1.58674 2.02858
1.66244 2.14036
0.65
1.22913 1.81551
1.28632 1.91811
1.34672 2.02550
1.41050 2.13797
1.47779 2.25577
0.7
1.06797 1.89720
1.11766 2.00442
1.17015 2.11664
1.22556 2.23417
1.28403 2.35727
0-75
0.90023 1.96720
0.94211 2.07837
0.98636 2.19473
1.03306 2.31660
1.08235 2.44424
0.8
0.72693 2.02507
0.76076 2.13951
0.79648 2.25930
0.83420 2.38474
0.87400 2.51614
0.85
0.54916 2.07045
0.57471 2.18745
0.60170 2.30993
0.63019 2.43818
0.66026 2.57253
0.9
0.36800 2.10307
0.38512 2.22191
0.40320 2.34632
0.42230 2.47659
0.44245 2.61306
0-95
0.18457 2.12272
0.19316 2.24268
0.20223 2.36824
0.21180 2.49973
0.22191 2.63747
I.O
o.oo 2.12928
o.oo 2.24961
o.oo 2.37557
0,00 2.50747
o.oo 2.64563
i. 05
0.18457 2.12272
0.19316 2.24268
0.20223 2.36824
o. 21180 2.49973
0.22191 2.63747
i.i
0.36800 2.10307
0.38512 2.22191
0.40320 2.34632
0.42230 2.47659
0.44245 2.61306
i-iS
0.54916 2.07045
0.57471 2.18745
0.60170 2.30993
0.63019 2.43818
0.66026 2.57253
1.2
0.72693 2.02507
0.76076 2.13951
0.79648 2.25930
0.83420 2.38474
0.87400 2.51614
I.2S
0.90023 1.96720
0.94211 2.07837
0.98636 2.19473
1.03306 2.31660
1.08235 2.44424
i-3
1.06797 1.89720
1.11766 2.00442
1.17015 2.11664
1.22556 2.23417
1.28403 2.35727
i-35
1.22913 1.81551
1.28632 1.91811
1.34672 2.02550
1.41050 2.13797
1.47779 2.25577
1.4
1.38271 1.72263
1.44704 1.81997
1.51500 1.92187
1.58674 2.02858
1.66244 2.14036
i-4S
1-52777 1.61912
1.59885 1.71062
1-67393 1.80640
1-75320 1.90669
1.83684 2.01175
i-5
1.66341 1.50563
1.74080 1.59071
1.82254 1.67978
1.90885 1.77305
1.99992 1.87074
i-SS
1.78879 1.38286
1.87201 1.46101
1.95992 1.54281
2.05274 1.62847
2.15067 1.71820
1.6
1.90314 1.25156
1.99169 1.32229
2.08522 1.39632
2.18396 1.47385
2.28816 1.55506
1.65
2.00576 1.11255
2.09908 1.17542
2.19765 1.24123
2.30173 1.31015
2.41154 1.38234
i-7
2.09601 0.96667
2.19353 1.02130
2.29654 1.07848
2.40529 1.13837
2.52005 1.20109
1-75
2-17334 0.81484
2.27446 0.86089
2.38127 0.90909
2.49404 0.95957
2.61302 1.01244
1.8
2.23727 0.65798
2.34137 0.69517
2.45131 0.73409
2.56740 0.77485
2.68989 0.81754
1.85
2.28742 0.49707
2.39384 0.52516
2.50625 0.55456
2.62494 0.58536
2.75017 0.61761
1.9
2.32345 0-33309
2.43155 0.35192
2-54573 0.37162
2.66629 0.39225
2.79350 0.41387
i. 95
2.34516 0.16706
2.45427 0.17650
2.56952 0.18639
2.69121 0.19673
2.81960 0.20757
2.35241 o.oo
2.46186 o.oo
2.57746 o.oo
2.69952 o.oo
2.82832 o.oo
Note.
Negative quantities are in heavy type.
Examples, cosh (1.6 + i 0.4) = 2.08522 -f- i 1.39632.
cosh (1.7 + * 1.2) = —0.87400 + i 2.51614.
[64]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + n>. CONTINUED
?
x =
i-75
x =
1.8
X =
1.85
X =
1.9
x =
i-95
o
2.96419
o.oo
3.10747
o.oo
3-25853
o.oo
3-41773
o.oo
3-58548
o.oo
0.05
2.95505
0.21893
3.09789
0.23084
3.24848
0.24332
3.40719
0.25642
3-57443
0.27015
O.I
2.92769
0.43652
3.06921
0.46026
3.21841
0.48627
3-37565
0.51125
3-54I34
0.53864
0.15
2.88229
0.65141
3.02161
0.68684
3.16850
0.72398
3-32330
0.76294
3.48641
0.80380
O.2
2.81911
0.86229
2.95538
0.90918
3.09904
0.95835
3-25045
1.00992
3.41000
1.06401
0.25
2.73855
1.06784
2.87093
1.12592
3.01049
1. 18681
3-I57S7
1.25067
3-31255
1.31766
o-3
2.64111
1.26682
2.76878
1-33572
2.90337
1.40796
3.04522
1.48372
3.19469
1.56318
0-35
2-52/39
1.45799
2.64956
1.53728
2.77835
1.62042
2.91409
1.90761
3-057I3
1.79907
0.4
2.39808
1.64016
2.51400
1.72937
2.63620
1.82289
2.76501
1.92098
2.90071
2.02387
0-45
2-25399
1.81223
2.36294
1.91079
2.47780
2.01413
2.59887
2.12250
2.72642
2.23618
o-5
2.09600
1.97312
2.19731
2.08043
2.30413
2.19294
2.41670
2.31094
2-53532
2.43471
Q-55
1.92509
2.12185
2.01814
2.23725
2.11624
2.35824
2.21964
2.48513
2.32858
2.61823
0.6
1.74231
2.25749
1.82653
2.38027
i-9I53i
2.50900
2.00889
2.64400
2.10749
2.78561
0.65
1.54^78
2.37922
1.62365
2.50862
1.70258
2.64429
1.78576
2.78657
r.87341
2.93582
0.7
I-3457I
2.48627
1.41076
2.62149
1-47934
2.76327
1-55162
2.91196
1.62777
3.06792
o-75
I-I3435
2.57800
1.18918
2.71821
1.24698
2.86522
1.30791
3.01939
1.37210
3.18111
0.8
0.91599
2-65384
0.96026
2.79817
1.00694
2.94950
1.05614
3.10821
1.10797
3.27468
0.85
0.69198
2.71331
0.72542
2.86088
0.76069
3.01560
0.79785
3.17787
0.83701
3-34807
0.9
0.46370
2.75606
0.48612
2.90595
0.50975
3.06311
0.53465
3-22793
0.56089
3.40081
o-95
0.23257
2.78181
0.24381
2.93310
0.25566
3-09173
0.26815
3.25809
0.28131
3.43259
I.O
o.oo
2.79041
o.oo
2.94217
o.oo
3.10129
o.oo
3.26816
0.00
3-44321
1.05
0.23257
2.78181
0.24381
2.93310
0.25566
2.09173
0.26815
3.25809
0.28131
3-43259
i.i
0.46370
2.75606
0.48612
2-90595
0.50975
3.06311
0.53465
3-22793
0.56089
3.40081
i-i5
0.69198
2.71331
0.72542
2.86088
0.76069
3.01560
0.79785
3.17787
0.83701
3.34807
1.2
0.91599
2.65384
0.96026
2.79817
1.00694
2.94950
1.05614
3.10821
1.10797
3-27468
1.25
.13435
2.57800
1.18918
2.71821
1.24698
2.86522
1.30791
3-OI939
1.37210
3.18111
i-3
.34571
2.48627
1.41076
2.62149
1.47934
2.76327
1.55162
2.91196
1.62777
3.06792
i-35
.54878
2.37922
1.62365
2.50862
1.70258
2.64429
1.78576
2.78657
1.87341
2.93582
1.4
.74231
2.25749
1.82653
2.38027
1-91531
2.50900
2.00889
2.64400
2.10749
2.78561
i-4S
.92509
2.12185
2.01814
2.23725
2.11624
2.35824
2.21964
2.48513
2.32858
2.61823
i-5
2.09600
1.97312
2.19731
2.08043
2.30413
2.19294
2.41670
2.31094
2.53532
2.43471
i-SS
2.25399
1.81223
2.36294
1.91079
2.47780
2.01413
2.59887
2.12250
2.72642
2.23618
1.6
2.39808
1.64016
2.51400
1.72937
2.63620
1.82289
2.76501
1.92098
2.90071
2.02387
1.65
2.52739
1-45799
2.64956
1-53728
2.77835
1.62042
2.91409
1.90761
3.05713
1.79907
1-7
2.64111
1.26682
2.76878
1-33572
2.90337
1.40796
3.04522
1-48372
3.19469
1.56318
i-75
2.73855
1.06784
2.87093
1.12592
3.01049
1. 18681
3.15757
1.25067
3.31255
1.31766
1.8
2.81911
0.86229
2.95538
0.90918
3.09904
0.95835
3-25045
1.00992
3.41000
1.06401
1.85
2.88229
0.65141
3.02161
0.68684
3.16850
0.72398
3-32330
0.76294
3.48641
0.80380
1.9
2.92769
0.43652
3.06921
0.46026
3.21841
0.48627
3.37565
0.51125
3.54134
0.53864
i-95
2.95505
0.21893
3.09789
0.23084
3.24848
0.24332
3.40719
0.25642
3.57443
0.27015
2.O 2.96419 O.OO
3.10747 o.oo
3.25853 o.oo
3.41773 o.oo
3.58548 o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (1.8 + « o. 2) = 2.95538 + i 0.90918.
cosh (1.8 + i 2.0) = —3.10747 + »o.
[65]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
9
x = 2.0
x = 2.05
X = 2.1
x = 2.15
X = 2.2
o
0.05
O.I
0.15
0.2
3.76220 o.oo
3.75059 0.28456
3.71587 0.56737
3.65825 0.84667
3.57806 1.12076
3.94832 o.oo
3.93615 0.29968
3.89971 0.59751
3.83923 0.89166
3.75507 1.18032
4.14431 o.oo
4.13154 0.31555
4.09329 0.62916
4.02981 0.93888
3.94148 1.24282
4.35067 o.oo
4.33726 0.33221
4.29711 0.66237
4.23046 0.98845
4-13773 1-30844
4.56791 o.oo
4.55382 0.34970
4.51167 0.69724
4.44170 1.04049
4-34433 1-37732
0.25
0-35
0.4
0-45
3.47581 1.38794
3.35214 1.64656
3.20780 1.89503
3.04368 2.13182
2.86080 2.35546
3.64777 1.46169
3.51798 1.73405
3.36649 1.99573
3.19426 2.24509
3.00232 2.48062
3.82885 1.53910
3.69261 1.82589
3.53361 2.10142
3.35283 2.36399
3.15137 2.61199
4.01950 1.62035
3.87647 1.92228
3.70956 2.21236
3.51977 2.48879
3.30828 2.74988
4.22019 1.70566
4.07003 2.02349
3.89478 2.32883
3.69552 2.61982
3-47347 2.89466
0-5
0-55
0.6
0.65
0-7
2.66027 2.56458
2-44335 2.75789
2.21136 2.93420
I-96574 3-09241
1.70800 3.23156
2.79188 2.70085
2.56423 2.90443
2.32076 3.09011
2.06299 3.23673
1.79250 3.40327
2.93048 2.84389
2.69152 3.05825
2-43597 3-25376
2.16540 3.42920
1.88148 3-5835I
3.07639 2.99402
2.82553 3.21970
2.55726 3-42553
2.27322 3.61024
1.97516 3-77269
3.23000 3.15165
2.96661 3.38921
2.68495 3.60588
2.38672 3.80031
2.07379 3.97I3I
0.75
0.8
0.85
0.9
0.95
1-43973 3-35078
1.16258 3-44935
0.87827 3.52666
0-58854 3.58221
0.29518 3.61568
1.51096 3.52883
I.22OIO 3.63264
0.92172 3.71404
0.61765 3.77256
0.30978 3.80781
1-58596 3-7I57I
1.28066 3.82501
0.96747 3-9I074
0.64831 3-97235
0.32516 4.00946
1.66493 3-91188
1-34443 4-02695
1.01564 4.11720
0.68059 4.18206
0.34135 4-22113
1.74806 4.11783
1.41156 4.23896
1.06636 4-33396
0.71458 4.40223
0.35839 4-44337
I.O
1.05
i.i
1.2
o.oo 3.62686
0.29518 3.61568
0.58854 3.58221
0.87827 3.52666
1.16258 3-44935
o.oo 3.81958
0.30978 3.80781
0.61765 3.77256
0.92172 3.71404
1. 22010 3.63264
o.oo 4.02186
0.32516 4.00946
0.64831 3-97235
0.96747 3-91074
1.28066 3.82501
o.oo 4.23419
0.34135 4-22113
0.68059 4.18206
1.01564 4.11720
1.34443 4-02695
0.00 4-457II
0.35839 4-44337
0.71458 4.40223
1.06636 4-33396
1.41156 4.23896
1.25
i-3S
1.4
MS
1-43973 3-35078
1.70800 3.23156
I-96574 3-09241
2.21136 2.93420
2.44335 2.75789
I.5I096 3.52883
1.79250 3-40327
2.06299 3.23673
2.32O76 3.OOOII
2.56423 2.90443
1.58596 3-71571
1.88148 3.58351
2.16540 3.42920
2.43597 3-25376
2.69152 3.05825
1.66493 3.91188
i.975i6 3.77269
2.27322 3.61024
2.55726 3.42553
2.82553 3-2I970
1.74806 4.11783
2.07379 3-97I3I
2.38672 3.80031
2.68495 3-60588
2.96661 3.38921
i-SS
.6
.65
•7
2.66027 2.56458
2.86080 2.35546
3.04368 2.13182
3.20780 1.89503
3.35214 1.64656
2.79188 2.70085
3.OO232 2.48062
3.19426 2.24509
3.36649 1.99573
3.51798 L73405
2.93048 2.84389
3.15137 2.61199
3.35283 2.36399
3.53361 2.10142
3.69261 1.82589
3.07639 2.99402
3.30828 2.74988
3.51977 2.48879
3.70956 2.21236
3.87647 1.92228
3.23000 3.15165
3-47347 2.89466
3.69552 2.61982
3.89478 2.32883
4.07003 2.02349
•75
.8
-85
1.9
3.47581 1.38794
3.57806 1.12076
3.65825 0.84667
3.71587 0.56737
3.75059 0.28456
3.64777 1.46169
3-75507 I.I8032
3.83923 0.89166
3.89971 0.59751
3.93615 0.29968
3.82885 1.53910
3.94148 1.24282
4.02981 0.93888
4.09329 0.62916
4.13154 0.31555
4.01950 1.62035
4-13773 1-30844
4.23046 0.98845
4.29711 0.66237
4.33726 0.33221
4.22019 1.70566
4-34433 1-37732
4.44170 1.04049
4.51167 0.69724
4.55382 0.34970
2.O
3.76220 o.oo
3.94832 o.oo
4.14431 o.oo
4.35067 o.oo
4.56791 o.oo
Note.
Negative quantities are in heavy type. •
Examples, cosh (2.1 + * 0.8) = 1.28066 + i 3.82501.
cosh (2.2 + i 1.25) = —1.74806 + i 4.11783.
[66]
TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED
2.25
x = 2.3
x = 2.35
2.4
x = 2.45
o
0.05
O.I
0.15
O.2
4.79657 o.oo
4.78178 0.36807
4-737SI 0.73386
4.66404 1.09513
4.56l8l 1.44965
5.03722 o.oo
5.02169 0.38735
4.97521 0.77231
4.89805 1.15251
4.79058 1.52560
5.29047 o.oo
5.27416 0.40760
5.22533 0.81269
5.14429 1.21277
5.03153 1.60537
5.55695 o.oo
5.53981 0.42888
5.48853 0.85511
5.40341 1.27607
5.28496 1.68916
5.83732 o.oo
5.81933 0.45122
5.76545 0.89966
5.67603 1.34255
5.55162 1.77716
0.25
o-3
o-3S
0.4
0-45
443145 1.79523
4.27377 2.12975
4.08975 2.45II3
3.88050 2.75740
3-64734 3-04677
4.65378 1.88930
4.48820 2.24134
4.29494 2.57956
4.07520 2.90188
3.83034 3.20630
4.88776 1.98808
4.71384 2.35853
4.51087 2.71443
4.28008 3.05360
4.02200 3.37395
5.13394 2.09184
4.95127 2.48162
4.73808 2.85610
4.49567 3.21297
4.22554 3.55003
5.39298 2.20082
5.20109 2.61091
4-97713 3-00490
4.72249 3.38036
443873 3-73499
o-5
0-55
0.6
0.65
0.7
3.39168 3.3I7I6
3.II5I2 3.567I-9
2.81935 3.79523
2.50620 3.99987
2.17760 4.17986
3.56185 3.49096
3.27141 3.75410
2.96081 3.99409
2.63194 4.20946
2.28685 4.39887
3-74093 3.67351
3.43588 3.95038
3.10966 4.20292
2.76426 4.42955
2.40182 4.62887
3-92935 3-86521
3.60894 4-15656
3.26629 4.42228
2.90349 4.66073
2.52280 4.87045
4.12761 4.06659
3.79104 4-373"
3.43109 4.65268
3.04999 4.90356
2.65009 5.12420
0-75
0.8
0.85
0.9
o-95
I-83557 4-33407
1.48222 4.46157
I.II974 4-56I5S
0.75035 4-6334I
0.37633 4.6767I
1.92766 4.56116
1.55659 4-69533
1.17592 4.80056
0.78799 4.87618
0.39522 4-92174
2.02457 4.79965
1.63485 4.94083
1.23504 5.05156
0.82761 5.13114
0.41509 5.17909
2.12655 5.05014
1.71719 5.19869
1.29724 5.31521
0.86930 5.39893
0.43599 5-44938
2-23385 5-3I325
1.80383 5-46955
1.36269 5.59213
0.91316 5.68022
0-45799 5-73330
I.O
1-05
i.i
"5
1.2
o.oo 4.69117
0.37633 4-67671
0.75035 4-63341
1.11974 4.56155
1.48222 4.46157
o.oo 4.93696
0.39522 4.92174
0.78799 4.87618
1.17592 4.80056
1.55659 4.69533
o.oo 5-I95Io
0.41509 5.17909
0.82761 5.13114
1.23504 5.05156
1.63485 4-94083
o.oo 5.46623
0.43509 5.44938
0.86930 5.39893
1.29724 5.31521
1.71719 5.19869
o.oo 5-75103
0.45799 5-73330
0.91316 5.68022
1.36269 5.59213
1.80383 5-46955
1.25
i-3
1-35
1.4
1-45
1.83557 4.33407
2.17760 4.17986
2.50620 3.99987
2.81935 3.79523
3.11512 3.56719
1.92766 4.56116
2.28685 4.39887
2.63194 4.20946
2.96081 3.99409
3.27141 3.75410
2.02457 4.79965
2.40182 4.62887
2.76426 4-42955
3.10966 4.20292
3.43588 3.95038
2.12655 5.05014
2.52280 4.87045
2.90349 4.66073
3.26629 4.42228
3.60894 4.15656
2.23385 5-3r325
2.65009 5.12420
3.04999 4-90356
3.43109 4.65268
3.79104 4.37311
i-5
i-55
1.6
1.65
i-7
3.39168 3.31716
3.64734 3-04677
3.88050 2.75740
4.08975 2.45113
4.27377 2.12975
3.56185 3.49096
3.83034 3.20630
4.07520 2.90188
4.29494 2.57956
4.48820 2.24134
3.74093 3-6735I
4.02290 3-37395
4.28008 3.05360
4.51087 2.71443
4.71384 2.35853
3.92935 3-86521
4.22554 3-55003
4.49567 3-21297
4.73808 2.85610
4.95127 2.48162
4.12761 4.06659
4.43873 3-73499
4.72249 3.38036
4-97713 3-00490
5.20109 2.61091
i-75
1.8
i.«5
1.9
i-95
4.43145 1.79523
4.56181 1.44965
4.66404 1.09513
4.73751 0.73386
4.78178 0.36807
4.65378 1.88930
4.79058 1.52560
4.89805 1.15251
4.97521 0.77231
5.02169 0.38735
4.88776 1.98808
5.03153 1-60537
5.14429 1.21277
5-22533 0.81269
5.27416 0.40760
5.13394 2.09184
5.28496 1.68916
5.40341 1.27607
5.48853 0.85511
5.53981 0.42888
5.39298 2.20082
5.55162 1.77716
5.67603 1.34255
5.76545 0.89966
5.81933 0.45122
2.O
4.79657 o.oo
5.03722 o.oo
5.29047 o.oo
5.55<595 o.oo
5.83732 o.oo
Note.
Negative quantities are in heavy type.
Examples, cosh (2.4 + 10.4) = 4-4956? + * 3-21297-
cosh (2.4 + i 1.5) = —3-92935 + * 3-86521
[67]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
q
x = 2.5
x = 2.55
X = 2.6
x = 2.65
x = 2.7
0
0.05
O.I
0.15
0.2
6.13229 o.oo
6.II339 0.47469
6.05680 0.94646
5.96287 I.4I239
5.83215 1.86961
6.44259 o.oo
6.42273 0.49935
6.36327 0.99563
6.26458 1.48577
6.12727 1.96674
6.76901 o.oo
6.74814 0.52526
6.68567 1.04729
6.58199 1.56285
6.43770 2.06878
7.11234 o.oo
7.09042 0.55249
7.02478 1.10156
6.91583 1.64385
6.76424 2.17600
7-47347 o.oo
7.45043 0.58109
7.38146 1.15859
7.26697 1.72896
7.10769 2.28866
0.25
0-3
0-35
0.4
0-45
5.66550 2.3I53I
5.46392 2.74674
5.22864 3.l6l22
4.96113 3.55622
4.66304 3.92929
5.95218 2.43559
5.74039 2.88943
5.49321 3.32545
5.21217 3.74097
4.89898 4-13342
6.25374 2.56196
6.03123 3.03934
5-77153 3-49799
5.47624 3.93506
5.14719 4-34788
6.57095 2.69474
6.33714 3-I9686
6.06427 3.67927
5.75401 4-13900
5.40827 4-57321
6.90458 2.83425
6.65891 3.36237
6.37218 3.86976
6.04606 4-35329
5.68287 4.80998
0-5
0-55
0.6
0.65
0.7
4.33619 4.27814
3.98260 4.60062
3.60448 4-89472
3.204II 5.15865
2.78401 5.39077
4.55560 4.50039
4.18413 4.83961
3.78686 5.14900
3.36624 5.42664
2.92488 5.67082
4.78641 4-73389
4.39612 5.09071
3.97872 5-41615
3.53680 5.70819
3.07306 5.96505
5.02919 4.97923
4.61910 5.35454
4.18053 5.69685
3.71619 6.00403
3.22894 6.27419
5.28454 5.23702
4-85363 5-63177
4.39279 5.99I79
3.90488 6.31488
3.39288 6.59903
0-75
0.8
0.85
0.9
0-95
2.34673 5.S8966
1.89498 5.75408
I.43I5S 5.88304
0-9593° 5-97572
0.48113 6.03155
2.46547 5.88004
1.99087 6.05301
1.50399 6.18866
1.00784 6.28615
0.50548 6.34489
2.59039 6.18512
2.09174 6.36706
1.58019 6.50976
1.05891 6.61231
0.53109 6.67409
2.72178 6.50567
2.19784 6.69705
1.66034 6.84713
1.11262 6.95500
0-55803 7.01998
2.85997 6.84249
2-30943 7-04377
1.74465 7.20163
1.16911 7.31508
0.58636 7.38343
I.O
1.05
i.i
i.iS
1.2
o.oo 6.05020
0.48113 6.03155
0.95930 5-97572
1.43155 5.88304
1.89498 5.75408
o.oo 6.36451
0.50548 6.34489
1.00784 6.28615
1.50399 6.18866
1.99087 6.05301
o.oo 6.69473
0.53109 6.67409
1.05891 6.61231
1.58019 6.50976
2.09174 6.36706
o.oo 7.04169
0.55803 7.01998
1.11262 6.95500
1.66034 6.84713
2.19784 6.69705
o.oo 7.40626
0.58636 7.38343
1.16911 7.31508
1.74465 7.20163
2.30943 7-04377
1.25
i-3
i-35
1.4
1-45
2.34673 5.58966
2.78401 5.39077
3.20411 5.15865
3.60448 4.89472
3.98260 4.60062
2.46547 5.88004
2.92488 5.67082
3.36624 5.42664
3.78686 5.14900
4.18413 4.83961
2.59039 6.18512
3.07306 5.96505
3.53680 5.70819
3.97872 5.41615
4.39612 5.09071
2.72178 6.50567
3.22894 6.27419
3.71619 6.00403
4.18053 5.69685
4.61910 5.35454
2.85997 6.84249
3.39288 6.59903
3.90488 6.31488
4-39279 5-99179
4-85363 5-63177
i-5
i-55
1.6
1.65
i-7
4.33619 4.27814
4.66304 3.92929
4.96113 3.55622
5.22864 3.16122
5.46392 2.74674
4.55560 4.50039
4.89898 4.13342
5.21217 3.74097
5.49321 3.32545
5.74039 2.88943
4.78641 4.73389
5.14719 4.34788
5.47624 3.93506
5-77153 3-49799
6.03123 3.03934
5.02919 4.97923
5.40827 4-57321
5.75401 4.13900
6.06427 3.67927
6.33714 3-19686
5.28454 5.23702
5.68287 4.80998
6.04606 4-35329
6.37218 3.86976
6.65891 3.36237
i-75
1.8
1.85
1.9
1-95
5.66550 2.31531
5.83215 1.86961
5.96287 1.41239
6.05680 0.94646
6.1 1339 0.47469
5.95218 2.43559
6.12727 1.96674
6.26458 1.48577
6.36327 0.99563
6.42273 0.49935
6.25374 2.56196
6.43770 2.06878
6.58199 1.56285
6.68567 1.04729
6.74814 0.52526
6.57095 2.69474
6.76424 2.17600
6.91583 1.64385
7.02478 1.10156
7.09042 0.55249
6.90458 2.83425
7.10769 2.28866
7.26697 1.72896
7.38146 1.15859
7.45043 0.58109
2.0
6.13229 0.00
6.44259 o.oo
6.76901 o.oo
7.11234 o.oo
7-47347 o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (2.7 + * i.oo) = o + i 7.40626.
cosh (2.6 + j 1.2) = —2.09174 + i 6.36706.
[68]
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
x = 2.75
x = 2.80
x = 2.85
x = 2.90
x = 2.95
o
0.05
O.I
0.15
O.2
7.85328 o.oo
7.82907 0.61115
7.75659 1.21852
7.63629 1.81839
7.46891 2.40704
8.25273 o.oo
8.22728 0.64273
8.15112 1.28150
8.02470 1.91237
7.84881 2.53144
8.67281 o.oo
8.64608 0.67592
8.56604 1.34768
8.43318 2.OIII2
8.24834 2.66217
9.11458 o.oo
9.08649 0.71081
9.00237 1.41723
8.86275 2.11491
8.66850 2.79956
9.57915 o.oo
9.54962 0.74747
9.46121 1.49032
9.31447 2.22399
9.11031 2.94395
0.25
0-3
o-35
0.4
0-45
7.25548 2.98086
6.99732 3.53629
6.69602 4.06993
6.35344 4-57847
5.97168 5.05878
7.62452 3.13491
7-35323 3-71905
7.03661 4.28026
6.67660 4.81509
6.27542 5.32022
8.01263 3.29681
7.72754 3-9HI2
7-39479 4-50I3I
7.01646 5.06375
6.59486 5.59498
8.42078 3.46694
8.12115 4.11295
7.77146 4-73361
7-37385 5-32508
6.93078 5.88371
8.84998 3.64575
8.53508 4.32508
8.16757 4-97774
7.74969 5.63048
7.28404 6.18717
o-S
0-55
0.6
0.65
0.7
5-55311 5-50790
5.10030 5.92307
4.61604 6.30172
4.10333 6.64151
3.56531 6.94036
5-83556 5-79256
5.35972 6.22918
4.85083 6.62740
4.31204 6.98476
3.74666 7.29905
6.13261 6.09170
5.63254 6.55088
5.09775 6.96966
4.53I53 7-34547
3.93738 7.67599
6.44498 6.40608
5.91945 6.88894
5-35742 7-32934
4.76236 7.72455
4.13793 8.07213
6.77348 6.73647
6.22116 7.24424
5.63048 7.70735
5.00509 8.12294
4.34884 8.48845
0-75
0.8
0.85
0.9
o-95
3.00532 7.19642
2.42680 7.40811
1-83331 7-57413
1.22852 7.69345
0.61616 7.76534
3.15818 7.56834
2.55023 7-79098
1.92656 7-96557
1.29101 8.09106
0.64750 8.16666
3.31894 7-959I9
2.68005 8.19332
2.02463 8.37694
I-35673 8.50891
0.68046 8.58841
3.48800 8.36994
2.81656 8.61616
2.12776 8.80924
1.42583 8.94802
0.71512 9.03163
3.66578 8.80162
2.96012 9.06053
2.23621 9.26358
1.49851 9.40952
0-75*57 9-49744
I.O
1.05
i.i
i-iS
1.2
o.oo 7-78935
0.61616 7-76534
1.22852 7.69345
1.83331 7-574I3
2.42680 7.40811
o.oo 8.19192
0.64750 8.16666
1.29101 8.09106
1.92656 7-96557
2.55023 7-79098
o.oo 8.61497
0.68046 8.58841
1-35673 8.50891
2.02463 8.37694
2.68005 8.19332
o.oo 9-05956
0.71512 9.03163
1.42583 8.94802
2.12776 8.80924
2.81656 8.61616
o.oo 9.52681
0.75157 9-49744
1.49851 9.40952
2.23621 9.26358
2.96012 9.06053
1-25
i-3
1-35
1.4
1-45
3.00532 7.19642
3-5653I 6.94036
4.10333 6.64151
4.61604 6.30172
5.10030 5.92307
3.15818 7-56834
3.74666 7.29905
4.31204 6.98476
4.85083 6.62740
5-35972 6.22918
3.31894 7.95919
3.93738 7-67599
4.53I53 7-34547
5.09775 6.96966
5.63254 6.55088
3.48800 8.36994
4-13793 8.07213
4.76236 7.72455
5-35742 7-32934
5.91945 6.88894
3.66578 8.80162
4.34884 8.48845
5.00509 8.12294
5.63048 7.70735
6.22116 7.24424
i-5
1-55
1.6
1.65
1-7
5-553II 5.50790
5.97168 5.05878
6-35344 4-57847
6.69602 4.06993
6-99732 3-53629
5.83556 5-79256
6.27542 5.32022
6.67660 4.81509
7.03661 4.28026
7.35323 3-71905
6.13261 6.09170
6.59486 5.59498
7.01646 5.06375
7.39479 4-50131
7.72754 3-91112
6.44498 6.40608
6.93078 5.88371
7-37385 5-32508
7.77146 4.73361
8.12115 4.11295
6.77348 6.73647
7.28404 6.18717
7.74969 5-63048
8.16757 4-97774
8.53508 4.32508
i-75
1.8
1.85
1.9
1-95
7.25548 2.98086
7.46891 2.40704
7.63629 1.81839
7.75659 1-21852
7.82907 0.61115
7.62452 3.1349!
7.84881 2.53144
8.02470 1.91237
8.15112 1.28150
8.22728 0.64273
8.01263 3.29681
8.24834 2.66217
8.43318 2.0III2
8.56604 1.34768
8.64608 0.67592
8.42078 3.46694
8.66850 2.79956
8.86275 2.11491
9.00237 1.41723
9.08649 0.71081
8.84998 3.64575
9.11031 2.94395
9.31447 2.22399
9.46121 1.49032
9.54962 0.74747
2.0
7.85328 o.oo
8.25273 o.oo
8.67281 o.oo
9.11458 o.oo
9.579I5 0.00
Note. Negative quantities are in heavy type.
Examples, cosh (2.95 + to) = 9.57915 + *'o.
cosh (2.8 + i 1.2) = —2.55023 + * 7-79098.
[69]
TABLE VIII. HYPERBOLIC COSINES, cosh (* + ig) = u + iv. CONTINUED
x = 3-0
3-°S
x = 3.10
= 3-iS
= 3.20
o
10.06766
o.oo
10.58135
o.oo
11.12150
o.oo
11.68946
o.oo
12.28665
0.00
0.05
10.03660
0.78599
10.54870
0.82649
11.08720
0.86905
11.65340
0.91378
12.24880
0.96080
O.I
9-94371
1.56714
10.45110
1.64788
10.98460
1.73274
11.54550
1.82193
12.13540
1.91568
0.15
9.78949
2.33863
10.28900
2.45911
10.81420
2.58575
11.36650
2.71885
11.94720
2.85874
O.2
9-57492
3-09569
10.06350
3.25518
10.57720
3.42281
11.11730
3.59900
11.68530
3.78419
0.25
9.30131
3.83368
9-77589
4.03120
10.27490
4.23878
10.79970
4-45696
11.35140
4.68630
o-3
8.97035
4.54806
9-42805
4-78233
9-90933
5.02860
10.41540
5-28745
10.94750
5-55952
o.3S
8.58409
5-23433
9.02209
5.50400
9.48264
5-78743
9.96690
6.08533
10.47610
6.39846
0.4
8.14491
5.88836
8.56049
6.19173
8.99749
6.51058
9-45697
6.84570
9.94011
.7-19795
0-45
7-655SI
6.50609
8.04612
6.84128
8.45686
7.19358
8.88873
7-56387
9.34284
7-95306
o-S
7.11891
7.08371
7.48215
7.44866
7.86409
7-83223
8.26570
8.23539
8.68797
8.65915
o-SS
6.53842
7.61765
6.87204
8.OIOH
7.22284
8.42260
7.59170
8.85615
7.97954
9.31184
0.6
5.91762
8.10463
6.21956
8.52218
6.53705
8.96104
6.87089
9.42230
7.22191
9.90712
0.65
5-26034
8.54164
5-52874
8.98171
5.81097
9.44423
6.10773
9-93036
6.41976
10.44130
0.7
4.57062
8.92599
4.80383
9.38586
5.04906
9.86919
5.30690
10.37720
5.57802
10.91120
0-75
3-85273
9-25531
4.04931
9.73216
4.25602
10.23330
4-47336
10.76010
4.70190
11.31370
0.8
3.11108
9-52757
3.26982
10.01840
3-43673
10.53430
3.61224
11.07660
3.79678
11.64650
0.85
2.35025
9.74108
2.47017
10.24290
2.59626
10.77040
2.72885
11.32480
2.86826
11.90750
0.9
1-57493
9-89454
1.65529
10.40430
1-73979
10.94010
1.82863
11.50320
1.92205
12.09510
Q-9S
o. 78990
9.98699
0.83020
10.50150
0.87258
11.04230
0.91714
11.61070
0.96400
12.20810
I.O
0.00
10.01787
o.oo
10.53399
0.00
11.07645
o.oo
11.64661
o.oo
12.24588
1.05
0.78990
9.98699
0.83020
10.50150
0.87258
11.04230
0.91714
11.61070
0.96400
12.20810
i.i
i -57493
9-89454
1.65529
10.40430
1.73979
10.94010
1.82863
11.50320
1.92205
12.09510
i-i5
2.35025
9.74108
2.47017
10.24290
2.59626
10.77040
2.72885
11.32480
2.86826
11.90750
1.2
3.11108
9-52757
3.26982
10.01840
3.43673
10.53430
3.61224
11.07660
3.79678
11.64650
1-25
3-85273
9-25531
4.04931
9.73216
4.25602
10.23330
4-47336
10.76010
4.70190
11.31370
i-3
4-57062
8.92599
4.80383
9.38586
5.04906
9.86919
5.30690
10.37720
5.57802
10.91120
1.35
5-26034
8.54164
5.52874
8.98171
5.81097
9-44423
6.10773
9.93036
6.41976
10.44130
1.4
5-91762
8.10463
6.21956
8.52218
6.53705
8.96104
6.87089
9.42230
7.22191
9.90712
i-45
6.53842
7.61765
6.87204
8.01011
7.22284
8.42260
7.59170
8.85615
7-97954
9.31184
i-S
7.11891
7.08371
7.48215
7.44866
7.86409
7.83223
8.26570
8.23539
8.68797
8.65915
i-5S
7.65551
6.50609
8.04612
6.84128
8.45686
7.19358
8.88873
7.56387
9.34284
7-95306
1.6
8.14491
5.88836
8.56049
6.19173
8.99749
6.51058
9.45697
6.84570
9.94011
7-19795
1.65
8.58409
5-23433
9.02209
5.50400
9.48264
5-78743
9.96690
6.08533
10.47610
6.39846
i-7
8.97035
4.54806
9.42805
4-78233
9.90933
5.02860
10.41540
5.28745
10.94750
5-55952
1-75
9-30131
3.83368
9.77589
4.03120
10.27490
4.23878
10.79970
4-45696
11.35140
4.68630
1.8
9-57492
3-09569
10.06350
3-255i8
10.57720
3.42281
11.11730
3-59900
11.68530
3.78419
1.85
9.78949
2.33863
10.28900
2.45911
10.81420
2.58575
11.36650
2.71885
11.94720
2.85874
1.9
9-94371
1.56714
10.45110
1.64788
10.98460
1.73274
11.54550
1.82193
12.13540
1.91568
1-95
10.03660
0.78599
10.54870
0.82649
11.08720
0.86905
11.65340
0.91378
12.24880
0.96080
2.0
10.06766
0.00
10.58135
o.oo
11.12150
o.oo
11.68946
o.oo
12.28665
o.oo
\
Note. Negative quantities are in heavy type.
Examples, cosh (3.10 + i 0.5) =
7.86409 +
* 7.83223.
cosh (3.10 + i 1.55) = -8.45686 + i 7.19358.
[7o]
TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED
= 3-25
x = 3.30
= 3-35
= 3-40
3-45
0
12.91456
o.oo
I3-57476
o.oo
14.26891
o.oo
14.99874
o.oo
15-76607
o.oo
0.05
12.87474
I.OIO22
I3-53290
1.06217
14.22498
1.11677
14.95250
1.17417
15.71746
1.23450
O.I
12.75555
2.OI422
13.40764
2.11780
14.09323
2.22666
14.81410
2.34110
I5-57I96
2.46139
0.15
12-55773
3-00579
13.19970
3.16036
13.87465
3.32282
14.58432
3-49359
I5-33045
3.673"
0.2
12.28247
3.97883
12.91035
4.18343
I3-57054
4.39850
14.26465
4.62455
14.99442
4.86217
0.25
11.93150
4-92735
I2.54U5
5.18072
13.18275
5-44705
13.85702
5.72700
14.56595
6.02127
o-3
11.50695
5.84548
I2.O952O
6.14670
12.71369*
6.46202
13-36397
6.79414
14.04764
7.14324
o-35
11.01147
6.72758
11.57440
7.07352
12.16623
7.43715
12.78852
7-81938
13.44278
8.22116
0.4
10.44810
7.56820
IO.98222
7.95737
"•54379
8.36643
12.13423
8.79642
12.75501
9.24840
o.45
9.82031
8.36217
10.32233
8.79215
10.85016
9.24413
11.40512
9.71923
11.98861
10.21863
0-5
9.13197
9.10456
9.59881
9.57273
10.08964
10.06483
10.60570
10.58212
11.14830
11.12585
o-SS
8.38733
9.79082
8.81610
10.29428
9.26691
10.82349
9.74090
"-37975
10.23924
11.96447
0.6
7.59099
10.41673
7-97905
10.95235
8.38705
11.51541
8.81604
12.10723
9.26706
12.72933
0.65
6.74784
10.9/841
7.09279
11.54294
7-45549
12.13633
7.83682
12.76006
8.23775
13.41572
0.7
5.86309
II.4724O
6.16381
12.06234
6-47795
12.68244
6.80928
I3-33423
7-I5765
14.01940
0-75
4.94218
11.89566
5-I9485
12.50736
5-46075
13-15033
5-73977
13.82620
6.03341
14.53662
0.8
3.99082
I2.2456O
4.19483
12.87530
4.40933
13-53717
4.63487
14.23291
4.87198
14.96423
0.85
3.01484
I2.52OO2
3.16897
13.16388
3-33IOI
13.84054
3-50I39
14.55188
3-68053
15.29960
0.9
2.02028
12.71726
2.21356
13.37120
2.23215
14.05851
2.34632
14.78111
2.46636
15.54061
0-95
1.01326
12.83610
1.06507
13.49615
I-II953
14.18950
1.17679
14.91923
1.23699
15.68581
I.O
0.00
12.87578
o.oo
13.53788
o.oo
14.23382
0.00
14.96536
0.00
15.73432
1.05
1.01326
12.83610
1.06507
13.49615
1.11953
14.18950
1.17679
14.91923
1.23699
15.68581
i.i
2.02028
12.71726
2.12356
13.37120
2.23215
14.05851
2.34632
14.78111
2.46636
15.54061
i-iS
3.01484
I2.52OO2
3-16897
13.16388
3.33101
13.84054
3.50139
14.55188
3.68053
15.29960
1.2
3.99082
12.24560
4.19483
12.87530
4-40933
13-53717
4.63487
14.23291
4.87198
14.96423
1-25
4.94218
11.89566
5.19485
12.50736
5-46075
13-15033
5-73977
13.82620
6.03341
14.53662
1-3
5.86309
II.4724O
6.16381
12.06234
6.47795
12.68244
6.80928
I3-33423
7.15765
14.01940
•35
6.74784
10.9/841
7.09279
11.54294
7-45549
12.13633
7.83682
12.76006
8.23775
13.41572
•4
7-59099
10.41673
7.97905
10.95235
8.38705
11-51541
8.81604
12.10723
9.26706
12.72933
•45
8.38733
9.79082
8. 81610
10.29428
9.26691
10.82349
9.74090
"•37975
10.23924
11.96447
•5
9-13197
9.10456
9.59881
9.57273
10.08964
10.06483
10.60570
10.58212
11.14830
11.12585
•55
9.82031
8.36217
10.32233
8.79215
10.85016
9.24413
11.40512
9.71923
11.98861
10.21863
.6
10.44810
7.56820
10.98222
7-95737
11-54379
8.36643
12.13423
8.79642
12.75501
9.24840
1.65
11.01147
6.72758
11.57440
7-07352
12.16623
7.43715
12.78852
7-81938
13.44278
8.22116
1-7
11.50695
5-84548
12.09520
6.14670
12.71369
6.46202
13-3639?
6.79414
14.04764
7.14324
1-75
11.93150
4.92735
12.54145
5.18072
13-18275
5-44705
13.85702
5.72700
14.56595
6.02127
1.8
12.28247
3.97883
12.91035
4-18343
I3-57054
4-39850
14.26465
4-62455
14.99442
4.86217
1.85
12.55773
3-00579
13.19970
3.16036
13-87465
3.32282
14.58432
3-49359
15.3304S
3.67311
1.9
12.75555
2.01422
13.40764
2.11780
14.09323
2.22666
14.81410
2.34110
15-57196
2.46139
1.95
12.87474
I.OIO22
I3-53290
1.06217
14.22498
1.11677
14.95250
1.17417
15.71746
1.23450
2.0 12.91456 o.oo
13.57476 o.oo
14.26891 o.oo
14.99874 o.oo
15.76607 o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (3.45 + i 0.05) = 15.71746 + i 1.23450.
cosh (3.25 + i 1.95) = —12.87474 + * 1.01022.
TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED
= 3-50
= 3-55
3.60
x = 3-65
x = 3.70
o
16.57282
o.oo
17.42102
o.oo
18.31278
o.oo
19.25033
o.oo
20.23601
o.oo
0.05
16.52173
1.29792
17.36731
1.36458
18.25632
1.43466
19.19100
1.50832
20.17362
1.58576
O.I
16.36878
2.58783
17.20653
2.72075
18.08732
2.86047
19.01331
3-00735
19.98688
3.16174
0.15
16.11491
3.86180
16.93970
4.06015
17.80680
4.26865
18.71843
4.48783
19.67690
4.71823
O.2
15.76170
5.11195
16.56840
5.37451
17.41650
5.65052
18.30814
5-94065
19.24560
6.24563
0.25
i5-3"3o
6.33059
16.09492
6.65574
16.91880
6.99754
17.78497
7.35683
18.69565
7-73453
0-3
14.76650
7.51020
15-52225
7.89594
16.31680
8.30143
17.1527.6
8.72766
18.03040
9-17573
o-3S
14.13065
8.64360
14.85386
9-08745
15.61421
9-55412
16.41360
10.04469
17.25404
10.56037
0.4
13.40770
9.72361
14.09390
10.22293
14-81535
10.74789
15-57383
11.29972
16.37127
11.87990
0-45
12.60210
10.74357
13.24705
11.29540
13-92515
11.87545
14.63806
12.48520
15.38760
13.12620
o.S
11.71820
11.69740
12.31852
12.29820
12.94910
12.92978
13.61203
13-59365
14.30902
14-29155
o-55
10.76320
12.57910
11.31405
13.22520
11.89320
13.90436
12.50208
14.61830
13.14223
15.36878
0.6
9.74126
13.38330
10.23982
14.07066
10.76400
14.79326
11.31505
15-55281
11.89444
16.35127
0.65
8.65928
14.10490
9.10246
14.82933
9.56840
15.59090
10.05827
16.39143
10.57330
17.23295
0.7
7-52391
14.73960
7.90898
15.49665
8.31383
16.29246
8.73946
17.12900
9.18696
18.00840
0-75
6.34215
15.28340
6.66674
16.06840
7.00800
16.89356
7.36678
17.76096
7-74399
18.67280
0.8
5.12129
15-73300
5-38339
16.54105
5.65896
17.39050
5.94868
18.28342
6.25327
19.22208
0.85
3.86885
16.08555
4.06686
16.91175
4-27503
17.78022
4.49390
18.69316
4.72401
19.65290
0.9
2.59256
16.33896
2.72525
17.17817
2.86475
18.06033
3.01142
18.98765
3.16561
19.96246
o-9S
1.30029
16.49163
1.36684
17-33870
1.43680
18.22900
1.51036
19.16506
1.58770
20.14900
I.O
o.oo
16.54263
0.00
17.39230
0.00
18.28546
o.oo
19.22434
0.00
20.21129
1.05
1.30029
16.49163
1.36684
17.33870
1.43680
18.22900
1.51036
19.16506
1.58770
20.14900
i.i
2.59256
16.33896
2.72525
17.17817
2.86475
18.06033
3.01142
18.98765
3.16561
19.96246
i-iS
3.86885
16.08555
4.06686
16.91175
4.27503
17.78022
4.49390
18.69316
4.72401
19.65290
1.2
5.12129
15-73300
5-38339
16.54105
5.65896
17.39050
5.94868
18.28342
6.25327
19.22208
I-2S
6.34215
15.28340
6.66674
16.06840
7.00800
16.89356
7.36678
17.76096
7-74399
18.67280
•3
7.52391
14.73960
7.90898
15.49665
8.31383
16.29246
8.73946
17.12900
9.18696
18.00840
•35
8.65928
14.10490
9.10246
14.82933
9.56840
15.59090
10.05827
16.39143
io.5733o
17.23295
•4
9.74126
13-38330
10.23982
14.07066
10.76400
14.79326
11.31505
15-55281
11.89444
16.35127
•45
10.76320
12.57910
11.31405
13.22520
11.89320
13.90436
12.50208
14.61830
13.14223
15.36878
•5
11.71820
11.69740
12.31852
12.29820
12.94910
12.92978
13.61203
13-59365
14.30902
14-29155
•55
12.60210
10.74357
13.24705
11.29540
13.92515
11-87545
14.63806
12.48520
15.38760
13.12620
.6
13.40770
9.72361
14.09390
10.22293
14.81535
10.74789
15.57383
11.29972
16.37127
11.87990
•65
14.13065
8.64360
14.85386
9.08745
15.61421
9-55412
16.41360
10.04469
17.25404
10.56037
•7
14.76650
7.51020
15.52225
7-89594
16.31680
8.30143
17.15216
8.72766
18.03040
9-!7573
i-7S
15.31130
6.33059
16.09492
6.65574
16.91880
6.99754
17.78497
7.35683
18.69565
7-73453
1.8
15.76170
S-n^S
16.56840
5-37451
17.41650
5-65052
18.30814
5-94065
19.24560
6.24563
1.85
16.11491
3.86180
16.93970
4.06015
17.80680
4.26865
18.71843
4.48783
19.67690
4.71823
1.9
16.36878
2.58783
17.20653
2.72075
18.08732
2.86047
19.01331
3-00735
19.98688
3.16174
i-95
16.52173
1.29792
17.36731
1.36458
18.25632
1.43466
19.19100
1.50832
20.17362
I-58576
2.O
16.57282
0.00
17.42102
0.00
18.31278
o.oo
19.25033
o.oo
20.23601
o.oo
Note. Negative quantities are in heavy type.
Examples, cosh (3.50 + * 0.70) =
7.52391 +
* 14.73960
.
cosh (3.60 + * 1.55) = -13-92515 + * ".87545-
[72]
TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED
* = 3-75
3-80
3-85
x = 3.90
3-95
o
21.27230
o.oo
22.36178
o.oo
23.50717
o.oo
24-71135
o.oo
25-97731
o.oo
0.05
21.20670
1.66716
22.29283
1-75273
23.43470
1.84268
24.63516
1.93724
25.89724
2.03665
O.I
21.01038
3-32404
22.08646
3-49465
23.21775
3.67400
24.40710
3.86254
25-65749
4.06074
0.15
20.68452
4.96043
21.74391
5-21503
22.85766
5-48267
24.02856
5.76402
25-25957
6.05979
O.2
20.23113
6.56624
21.26731
6.90325
22.35664
7.25754
23.50188
7.62997
24.70590
8.02149
0.25
19.65301
8.13156
20.65958
8.54892
21.71778
8.98766
22.83030
9.44887
23.99991
9-93373
0-3
i8.95373
9.64674
19.92448
10.14188
20.94503
10.66233
22.01797
11.20952
23-14597
11.78474
o-3S
18.13760
11.10246
19.06655
11.67230
20.04315
12.27134
21.06988
12.90106
22.14931
13-56305
0.4
17.20963
12.48971
18.09105
13.13076
19.01770
13.80465
19.99190
14-51307
21.01610
I5-25776
0.45
16.17556
13.79998
1 7.00402
14.50828
17.87500
15.25286
18.79065
16.03558
19.75331
16.85842
o-S
15.04177
15.02516
15.81216
15-79634
16.62208
16.60702
17-47355
17.45924
18.36873
18.35512
o-SS
13.81524
16.15770
14.52281
16.98701
15.26668
17.85880
16.04874
18.77526
16.87098
19.73867
0.6
12.50353
17.19062
13-14392
18.07296
13.81716
19.00048
14.52497
19-97556
15.26910
21.00052
0.65
11.11473
18.11756
11.68400
19.04746
12.28246
20.02501
12.91164
21.05272
13-57315
22.13290
0.7
9.65741
18.93280
10.15203
19.90455
10.67203
20.92608
11.21871
21.99993
11.79366
23.12881
o.7S
8.14055
19.63131
8.55748
20.63891
8.99580
21.69813
9.45662
22.81160
9.94109
23.98212
0.8
6.57350
20.20879
6.91017
21.24603
7.26411
22.33640
7.63623
23.48262
8.02744
24.68760
0.85
4.96592
20.66167
5.22025
21.72216
5.48764
22.83696
5.76875
24.00888
6.06429
25.24084
0.9
3.32772
20.98716
3-49815
22.06437
3-67733
23.19673
3-86571
24.38710
4.06375
25.63849
0-95
1.66900
21.18327
1.75448
22.27052
1-84435
23-41348
1.93883
24.61500
2.03816
25-87805
I.O
o.oo
21.24878
0.00
22.33941
o.oo
23.48589
0.00
24.69110
o.oo
25.95806
•05
1.66900
21.18327
1.75448
22.27052
1.84435
23.41348
1.93883
24.61500
2.03816
25.87805
.1
3.32772
20.98716
3.49815
22.06437
3.67733
23.19673
3.86571
24.38710
4.06375
25.63849
•IS
4.96592
20.66167
5-22025
21.72216
5.48764
22.83696
5.76875
24.00888
6.06429
25.24084
.2
6.57350
20.20879
6.91017
21.24603
7.26411
22.33640
7.63623
23.48262
8.02744
24.68760
•2S
8.14055
19-63131
8.55748
20.63891
8.99580
21.69813
9.45662
22.81160
9.94109
23.98212
•3
9.65741
18.93280
10.15203
19.90455
10.67203
20.92608
11.21871
21.99993
11.79366
23.12881
i-3S
11.11473
18.11756
11.68400
19.04746
12.28246
20.02501
12.91164
21.05272
13.57315
22.13290
1.4
12.50353
17.19062
13.14392
18.07296
13.81716
19.00048
14.52497
19-97556
15.26910
21.00052
MS
13.81524
16.15770
14.52281
16.98701
15.26668
17.85880
16.04874
18.77526
16.87098
19.73867
*•$
15.04177
15.02516
15.81216
15-79634
16.62208
16.60702
17.47355
17.45924
18.36873
18.35512
x-55
16.17556
13.79998
17.00402
14.50828
17.87500
15.25286
18.79065
16.03558
19.75331
16.85842
1.6
17.20963
12.48971
18.09105
13.13076
19.01770
13.80465
19.99190
14.51307
21.01610
15-25776
1.65
18.13760
11.10246
19.06655
11.67230
20.04315
12.27134
21.06988
12.90106
22.14931
13-56305
«.?
18.95373
9.64674
19.92448
10.14188
20.94503
10.66233
22.01797
11.20952
23.14597
11.78474
1-75
19.65301
8.13156
20.65958
8.54892
21.71778
8.98766
22.83030
9.44887
23.99991
9-93373
1.8
20.23113
6.56624
21.26731
6.90325
22.35664
7.25754
23.50188
7.62997
24.70590
8.02149
1.85
20.68452
4.96043
21.74391
5-21503
22.85766
5.48267
24.02856
5.76402
25.25957
6.05979
1.9
21.01038
3-32404
22.08646
3-49465
23.21775
3.67400
24.40710
3.86254
25.65749
4.06074
i-9S
21.20670
1.66716
22.29283
1-75273
23.43470
1.84268
24.635*6
1.93724
25.89724
2.03665
2.0
21.27230
o.oo
22.36178
o.oo
23.50717
o.oo
24.71135
o.oo
35.97731
0.00
Note. Negative quantities are in heavy type.
Examples, cosh (3.90 + i 0.25) =
22.83030 + i 9.44887.
cosh (3.75 + i 1.25) = -8.14055 + i 19.63131.
[73]
TABLE IX. HYPERBOLIC TANGENTS, tanh (* + iq) = u + iv
x = 0.05
x = 0.15
X = 0.2
0
o.oo
o.oo
0.04996
o.oo
0.09967
o.oo
0.14889
o.oo
0.19738
o.oo
0.05
o.oo
0.07870
0.05027
0.07850
0.10028
0.07792
0.14979
0.07695
0.19855
0.07562
O.I
o.oo
0.15838
0.05121
0.15798
0.10214
0.15677
0.15254
0.15479
„ 0.20213
0.15207
0.15
o.oo
0.24008
0.05283
0.23944
0.10535
0.23755
0.15727
0.23446
0.20828
0.23021
0.2
o.oo
0.32492
0.05522
0.32402
0.11008
0.32136
0.16422
0.31698
0.21732
0.31098
0.25
o.oo
0.41421
0.05850
0.41300
0.11657
0.40940
0.17377
0.40350
0.22970
0-39543
o-3
0.00
0.50953
0.06289
0.50792
0.12522
0.50317
0.18647
0.49538
0.24613
0.48477
0-35
o.oo
0.61280
0.06865
0.61070
0.13659
0.60446
0.20310
0.59427
0.26758
0.58044
0.4
o.oo
0.72654
0.07623
0.72378
0.15149
0.71557
0.22484
0.70222
0.29549
0.68417
0-45
o.oo
0.85408
0.08624
0.85040
0.17113
0.83951
0.25339
0.82186
0.33192
0.79813
o.S
o.oo
I.OOOO
0.09967
0.99503
0.19738
0.98033
0.29131
0.95663
0.37995
0.92501
0-55
o.oo
1.17085
0.11804
1-16395
0.23313
1.14365
0.34258
1. 11113
0.44423
1.06819
0.6
o.oo
1.37638
0.14392
1.36649
0.28315
1-33754
0.41357
1.29164
0.53203
1-23185
0.65
o.oo
1.63185
0.18179
1.61702
0.35567
1.57401
0.51496
1.50674
0.65502
1.42088
0.7
o.oo
1.96261
0.24007
1.93900
0.46575
1.87150
0.66202
1.75880
0.83268
1.64005
0-75
o.oo
2.41421
0.33624
2.37365
0.64333
2.25941
0.90034
2.09061
1.09837
1.89083
0.8
o.oo
3.07768
0.51109
2.99911
0.95397
2.78504
1.28858
2.48723
1.50982
2.16055
0.85
o.oo
4.16530
0.87867
3.98246
1.56000
3-51765
1.97316
2.94167
2.16111
2.38860
0.9
o.oo
6.31375
1.85674
5.72808
2.91746
4.47780
3.22989
3-27758
3-15925
2.37676
0-95
o.oo
12.70620
5.79801
9.05499
6.21808
4-83133
5.28217
2.71349
4.39854
1.67517
I.O
o.oo
00
20.01667
o.oo
10.03331
o.oo
6.71659
o.oo
5.06649
o.oo
1.05
o.oo
12.70620
5.79801
9.05499
6.21808
4.83133
5.28217
2.71349
4.39854
1.67517
i.i
o.oo
6.31375
1.85674
5.72808
2.91746
4.47780
3.22989
3.27758
3-15925
2.37676
i.iS
o.oo
4.16530
0.87867
3.98246
1.56000
3.51765
1.97316
2.94167
2.16111
2.38860
1.2
o.oo
3.07768
0.51109
2.99911
0.95397
2.78504
1.28858
2.48723
1.50982
2.16055
I-2S
o.oo
2.41421
0.33624
2.37365
0.64333
2.25941
0.90034
2.09061
1.09837
1.89083
1-3
o.oo
1.96261
0.24007
1.93900
0.46575
1.87150
0.66202
1.75880
0.83268
1.64005
i-3S
o.oo
1.63185
0.18179
1.61702
0.35567
1.57401
0.51496
1.50674
0.65502
1.42088
1.4
o.oo
1.37638
0.14392
1.36649
0.28315
1-33754
0.41357
1.29164
0.53203
1.23185
1-45
0.00
1.17085
0.11804
1.16395
0.23313
1.14365
0.34258
I. IXII3
0.44423
1.06819
i-S
o.oo
I.OOOO
0.09967
0.99503
0.19738
0.98033
0.29131
0.95663
0-37995
0.92501
i-S5
o.oo
0.85408
0.08624
0.85040
0.17113
0.83951
0.25339
0.82186
0.33192
0.79813
1.6
o.oo
0.72654
0.07623
0.72378
0.15149
0.71557
0.22484
0.70222
0.29549
0.68417
1.65
o.oo
0.61280
0.06865
0.61070
0.13659
0.60446
0.20310
0.59427
0.26758
0.58044
i-7
o.oo
0.50953
0.06289
0.50792
0.12522
0.50317
0.18647
0.49538
0.24613
0.48477
i-75
o.oo
0.41421
0.05850
0.41300
0.11657
0.40940
0.17377
0.40350
0.22970
0.39543
1.8
o.oo
0.32492
0.05522
0.32402
0.11008
0.32136
0.16422
0.31698
0.21732
0.31098
1.85
o.oo
0.24008
0.05283
0.23944
0.10535
0.23755
0.15727
0.23446
0.20828
0.23021
1.9
o.oo
0.15838
0.05121
0.15798
0.10214
0.15677
0.15254
0.15479
0.20213
0.15207
i-9S
o.oo
0.07870
0.05027
0.07850
0.10028
0.07792
0.14979
0.07695
0.19855
0.07562
o.oo o.oo
0.04996 o.oo
0.09967 o.oo
0.14889 o.oo
0.19738 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (o + i 0.95) = o 4- i 12.70620.
tanh (o 4- i 1.45) = o — i 1.17085.
[74]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 0.25
= 0.3
o-35
x = 0.4
x = 0.45
o
0.05
O.I
0.15
O.2
0.24492 o.oo
0.24635 0.07395
0.25069 0.14866
0.25814 0.22490
0.26907 0.30351
0.29131 o.oo
0.29296 0.07199
0.29799 0.14464
0.30660 0.21864
0.31921 0.29471
0.33638 o.oo
0.33322 0.06975
0.34384 0.14007
0.35346 0.21153
0.36750 0.28475
0-37995 o.oo
0.38196 0.06728
0.38808 0.13503
0-39853 0.20373
0.41376 0.27384
0.42190 o.oo
0.42405 0.06462
0.43056 0.12961
0.44169 0.19534
0.45784 0.26216
0.25
0-3
0-35
0.4
0-45
0.28402 0.38540
0.30377 0.47162
0.32947 0.56335
0.36272 0.66200
0.40582 0.76919
0.33640 0.37362
0.35903 0.45623
0.38833 0.54348
0.42600 0.63638
0.47444 0.73604
0.38658 0.36035
0.41161 0.43898
0.44383 0.52131
0.48497 0.60802
0-53739 0.69969
0.43438 0.34585
0.46130 0.42022
0-49575 0.49737
0.53941 0.57764
0.59450 0.66116
0.47964 0.33039
0.50796 0.40033
0-54397 0.47216
0.58924 0.54593
0.64580 0.62138
o-S
o-SS
0.6
0.65
0.7
0.46212 0.88682
0.53655 1.01699
0.63656 1. 16180
0.77356 1.32269
0.96528 1.49862
0.53705 0.84355
0.61869 0.95983
0.72640 1.08513
0.87037 1. 21810
1.06521 1.35360
0.60437 0.79671
0.69041 0.89893
0.80176 1.00519
0.94684 1. 1 1 212
1.13666 1. 21222
0.66404 0.74770
0.75200 0.83632
0.86357 0.92478
1.00528 1.00856
1.18469 1.07919
0.71630 0.69779
0.80407 0.77365
0.91321 0.84608
1.04844 0.91003
1.21438 0.95708
o.7S
0.8
0.85
0.9
o-9S
1.23914 1.68151
1-63553 1-84484
2.20223 1.91863
2.95122 1.75011
3.72316 1.11789
1.33091 1.47820
1.69121 1.56140
2.16210 1.54177
2.71602 1.31829
3.21905 0.79096
1.38412 I.29O2O
1.70028 I.3I745
2.08309 1.24667
2.49443 I.OI6I3
2.83603 0.58484
1.40896 1. 12181
1.68068 1.11235
1.98936 1.01695
2.29855 0.79978
2.53928 0.44728
1.41397 0.97400
1.64487 0.94186
1.89366 0.83750
2.12957 0.64107
2.30471 0.35122
I.O
1.05
i.i
i-iS
1.2
4.08299 o.oo
3.72316 1.11789
2.95122 1.75011
2.20223 1.91863
1-63553 1.84484
3.43274 o.oo
3.21905 0.79096
2.71602 1.31829
2.16210 1.54177
1.69121 1.56140
2.97287 o.oo
2.83603 0.58484
2.49443 1.01613
2.08309 1.24667
1.70028 1.31745
2.63193 o.oo
2.53928 0.44728
2.29855 0.79978
1.98936 .01695
1.68068 .11235
2.37024 o.oo
2.30471 0.35122
2.12957 0.64107
1.89366 0.83750
1.64487 0.94186
1-25
i-3
1-35
1.4
1-45
1.23914 1.68151
0.96528 1.49862
0.77356 1.32269
0.63656 1. 16180
0.53655 1-01699
1.33091 1.47820
1.06521 1.35360
0.87037 1. 21810
0.72640 1.08513
0.61869 0.95983
1.38412 1.29020
1.13666 1. 21222
0.94684 I.II2I2
0.80176 I.OO5I9
0.69041 0.89893
1.40896 .12181
1.18469 .07919
1.00528 .00856
0.86357 0.92478
0.75200 0.83632
1.41397 0.97400
1.21438 0.95708
1.04844 0.91003
0.91321 0.84608
0.80407 0.77365
i-5
i-55
1.6
1.65
i-7
0.46212 0.88682
0.40582 0.76919
0.36272 0.66200
0.32947 0.56333
0.30377 0.47162
0.53705 0.84355
0.47444 0.73604
0.42600 0.63638
0.38833 0.54348
0-35903 0.45623
O.6O437 0.79671
0-53739 0.69969
0.48497 0.60802
0.44383 0.52131
0.41161 0.43898
0.66404 0.74770
0.59450 0.66 1 16
0.53941 0.57764
0.49575 0.49737
0.46130 0.42022
0.71630 0.69779
0.64580 0.62138
0.58924 0.54593
0.54397 0.47216
0.50796 0.40033
i-7S
1.8
1.85
1.9
J-95
0.28402 0.38540
0.26907 0.30351
0.25814 0.22490
0.25069 0.14866
0.24635 0.07395
0.33640 0.37362
0.31921 0.29471
0.30660 0.21864
0.29799 0.14464
0.29296 0.07199
0.38658 0.36035
0.36750 0.28475
0.35346 0.21153
0.34384 0.14007
0.33822 0.06975
0.43438 0.34585
0.41376 0.27384
0-39853 0.20373
0.38808 0.13503
0.38196 0.06728
0.47964 0.33039
0.45784 0.26216
0.44169 0.19534
0.43056 0.12961
0.42405 0.06462
2.O
0.24492 o.oo
0.29131 o.oo
0-33638 o.oo
0-37995 0.00
0.42190 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (0.4 + i 0.4) = 0.53941 + i 0.57764.
tanh (0.45 + i 1.75) = 0.47964 - i 0.33039.
[75]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = « + w. CONTINUED
o-5
= 0.55
x = 0.6
x = 0.65
= 0.7
o
0.05
O.I
0.15
0.2
0.46211 o.oo
0.46436 0.06181
0.47119 0.12390
0.48281 0.18651
0.49964 0.24990
0.50052 o.oo
0.50284 0.05889
0.50987 0.11796
0.52183 0.17737
0.53910 0.23725
0.53704 o.oo
0.53941 0.05590
0.54657 0.11189
0.55872 0.16804
0.57620 0.22437
0.57167 o.oo
0.57405 0.05288
0.58125 0.10576
0.59344 0.15863
0.61094 0.21144
0.60437 o-°°
0.60674 0.04984
0.61390 0.09962
0.62602 0.14925
0.64336 0.19858
0.25
0-3
0-35
0.4
0-45
0.52227 0.31424
0.55151 0.37967
0.58846 0.44616
0.63452 0.51350
0.69149 0.58116
0.56223 0.29765
0.59196 0.35856
0.62928 0.41979
0.67541 0.48093
0.73188 0.54121
0-59953 0.28085
0,62935 0.33731
0.66653 0.39344
0.71212 0.44869
0.76736 0.50211
0.63419 0.26404
0.66377 0.31618
0.70039 0.36748
0.74493 0.41714
0.79836 0.46428
0.66631 0.24741
0.69534 0.29540
0.73105 0.34205
0.77413 0.38662
0.82533 0.42806
o-S
0-55
0.6
0.65
0.7
0.76159 0.64805
0.84752 0.71229
0.95230 0.77067
1.07907 0.81812
1.23020 0.84688
0.80050 0.59933
0.88332 0.65320
0.98247 0.69957
1.09973 0.73362
1.23587 0.74858
0.83365 0.55229
0.91249 0.59707
1.00521 0.63346
1.11262 0.65676
1.23436 0.66157
0.86173 0.50738
0.93607 0.54434
1.02195 0.57227
1.11962 0.58738
1.22793 0.58492
0.88535 0.46492
0.95480 0.49522
1.03389 0.51635
1. 12222 0.52508
I.2I827 0.51757
0-75
0.8
0.85
0.9
o-9S
1.40579 0.84585
1.60095 0.80073
1.80225 0.69623
1.98505 0.52197
2.11599 0.28167
1.38926 0.73549
1.55398 0.68387
1.71785 0.58390
1.86183 0.43071
1.96180 0.22977
1.36782 0.64076
1.50695 0.58681
1.64135 0.49366
1.75601 0.35949
1.83417 0.19009
1.34386 0.55951
1.46173 0.50588
1.57271 0.42040
1.66532 0.30300
1.72736 0.15910
1.31896 0.48976
1.41909 0.43803
I.5II48 0.36034
1.58713 0.25755
I.637II 0.13449
I.O
1.05
i.i
i-iS
1.2
2.16395 °-o°
2.11599 0.28167
1.98505 0.52197
1.80225 0.69623
1.60095 0.80073
1.99792 o.oo
1.96180 0.22977
1.86163 0.43071
1.71785 0.58390
1-55398 0.68387
1.86202 o.oo
1.83417 0.19009
1.75601 0.35949
1.64135 0.49366
1.50695 0.58681
1.74926 o.oo
1.72736 0.15910
1.66532 0.30300
1.57271 0.42040
1.46173 0.50588
1.65462 o.oo
1.63711 0.13449
1.58713 0.25755
1.51148 0.36034
1.41909 0.43803
I.2S
i-3
i-3S
1.4
i. 45
1.40579 0.84585
1.23020 0.84688
1.07907 0.81812
0.95230 0.77067
0.84752 0.71229
1.38926 0.73549
1.23587 0.74858
1.09973 0.73362
0.98247 0.69957
0.88332 0.65320
1.36782 0.64076
1.23436 0.66157
1.11262 0.65676
1.00521 0.63346
0.91249 0.59707
1.34386 0.55951
1.22793 0.58492
1.11962 0.58738
1.02195 0.57227
0.93607 0.54434
1.31896 0.48976
1.21827 0.51757
1. 12222 0.52508
1.03389 0.51635
0.95480 O.49522
i-5
i-55
1.6
1.65
i-7
0.76159 0.64805
0.69149 0.58116
0.63452 0.51350
0.58846 0.44616
o-SS^i 0.37967
0.80050 0.59933
0.73188 0.54121
0.67541 0.48093
0.62928 0.41979
0.59196 0.35856
0.83365 0.55229
0.76736 0.50211
0.71212 0.44869
0.66653 0.39344
0.62935 0.33731
0.86173 0.50738
0.79836 0.46428
0.74493 0.41714
0.70039 0.36748
0.66377 0.31618
0.88535 0.46492
0.82533 0.42806
0.77413 0.38662
0.73105 0.34205
0.69534 0.29540
1-75
1.8
1-85
1.9
i-9S
0.52227 0.31424
0.49964 0.24990
0.48281 0.18651
0.47119 0.12390
0.46436 o. 06181
0.56223 0.29765
0.53910 0.23725
0.52183 0.17737
0.50987 0.11796
0.50284 0.05889
0-59953 0.28085
0.57620 0.22437
0.55872 0.16804
0.54657 0.11189
0.53941 0.05590
0.63419 0.26404
0.61094 0.21144
0.59344 0.15863
0.58125 0.10576
0.57405 0.05288
0.66631 O.2574I
0.64336 0.19858
O.626O2 O.I4925
0.61390 0.09962
0.60674 0.04984
2.O
0.46211 o.oo
0.50052 o.oo
0-53704 o.oo
0.57167 o.oo
0.60437 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (0.6 + i 0.6) = 1.00521 + i 0.63346.
[76]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 0.75
= 0.85
* = 0.9
x = 0.95
o
0.05
O.I
0.15
0.2
0.63515 o.oo
0.63749 0.04684
0.64456 0.09354
0.65649 0.13997
0.67352 0.18592
0.66403 o.oo
0.66633 0.04388
0.67325 0.08758
0.68490 0.13089
0.70148 0.17357
0.69107 o.oo
0.69330 0.04099
0.70002 0.08176
0.71132 0.12206
0.72736 0.16160
0.71629 o.oo
0.71845 0.03820
0.72494 0.07614
0.73582 0.11354
0.75123 0.15008
0.73978 o.oo
0.74185 0.03551
0.74807 0.07073
0.75850 0.10537
0.77321 0.13906
0.25
0-3
o-3S
0.4
o-45
0.69595 0.23112
0.72420 0.27516
0.75872 0.31749
0.80005 0.35735
0.84871 0.39368
0.72325 0.21528
0.75051 0.25559
0.78364 0.29392
0.82300 0.32949
0.86893 0.36128
0.74832 O.2OOOI
0.77446 0.23683
0.80603 0.27146
0.84328 0.30314
0.88638 0.33091
0.77130 0.18537
0.79620 0.21893
0.82611 0.25018
0.86117 0.27837
0.90143 0.30261
0.79231 0.17143
0.81592 0.20198
0.84411 0.23013
0.87695 0.25520
0.91438 0.27634
0-5
0-55
0.6
0.65
0.7
0.90515 0.42510
0.96963 0.44978
1.04203 0.46543
1. 12161 0.46934
1.20665 0.45847
0.92167 0.38798
0.98122 0.40796
1.04722 0.41926
1.11872 0.42057
1.19395 0.40661
0.93541 0.35357
0.99018 0.36966
1.05015 0.37751
1.11427 0.37527
1. 18081 0.36108
0.94681 0.32181
0.99700 0.33469
1.05136 0.33985
1.10885 0.33580
1.16765 0.32108
0.95624 0.29259
I.O02II 0.30285
1.05129 0.30593
1.10271 0.30064
1.15485 0.28588
0-75
0.8
0.85
0.9
0-95
1.29416 0.42977
1.37961 0.38084
1.45701 0.31065
1.51945 0.22051
1.56023 0.11463
1.27012 0.37806
1.34331 0.33238
1.40862 0.26920
1.46062 0.19000
1.49428 0.09840
1.24723 0.33335
1.31017 0.29108
1.36562 0.23434
1.40931 0.16461
1-43735 0.08499
1.22572 0.29458
1.28006 0.25573
1.32742 0.20483
1.36438 0.14330
1.38796 0.07380
1.20569 0.26087
1.25279 0.22532
1.29344 0.17968
1.32493 0.12528
1.34490 0.06438
I.O
1.05
i.i
i-iS
1.2
1.57443 o.oo
1.56023 0.11463
1.51945 0.22051
1.45701 0.31065
1.37961 0.38084
1.50594 o.oo
1.49428 0.09840
1.46062 0.19000
1.40862 0.26920
1.34331 0.33238
1.44703 o.oo
1-43735 0.08499
1.40931 0.16461
1.36562 0.23434
1.31017 0.29108
1.39606 o.oo
1.38796 0.07380
1.36438 0.14330
1.32742 0.20483
1.28006 0.25573
1.35175 o.oo
1.34490 0.06438
1.32493 0.12528
1.29344 0.17968
1.25279 0.22532
1-25
i-3
i-3S
1.4
I-4S
1.29416 0.42977
1.20665 0.45847
1. 12161 0.46934
1.04203 0.46543
0.96963 0.44978
1.27012 0.37806
1.19395 0.40661
1.11872 0.42057
1.04722 0.41926
0.98122 0.40796
1.24723 0.33335
1. 18081 0.36108
1.11427 0.37527
1.05015 0.37751
0.99018 0.36966
1.22572 0.29458
1.16765 0.32108
1.10885 0.33580
1.05136 0.33985
0.99700 0.33469
1.20569 0.26087
1.15485 0.28588
1.10271 0.30064
1.05129 0.30593
I.OO2II 0.30285
i-5
i-5S
1.6
1.65
i-7
0.90515 0.42510
0.84871 0.39368
0.80005 0.35735
0.75872 0.31749
0.72420 0.27516
0.92167 0.38798
0.86893 0.36128
0.82300 0.32949
0.78364 0.29392
0.75051 0.25559
0.93541 0.35357
0.88638 0.33091
0.84328 0.30314
0.80603 0.27146
0.77446 0.23683
0.94681 0.32181
0.90143 0.30261
0.86117 0.27837
0.82611 0.25018
0.79620 0.21893
0.95624 0.29259
0.91438 0.27634
0.87695 0.25520
0.84411 0.23013
0.81592 0.20198
i-75
1.8
1.85
1.9
I-9S
0.69595 0.23112
0.67352 0.18592
0.65649 0.13997
0.64456 0.09354
0.63749 0.04684
0.72325 0.21528
0.70148 0.17357
0.68490 0.13089
0.67325 0.08758
0.66633 0.04388
0.74832 O.200OI
0.72736 0.16160
0.71132 0.12206
0.70002 0.08176
0.69330 0.04099
0.77130 0.18537
0.75123 0.15008
0.73582 0.11354
0.72494 0.07614
0.71845 0.03820
0.79231 0.17143
0.77321 0.13906
0.75850 0.10537
0.74807 0.07073
0.74185 0.03551
2.0
0.63515 o.oo
0.66403 o.oo
0.69107 o.oo
0.71629 o.oo
0.73978 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (0.95 + * o) = 0.73978 + *o.
tanh (0.9 + i 1.9) = 0.72494 — 10.07614.
[77]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 1.05
x = 1.15
0
0.05
O.I
0.15
O.2
0.76159 o.oo
0-76357 0.03293
0.76950 0.06556
0.77943 0.09757
0.79341 0.12858
0.78181 o.oo
0.78368 0.03048
0.78932 0.06065
0.79873 0.09016
0.81195 0.11867
0.80050 o.oo
0.80227 0.02816
0.80760 0.05599
0.81648 0.08317
0.82893 0.10932
0.81775 o.oo
0.81943 0.02597
0.82444 0.05160
0.83279 0.07658
0.84447 0.10054
0.83365 o.oo
0.83522 0.02390
0.83992 0.04748
0.84775 0.07041
0.85867 0.09233
0.25
o-3
Q-3S
0.4
0-45
0.81151 0.15821
®-833 7 7 0.18598
0.86022 0.21133
0.89086 0.23361
0.92554 0.25205
0.82901 0.14575
0.84991 0.17096
0.87464 0.19377
0.90311 0.21356
o-935J5 0.22966
0.84495 0.13405
0.86450 0.15692
0.88753 0.17742
0.91392 0.19501
0.94344 0.20906
0.85945 0.12310
0.87768 0.14383
0.89907 0.16226
0.92345 0.17789
0.95058 0.19017
0.87264 0.11288
0.88958 0.13166
0.90938 0.14823
0.93186 0.16213
0.95674 0.17287
0-5
o.SS
0.6
0.65
0.7
0.96403 0.26580
1.00585 0.27392
1.05030 0.27542
1.09632 0.26933
1.14516 0.25486
0.97045 0.24130
1.00852 0.24767
1.04864 0.24798
1.08984 0.24144
1.13084 0.22747
0.97574 0.21892
1.01034 0.22389
1.04654 0.22331
1.08342 0.21658
1.11984 0.20326
0.98010 0.19852
1.01151 0.20236
1.04415 0.20115
1.07718 0.19441
1.10957 0.18182
0.98368 0.17996
1.01217 0.18288
1.04160 0.18123
1.07119 0.17461
1.10003 0.16281
0.75
0.8
0.85
0.9
0-95
1.18715 0.23145
1.22812 0.19904
1.26319 0.15812
1.29017 0.10993
1.30721 0.05638
1.17009 0.20572
1.20585 0.17623
1.23624 0.13955
1.25949 0.09677
1.27410 0.04956
1.15445 0.18315
1.18575 0.15637
1.21219 0.12347
1.23231 0.08544
1.24492 0.04369
1.14015 0.16330
1.16763 0.13902
1.19346 0.10950
1.20821 0.07563
1.21914 0.03863
1.12709 0.14580
1.15129 0.12380
1.17152 0.09730
1. 1 8680 0.06709
1.19631 0.03424
I.O
i. 05
i.i
i-iS
1.2
1.31304 o.oo
1.30721 0.05638
1.29017 0.10093
1.26319 0.15812
1.22812 0.19904
1.27908 o.oo
1.27410 0.04956
1.25949 0.09677
1.23624 0.13955
1.20585 0.17623
1.24922 o.oo
1.24492 0.04369
1.23231 0.08544
1.21219 0.12347
1.18575 0.15637
1.22286 o.oo
1.21914 0.03863
1.20821 0.07563
1.19346 0.10950
1.16763 0.13902
1.19954 o.oo
1.19631 0.03424
1. 18680 0.06709
1.17152 0.09730
1.15129 0.12380
1-25
J-3
1-35
1.4
i-45
1.18715 0.23145
1.14516 0.25486
1.09632 0.26933
1.05030 0.27542
1.00585 0.27392
1.17009 0.20572
1.13084 0.22747
1.08984 0.24144
1.04864 0.24798
1.00852 0.24767
1.15445 0.18315
1.11984 0.20326
1.08342 0.21658
1.04654 0.22331
1.01034 0.22389
1.14015 0.16330
1.10957 0.18182
1.07718 0.19441
1.04415 0.20115
1.01151 0.20236
1.12709 0.14580
1.10003 0.16281
1.07119 0.17461
1.04160 0.18123
1.01217 0.18288
i-5
i-SS
1.6
1.65
i-7
0.96403 0.26580
0.92554 0.25205
0.89086 0.23361
0.86022 0.21133
0.83377 0.18598
0.97045 0.24130
0.93515 0.22966
0.90311 0.21356
0.87464 0.19377
0.84991 0.17096
0.97574 0.21892
0.94344 0.20906
0.91392 0.19501
0.88753 0.17742
0.86450 0.15692
0.98010 0.19852
0.95058 0.19017
0.92345 0.17789
0.89907 0.16226
0.87768 0.14383
0.98368 0.17996
0.95674 0.17287
0.93186 0.16213
0.90938 0.14823
0.88958 0.13166
1-75
1.8
1.85
1.9
i-95
0.81151 0.15821
0.79341 0.12858
0.77943 0.09757
0.76950 0.06556
0.76357 0.03293
0.82901 0.14575
0.81195 0.11867
0-79873 0.09016
0.78932 0.06065
0.78368 0.03048
0.84495 0.13405
0.82893 0.10932
0.81648 0.08317
0.80760 0.05599
0.80227 0.02816
0.85945 0.12310
0.84447 0.10054
0.83279 0.07658
0.82444 0.05160
0.81943 0.02597
0.87264 0.11288
0.85867 0.09233
0.84775 0.07041
0.83992 0.04748
0.83522 0.02390
2JO
0.76159 o.oo
0.78181 o.oo
0.80050 o.oo
0.81775 o.oo
0.83365 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (1.2 + to. 75) = 1.12709 + 1*0.14580.
[78]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u+ iv. CONTINUED
* = 1.25
* = 1.3
* = i-35
x = 1.4
* = 1.45
0
0.84828 o.oo
0.86172 o.oo
0.87405 o.oo
0.88535 o.oo
0.89569 o.oo
0.05
0.84975 0.02197
0.86309 0.02017
0.87533 0.01849
0.88653 0.01693
0.89678 0.01549
O.I
0.85414 0.04363
0.86719 0.04003
0.87913 0.03668
0.89006 0.03357
0.90005 0.03070
0.15
0.86145 0.06464
0.87398 0.05927
0.88544 0.05428
0.89591 0.04965
0.90545 0.04537
0.2
0.87162 0.08468
0.88344 0.07756
0.89421 0.07097
0.90401 0.06486
0.91293 0.05923
0.25
0.88461 0.10339
0.89548 0.09458
0.90535 0.08644
0.91429 0.07892
0.92240 0.07199
o-3
0.90032 0.12039
0.90791 0.10997
0.91875 0.10036
0.92663 0.09151
0-93375 0.08338
0-35
0.91861 0.13528
0.92686 0.12336
0.93425 0.11240
0.94087 0.10234
0.94680 0.09312
0.4
0.93928 0.14765
0.94585 0.13437
0.95166 O.I222I
0.95680 0.11108
0.96137 0.10092
0-45
0.96207 0.15706
0.96669 0.14262
0.97069 O.I2945
0.97417 0.11745
0.97719 0.10654
o-S
0.98661 0.16307
0.98903 0.14773
O.99IOI 0.13381
0.99263 0.12117
0.99396 0.10971
0-55
1.01244 0.16528
1.01243 0.14937
I.OI2I9 0.13499
1.01181 0.12199
1.01133 0.11026
0.6
1.03897 0.16332
1.03634 0.14722
1-03375 0.13275
1.03125 0.11972
1.02885 0.10801
0.65
1.06550 0.15691
1.06013 0.14109
I.055IO 0.12693
1.05042 0.11425
1.04607 0.10288
0.7
1.09121 0.14591
1.08308 0.13088
1.07560 O.II749
1.06875 0.10555
1.06248 0.09488
o-7S
1.11521 0.13034
1.10439 0.11665
1.09457 0.10450
1.08565 0.09371
1.07756 0.08411
0.8
1.13656 0.11042
1.12328 0.09862
I.III3I 0.08820
1.10052 0.07896
1.09078 0.07077
0.85
1.15434 0.08662
1.13895 0.07724
I.I25I2 0.06897
1.11277 0.06167
1. 10166 0.05521
0.9
1.16772 0.05964
1.15070 0.05311
I.I355I 0.04738
1.12192 0.04232
1.10976 0.03785
0-95
1.17603 0.03041
1.15799 0.02706
I.I4I92 O.O24I2
1.12758 0.02153
1.11476 0.01925
I.O
1.17885 o.oo
1.16047 o.oo
1.14410 o.oo
1.12950 o.oo
1.11646 o.oo
1.05
1.17603 0.03041
1.15799 0.02706
1.14192 0.02412
1.12758 0.02153
1.11476 0.01925
i.i
1.16772 0.05964
1.15070 0.05311
1.13551 0.04738
1.12192 0.04232
1.10976 0.03783
i-iS
1.15434 0.08662
1.13895 0.07724
1.12512 0.06897
1.11277 0.06167
1. 10166 0.05521
1.2
1.13656 0.11042
1.12328 0.09862
1.11131 0.08820
1.10052 0.07896
1.09078 0.07077
1-25
1.11521 0.13034
1.10439 0.11665
1.09457 0.10450
1.08565 0.09371
1.07756 0.08411
i-3
1.09121 0.14591
1.08308 0.13088
1.07560 0.11749
1.06875 0.10555
1.06248 0.09488
i-35
1.06550 0.15691
1.06013 0.14109
1.05510 0.12693
1.05042 0.11425
1.04607 0.10288
1.4
1.03897 0.16332
1.03634 0.14722
1-03375 0.13275
1.03125 0.11972
1.02885 0.10801
1-45
1.01244 0.16528
1.01243 0.14937
1.01219 0.13499
1.01181 0.12199
1.01133 0.11026
i-5
0.98661 0.16307
0.98903 0.14773
0.99101 0.13381
0.99263 0.12117
0.99396 0.10971
i-55
0.96207 0.15706
0.96669 0.14262
0.97069 0.12945
0.97417 0.17745
0.97719 0.10654
1.6
0.93928 0.14765
0.94585 0.13437
0.95166 O.I222I
0.95680 0.11108
0.96137 0.10092
1.65
0.91861 0.13528
0.92686 0.12336
0.93425 O.II240
0.94087 0.10234
0.94680 0.09312
i-7
0.90032 0.12039
0.90791 0.10997
0.91875 O.IO036
0.92663 0.09151
0-93375 0.08338
i-7S
0.88461 0.10339
0.89548 0.09458
0.90535 0.08644
0.91429 0.07892
0.92240 0.07199
1.8
0.87162 0.08468
0.88344 0.07756
0.89421 O.O7O97
0.90401 0.06486
0.91293 0.05923
1.85
0.86145 0.06464
0.87398 0.05927
0.88544 O.O5428
0.89591 0.04965
0.90545 0.04537
1.9
0.85414 0.04363
0.86719 0.04003
0.87913 0.03668
0.89006 0.03357
0.90005 0.03070
I- 95
0.84975 0.02197
0.86309 0.02017
0.87533 0.01849
0.88653 0.01693
0.89678 0.01549
2.0 0.84828 0.00
0.86172 o.oo
0.87405 o.oo
0.88535
0.89569 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (1.4 + i 0.8) = 1.10052 + i 0.07896.
tanh (1.3 + * 1.3) = 1.08308 — * 0.13088.
[79]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
- 1.5
= i-55
x = 1.6
= 1.65
x = 1.7
0
0.05
O.I
0.15
O.2
0.90515 o.oo
0.90616 O.OI4I5
0.90917 O.O28O4
0.9I4I5 0.04143
0.92104 0.05404
0.91379 o.oo
0.91471 0.01292
0.91749 0.02560
0.92208 0.03779
0.92842 0.04927
0.92167 o.oo
0.92253 0.01178
0.92508 0.02334
0.92929 0.03445
0.93511 0.04488
0.92886 o.oo
0.92964 0.01074
0.93199 0.02127
0.93586 0.03138
0.94119 0.04086
0.93541 o.oo
0.93613 0.00979
0.93828 0.01937
0.94183 0.02857
0.94671 0.03718
O.25
o-3
o-35
0.4
o-45
0.92975 0.06563
0.94017 0.07593
O.952I2 0.08468
0.96542 0.09165
0.97983 0.09660
0.93641 0.05978
0.94595 0.06909
0.95689 0.07697
0.96903 0.08320
0.98214 0.08758
0.94245 0.05442
0.95118 0.06284
0.96117 0.06993
0.97223 0.07551
0.98415 0.07938
0.94791 0.04951
0.95589 0.05712
0.96501 0.06351
0.97509 0.06850
0.98592 0.07193
0.95285 0.04502
0.96015 0.05190
0.96846 0.05766
0.97763 0.06213
0.98748 0.06517
0-5
0-55
0.6
0.65
0.7
0.99506 0.09933
I.OIO76 0.09965
I.O2657 0.09746
I.O42O4 0.09268
1.05675 0.08534
0.99595 0.08092
1.01016 0.09008
1.02441 0.08796
1.03834 0.08353
1.05152 0.07680
0.99668 0.08139
1.00954 0.08142
1.02240 0.07940
1.03492 0.07530
1.04676 0.06915
0.99728 0.07367
1.00891 0.07361
1.02052 0.07169
1.03179 0.06791
1.04242 0.06230
0-99777 0.06667
1.00829 0.06655
1.01877 0.06474
1.02892 0.06126
1.03847 0.05614
0-75
0.8
0.85
0.9
0-95
1.07022 0.07554
I.O82OO 0.06349
1.09167 0.04947
1.09886 O.O339O
I.IO329 O.OI723
1.06357 0.06790
1.07408 0.05700
1.08269 0.04438
1.08909 0.03038
1.09302 0.01544
1-05755 0.06107
1.06693 0.05121
1.07461 0.03984
1.08030 0.02726
1.08380 0.01385
1.05209 0.05495
1.06049 0.04604
1.06734 0.03579
1.07241 0.02448
1.07554 0.01243
1.04714 0.04948
1.05466 0.04142
1.06079 0.03218
1.06533 O.02200
1.06811 0.01117
I.O
1.05
i.i
i.i5
1.2
I.I0479 °'0°
I.I0329 0.01723
1.09886 O.O339O
1.09167 O.O4947
I.O82OO O.OO349
1.09436 o.oo
1.09302 0.01544
1.08909 0.03038
1.08269 0.04438
1.07408 0.05700
1.08500 o.oo
1.08380 0.01385
1.08030 0.02726
1.07461 0.03984
1.06693 0.05121
1.07659 o.oo
1.07554 0.01243
1.07241 0.02448
1.06734 0.03579
1.06049 0.04604
1.06906 o.oo
1.06811 0.01117
1.06533 0.02200
1.06079 0-03218
1.05466 0.04142
1.25
i-3
i-35
1.4
1-45
I.O7O22 O.O75S4
1.05675 0.08534
I.O4204 0.09268
I.O2657 0.09746
I.OIO76 O.O9905
1.06357 0.06790
1.05152 0.07680
1.03834 0.08353
1.02441 0.08796
1.01016 0.09008
1.05755 0.06107
1.04676 0.06915
1.03492 0.07530
1.02240 0.07940
1.00954 0.08142
1.05209 0.05495
1.04242 0.06230
1.03179 0.06791
1.02052 0.07169
1.00891 0.07361
1.04714 0.04948
1.03847 0.05614
1.02892 0.06126
1.01877 0.06474
1.00829 0.06655
i-5
i-55
1.6
1.65
i-7
0.99506 0.09933
0.97983 0.09660
0.96542 O.O9I65
O.952I2 0.08468
0.94017 0.07593
0.99595 0.08992
0.98214 0.08758
0.96903 0.08320
0.95689 0.07697
0.94595 0.06909
0.99668 0.08139
0.98415 0.07938
0.97223 0.07551
0.96117 0.06993
0.95118 0.06284
0.99728 0.07367
0.98592 0.07193
0.97509 0.06850
0.96501 0.06351
0.95589 0.05712
0-99777 0.06667
0.98748 0.06517
0.97763 0.06213
0.96846 0.05766
0.96015 0.05190
i-75
1.8
1.85
1.9
i-95
0.92975 0.06563
O.92IO4 O.O5404
0.91415 0.04143
0.90917 O.O28O4
0.90616 O.OI4I5
0.93641 0.05978
0.92842 0.04927
0.92208 0.03779
0.91749 0.02560
0.91471 0.01292
0.94245 0.05442
0.93511 0.04488
0.92929 0.03445
0.92508 0.02334
0.92253 0.01178
0.94791 0.04951
0.94119 0.04086
0.93586 0.03138
0.93199 0.02127
0.92964 0.01074
0.95285 0.04502
0.94671 0.03718
0.94183 0.02857
0.93828 0.01937
0.93613 0.00979
2.0
0.90515 o.oo
0.91379 o.oo
0.92167 o.oo
0.92886 o.oo
0.93541 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (1.7 + *Q-7) = 1.03847 + i 0.05614.
tanh (1.7 + i Q-7)
tanh (1.6 + i 1.6)
0.97223 — » 0.07551.
[80]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 1.75
x = 1.85
x = 1.9
x = 1.95
o
0.05
O.I
0.15
0.2
0.94138 o.oo
0.94204 0.00891
0.94400 0.01763
0.94725 0.02600
0.95172 0.03382
0.94681 o.oo
0.94741 0.00811
0.94921 0.01604
0.95218 0.02364
0.95626 0.03074
0.95175 o.oo
0.95230 0.00737
0.95394 0.01459
0.95666 0.02149
0.96038 0.02793
0.95624 o.oo
0.95674 0.00670
0.95825 0.01326
0.96072 0.01952
0.96412 0.02537
0.96032 o.oo
0.96078 0.00609
0.96215 0.01204
0.96441 0.01773
0.96750 0.02303
0.2S
0-3
o-3S
0.4
0-45
Q-95733 0.04092
0.96399 0.04714
0.97156 0.05233
0.97991 0.05634
0.98884 0.05904
0.96139 O.037j8
0.96746 0.04280
0.97435 0.04748
0.98194 0.05107
0.99005 0.05348
0.96506 0.03376
0.97059 0.03885
0.97686 0.04307
0.98376 0.04629
0.99111 0.04843
0.96838 0.03065
0.97342 0.03525
0.97912 0.03905
0.98538 0.04195
0.99206 0.04386
0.97138 0.02782
0.97597 0.03198
0.98116 0.03541
0.98684 0.03801
0.99289 0.03972
o-5
o-5S
0.6
0.65
0.7
0.99818 0.06034
1.00769 0.06017
1.01714 0.05848
1.02629 0.05528
1.03488 0.05061
0.99851 0.05461
1.00711 0.05440
1.01565 0.05283
1.02389 0.04989
1.03162 0.04564
0.99878 0.04942
1.00656 0.04919
1.01427 0.04773
1.02170 0.04504
1.02866 0.04118
0.99900 0.04472
1.00604 0.04448
1.01300 0.04313
1.01970 0.04067
1.02596 0.03716
0.99918 0.04047
1.00555 0.04022
1.01184 0.03897
1.01788 0.03673
1.02352 0.03354
0-75
0.8
0.85
0.9
0-95
1.04266 0.04457
1.04940 0.03729
1.05489 0.02895
1.05895 0.01978
1.06143 0.01004
1.03861 0.04016
1.04466 0.03358
1.04958 0.02606
1.05320 0.01780
1.05543 0.00903
1.03494 0.03621
1.04037 0.03026
1.04478 0.02347
1.04804 0.01602
1.05003 0.00813
1.03162 0.03265
1.03650 0.02727
1.04046 0.02115
1.04337 0.01443
1.04516 0.00732
1.02862 0.02946
1.03300 0.02459
1.03656 0.01906
1.03918 0.01301
1.04078 0.00659
I.O
1.05
i.i
1.13
1.2
1.06228 o.oo
1.06143 0.01004
1.05895 0.01978
1.05489 0.02895
1.04940 0.03729
1.05619 o.oo
1.05543 0.00903
1.05320 0.01780
1.04958 0.02606
1.04466 0.03358
1.05070 o.oo
1.05003 0.00813
1.04804 0.01602
1.04478 0.02347
1.04037 0.03026
1.04576 o.oo
1.04516 0.00732
1.04337 0.01443
1.04046 0.02115
1.03650 0.02727
1.04131 o.oo
1.04078 0.00659
1.03918 0.01301
1.03656 0.01906
1.03300 0.02459
I.2S
i-3
i-35
1.4
1-45
1.04266 0.04457
1.03488 0.05061
1.02629 0.05528
1.01714 0.05848
1.00769 0.06017
1.03861 0.04016
1.03162 0.04564
1.02389 0.04989
1.01565 0.05283
1.00711 0.05440
1.03494 0.03621
1.02866 0.04118
1.02170 0.04504
1.01427 0.04773
1.00656 0.04919
1.03162 0.03265
1.02596 0.03716
1.01970 0.04067
1.01300 0.04313
1 .00604 0.04448
1.02862 0.02946
1.02352 0.03354
1.01788 0.03673
1.01184 o.o'3897
1.00555 0.04022
i-5
i-5S
1.6
1.65
i-7
0.99818 0.06034
0.98884 0.05904
0.97991 0.05634
0.97156 0.05233
0.96399 0.04714
0.99851 0.05461
0.99005 0.05348
0.98194 0.05107
0.97435 0.04748
0.96746 0.04280
0.99878 0.04942
0.99111 0.04843
0.98376 0.04629
0.97686 0.04307
0.97059 0.03885
0.99900 0.04472
0.99206 0.04386
0.98538 0.04195
0.97912 0.03905
0.97342 0.03525
0.99918 0.04047
0.99289 0.03972
0.98684 0.03801
0.98116 0.03541
0-97597 0.03198
i-7S
1.8
1.85
1.9
i-95
0-95733 0.04092
0.95172 0.03382
0.94725 0.02600
0.94400 0.01763
0.94204 0.00891
0.96139 0.03718
0.95626 0.03074
0.95218 0.02364
0.94921 0.01604
0.94741 0.00811
0.96506 0.03376
0.96038 0.02793
0.95666 0.02149
0.95394 0.01459
0.95230 0.00737
0.96838 0.03065
0.96412 0.02537
0.96072 0.01952
0.95825 0.01326
0.95674 0.00670
0.97138 0.02782
0.96750 0.02303
0.96441 0.01773
0.96215 0.01204
0.96078 0.00609
2.0 0.94138 o.oo 0.94681 o.oo 0.95175 o.oo
0.95624 o.oo
0.96032 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (1.85 + i 0.85) = 1.04478 + i 0.02347.
tanh (1.95 + * 1.25) = 1.02862 — i 0.02946.
[81]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 2.05
x = 2.15
0
0.05
O.I
0.15
0.2
0.96403 o.oo
0.96445 0.00553
0.96570 O.OIO94
0-96775 0.01610
0.97058 0.02090
0.96740 o.oo
0.96778 0.00502
0.96892 0.00993
0.97079 0.01461
0.97336 0.01897
0.97045 o.oo
0.97080 0.00456
0.97184 0.00901
0.97354 0.01326
0.97588 0.01721
0.97323 o.oo
0-97354 0.00413
0.97449 0.00817
0.97604 0.01203
0.97816 0.01561
0.97574 o.oo
0.97603 0.00375
0.97689 0.00741
0.97830 0.01091
0.98023 0.01415
0.2S
o-3
°-35
0.4
o-4S
0.97411 0.02524
0.97827 0.02900
0.98299 0.03209
0.98815 0.03444
0.99364 0.03596
0.97657 0.02289
0.98036 0.02630
0.98464 0.02909
0.98932 0.03120
0.99430 0.03256
0.97880 0.02076
0.98224 0.02384
0.98613 0.02636
0.99037 0.02826
0.99489 0.02948
0.98081 0.01882
0.98394 0.02161
0.98747 0.02388
0.99132 0.02559
0.99541 0.02669
0.98264 0.01706
0.98548 0.01958
0.98868 0.02163
0.99217 0.02317
0.99587 0.02416
o-S
o-SS
0.6
0.65
0.7
0.99933 0.03662
1.00509 0.03638
1.01077 0.03523
1.01623 0.03318
1.02131 0.03028
0.99945 0.03314
1.00466 0.03290
1.00979 0.03184
1.01471 0.02998
1.01930 0.02734
0.99955 0.02998
1.00426 0.02976
.00890 0.02878
.01334 0.02709
.01748 0.02469
0.99963 0.02713
1.00389 0.02691
.00808 0.02602
.OI2IO 0.02448
.01583 0.02231
0.99970 0.02455
1.00355 0.02434
1.00734 0.02353
I.OI097 O.O22I2
I.OI434 O.O2OI5
0-75
0.8
0.85
0.9
Q-95
1.02589 0.02658
1.02984 0.02218
1.03304 0.01719
1-03539 0.01172
1.03683 0.00594
1.02343 0.02399
1.02698 0.020OI
1.02986 O.OI55O
1.03197 0.01057
1.03327 0.00536
.02120 0.02166
.02440 0.01806
1.02699 0.01399
1.02889 0.00954
1.03005 0.00483
.01919 O.OI956
.022O7 O.OI63I
.0244O O.OI262
.02611 O.OO86I
.02715 0.00436
1.01736 0.01767
1.01996 O.OI472
1.02206 O.OII40
1.02360 O.O0777
1.02454 0.00394
I.O
1.05
i.i
i-iS
1.2
1.03731 o.oo
1.03683 0.00594
1.03539 0.01172
1.03304 0.01719
1.02984 0.02218
1.03370 o.oo
1.03327 0.00536
1.03197 0.01057
1.02986 0.01550
1.02698 O.O2OOI
1.03045 o.oo
1.03005 0.00483
1.02889 0.00954
1.02699 0.01399
1.02440 0.01806
1.02751 o.oo
1.02715 0.00436
1.02611 0.00861
1.02440 0.01262
1.02207 0.01631
1.02486 o.oo
1.02454 0.00394
1.02360 0.00777
1.02206 0.01140
1.01996 0.01472
1-25
!-3
i-3S
1.4
i. 45
1.02589 0.02658
1.02131 0.03028
1.01623 0.03318
1.01077 0.03523
1.00509 0.03638
1.02343 0.02399
I.OI930 0.02734
I.OI47I 0.02998
I.O0979 0.03184
I.OO466 0.0329O
I.O2I2O O.O2I66
1.01748 0.02469
I.OI334 0.02709
1.00890 0.02878
I.OO426 O.02976
1.01919 0.01956
1.01583 0.02231
I.OI2IO 0.02448
I.Oo8o8 0.02602
1.00389 O.O269I
1.01736 0.01767
1.01434 0.02015
I.OIO97 O.022I2
1.00734 0.02353
1.00355 0.02434
i-5
i-SS
1.6
1.65
i-7
0-99933 0.03662
0.99364 0.03596
0.98815 0.03444
0.98299 0.03209
0.97827 0.02900
0.99945 0.03314
0.99430 0.03256
0.98932 O.O3I2O
0.98464 O.02909
0.98036 O.O263O
0.99955 0.02998
0.99489 0.02948
0.99037 O.O2826
0.98613 0.02636
0.98224 0.02384
0.99963 0.02713
0.99541 0.02669
0.99132 0.02559
0.98747 O.02388
0.98394 0.02161
0.99970 0.02455
0.99587 O.O24I6
0.99217 O.O23I7
0.98868 O.02I63
0.98548 O.OI958
1-75
1.8
1.85
1.9
i -95
0.97411 0.02524
0.97058 0.02090
0.96775 0.01610
0.96570 0.01094
0.96445 0.00553
0-97657 0.02289
0.97336 0.01897
0.97079 O.OI46I
0.96892 O.OO993
0.96778 O.OO5O2
0.97880 O.O2O76
0.97588 O.OI72I
0.97354 0.01326
0.97184 jo. 0090 1
0.97080 0.00456
0.98081 0.01882
0.97816 0.01561
0.97604 0.01203
0.97449 0.00817
0.97354 0.00413
0.98264 O.OI7O6
0.98023 O.OI4I5
0.97830 O.OIO9I
0.97689 O.0074I
0.97603 0.00375
2.0
0.96403 o.oo
0.96740 o.oo
0.97045 o.oo
0.97323 o.oo
0.97574 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (2.2 + * o) = o 97574 + i o.
tanh (2.15 -f i 1.15) = 1.02440 — £0.01262.
[82]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 2.25
x = 2.3
= 2.35
2.4
* = 2.45
o
0.05
O.I
0.15
O.2
0.97803 o.oo
0.97829 0.00340
0.97907 0.00672
0.98035 0.00989
0.98210 0.01283
0.98010 o.oo
0.98034 0.00308
0.98105 0.00610
0.98221 0.00897
0.98380 0.01163
0.98197 o.oo 0.98367 o.oo
0.98219 0.00280 0.98387 0.00253
0.98283 0.00553 0.98446 0.00501
0.98389 0.00813 0.98541 0.00736
0.98534 0.01054 0.98673 0.00955
0.98522 o.oo
0.98540 0.00230
0.98592 0.00454
0.98680 0.00667
0.98799 0.00865
0.25
°-3
°-35
0.4
0-45
0.98429 0.01547
0.98687 0.01774
0.98977 0.01960
0.99294 0.02098
0.99629 0.02187
0.98579 0.01401
0.98813 0.01607
0.99076 0.01775
0.99363 0.01900
0.99667 0.01979
0.98714 0.01270 0.98836 0.01150
0.98926 0.01456 0.99029 0.01319
0.99165 0.01607 0.99245 0.01456
0.99425 0.01720 0.99481 0.01557
0.99700 0.01791 0.99730 0.01621
0.98947 0.01042
0.99121 0.01194
0.99317 0.01318
0.99531 0.01410
O-99757 0.01468
o-5
0-55
0.6
0.65
0.7
O.99975 O.O2222
1.00324 O.O22O2
I.OO666 O.O2I27
I.OO994 O.O2OOO
I.OI298 O.OI82I
0.99980 O.O2OIO
1.00295 0.01992
I.O06O4 O.OI924
1.00900 O.Ol8o8
I.OII75 0.01646
0.99984 0.01819 0.99986 0.01646
1.00269 0.01802 1.00245 0.01630
.00549 0.01740 1.00498 0.01573
.00816 0.01634 1.00739 0.01478
.01064 0.01487 1.00963 0.01345
0.99989 0.01489
1.00222 0.01474
1.00451 0.01423
1.00693 0.01336
1.00872 0.01216
0-75
0.8
0.85
0.9
0-95
I.OI57I 0.01596
I.OI8O5 O.OI33O
I.OI994 O.OIO29
I.02I33 O.OO7OI
I.022I8 0.00355
I.OI42I O.OI442
I.OI632 O.OI2OI
I.OI8O3 0.00929
I.OI928 0.00633
I.O2006 O.OO322
.01286 0.01303 1.01164 0.01178
.01477 0.01085 1.01336 0.00981
1.01631 0.00839 1-OI475 0.00758
1.01744 0.00572 1.01576 0.00517
1.01812 0.00290 1.01639 0.00262
1.01053 0.01064
1.01208 0.00886
1.01333 0.00685
1.01425 0.00467
1.01482 0.00237
.0
•°5
.1
•IS
.2
1.02247 o.oo
1.02218 0.00355
1.02133 0.00701
1.01994 0.01029
1.01805 0.01330
1.02031 o.oo
I.O20O6 O.OO322
I.OI928 O.OO633
I.OI8O3 O.OO929
I.OI632 O.OI2OI
1.01836 o.oo 1.01659 o.oo
1.01812 0.00290 1.01639 0.00262
1.01744 0.00572 1.01576 0.00517
1.01631 0.00839 1.01475 0.00758
1.01477 0.01085 1.01336 0.00981
1.01500 o.oo
1.01482 0.00237
1.01425 0.00467
1.01333 0.00685
1. 01208 0.00886
•25
•3
•35
•4
J-45
1.01571 0.01596
1.01298 0.01821
I.OO994 O.O2OOO
I.00666 0.02127
1.00324 O.O22O2
I.OI42I O.OI442
I.OII75 0.01646
I.OO9OO O.OlSoS
1.00604 O.OI924
I.OO295 O.OI992
1.01286 0.01303 1.01164 0.01178
1.01064 0.01487 1.00963 0.01345
1.00816 0.01634 1.00739 0.01478
1.00549 0.01740 1.00498 0.01573
1.00269 0.01802 1.00245 0.01630
1.01053 0.01064
1.00872 0.01216
1.00693 0.01336
1.00451 0.01423
1.00222 O.OI474
i-5
i-55
1.6
1.65
i-7
0-99975 0.02222
0.99629 O.02I87
0.99294 O.O2098
0.98977 O.OI96O
0.98687 O.OI774
0.99980 O.O20IO
0.99667 O.OI979
0.99363 O.OI9OO
0.99076 O.OI775
0.98813 O.OI6O7
0.99984 0.01819 0.99986 0.01646
0.99700 0.01791 0.99730 0.01621
0.99425 0.01720 0.99481 0.01557
0.99165 0.01607 0.99245 0.01456
0.98926 0.01456 0.99029 0.01319
0.99989 0.01489
0.99757 0.01468
0.99531 O.OI4IO
0.99317 O.OI3I8
O.99I2I O.OII94
1-75
1.8
1.85
1.9
1-95
0.98429 O.OI547
0.98210 O.OI283
0.98035 0.00989
0.97907 O.OO672
0.97829 O.OO340
0.98579 O.OI4OI
0.98380 O.OII63
0.98221 O.O0897
0.98105 O.OO6IO
0.98034 O.OO3O8
0.98714 0.01270 0.98836 0.01150
0.98534 0.01054 0.98673 0.00955
0.98389 0.00813 0.98541 0.00736
0.98283 0.00553 0.98446 0.00501
0.98219 0.00280 0.98387 0.00253
0.98947 O.OI042
0.98799 0.00865
0.98680 0.00667
0.98592 O.OO454
0.98540 O.OO23O
2.0
0.97803 o.oo
0.98010 o.oo
0.98197 o.oo 0.98367 o.oo
0.98522 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (2.25 + £0.25) = 0.98429 + i 0.01547.
tanh (2.45 + * 1.45) = 1.00222 — * 0.01474.
[83]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
= 2.5
* = 2.55
* = 2.6
x = 2.65
x = 2.7
0
0.05
O.I
0.15
O.2
o. 8661 o.oo
0.98678 O.O0208
0.98726 O.004II
0.98805 O.006O5
0.98913 0.00784
0.98788 o.oo
0.98803 0.00189
0.98846 0.00373
0.98918 0.00548
0.99016 0.00710
0.98903 o.oo
0.98916 O.OOI7I
0.98956 0.00337
O.9902I 0.00496
0.99109 0.00643
0.99007 o.oo
0.99019 0.00155
0.99055 0.00306
0.99113 0.00449
0.99194 0.00582
0.99101 o.oo
0.99112 0.00140
0.99144 0.00277
0.99198 0.00407
0.99270 0.00527
0.25
o-3
o-3S
0.4
Q-4S
0.99047 O.00944
0.99205 0.01082
0.99383 O.OII93
0.99576 O.OI276
0.99781 0.01328
0.99138 0.00855
0.99281 0.00979
0.99442 0.01080
0.99617 0.01155
0.99802 O.OI2O2
O.9922O O.OO774
0.99350 0.00887
0.99495 0.00978
0.99654 O.OI046
0.99822 O.OI088
0.99294 0.00701
0.99412 0.00803
0.99544 0.00886
0.99688 0.00947
0.99839 0.00985
0.99361 0.00635
0.99468 0.00727
0.99588 0.00802
0.99718 0.00857
0.99855 0.00891
o-5
o-SS
0.6
0.65
0.7
0.99991 0.01348
1.00202 O.OI334
I.004O9 O.OI287
I.Oo6o6 O.OI2O8
1.00789 O.OIO99
0.99993 O.OI2I9
I.OOI84 O.OI207
I.OO37I O.OII64
1.00549 0.01093
I.OO7I4 0.00994
0.99994 O.OIIO3
I.OOI67 O.OIO92
1.00336 0.01053
1.00498 0.00988
1.00647 0.00898
0.99995 0.00998
1.00151 0.00988
1.00304 0.00952
1.00450 0.00894
1.00585 0.00812
0.99996 0.00903
1.00137 0.00893
1.00276 0.00862
1.00408 0.00808
1.00530 0.00735
0-75
0.8
0.85
0.9
o-9S
1.00953 0.00962
I.OIO93 O.00§01
I. OI2O6 0.00619
I.OI289 0.00422
I.OI34O O.O02I4
I.O0862 O.OO87O
1.00989 O.OO726
I.OIOgi 0.00560
1.01166 0.00381
i.oi2ii 0.00193
I.O078o O.00786
1.00895 0.00654
I.OOgS? 0.00506
I.OI054 0.00345
1.01096 O.OOI75
1.00706 0.00711
1.00809 0.00592
1.00893 0-°°457
1.00954 0.00312
1.00991 0.00158
1.00639 0.00643
1.00732 0.00535
1.00807 0.00413
1.00862 0.00282
1.60896 0.00143
I.O
1.05
i.i
i-iS
1.2
1.01357 o.oo
I.OI34O O.OO2I4
1.01289 O.00422
1.01206 0.00619
1.01093 0.00801
1.01227 °-°°
1.01211 O.OOI93
I.OII66 O.0038I
I.OIOgi O.OO56O
1.00989 O.OO726
I. OHIO O.OO
1.01096 0.00175
1.01054 0.00345
1.00987 0.00506
1.00895 0.00654
1.01003 o.oo
1.00991 0.00158
1.00954 0.00312
1.00893 0.00457
1.00809 0.00592
1.00907 o.oo
1.00896 0.00143
1.00862 0.00282
1.00807 0.00413
1.00732 0.00535
1-25
i-3
i-3S
1.4
145
1.00953 0.00962
1.00789 0.01099
1.00606 0.01208
1.00409 0.01287
1.00202 O.OI334
I.OO862 O.0087O
I.OO7I4 O.OO994
I.OO549 O.OIO93
I.O037I O.OII64
I.OOl84 O.O12O7
1.00780 0.00786
1.00647 0.00898
1.00498 0.00988
1.00336 0.01053
1.00167 0.01092
1.00706 0.00711
1.00585 0.00812
1.00450 0.00894
1.00304 0.00952
1.00151 0.00988
1.00639 0.00643
1.00530 0.00735
1.00408 0.00808
1.00276 0.00862
1.00137 0.00893
i-5
i-55
1.6
1.65
i-7
0.99991 O.OI348
0.99781 O.OI328
0.99576 O.OI276
0-99383 O.OII93
0.99205 O.OI082
O-99993 O.OI2I9
0.99802 O.OI2O2
0.99617 O.OII55
0.99442 O.OI08O
0.99281 O.OO979
0.99994 0.01103
0.99822 0.01088
0.99654 0.01046
0.99495 0.00978
0.99350 0.00887
0.99995 0.00998
0.99839 0.00985
0.99688 0.00947
0.99544 0.00886
0.99412 0.00803
0.99996 0.00903
0.99855 0.00891
0.99718 0.00857
0.99588 0.00802
0.99468 0.00727
i-7S
1.8
1.85
1.9
i-9S
0.99047 0.00944
0.98913 0.00784
0.98805 O.00605
0.98726 O.OO4II
0.98678 O.00208
0.99138 0.00855
0.99016 O.OO7IO
0.98918 O.OO548
0.98846 O.00373
0.98803 O.OOI89
0.99220 0.00774
0.99109 0.00643
0.99021 0.00496
0.98956 0.00337
0.98916 0.00171
0.99294 0.00701
0.99194 0.00582
0.99113 0.00449
0.99055 0.00306
0.99019 0.00155
0.99361 0.00635
0.99270 0.00527
0.99198 0.00407
0.99144 0.00277
0.99112 0.00140
2.O
0.98661 o.oo
0.98788 o.oo
0.98903 o.oo
0.99007 o.oo
0.99101 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (2.60 + i 0.35) = 0.99495 + * 0.00978.
tanh (2.70 + * 1.35) = 1.00408 — i 0.00808.
[84]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
x = 2.75
x = 2.80
x = 2.85
= 2.90
x = 2.95
0
0.99186 o.oo
0.99263 o.oo
0-99333 o.oo
0.99396 o.oo
0.99454 o.oo
0.05
0.99196 0.00127
0.99272 0.00115
0.99341 0.00104
0.99404 0.00094
0.99460 0.00085
O.I
0.99225 0.00251
0.99299 0.00227
0.99365 0.00206
0.99426 0.00186
0.99480 0.00168
0.15
0.99274 0.00368
0.99343 0.00334
0.99405 0.00302
0.99462 0.00273
0.99513 0.00248
0.2
0.99340 0.00477
0.99403 0.00432
0.99459 0.00391
0.99511 0.00354
0-99557 0.00321
0.25
0.99422 0.00575
0.99477 0.00519
0.99527 0.00471
0.99572 0.00426
0.99613 0.00386
0-3
0.99519 0.00658
0.99564 0.00596
0.99606 0.00539
0.99644 0.00488
0.99678 0.00442
o-3S
0.99627 0.00726
0.99663 0.00657
0.99695 0.00594
0.99724 0.00538
0.99750 0.00487
0.4
0.99745 0.00775
0.99769 0.00702
0.99791 0.00635
0.99811 0.00575
0.99830 0.00520
0-45
0.99869 0.00806
0.99882 0.00730
0.99893 0.00660
0.99904 0.00598
0.99913 0.00541
o-5
0.99997 0.00817
0.99997 0.00740
0.99998 0.00669
0.99998 0.00606
0.99999 0.00548
o-55
1.00125 0.00808
1.00113 0.00731
1.00103 0.00662
1.00093 0.00599
1.00084 0.00542
0.6
1.00250 0.00779
1.00226 0.00705
1.00205 0.00638
1.00186 0.00577
1.00168 0.00522
0.65
1.00369 0.00731
1.00334 0.00661
1.00303 0.00598
1.00274 0.00541
1.00248 0.00489
0.7
1.00479 0.00664
1.00434 0.00601
1.00393 0.00544
I-0035S 0.00480
1.00322 0.00445
o.7S
1.00578 0.00581
1.00523 0.00525
1.00473 0.00475
1.00428 0.00430
1.00387 0.00389
0.8
1.00662 0.00484
1.00599 0-00437
1.00542 0.00396
1.00490 0.00358
1.00444 0.00324
0.85
.00730 0.00374
1. 0066 1 0.00338
1.00598 0.00306
1.00540 0.00276
1.00489 0.00250
0.9
.00780 0.00255
1.00706 0.00230
1.00638 0.00208
1.00577 0.00188
1.00522 0.00170
o-95
.00810 0.00129
1.00733 0.00117
1.00663 0.00105
1.00600 0.00095
1.00542 0.00086
I.O
.00821 o.oo
1.00742 o.oo
1.00671 o.oo
1.00607 o.oo
1.00549 o.oo
'1.05
.00810 0.00129
1.00733 0.00117
1.00663 0.00105
1.00600 0.00095
1.00542 0.00086
i.i
.00780 0.00255
1.00706 0.00230
1.00638 0.00208
1.00577 0.00188
1.00522 0.00170
i-iS
.00730 0.00374
1. 00661 0.00338
1.00598 0.00306
1.00540 0.00276
1.00489 0.00250
1.2
1.00662 0.00484
1.00599 0.00437
1.00542 0.00396
1.00490 0.00358
1.00444 0.00324
I-2S
1.00578 0.00581
1.00523 0.00525
1.00473 0.00475
1.00428 0.00430
1.00387 0.00389
i-3
1.00479 0.00664
1.00434 0.00601
1.00393 0.00544
1 -003 5 5 0.00480
1.00322 0.00445
i-35
1.00369 0.00731
1.00334 0.00661
1.00303 0.00598
1.00274 0.00541
1.00248 0.00489
1.4
1.00250 0.00779
1.00226 0.00705
1.00205 0.00638
1.00186 0.00577
i .00168 0.00522
i -45
1.00125 0.00808
1.00113 0.00731
1.00103 0.00662
1.00093 0.00599
1.00084 0.00542
i-5
0.99997 0.00817
0.99997 0.00740
0.99998 0.00669
0.99998 0.00606
0.99999 0.00548
i-55
0.99869 0.00806
0.99882 0.00730
0.99893 0.00660
0.99904 0.00598
0.99913 0.00541
1.6
0.99745 0.00775
0.99769 0.00702
0.99791 0.00635
0.99811 0.00575
0.99830 0.00520
1.65
0.99627 0.00726
0.99663 0.00657
0.99695 0.00594
0.99724 0.00538
0.99750 0.00487
i-7
0.99519 0.00658
0.99564 0.00596
0.99606 0.00539
0.99644 0.00488
0.99678 0.00442
1-75
0.99422 0.00575
0.99477 0.00519
0.99527 0.00471
0.99572 0.00426
0.99613 0.00386
1.8
0.99340 0.00477
0.99403 0.00432
0.99459 0.00391
0.99511 0.00354
0-99557 0.00321
1.85
0.99274 0.00368
0-99343 0.00334
0.99405 0.00302
0.99462 0.00273
0.99513 0.00248
1.9
0.99225 0.00251
0.99299 0.00227
0.99365 0.00206
0.99426 0.00186
0.99480 0.00168
1-95
0.99196 0.00127
0.99272 0.00115
0.99341 0.00104
0.99404 0.00094
0.99460 0.00085
2.O 0.99186 O.OO
0.99263 o.oo
0-99333 o.oo
0.99396 o.oo
0.09454 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (2.9 + i 0.9) = 1.00577 + * 0.00188.
tanh (2.95 + i 1.95) = 0.99460 — * 0.00085.
[85]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
= 3-0
* = 3-°5
x = 3.10
x = 3.20
0
0.05
O.I
0.15
O.2
0.99505 o.oo
0.99512 O.OOO77
0.99530 0.00153
0.99559 0.00224
0.99599 O.O029O
0.99552 o.oo
0.99558 0.00070
0.99574 0.00138
0.99601 0.00203
0.99637 0.00263
0.99595 o.oo
0.99600 0.00063
0.99615 0.00125
0.99639 0.00184
0.99672 0.00238
0.99633 o.oo
0.99638 0.00057
0.99651 0.00113
0.99673 0.00166
0.99703 0.00215
0.99668 o.oo
0.99672 0.00052
0.99684 O.OOIO2
0.99704 0.00150
0.99731 0.00195
0.25
0-3
0-35
0.4
0-45
0.99649 0.00349
0.99708 O.O040O
0.99774 O.OO44I
0.99846 O.0047I
0.99921 0.00489
0.99683 0.00316
0.99736 0.00362
0.99796 0.00399
0.99861 0.00426
0.99929 0.00443
0.99713 0.00286
0.99761 0.00328
0.99815 0.00361
0.99874 0.00386
0.99936 0.00401
0.99740 0.00259
0.99784 0.00297
0.99833 0.00327
0.99886 0.00349
0.99942 0.00363
0.99765 0.00234
0.99805 0.00268
0.99849 0.00296
0.99897 0.00316
0.99947 0.00328
0-5
0-55
0.6
0.65
0.7
0.99999 0.00496
I.OOO76 0.00490
I.OOI52 0.00472
1.00224 0.00443
I.OO29I O.OO4O2
0.99999 0.00449
1.00069 0.00443
1.00138 0.00427
1.00203 0.00401
1.00263 0.00364
0.99999 0.00406
1.00063 0.00401
1.00125 0.00387
1.00184 0.00362
1.00238 0.00329
0.99999 0.00367
1.00057 0.00363
1.00113 0.00350
1.00166 0.00328
1.00216 0.00298
0.99999 0.00332
1.00051 0.00328
I.OOI02 O.OO3I6
1.00151 0.00297
1.00195 0.00269
0-75
0.8
0.85
0.9
0-95
I.0035I 0.00352
I.OO4OI O.OO293
1.00443 O.00226
1.00473 0.00154
1.00491 0.00078
1.00317 0.00318
1.00363 0.00265
1.00400 0.00205
1.00426 0.00139
1.00444 0.00071
1.00287 0.00288
1.00329 0.00239
1.00362 0.00185
1.00387 0.00126
1.00402 0.00064
1.00260 0.00260
1.00297 0.00217
1.00328 0.00167
1.00350 0.00114
1.00363 0.00058
1.00235 0.00236
1.00269 0.00196
1.00297 0.00151
1.00317 0.00103
1.00329 0.00052
.0
•OS
.1
•IS
.2
1.00497 o.oo
1.00491 0.00078
1.00473 0.00154
1.00443 0.00226
1.00401 0.00293
1.00450 o.oo
1.00444 0.00071
1.00426 0.00139
1.00400 0.00205
1.00363 0.00265
1.00407 o.oo
1.00402 0.00064
1.00387 0.00126
1.00362 0.00185
1.00329 0.00239
1.00368 o.oo
1.00363 0.00058
1.00350 0.00114
1.00328 0.00167
1.00297 0.00217
1.00333 o.oo
1.00329 0.00052
1.00317 0.00103
1.00297 0.00151
1.00269 0.00196
•25
•3
•35
1.4
i-45
1.00351 0.00352
1.00291 0.00402
1.00224 0.00443
1.00152 0.00472
1.00076 0.00490
1.00317 0.00318
1.00263 0.00364
1.00203 0.00401
1.00138 0.00427
1.00069 0.00443
1.00287 0.00288
1.00238 0.00329
1.00184 0.00362
1.00125 0.00387
1.00063 0.00401
1.00260 0.00260
1. 00216 0.00298
1.00166 0.00328
1.00113 0.00350
1.00057 0.00363
1.00235 0.00236
1.00195 0.00269
1.00151 0.00297
I.OOI02 O.O03I6
1.00051 0.00328
i-5
i-SS
1.6
1.65
i-7
0.99999 0.00496
0.99921 0.00489
0.99846 0.00471
0.99774 0.00441
0.99708 0.00400
0.99999 0.00449
0.99929 0.00443
0.99861 0.00426
0.99796 0.00399
0.99736 0.00362
0.99999 0.00406
0.99936 0.00401
0.99874 0.00386
0.99815 0.00361
0.99761 0.00328
0.99999 0.00367
0.99942 0.00363
0.99886 0.00349
0.99833 0.00327
0.99784 0.00297
0.99999 0.00332
0.99947 0.00328
0.99897 0.00316
0.99849 0.00296
0.99805 0.00268
i-75
1.8
1.85
1.9
i-95
0.99649 0.00349
0.99599 0.00290
0-99559 0.00224
0.99530 0.00153
0.99512 0.00077
0.99683 0.00316
0.99637 0.00263
0.99601 0.00203
0.99574 0.00138
0.99558 0.00070
0.99713 0.00286
0.99672 0.00238
0.99639 0.00184
0.99615 0.00125
0.99600 0.00063
0.99740 0.00259
0.99703 0.00215
0.99673 0.00166
0.99651 0.00113
0.99638 0.00057
0.99765 0.00234
0.99731 0.00195
0.99704 0.00150
0.99684 O.OOI02
0.99672 0.00052
2.O
0-99505 o.oo
0.99552 o.oo
0.99595 o-o°
o-99633 o-00
0.99668 o.oo
Note. Negative quantities are in heavy type.
Examples.
tanh (3.0 + * i.oo)
tanh (3.0 + * 1.50)
1.00497 + io.
0.99999 ~~ » 0.00496.
[86]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) =u+ iv. CONTINUED
= 3-25
* = 3-30
* = 3-35
x = 3.40
= 3-45
o
0.99700 o.oo
0.99728 o.oo
0.99754 o.oo
0.99777 o.oo
0.99799 o.oo
0.05
0.99704 0.00047
0.99732 0.00042
0.99757 0.00038
0.99780 O.OOO35
0.99801 0.00031
O.I
0.99715 0.00093
0.99742 0.00084
0.99766 0.00076
0.99788 0.00069
0.99809 0.00062
0.15
0.99732 0.00136
0.99758 0.00123
0.99781 O.OOII2
0.99802 o.ooioi
0.99821 0.00091
0.2
0.99757 0.00176
0.99780 0.00160
0.99801 0.00144
0.99820 0.00131
0.99837 0.00118
0.25
0.99787 O.O02I2
0.99808 0.00192
0.99826 O.OOI74
0.99843 0.00157
0.99857 0.00142
0-3
0.98823 0.00243
0.99840 O.OO22O
0.99855 0.00199
0.99869 0.00180
0.99881 0.00163
0-35
0.99863 0.00268
0.99876 O.O0242
0.99888 O.OO2I9
0.99899 0.00198
0.99908 0.00179
0.4
0.99907 O.O0286
0.99916 O.OO259
0.99924 O.OO234
O.9993I 0.00212
0.99938 0.00192
0-45
0.99953 0.00297
O.99957 O.O0269
0.99961 O.OO243
0.99965 O.OO22O
0.99968 O.OO2OO
0-5
.OOOOO O.OO3OI
i.ooooo 0.00272
i.ooooo 0.00246
i.ooooo 0.00223
I.OOOOO O.002O2
0-55
.00047 O.O0297
1.00042 0.00269
1.00038 0.00243
I.OOO35 O.OO220
I.OO03I O.OO2OO
0.6
.OOO9I O.OO286
1.00084 O.O0259
1.00076 0.00234
1.00069 O.OO2I2
I.OOO62 O.OOI92
0.65
.00136 O.OO268
I.OOI23 O.OO243
I.OOII2 O.OO22O
i.ooioi 0.00199
I.OOOgi O.OOlSo
0.7
.OOI77 O.00244
I.OOI6O O.OO22O
I.OOI45 O.OOI99
I.OOI3I O.OOlSo
1.00118 0.00163
0.75
I.OO2I3 O.OO2I3
I.OOI92 O.OOI93
I.OOI74 O.OOI74
.OOI58 O.OOI58
I.OOI43 O.OOI43
0.8
I.OO244 O.OOI77
I.OO222 O.OOl6o
I.OOI99 O.OOI45
.OOlSo O.OOI3I
1.00163 O.OOIig
0.85
1.00268 0.00137
I.OO242 O.OOI24
I.OO22O O.OOII2
.OOI99 O.OOIO3
I.OOlSo 0.00092
0.9
I.OO286 O.OOO93
1.00260 0.00084
I.OO234 O.OOO76
.OO2I2 O.OOO69
I.OOI92 0.00062
o-9S
I.OO298 0.00047
1.00269 O.OOO43
I.OO243 O.OOO39
.OO22O O.OOO35
I.OO20O O.00032
I.O
1.00301 o.oo
1.00273 o.oo
1.00246 o.oo
.OO223 °-°°
.00202 o.oo
1.05
1.00298 0.00047
1.00269 O.OOO43
I.OO243 O.OOO39
.OO22O 0.00035
.00200 0.00032
I.I
1.00286 0.00093
I.0026O 0.00084
I.OO234 O.OO076
.OO2I2 O.OOO69
.00192 0.00062
1-15
1.00268 0.00137
I.OO242 O.OOI24
I.OO22O O.OOII2
I.OOI99 O.OOIO3
.00180 0.00092
1.2
1.00244 0.00177
I.OO222 O.OOI6O
1.00199 O.OOI45
I.OOlSo O.OOI3I
.00163 0.00119
1-25
1.00213 0.00213
I.OOI92 O.OOI93
I.OOI74 O.OOI74
I.OOI58 O.OOI58
.00143 0.00143
'•3
1.00177 0.00244
I.OOl6o O.0022O
I.OOI45 O.OOI99
I.OOI3I O.OOlSo
.00118 0.00163
i-35
1.00136 0.00268
I.OOI23 O.O0243
I.OOII2 O.OO22O
i.ooioi 0.00199
.00091 0.00180
1.4
1.00091 0.00286
1.00084 O.O0259
1.00076 O.OO234
1.00069 O.OO2I2
.00062 0.00192
i -45
1.00047 0.00297
I.OOO42 O.OO269
1.00038 O.OO243
I.OOO35 O.OO22O
.OOO3I O.OO2OO
i-5
i. ooooo 0.00301
i.ooooo 0.00272
i.ooooo 0.00246
i.ooooo 0.00223
I.OOOOO O.OO2O2
i-SS
0.99953 0.00297
O.99957 O.OO269
0.99961 O.OO243
0.99965 O.OO22O
0.99968 O.OO2OO
1.6
0.99907 0.00286
0.99916 O.OO259
0.99924 O.OO234
0.99931 O.OO2I2
0.99938 O.OOI92
1.65
0.99863 0.00268
0.99876 O.OO242
0.99888 O.OO2I9
0.99899 O.OOI98
0.99908 O.OOI79
i-7
0.99823 0.00243
0.99840 O.OO22O
0.99855 O.OOI99
0.99869 O.OOlSo
0.99881 0.00163
1-75
0.99787 O.O02I2
0.99808 O.OOI92
0.99826 O.OOI74
0.99843 O.OOI57
0.99857 O.OOI42
1.8
O-99757 O.OOI76
0.99780 O.OOI6O
0.99801 O.OOI44
0.99820 O.OOI3I
0.99837 O.OOIlS
1.85
O.99732 O.OOI36
0.99758 O.OOI23
O.9978l O.OOII2
0.99802 o.ooioi
0.99821 O.OOO9I
i. 9
0.99715 0.00093
O.99742 O.O0084
0.99766 O.OO076
0.99788 O.O0069
0.99809 O.OOO62
i-9S
0.99704 O.O0047
O.99732 O.O0042
O.99757 O.O0038
0.99780 0.00035
0.99801 O.O003I
2.O O.997OO O.OO
0.99728 o.oo
0.99754 o.oo
0-99777 °'°°
0.99799 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (3.25 + to. 75) = 1.00213 + i 0.00213.
tanh (3.30 + » 1.50) = i.ooooo — i 0.00272.
[87]
TABLE IX HYPERBOLIC TANGENTS, tanh (* + iq) =u + iv. CONTINUED
- 3-50
= 3-55
x = 3.60
x = 3.70
0
0.05
O.I
0.15
O.2
0.99818
0.99820
0.99827
0.99837
0.99853
o.oo
0.00028
0.00056
0.00083
0.00107
0.99835
0.99837
0.99843
0-99853
0.99867
o.oo
0.00026
0.00051
0.00075
0.00097
0.99851
0.99853
0.99858
0.99867
0.99879
o.oo
0.00023
0.00046
0.00068
0.00088
0.99865
0.99867
0.99872
0.99880
0.99891
O.OO
O.OOO2I
0.00042
O.OOo6l
O.OOO79
0.99878
0.99879
0.99884
0.99891
0.99901
o.oo
0.00019
0.00038
0.00055
0.00072
0.25
o-3
0.35
0.4
0.45
0.99871
0.99893
0.99917
0.99944
0.99971
0.00129
0.00147
0.00162
0.00173
0.00180
0.99883
0.99903
0.99925
0.99949
0.99974
0.00117
0.00133
0.00147
0.00157
0.00163
0.99894
0.99912
0.99932
0.99954
0.99977
0.00105
O.OOI2I
0.00133
0.00142
0.00147
0.99904
0.99921
0-99939
0.99958
0-99979
0.00095
O.OOI09
O.OOI2O
0.00128
0.00133
0.99914
0.99928
0.99944
0.99962
0.99981
0.00086
0.00099
0.00109
0.00116
O.OOI2I
o-S
o-5S
0.6
0.65
0.7
I.OOOOO
I 00028
1.00056
1.00083
1.00107
0.00182
0.00180
0.00174
0.00163
0.00148
I.OOOOO
1.00026
1.00051
1.00075
1.00097
0.00165
0.00163
0.00157
0.00147
0.00134
I.OOOOO
1.00025
1.00048
1.00068
1.00088
0.00149
0.00148
0.00142
0.00133
O.OOI2I
I.OOOOO
I.OO02I
I.OOO42
I. OOO6 1
I.OOO79
0.00135
0.00133
O.OOI29
O.OO I 2O
0.00109
I.OOOOO
1.00019
1.00038
1.00055
1.00072
O.OOI22
O.OOI2I
0.00116
0.00109
0.00099
o.75
0.8
0.85
0.9
0-95
1.00129
1.00148
1.00163
1.00174
1.00180
0.00129
0.00107
0.00083
0.00056
0.00029
1.00117
1.00133
1.00147
1.00157
1.00163
0.00117
0.00097
0.00075
0.00051
0.00026
1.00106
I.OOI2I
1.00133
1.00142
1.00148
0.00106
0.00088
0.00068
0.00046
0.00023
I.OOO96
I.OOI08
I.OOI2O
I.OOI29
I.OOI34
0.00096
0.00080
0.00061
0.00042
O.OOO2I
1.00086
1.00099
1.00109
1.00116
I.OOI2I
0.00086
0.00072
0.00056
0.00038
0.00019
I.O
1.05
i.i
i-iS
1.2
1.00183
1.00180
1.00174
1.00163
1.00148
0.00
0.00029
0.00056
0.00083
0.00107
1.00165
1.00163
1.00157
1.00147
1.00133
o.oo
0.00026
0.00051
0.00075
0.00097
1.00149
1.00148
1.00142
1.00133
I.OOI2I
0.00
0.00023
0.00046
0.00068
0.00088
I.OOI35
I.OOI34
1.00129
I.OOI2O
I.OOI08
O.OO
O.OOO2I
O.OOO42
O.OOO6l
O.OOO8O
I.OOI22
I.OOI2I
1.00116
1.00109
1 .00099
o.oo
0.00019
0.00038
0.00056
0.00072
1-25
i-3
i-35
1.4
1-45
1.00129
1.00107
1.00083
1.00056
1.00028
0.00129
0.00148
0.00163
0.00174
0.00180
1.00117
1.00097
1.00075
1.00051
1.00026
0.00117
0.00134
0.00147
0.00157
0.00163
1.00106
1.00088
1.00068
1.00048
1.00025
0.00106
O.OOI2I
0.00133
0.00142
0.00148
I.OOO96
I.OOO79
1. 000(5 1
I.OOO42
I.OO02I
O.OOO96
O.OOIO9
O.OOI2O
O.OOI29
O.OOI33
1 .00086
1.00072
1.00055
1.00038
1.00019
0.00086
0.00099
0.00109
o. 00116
O.OOI2I
i-5
I-5S
1.6
1.65
i-7
I.OOOOO
0.99971
0.99944
0.99917
0.99893
0.00182
0.00180
0.00173
0.00162
0.00147
I.OOOOO
0.99974
0.99949
0.99925
0.99903
0.00165
0.00163
0.00157
0.00147
0.00133
I.OOOOO
0.99977
0.99954
0.99932
0.99912
0.00149
0.00147
0.00142
0.00133
O.OOI2I
I.OOOOO
0.99979
0.99958
0.99939
0.99921
0.00135
0.00133
O.OOI28
O.OOI20
O.OOIO9
I.OOOOO
0.99981
0.99962
0-99944
0.99928
O.OOI22
O.OOI2I
o. 00116
O.OOIO9
O.OOO99
1-75
1.8
1.85
1.9
i-95
0.99871
0.99853
0.99837
0.99827
0.99820
0.00129
0.00107
0.00083
0.00056
0.00028
0.99883
0.99867
0-99853
0.99843
0.99837
0.00117
0.00097
0.00075
0.00051
0.00026
0.99894
0.99879
0.99867
0.99858
0.99853
0.00105
0.00088
0.00068
0.00046
0.00023
0.99904
0.99891
0.99880
0.99872
0.99867
O.OOO95
O.OOO79
0.00061
O.OOO42
O.OO02I
0.99914
0.99901
0.99891
0.99884
0.99879
O.OO086
O.OOO72
0.00055
O.OOO38
O.OOOI9
2.O
0.99818
O.OO
0.99835
o.oo
0.99851
o.oo
0.99865
O.OO
0.99878
0.00
Note.
Negative quantities are in heavy type.
Examples, tanh (3.60 +
i 0.80) =
I.OOI2I + i O.OOO88.
tanh (3.70 -+• * 1.70) = 0.99928 — i 0.00099.
[88]
TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED
= 3-75
x = 3.80
= 3-85
3-90
= 3-95
o
0.99889
o.oo
0.99900
o.oo
0.99909
o.oo
0.99918
o.oo
0.99926
o.oo
0.05
0.99891
0.00017
0.99901
0.00016
0.99911
0.00014
0.99919
0.00013
0.99927
O.OOOI2
O.I
0.99895
0.00034
0.99905
0.00031
0.99914
0.00028
0.99922
0.00025
0.99930
0.00023
0.15
0.99901
0.00050
0.99911
0.00045
0.99919
0.00041
0.99927
0.00037
0.99934
0.00034
0.2
0.99911
0.00065
0.99919
0.00059
0.99927
0.00053
0.99934
0.00048
0.99940
0.00044
0.25
0.99922
0.00078
0.99929
0.00071
0.99936
0.00064
0.99942
0.00058
0.99948
0.00052
°-3
0-99935
0.00089
0.99941
0.00081
0.99947
0.00073
0.99952
0.00066
0.99956
0.00060
0-35
0.99950
0.00099
0-99955
0.00089
0.99959
0.00081
0.99963
0.00073
0.99966
0.00066
0.4
0.99966
0.00105
0.99969
0.00095
0.99972
0.00086
0.99975
0.00078
0.99977
0.00071
0-45
0.99983
0.00109
0.99984
0.00099
0.99986
0.00089
0.99987
0.00081
0.99988
0.00073
o.S
1. 00000
O.OOIII
I.OOOOO
O.OOIOO
I.OOOOO
0.00091
I.OOOOO
0.00082
I.OOOOO
0.00074
0-55
1.00017
0.00109
1.00016
0.00099
1.00014
0.00089
1.00013
0.00081
I.OOOI2
0.00073
0.6
1.00034
0.00105
1.00031
0.00095
1.00028
0.00086
1.00025
0.00078
1.00023
0.00071
0.65
1.00050
0.00099
1.00045
0.00089
1.00041
0.00081
1.00037
0.00073
1.00034
0.00066
0.7
1.00065
0.00090
1.00059
0.00081
1.00053
0.00073
1.00048
0.00066
1.00044
0.00060
0-75
1.00078
0.00078
1.00071
0.00071
1.00064
0.00064
1.00058
0.00058
1.00052
0.00052
0.8
1.00089
0.00065
1.00081
0.00059
1.00073
0.00053
1.00066
0.00048
1.00060
0.00044
0.85
1.00099
0.00050
1.00089
0.00045
1.00081
0.00041
1.00073
0.00037
1.00066
0.00034
0.9
1.00105
0.00034
1.00096
0.00031
1.00086
0.00028
1.00078
0.00025
1.00071
0.00023
o-9S
1.00109
0.00017
1.00099
0.00016
1.00089
0.00014
1.00081
0.00013
1.00073
O.OOOI2
I.O
I.OOIII
o.oo
I.OOIOO
o.oo
1.00090
o.oo
1.00082
o.oo
1.00074
O.OO
1.05
1.00109
0.00017
1.00099
0.00016
1.00089
0.00014
1.00081
0.00013
1.00073
0.00012
i.i
1.00105
0.00034
1.00096
0.00031
1.00086
0.00028
1.00078
0.00025
1.00071
O.OO023
i-iS
1.00099
0.00050
1.00089
0.00045
1.00081
0.00041
1.00073
0.00037
1.00066
O.OOO34
1.2
1.00089
0.00065
1.00081
0.00059
1.00073
0.00053
1.00066
0.00048
1.00060
O.OOO44
I.2S
1.00078
0.00078
1.00071
0.00071
1.00064
0.00064
1.00058
0.00058
1.00052
O.OO052
i-3
1.00065
0.00090
1.00059
0.00081
1.00053
0.00073
1.00048
0.00066
1.00044
O.OO06O
i-35
1.00050
0.00099
1.00045
0.00089
1.00041
0.00081
1.00037
0.00073
1.00034
O.OO066
1.4
1.00034
0.00105
1.00031
0.00095
1.00028
0.00086
1.00025
0.00078
1.00023
O.OOO7I
1-45
1.00017
0.00109
1.00016
0.00099
1.00014
0.00089
1.00013
0.00081
I.OOOI2
O.OOO73
i-5
I.OOOOO
O.OOIII
I.OOOOO
O.OOIOO
I.OOOOO
0.00091
I.OOOOO
0.00082
I.OOOOO
O.OO074
i-SS
0.99983
o.ooiog
0.99984
0.00099
0.99986
0.00089
0.99987
0.00081
0.99988
O.00073
1.6
0.99966
0.00105
0.99969
0.00095
0.99972
0.00086
0-99975
0.00078
0.99977
O.OOO7I
1.65
0.99950
0.00099
0.99955
00089
0.99959
0.00081
0.99963
0.00073
0.99966
O.OOO66
i-7
0-99935
0.00089
0.99941
0.00081
0.99947
0.00073
0-99952
0.00066
0.99956
O.OOO6O
i-75
0.99922
0.00078
0.99929
0.00071
0.99936
0.00064
0.99942
0.00058
0.99948
O.OOO52
1.8
0.99911
0.00065
0.99919
0.00059
0.99927
0.00053
0.99934
0.00048
0.99940
O.OOO44
1.85
0.99901
0.00050
0.99911
0.00045
0.99919
0.00041
0.99927
0.00037
0.99934
O.OOO34
1.9
0.99895
0.00034
0.99905
0.00031
0.99914
0.00028
0.99922
0.00025
0.99930
O.OOO23
J-95
0.99891
0.00017
0.99901
0.00016
0.99911
0.00014
0.99919
0.00013
0.99927
O.OOOI2
2.O 0.99889 O.OO
0.99900 o.oo
0.99909 o.oo
0.99918 o.oo
0.99926 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (3.95 + i 0.95) = 1.00073 + » 0.00012.
tanh (3.95 -f i 1.05) = 1.00073 — i 0.00012.
[89]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y_
x = 0.05
x = 0.15
o
o.ooo
9°
0.05002
o.ooo
0.10017
o.ooo
0.15056
o.ooo
0.20134
o.ooo
0.05
0.07846
90
0.09305
57-593
0.12724
38.300
0.16978
27.861
0.21608
21.739
O.I
0.15643
90
0.16424
72.493
0.18576
57-819
0.21712
46.771
0.25497
38.746
0.15
0-23345
9°
0.23874
78-245
0.25403
67-455
0.27779
58.195
0.30827
50.576
O.2
0.30902
90
0.31304
81.259
0.32485
72.947
0-34375
65.382
0.36882
58.723
O.25
0.38268
9°
0.38594
83-123
0.39558
76.471
0.41124
70.229
0.43422
64.522
°-3
0.45399
90
0.45672
84.400
0.46491
78.932
0.47831
73-7"
0.49663
68.825
0.35
0.52250
90
0.52487
85-339
0.53201
80.762
0.54376
76-344
0-55995
72.147
0.4
0.58778
9°
0.58989
86.066
0.59626
82.189
0.60676
78.419
0.62131
74.802
0.45
0.64944
90
0-65135
86.652
0.65713
83.344
0.66667
80.111
0.67994
76.988
°-5
0.70711
90
0.70803
87-I39
0.71417
84.308
0.72296
81.532
0.73521
78.835
o-55
0.76041
90
0.76202
87-557
0.76698
85-134
0.77527
82.753
0.78661
80.431
0.6
0.80902
00
0.81053
87.921
0.81520
85.858
0.82291
83.826
0.83370
81.839
0.65
0.85264
90
0.85407
88.246
0.85851
86.505
0.86583
84.787
0.87609
83.103
0.7
0.89101
90
0.89237
88.542
0.89662
87.093
0.90364
85.662
0.91347
84.257
o-7S
0.92388
90
0.92523
88.814
0.92930
87.636
0.93607
86.471
0.94556
85.326
0.8
0.95106
90
0.95237
89.070
0.95632
88.145
0.96290
87-231
0.97213
86.331
0.85
0.97237
90
0.97366
89-313
0.97752
88.629
0.98396
87-953
0.99300
87.287
0.9
0.98769
90
0.98895
89-547
0.99275
89.095
0.99909
88.649
1.00800
88.209
0-95
0.99692
90
0.99817
89-775
1.00194
89.550
1.00822
89.329
1.01704
89.110
I.O
I.OOOOO
90
1.00125
90.000
1.00500
90.000
1.01127
90.000
1.02007
90.000
1.05
0.99692
90
0.99817
90.225
1.00194
90.450
1.00822
90.671
1.01704
90.890
i.i
0.98769
90
0.98895
90-453
0.99275
90.905
0.99909
9i-35i
1.00800
91.791
"5
0.97237
90
0.97366
90.687
0.97752
9*-37i
0.98396
92.047
0.99300
92.713
1.2
0.95106
90
0.95237
90.930
0.95632
9I-855
0.96290
92.769
0.97213
93.669
I-2S
0.92388
90
0.92523
91.186
0.92930
92-364
0.93607
93-529
0.94556
94.674
i-3
0.89101
90
0.89237
91.458
0.89662
92.907
0.90364
94.338
0.91347
95-743
i-35
0.85264
90
0.85407
91-754
0.85851
93-495
0.86583
95-213
0.87609
96.897
1.4
0.80902
90
0.81053
92.079
0.81520
94.142
0.82291
96.174
0.83370
98.161
i -45
0.76041
90
0.76202
92.443
0.76698
94.866
0.77527
97.247
0.78661
99.569
1-5
0.70711
90
0.70803
92.860
0.71417
95.692
0.72296
98.468
0.73521
101.165
1-55
0.64944
90
0.65135
93-348
0.65713
96.656
0.66667
99.889
0.67994
103.012
1.6
0.58778
90
0.58989
93-934
0.59626
97.811
0.60676
101.581
0.62131
105.198
1.65
0.52250
90
0.52487
94.661
0.53201
99-238
0.54376
103.656
0-55995
107-853
i-7
0.45399
90
0.45672
95.600
0.46491
101.068
0.47831
106.289
0.49663
111.175
i-7S
0.38268
90
0.38594
96.877
0.39558
103.529
0.41124
109.771
0.43242
115.478
1.8
0.30902
90
0.31304
98.741
0.32485
107.053
0-34375
114.618
0.36882
121.277
1.85
0.23345
90
0.23874
101.755
0.25403
112.545
0.27779
121.805
0.30827
129.424
1.9
0.15643
90
0.16424
107.507
0.18576
122.181
0.21712
133.229
0.25497
141.254
I- 95
0.07846
90
0.09305
122.407
0.12724
141.700
0.16978
152-139
0.21608
158.261
2.O
o.oo
90
0.05002
180.000
0.10017
180.000
0.15056
180.000
0.20134
180.000
Example. sinh (0.15 + to. 15) = 0.27779
= 0.27779 /58°.ii/.42*.
TABLE X. HYPERBOLIC SINES, sinh (at + iq) = r /T. CONTINUED
= 0.25
x = 0.3
= 0.35
* = 0.4
x = 0.45
0.25261
o.ooo
0.30452
0.000
0.35719
o.ooo
0.41075
o.ooo
0.46534
o.ooo
0.26452
17.814
0.31447
15.118
0.36571
13.169
0.41818
11.703
0.47191
10.567
0.29782
32.890
0.34235
28.533
0.38994
25.214
0-43953
22.629
0.49093
20.576
0.34396
44-943
0.38370
39-493
0.42671
35.516
0.47246
32.288
0.52062
29.642
o.399J3
52.992
0.43385
48.122
0.47231
44.008
0.51401
40.536
0.55860
37.601
0.45854
59-405
0.48906
54.882
0.52348
50.921
0.56140
47.471
0.60249
44-473
0.51954
64.327
0.54666
60.242
0.57766
56.843
0.61223
53-288
0.65012
50-374
0.58036
68.215
0.60476
64-575
0.63292
61.237
0.66462
58.200
0.69968
55-454
0.63977
7I-37I
0.66199
68.151
0.68781
65.157
0.71708
62.393
0.74969
59-856
0.69685
73-999
0.71730
71.166
0.74II9
68.503
0.76844
66.018
0.79895
63.712
0.75088
76.238
0.76989
73-759
0.79220
71.408
0.81775
69.196
0.84649
67.125
0.80127
78.185
0.81912
76.028
O.840I2
73.971
0.86426
72.021
0.89149
70.184
0.84754
79.911
0.86443
78.049
0.88436
76.267
0.90732
74.568
0-93330
72.958
0.88927
81.464
0-90539
79-878
0.92444
78.353
0.94642
76.893
0.97136
75.504
0.92612
82.887
0.94161
8i.557
0.95994
80.274
0.98113
79.043
1.00521
77.868
0-95779
84.207
0.97277
83.119
0.99052
82.068
1.01107
81.056
1.03446
80.087
0.98403
85-450
0.99862
84-593
I.OI592
83.762
1.03597
82.962
1.05880
82.194
1.00464
86.635
1.01894
85.999
1.03590
85.383
1-05557
84.788
1.07798
84.216
1.01949
87-779
1-03357
87-358
I.05O29
86.951
1.06970
86.556
1.09182
86.177
1.02843
88.896
1.04239
88.687
1.05898
88.484
1.07822
88.287
1.10017
88.098
1.03141
90.000
1-04534
90.000
i. 06 1 88
90.000
1.08107
90.000
1.10297
90.000
1.02843
91.104
1.04239
91.313
1.05898
91.516
1.07822
91-713
1.10017
91.902
1.01949
92.221
1-03357
92.642
1.05029
93.050
1.06970
93-444
1.09182
93-823
i .00464
93-365
1.01894
94.001
1.03590
94.617
1-05557
95-212
1.07798
0.98403
94-550
0.99862
95.407
1.01592
96.238
1-03597
97.038
1.05880
97.806
0-95779
95-793
0.97277
96.881
0.99052
97.932
1.01107
98.944
1.03446
99-9J3
0.92612
0.94161
98.443
0.95994
99.726
0.98113
100.957
1.00521
102.132
0.88927
98*536
0-90539
IOO.I22
0.92444
101.647
0.94642
103.107
0.97136
104.496
0-84754
100.090
0.86443
IOI.95I
0.88436
103.733
0.90732
105.432
0-93330
107.042
0.80127
101.815
0.81912
103.972
0.84012
106.029
0.86426
107.979
0.89149
109.816
0.75088
103.762
0.76989
IO6.24I
0.79220
108.592
0.81775
110.804
0.84649
112.875
0.69685
106.001
0.71730
108.834
0.74119
111.497
0.76844
113.982
0.79895
116.288
0.63977
108.629
0.66199
III.849
0.68781
114.843
0.71708
117.607
0.74969
120.144
0.58036
111.785
0.60476
115425
0.63292
118.763
0.66462
121.800
0.69968
124.546
0.51954
0.54666
119.758
0.57766
123.157
0.61223
126.712
0.65012
129.626
0-45854
120.595
0.48906
I25.II8
0.52348
129.079
0.56140
132.529
0.60249
I35.527
0.39913
127.008
0-43385
131.878
0.47231
135.992
0.51401
139.464
0.55860
142.399
0.34396
I35-057
> 0.38370
140.507
0.42671
144.484
0.47246
147.712
0.52062
150.358
0.29782
147.110
0.34235
151.467
0.38994
154.786
0-43953
I57-37I
0.49093
159.424
0.26452
162.186
0.31447
164.882
0.36571
166.831
0.41818
168.297
0.47191
169.433
0.25261
180.000
0.30452
iSo.OOO
0.35719
180.000
0.41075
180.000
0.46534
180.000
Example. sinh (0.40 + £0.25) = 0.56140 (
= 0.56140 /47°.28'.i6".
[90
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r
CONTINUED
x = 0.5
= °-55
x = 0.6
* = 0.65
0
0.05
O.I
0.15
0.2
0.52110 o.ooo
0.52697 9.665
0.54407 18.918
0.57100 27.453
0.60583 35.111
0.57815 o.ooo
0.58345 8.936
0.59894 17.559
0.62350 25.625
0.65555 32.990
0.63665 o.ooo
0.64147 8.337
0.65559 16.432
0.67810 24.086
0.70769 31.174
0.69675 o.ooo
0.70115 7.839
0.71409 15.486
0.73482 22.781
0.76220 29.613
0.75858 o.ooo
0.76263 7.419
0.77455 14.685
0.79369 21.665
0.81911 28.263
0.25
o-3
0-35
0.4
0-45
0.64652 41.871
0.69112 47-793
0-73793 52-980
0.78552 57-542
0.83266 61.584
0.69333 39-610
0.73510 45.511
0.77927 50.759
0.82447 55-437
0.86951 59.628
0.74282 37.642
0.78194 43-494
0.82361 48.769
0.86650 53-529
0.90946 57.838
0.79492 35.926
0.83160 41.710
0.87090 46.989
0.91156 51.803
0.95249 56.204
0.84965 34.426
0.88406 40.133
0.92112 45-397
0.95966 50.245
0.99861 54.716
0-55
0.6
0.65
0.7
0.87837 65.198
0.92182 68.462
0.96232 71-441
0.99927 74-189
1.03220 76-751
0.91338 63.411
0.95524 66.854
0-99437 70.016
1.03004 72.948
1.06214 75.693
0.95149 61.762
0.99171 65.360
1.02948 68.685
1.06411 71.784
1.09509 74.696
0.99270 60.245
.03134 63.976
.06769 67.445
.101 1 1 70.694
.13108 73.760
1.03704 58.853
1.07409 62.698
1.10904 66.294
1.14125 69.677
1.17019 72.884
0-75
0.8
0.85
0.9
0-95
1.06070 79.164
1.08446 81.461
1.10320 83.669
1.11672 85.814
1.12489 87.917
1.08987 78.287
1.11300 80.763
1.13127 83.148
1.14446 85.467
1.15244 87.744
1. 12200 77-459
1.14448 8O.IO2
1.16226 82.653
I.I75IO 85.121
I.I8287 87.580
.15715 76.678
.17892 79-477
.19623 82.185
.20871 84.826
1.21626 87.424
1.19541 75-946
1.21653 78.890
1.23327 81.744
1.24538 84.532
1.25271 87.277
I.O
.1
•IS
.2
1.12763 90.000
1.12489 92.083
1.11672 94.186
1.10320 96.331
1.08446 98.539
1.15510 90.000
1.15244 92.256 (
1.14446 94-533
1.13127 96.852
1.11300 99.237
1.18547 9O.OOO
1.18287 92.420
I.I75IO 94.879
I.I6226 97-347
1.14448 99.898
1.21879 90.000
1.21626 92.576
1.20871 95.174
1.19623 97.815
1.17892 100.523
1.25517 90.000
1.25271 92.723
1.24538 95-468
1.23327 98.256
1.21653 ioi.no
•25
•3
•35
1.4
1-45
1.06070 100.836
1.03220 103.249
0.99927 105.811
0.96232 108.559
0.92182 111.538
1.08987 101.713
1.06214 104.307
1.03004 107.052
0.99437 109.984
0.95524 113.146
I.I22OO 102.541
1.09509 105.304
I.O64H IO8.2I6
1.02948 III.3I5
0.99174 114.640
1.15715 103.322
1.13108 106.240
i.ioni 109.306
1.06769 112.555
1.03134 116.024
1.19541 104.054
1.17019 107.116
1.14125 110.323
1.10904 113.706
1.07409 117.302
i-SS
1.6
1.65
0.87837 114.803
0.83266 118.416
0.78552 122.458
0.73793 127.020
0.69112 132.207
0.91338 116.589
0.86951 120.372
0.82447 124.563
0.77927 129.241
0.73510 134-489
0.95149 II8.238
0.90946 I22.I62
0.86650 126.471
0.82361 131.231
0.78194 136.506
0.99270 119.755
0.95249 123.796
0.91156 128.197
0.87090 133.011
0.83160 138.290
1.03704 121.148
0.99861 125.284
0.95966 129.755
0.92112 134.603
0.88406 139.867
1.8
1.85
1.9
0.64652 138.129
0.60583 144.889
0.57100 152.547
0.54407 161.082
0.52697 170.335
0-69333 140-390
0.65555 147-010
0.62350 154-375
0.59894 162.441
0.58345 171.064
0.74282 142.358
0.70769 148.826
0.67810 155.914
0.65559 163.568
0.64147 171.663
0.79492 144-074
0.76220 150.387
0.73482 157.219
0.71409 164.514
0.70115 172.161
0.84965 145.574
0.81911 151.737
0.79369 158.335
0.77455 165.315
0.76263 172.581
2.0
0.52110 180.000
0.57815 180.000
0.63665 iSo.OOO
0.69675 180.000
0.75858 180.000
Example. sinh (0.70 + i 1.70) = 0.88406
= 0.88406 / 130°. $2'. 01*.
[92]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
x =
0-75
x =
0.8
x =
0.85
x =
0.9
x =
0.95
9
r
7
r
y
r
y
r
y
r
y
0
o
0
o
o
o.oo
0.82232
o.ooo
0.88811
0.000
0.95612
O.OOO
1.02652
o.oob
1.09948
O.OOO
0.05
0.82605
7.064
0.89157
6-759
0-95933
6.497
1.02951
6.270
1.10227
6.073
O.IO
0.83706
14.002
0.90178
I3-4I5
0.96883
12.909
1-03837
12.468
1.11056
12.085
0.15
0.85481
20.706
0.91828
19.877
0.98420
19.158
1.05273
18.529
1.12400
17.980
O.2
0.87846
27.093
0.94033
26.073
1.00481
25.l8l
1.07202
24-399
1.14209
23.712
0.25
0.90700
33-1"
0.96705
3I-955
1.02985
30.938
1-09553
30-039
1.16418
29.245
0-3
0.93932
38.737
0.99742
37-500
1.05843
36.401
1.12243
35-426
1.19227
34-557
0.35
0.97427
43-974
1.03041
42.702
1.08957
4I.565
1.15184
40-547
1.21732
39-637
0.4
1.01079
48.840
1.06500
47-574
1.12234
46.433
1.18289
45-407
1.24674
44.483
Q-45
1.04785
53-363
1.10024
52.135
I.I5583
5I.O22
1.21471
50.014
1.27697
49.102
0-5
1.08453
57-578
1.13522
56.414
1.18918
55-353
1.24649
54.386
1.30723
53-507
o.SS
I.I200I
61.522
1.16917
60.441
1.22163
59-450
1.27748
58.543
1.33682
57.714
0.6
I-I5356
65.228
1.20135
64.245
1.25246
63-399
1.30700
62.507
1-36505
6i.743
0.65
1.18457
68.733
1.23115
67-857
1.28107
67.048
1-33444
66.301
1.39136
65.613
0.7
I.2I248
72.067
1.25802
71.307
1.30693
70.602
1.35927
69-949
1.41519
69-347
0-75
1.23689
75.260
1.28152
74.621
I-32955
74.026
1.38105
73-474
1.43611
72.963
0.8
1.25726
78.339
1.30124
77-825
1-34858
77-344
1-39937
76.898
I-4537I
76.484
0.85
1.27346
81.330
1.31691
80.942
1.36369
80.580
I-4I395
80.242
1.46778
79-929
0.9
1.28520
84-255
1.32825
83.996
1.37466
83-754
1.42452
83-527
1.47797
83-316
o-9S
1.29230
87-138
I-335I3
87.008
1.38130
86.887,
1.43094
86.773
1.48415
86.668
I.O
1.29468
90.000
1-33743
90.000
I.38353
90.000
I-43309
90.000
1.48623
90.000
1.05
1.29230
92.862
I-335I3
92.992
1.38130
93-II3
1.43094
93.227
1.48415
93-332
i.i
1.28520
95-745
1.32825
96.004
1.37466
96.246
1.42452
96-473
1.47797
96.684
i-iS
1.27346
98.670
1.31691
99.058
1.36369
99.420
I-4I395
99.758
1.46778
100.071
1.2
1.25726
101.661
1.30124
102.175
1.34858
102.656
1-39937
103.102
I-4537I
103.516
I.2S
1.23689
104.740
1.28152
105-379
I.32955
105.974
1.38105
106.526
1.43611
107.037
i-3
I.2I248
107.933
1.25802
108.693
1.30693
109.398
I-35927
110.051
1.41519
110.653
i-35
I.I8457
111.267
1-23115
112.143
1.28107
112.952
1.33444
113.699
1.39136
114.387
•4
I-I5356
114.772
1.20135
"5-755
1.25246
116.601
1.30700
117-493
1-36505
118.257
•45
I.I2OOI
118.478
1.16917
119-559
1.22163
120.550
1.27748
121-457
1.33682
122.286
•5
1.08453
122.422
1.13522
123.586
1.18918
124.647
1.24649
125.614
1.30723
126.493
•55
1.04785
126.637
1.10024
127.865
1-15583
128.978
1.21471
129.986
.27697
130.898
.6
I.OIO79
131.160
1.06500
132.426
1.12234
I33-567
1.18289
134-593
.24674
I35-5I7
•6S
0.97427
136.026
1.03041
137.298
1.08957
138.435
1.15184
139-453
.21732
140.363
•7
0.93932
141.263
0.99742
142.500
1.05843
143-599
1.12243
144-574
19227
145-443
•75
O.907OO
146.889
0.96705
148.045
1.02985
149.062
1.09553
149.961
.16418
150.755
.8
0.87846
152.907
0.94033
153-927
1.00481
154.819
1.07202
155.601
.14209
156.288
1.85
0.85481
159.294
0.91828
160.123
0.98420
160.842
1-05273
161.471
.12400
162.020
1.9
0.83706
165.998
0.90178
166.585
0.96883
167.091
1.03837
167.532
.11056
167.915
i-95
0.82605
172.936
0.89157
173.241
0-95933
I73-503
1.02951
I73.730
.10227
173.927
2.0 0.82232 ISO.OOO 0.888II iSo.OOO 0.95612 iSo.OOO 1.02652 iSo.OOO 1.09948 iSo.OOO
Example. sinh (0.90 + i i.o) = 1.43309 /go0.
[93]
TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r /r CONTINUED
X = I.O
x = 1.05
X — I.I
x = 1.15
X = 1.2
9
f y
f y
r y
r y
r y
0
0.05
O.I
0.15
O.2
0
1.17520 o.ooo
1.17782 5.900
1.18552 11.748
1.19816 17.496
1.21515 23.105
0
1.25386 o.ooo
I.2563I 5.748
1.26358 11-453
1.27540 17.071
I.29I37 22.568
o
L33565 o.ooo
1-33795 5-615
1.34478 11.192
1-35590 16.694
1.37093 22.092
0
1.42078 o.ooo
1.42294 5.497
1.42936 10.961
1.43983 16.361
1.45399 21.670
o
1.50946 o.ooo
I.5II50 5.393
I.5I755 10.757
I.5274I 16.065
1.54077 21.294
0.25
0-35
0.4
0-45
1.23594 28.541
1.25984 33.784
1.28612 38.821
1.31400 43-651
1.34272 48.276
I.3I095 27.915
I-3335I 33-093
I-35837 38.090
1.38479 42.902
I.4I207 47-530
1.38939 27.359
1.41070 32.477
1.43421 37-435
1.45926 42.227
1.48517 46.855
1.47141 26.864
i.49x55 3I-926
1.51381 36.847
1.53756 41.620
1.56217 46.245
I.5572I 26.421
1.57626 31.433
1-59733 36-3J9
1.61987 41-073
1.64325 45-693
0-5
0-55
0.6
0.65
0.7
I.37I53 52-707
1.39976 56.957
1.42675 61.043
1.45193 64.981
1.47479 68.791
1.43950 51.981
1.46641 56.268
1.49220 60.403
1.51630 64.401
1.53820 68.280
1.51128 51.323
I.53694 55-640
1.56156 59.818
1.58460 63.870
1.60558 67.811
1.58701 50.725
1.61147 55.069
1.63497 59.284
1.65699 63.384
1.67705 67.380
1.66688 50.184
1.69018 54-549
1.71260 58.797
I.73363 62.939
1.75282 66.986
0-75
0.8
0.85
0.9
0-95
1.49487 72-491
1.51182 76.101
1.52532 79.638
I-535I3 83.122
1.54108 86.724
1-55747 72.056
J-57374 75-747
1.58671 79-369
1.59614 82.941
1.60187 86.479
1.62404 71.656
1-63965 75-421
1.65211 79.121
1.66117 82.774
1.66667 86.395
1.69472 71.287
1.70971 75.120
1.72166 78.893
1.73036 82.620
1.73564 86.317
I.76975 70-950
1.78409 74.844
1-79555 78-682
1.80389 82.478
1.80895 86.246
1.05
i.i
1.2
1.54308 90.000
1.54108 93.276
I-535I3 96-878
1.52532 100.362
1.51182 103.899
1.60379 90.000
1.60187 93-521
1.59614 97-059
1.58671 100.631
1.57374 104.253
1.66852 90.000
1.66667 93-605
1.66117 97.226
1.65211 100.879
1.63965 104.579
1.73741 90.000
I.73564 93-683
1.73036 97.380
1.72166 101.107
1.70971 104.880
i. 81066 90.000
1.80895 93-754
1.80389 97-522
1-79555 101.318
1.78409 105.156
1.25
1.4
i-45
1.49487 107.509
1.47479 111.209
1.45193 115.019
1.42675 118.957
1.39976 123.043
1.55747 107.944
1.53820 111.720
1.51630 115.599
1.49220 119.597
1.46641 123.732
1.62404 108.344
1.60558 112.189
1.58460 116.130
1.56156 120.182
1.53694 124.360
1.69472 108.713
1.67705 112.620
1.65699 116.616
1.63497 120.716
1.61147 124.931
1.76975 109.050
1.75282 113.014
1-73363 117-061
1.71260 121.203
1.69018 125.451
i-SS
1.6
1.65
1.37153 127.293
1.34272 131.724
1.31400 136.349
1.28612 141.179
1.25984 146.216
1.43950 128.019
1.41207 132.470
1.38479 137.098
1.35837 141.910
1 -3335 1 146.907
1.51128 128.677
1.48517 133.145
1.45926 137.773
1.43421 142.565
1.41070 147.523
1.58701 129.275
1.56217 133-755
I.53756 138.380
1.51381 143-153
1.49155 148.074
1.66688 129.816
1.64325 134-307
1.61987 138.927
1-59733 143-681
1.57626 148.567
1-75
1.8
1.85
1.9
1.23594 151.459
1.21515 156.895
1.19816 162.504
1.18552 168.252
1.17782 174.100
1.31095 152.085
1.29137 157.432
1.27540 162.929
1.26358 168.547
1.25631 174-252
1.38939 152.641
I-37093 I57-908
1.35590 163.306
1.34478 168.808
1-33795 174-385
1.47141 153.136
1.45399 158.330
1.47983 163.639
1.41936 169.039
1.42294 174-503
1.55721 153.579
1.54077 158.706
1.52741 163.935
1.51755 169.243
1.51150 174.607
2.00
1.17520 180.000
1.25386 180.000
I-33565 180.000
1.42078 180.000
1.50946 180.000
Example.
sinh (1.20 + i 1.25)
= 1.76975^109^.050
= 1.76995 /i090.03'.oo*.
[94]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
= 1.25
x = 1.3
* = i-3S
x = 1.4
x = 1.45
o
0.05
O.I
0.15
O.2
1.60192 o.ooo
1.60384 5.301
1.60954 10.576
1.61884 15.803
1.63145 20.958
1.69838 o.ooo
1.70019 5.218
1.70557 10.415
1.71435 15.568
1.72627 20.659
1.79909 o.ooo
1.80080 5.145
1.80588 10.271
1.81417 15-359
1.82544 20.392
1.90430 o.ooo
1.90592 5.080
1.91072 10.143
1.91856 15.172
1.92921 20.153
2.01427 o.ooo
2.OI58O 5-021
2.02034 IO.028
2.02775 15.005
2.03784 19-939
0.25
o-35
0.4
0-45
1.64699 26.026
1.66501 30.991
1.68497 35.845
1.70635 40.580
1.72856 45-195
1.74096 25.673
1.75802 30.595
1.77694 35.418
1.79722 40.135
1.81832 44-745
1.83934 25.356
I.85549 30.240
I.87343 35-035
1.89268 39.735
1.91273 44-338
1.94237 25.073
1.95767 29.921
1.97468 34.689
1.99295 39.373
2.OI2OO 43.97O
2.O5O3O 24.8l8
2.06479 29.634
2.08093 34-379
2.09828 39.O47
2.II637 43.638
o-S
0-55
0.6
0.65
0.7
1.75104 49-693
1.77323 54-077
1.79462 58.354
1.81470 62.533
1.83304 66.625
1.83970 49.248
1.86084 53.648
1.88123 57.950
1.90040 62.163
1.91792 66.295
1.93306 48.845
i-953I9 53-258
1.97262 57-583
1.99091 61.825
2.00764 65.994
2.03135 48.480
2.05051 52.905
2.06903 57-249
2.08647 61.518
2.10244 65.719
2.13478 48.150
2.15302 52.584
2.17067 56.945
2.I873I 61.238
2.20254 65.469
0-75
0.8
0.85
0.9
o-95
1.84924 70.640
1.86297 74.590
1.87394 78.489
1.88193 82.348
1.88679 86.180
1.93341 70.356
1.94654 74-358
1.95704 78.311
1.96470 82.228
1.96935 86.120
2.02245 70.097
2.03500 74-145
2.04505 78.149
2.05238 82.118
2.05684 86.065
2.11658 69.861
2.12858 73.951
2.13819 78.00O
2.I452O 82.0l8
2.14947 86.014
2.2I6O4 69.645
2-22751 73-774
2.23669 77.864
2.24334 81.925
2.24747 85.968
I.O
1.05
i.i
i.i5
1.2
1.88842 90.000
1.88679 93.820
1.88193 97.652
1.87394 101.511
1.86297 105.410
1.97091 90.000
1.96935 93.880
1.96470 97-772
1.95704 101.689
1.94654 105.642
2.05833 90.000
2.05684 93-935
2.05238 97.882
2.04505 101.851
2.03500 105.855
2.15090 9O.OOO
2.14947 93.986
2.14520 97-982
2.13819 I02.OOO
2.12858 IO6.O49
2.24884 90.000
2.24747 94-032
2.24334 98-075
2.23669 102.136
2.22751 106.226
1.25
i-35
1.4
1-45
1.84924 109.360
1.83304 113.375
1.81470 117.467
1.79462 121.646
1.77323 125.923
1.93341 109.644
1.91792 113.705
1.90040 117.837
1.88123 122.050
1.86084 126.352
2.02245 109.903
2.00764 114.006
1.99091 118.175
1.97262 122.417
1.95319 126.742
2.11658 IIO.I39
2.IO244 II4.28I
2.08647 118.482
2.06903 122.751
2.05051 127.095
2.21604 110.355
2.20254 H4-531
2.18731 118.762
2.17067 123.055
2.15302 127.416
1-55
1.6
1.65
1.75104 130.308
1.72856 134.805
1.70635 139.420
1.68497 144.155
1.66501 149.009
1.83970 130.752
1.81832 135.255
1.79722 139.865
1.77694 144.582
1.75802 149-405
1.93306 131.155
1.91273 135.662
1.89268 140.265
1.87343 144-965
1.85549 149.760
2.03135 131.520
2.OI20O 136.030
1.99295 140.627
1.97468 145-3"
L95767 150.079
2.13478 131.851
2.11637 136.362
2.09828 140.953
2.08093 I45-62I
2.06479 150.366
1-75
1.8
1.85
1.9
1-95
1.64699 153.974
1.63145 159.042
1.61884 164.197
1.60954 169.424
1.60384 174.699
1.74096 154-327
1.72627 159-341
1.71435 164.432
1.70557 169.585
1.70019 174.782
1.83934 154-644
1.82544 159.608
1.81417 164.641
1.80588 169.729
1.80080 174.855
1.94237 I54.927
I.9292I 159.847
1.91856 164.828
1.91072 169.857
1.90592 174.920
2.05030 155.182
2.03784 160.061
2.02775 164.995
2.02034 169.972
2.01580 174.979
2.O
1.60192 180.000
1.69838 180.000
1.79909 180.000
1.90430 iSo.OOO
2.01427 180.000
Example.
sinh (1.45 + i 1.70)
= 2.06479 /I50°.366
= 2.06479 /I SO0. 2,2
'.01'.
[95]
TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r /V CONTINUED
x = 1.50
X = 1 .60
x = 1.65
a; = 1.70
o
2.12928
o.ooo
2.24961
o.ooo
2-37557
o.ooo
2.50746
o.ooo
2.64563
O.OOO
0.05
2.13073
4.969
2.25098
4-923
2.37686
4.881
2.50869
4.843
2.64679
4.809
O.I
2.13502
9-925
2.25504
9.833
2.38071
9-751
2.51234
9.677
2.65025
9.610
0.15
2.14204
14-855
2.26169
14.721
2.38701
14.600
2.51831
14.492
2.65591
14-395
0.2
2-15159
19.746
2.27074
19-574
2.39558
19.419
2.52644
19.280
2.66361
19.155
0.25
2.16340
24.590
2.28193
24-385
2.40619
24.200
2.53650
24.034
2.67316
23.884
°-3
2.17714
29.376
2.29496
29.144
2.41856
28.935
2.54824
28.747
2.68430
28.578
0-35
2.19245
34.099
2.30949
33-846
2-43235
33-6l9
2.56133
33-415
2.69673
33.229
0.4
2.20892
38.753
2.32513
38.488
2-44721
38.248
2-57544
38.032
2.71014
37.837
0-45
2.22612
43-337
2.34148
43.066
2.46274
42.820
2.59021
42.599
2.72418
42.398
0-5
2.24362
47.850
2.35812
47.580
2-47857
47-334
2.60527
47.113
2.73850
46.911
o-SS
2.26098
52-294
2.37465
52.030
2.49430
Si-791
2.62024
51.574
2.75274
51-378
0.6
2.27779
56.670
2.39066
56-420
2-50955
56.192
2.63476
55.986
2.76657
55-799
0.65
2.29365
60.984
2.40577
60.752
2-52395
60.542
2.64847
60.351
2.77963
60.178
0.7
2.30819
65.241
2.41976
65-033
2.53717
64.845
2.66108
64.673
2.79164
64-517
0-75
2.32107
69.448
2.43*93
69.268
2.54890
69.105
2.67226
68.956
2.80230
68.821
0.8
2.33202
73.611
2.44238
73-464
2-55887
73-329
2.68178
73.206
2.81138
73-094
0.85
2.34080
77.740
2.45076
77.626
2.56687
77-523
2.68941
77.429
2.81867
77-343
0.9
2.34720
81.842
2.45688
81.765
2.57272
81.695
2.69499
81.631
2.82399
Si-573
o-95
2.35110
85-925
2.46061
85-887
2.57627
85-851
2.69839
85.819
2.82723
85.790
I.O
2.35241
90.000
2.46186
90.000
2.57746
90.000
2.69951
90.000
2.82832
90.000
1.05
2.35110
94-075
2.46061
94-113
2.57627
94.149
2.69839
94.181
2.82723
94.210
i.i
2.34720
98.158
2.45688
98.235
2.57272
98-305
2.69499
98.369
2.82399
98.427
i-iS
2.34080
102.260
2.45076
102.374
2.56687
102.477
2.68941
102.571
2.81867
102.657
1.2
2.33202
106.389
2.44238
106.536
2.55887
106.671
2.68178
106.794
2.81138
106.906
1-25
2.32107
110.552
2.43*93
110.732
2.54890
110.895
2.67226
111.044
2.80230
111.179
i-3
2.30819
114-759
2.41976
114.967
2.53717
115-155
2.66108
"5-327
2.79164
115-483
i-3S
2.29365
119.016
2.40577
119.248
2-52395
119.458
2.64847
.119.649
2.77963
119.822
1.4
2.27779
123.330
2.39066
123.580
2-50955
123.808
2.63476
124.014
2.76657
124.201
1-45
2.26098
127.706
2.37465
127.970
2.49430
128.209
2.62024
128.426
2.75274
128.622
i-5
2.24362
132.150
2.35812
132.421
2.47857
132.666
2.60527
132.887
2.73850
133-089
i-55
2.22612
136.663
2.34148
136.934
2.46274
137.180
2.59021
137.401
2.72418
137.602
1.6
2.20892
141.247
2.32513
141.512
2.44721
141.752
2-57544
141.968
2.71014
142.163
1.65
2.19245
145.901
2.30949
146.154
2-43235
146.381
2-56133
146.585
2.69673
146.771
i-7
2.17714
150.624
2.29496
150.856
2.41856
151.065
2.54824
I5L253
2.68430
151.422
i-7S
2.16340
I55.4IO
2.28193
I55-6I5
2.40619
155.800
2.53650
155.966
2.67316
156.116
1.8
2-15159
160.254
2.27074
160.426
2.39558
160.581
2.52644
160.720
2.66361
160.845
1.85
2.14204
165.145
2.26169
165.279
2.38701
165.400
2.51831
165.508
2.65591
165.605
1.9
2.13502
170.075
2.25504
170.167
2.38071
170.249
2.51234
170.323
2.65025
170.390
i-95
2.13073
I75-03I
2.25098
175.077
2.37686
175-119
2.50869
I75-I57
2.64679
I75-I9I
2.O
2.12928
180.000
2.24961
180.000
2-37557
180.000
2.50746
180.000
2.64563
180.000
Example.
sinh (1.55
+ i 0.60)
= 2.39066 /56°.42O
= 2.39066 /56°. 25'.
12*.
[96]
TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r fy. CONTINUED
x =
i-75
* =
1.8
x =
1.85
x =
1.9
x =
i-95
9
r
7
r
V
r
T
r
•y
r
7
o
o
o
0
0
0
2.79041
o.oo
2.94217
o.oo
3
.10129
o.oo
3.26816
0.00
3-44321
o.oo
0.05
2.79151
4-779
2.94322
4.752
3
.10228
4.727
3.26911
4-705
3.44410
4.685
O.I
2.79479
9-551
2-94633
9-497
3
10523
9.448
3.27191
9-405
3-44675
9-365
0.15
2.80016
I4-307
2.95142
14.228
3
,11006
14.158
3.27649
14.094
3-45H2
14.036
0.2
2.80747
19.042
2-95835
18.941
3
11665
18.850
3.28274
!8.767
3-45702
18.693
0.25
2.81653
23-750
2.96695
23.629
3'
12481
23.520
3.29049
23.421
3.46441
23-332
o-3
2.82710
28.425
2-97699
28.287
3'
!3434
28.163
3-29955
28.051
3-47301
27.950
o-3S
2.83891
33-063
2.98821
32.912
3-
14500
32.776
3.30967
32-654
3.48263
32.543
0.4
2.85165
37.661
3.00031
37-501
3-
15650
37-357
3.32060
37-227
3-49302
37.110
0-45
2.86499
42.216
3.01300
42-053
3-
16856
41.904
3-33207
41.770
3-50393
41.649
o-5
2.87861
46.730
3-02595
46-565
3-
18088
46.416
3-34378
46.281
3-5J507
46.160
o-SS
2.89217
51.200
3-03885
5I-039
3-
I93I5
50-893
3-35546
50.761
3.52617
50.642
0.6
2.90532
55-630
3-05138
55-4/6
3'
20507
55-337
3.36681
55-211
3-53698
55.096
0.65
2.91777
60.020
3.06323
59-877
3-
21636
59-748
3-37756
59-63I
3-54721
59.524
0.7
2.92921
64-375
3-07413
64.246
3-
22675
64.1 29
3-38745
64.023
3-55663
63.927
0-75
2-93938
68.698
3.08382
68.586
3'
23598
68.484
3-39624
68.392
3-56500
68.308
0.8
2.94804
72.992
3.09206
72.900
3-
24384
72.816
3-40373
72.740
3-572I3
72.670
0.85
2.95498
77.265
3-09869
77-194
3-
25015
77.129
3-40975
77-071
3-57788
77.017
0.9
2.96006
81.520
3-J0353
81.471
3-
25477
81.428
3-4i4i5
81.388
3.58207
81.352
0-95
2.96315
85-763
3.10648
85-738
3-
25759
85-716
3.41683
85.696
3.58462
85.678
I.O
2.96419
90.000
3.10747
90.000
3-
25853
90.000
34I773
90.000
3-58548
90.000
1.05
2.96315
94-237
3.10648
94.262
3'
25759
94.284
3.41683
94-304
3.58462
94.322
i.i
2.96006
98.480
3-10353
98.529
3.
25477
98.572
3-41415
98.612
3.58207
98.648
i-i5
2.95498
102.735
3.09869
102.806
3
25015
102.871
340975
102.929
3-57788
102.983
1.2
2.94804
107.008
3.09206
107.100
3
24384
107.184
3.40373
107.260
3-572I3
107.330
I-2S
2-93938
111.302
3.08382
111.414
3
•23598
111.516
3-39624
111.608
3-56500
111.692
i-3
2.92921
115-625
3-074I3
115-754
3
.22675
115.871
3-38745
"5-977
3-55663
116.073
i-3S
2.91777
119.980
3-06323
120.123
3
.21636
120.252
3-37756
120.369
3-54721
120.476
i-4
2.90532
124.370
3-05138
124.524
3
.20507
124.663
3.36681
124.789
3-53698
124.904
i -45
2.89217
128.800
3-03885
128.961
3
•I93I5
129.107
3-35546
129.239
3.52617
129.358
i-5
2.87861
133.270
3-02595
133-435
3
.18088
133-584
3-34378
I33-7I9
3-5I507
133.840
i-SS
2.86499
I37-784
3.01300
137-947
3
.16856
138.096
3-33207
138.230
3-50393
138.351
1.6
2.85165
142.339
3.00031
142.499
3
•15650
142.643
3.32060
142.773
3-49302
142.890
1.65
2.83891
146.937
2.98821
147.088
3
.14500
147.224
3-30967
147.346
3.48263
147-457
i-7
2.82710
I5I-575
2-97699
I5L7I3
3
•13434
151-837
3-29955
151.949
3-47301
152.050
i-7S
2.81653
156.250
2-96695
156.371
3
.12481
156.480
3.29049
156.579
3.46441
156.668
1.8
2.80747
160.958
2-95835
161.059
3
.11665
161.150
3.28274
161.233
3-45702
161.307
1.85
2.80016
165.693
2.95142
165.772
3
.11006
165.842
3-27649
165.906
3-45ii2
165.964
1.9
2.79479
170.449
2-94633
170.503
3
•10523
170.552
3.27191
170.595
3-44675
170.635
1-95
2.79151
175.221
2.94322
175.248
3
.10228
I75.273
3.26911
I75.295
3.44410
J75-3I5
2.O 2.79041 iSo.OOO 2.94217 iSo.OOO 3.IOI29 iSo.OOO 3.26816 iSo.OOO 3.44321 iSo.OOO
Example. sinh (1.90 + * 2.0) = 3.26816 /i8o°.o = 3.26816 \i8o°.o.
[97]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
X = 2.O
* = 2.05
* = 2.1
* = 2.15
* = 2.2
g
r i
f 7
r 7
r V
f V
o
o
o
o
o
0
3.62686 o.ooo
2.81950 o.ooo
4.02186 o.ooo
4.23419 o.ooo
4.45711 o.ooo
0.05
3.62771 4-667
3.82039 4.651
4.02263 4.636
4.23491 4.623
4.45779 4.611
O.I
3.63023 9.330
3.82279 9.298
4.02490 9.269
4.23707 9.243
4.45985 9.220
0.15
3.63437 13.984
3.82671 13.938
4.02863 13.895
4.24061 13.857
4.46322 13.823
0.2
3.64000 18.626
3.83206 18.566
4.03371 18.511
4.24544 18.462
4.46780 18.418
0.25
3.64699 23.252
3.83870 23.180
4.04002 23.114
4.25145 23.055
4.47350 23.002
o-3
3.65517 27.858
3.84648 27.776
4.04740 27.701
4.25846 27.634
4.48017 27.573
0-35
3.66430 32.443
3.85515 32.353
4.05566 32.271
4.26630 32.197
4.48763 32.130
0.4
3.67418 37.004
3.86454 36.908
4.06459 36.821
4-27479 36.743
4.49570 36.671
0-45
3.68455 41.539
3.87440 41.440
4.07396 41-351
4.28371 41.270
4.50417 41.196
0-5
3.69515 46.049
3.88448 45-950
4.08355 45.859
4.29282 45.778
4.51285 45.703
o-SS
3.70572 50.533
3-89453 50-435
4.09311 50.347
4-3OI93 50-266
4.52150 50.193
0.6
3.71600 54-992
3.90432 54.898
4.10243 54.813
4.31079 54.736
4-52993 54-666
0.65
3.72574 59-427
3-91359 59-340
4.11125 59.260
4.31918 59.188
4-53793 59-123
0.7
3-73470 63.840
3.92213 63.761
4.11938 63.689
4.32693 63.624
4.54529 63.565
0-75
3.74268 68.232
3.92972 68.164
4.12661 68.101
4.33381 68.044
4.55184 67.993
0.8
3.74948 72.608
3.93620 72.551
4.13278 72-499
4.33968 72-452
4-55744 72.409
0.85
•3-75495 76-969
3.94142 76.925
4.13774 76.885
4.34440 76.849
4.56167 76.816
0.9
3.75894 81.319
3.94521 81.289
4.14136 81.262
4.34785 81.237
4.56523 81.215
o.QS
3.76137 85.661
3-94753 85.646
4.14357 85.632
4.34996 85.620
4-56723 85.609
1.0
3.76220 oo.ooo
3.94832 90.000
4.14431 90.000
4-35o67 go-000
4.56791 90.000
1.05
3.76137 94.339
3-94753 94-354
4-14357 94-368
4.34996 94-38o
4.56723 94.391
.1
3.75894 98.681
3.94521 98.711
4.14136 98-738
4.34785 98.763
4-56523 98-785
•IS
3.75495 103.031
3.94142 103.075
4.13774 103.115
4.34440 103.151
4.56167 103.184
.2
3.74948 107.392
3.93620 107.449
4.13278 107.501
4.33968 107.548
4-55744 107.59!
•25
3.74268 111.768
3.92972 111.836
4.12661 111.899
4.33381 111.956
4.55184 112.007
•3
3.73470 116.160
3.92213 116.239
4.11938 116.311
4.32693 116.376
4.54529 116.435
•35
3.72574 120.573
3.91359 120.660
4.11125 120.740
4.31918 120.812
4-53793 120.877
•4
3.71600 125.008
3.90432 125.102
4.10243 125.187
4.31079 125.264
4.52993 125.334
145
3.70572 129.467
3.89453 129.565
4.09311 129.653
4.30193 129.734
4.52150 129.807
i-5
3.69515 133.951
3.88448 134-051
4-08355 134-141
4.29282 134.223
4.51285 134-297
i-SS
3.68455 138.461
3.87440 138.560
4.07396 138.649
4.28371 138.730
4.50417 138.804
1.6
3.67418 142.996
3.86454 143.092
4.06459 143-179
4.27479 143-257
4.49570 143-329
1.65
3.66430 147-557
3.85515 147-647
4.05566 147-729
4.26630 147.803
4.48763 147.870
i-7
3.65517 152.142
3.84648 152.224
4.04740 152.299
4.25846 152.366
4.48017 152.427
•75
3.64699 156.748
3.83870 156.820
4.04002 156.886
4.25145 156.945
4.47350 156.998
.8
3.64000 161.374
3.83206 161.434
4.03371 161.489
4.24544 161.538
4.46780 161.582
•85
3.63437 166.016
3.82671 166.062
4.02863 166.105
4.24061 166.143
4.46322 166.177
•9
3.63023 170.670
3.82279 170.702
4.02490 170.731
4.23707 170.757
445985 170.780
•95
3.62771 175.333
3.82039 175-349
4.02263 175.364
4.23491 175-377
4-45779 175-389
2.0
3.62686 180.000
3.81958 180.000
4.02186 180.000
4.23419 180.000
4.45711 180.000
Example.^ sinh (2.0 + i i.o) = 3.76220 /go0.
[98]
TABLE X. HYPERBOLIC SINES, sinh (x + ig) = r fy. CONTINUED
* = 2.25
x = 2.3
* = 2.35
x = 2.4
x = 245
0
4.69117
o.ooo
4.93696
o.ooo
5
19510
o.ooo
5.46623
o.ooo
5-75*03
o.ooo
0.05
4.69182
4.601
4-93759
4-591
5
19569
4.582
5.46679
4-574
5.75I56
4-567
O.I
4-69377
9.199
4-93944
9.180
5
I974S
9.163
5.46847
9.147
5-753i6
9-133
0.15
4.69697
13-792
4.94249
13-764
5-
20035
13.739
5.47122
13.716
5-75576
13-695
0.2
4.70134
18.378
4.94662
18.341
5
20428
18.309
5-47495
18.279
5-75932
18.252
0.2S
4.70675
22.954
4-95I77
22.910
5
20917
22.871
5.47961
22.835
5-76375
22.803
o-3
4.71308
27.518
4.95780
27.469
•5
21490
27.424
5-48505
27-383
5-76892
27-347
0-35
4.72017
32.070
4.96454
32.016
5
22131
31.966
549II5
31.922
5.77472
31.881
0.4
4.72784
36.608
4.97183
36.549
5
22825
36-497
549775
36449
5.78099
36.407
°4S
4-73592
41.130
4-97950
41.070
5
23553
41.016
5.50468
40.966
5.78758
40.922
°S
4.74415
45-636
4-98734
45-576
5
23728
45.521
5.5"77
45471
5-79434
45427
o-SS
4.75240
50.127
4.99518
50.068
5
25046
50.014
5-51887
49.965
5.80108
49.921
0.6
4.76042
54-603
5.00281
54-546
5
25772
54-494
5.52578
54-447
5.80765
54405
0.65
4.76802
59.064
5.01005
59-on
5
26461
58.962
5.53233
58.919
5-81389
58.879
0.7
4-77503
63.512
5.01672
63-463
5
27096
63.419
5.53837
63.380
5.81964
63-344
0-75
4.78127
67.947
5.02265
67.904
5
27662
67.866
5-54375
67-831
5.82476
67.800
0.8
4.78661
72.371
5-02773
72.336
5
28143
72.304
5-54834
72.275
5-82913
72.249
0.85
4.79088
76.786
5.03181
76.759
5
28531
76.735
5-55204
76.712
5-83265
76.692
0.9
4.79402
81.195
5-03479
81.176
5
28816
81.160
5-55474
81.144
5-83522
81.131
o-95
4-79593
85-599
5.03661
85-589
5
28989
85.581
5.55639
85-573
5-83680
85.566
1.0
4-79657
90.000
5.03722
90.000
5
29047
90.000
5.55695
90.000
5.83732
90.000
1.05
4-79593
94.401
5.03661
94.411
5
28989
94.419
5.55639
94.427
5.83680
94434
i.i
4.79402
98.805
5-03479
98.824
5
28816
98.840
5-55474
98.856
5-83522
98.869
i-iS
4.79088
103.214
5.03181
103.241
5
28531
103.265
5-55204
103.288
5-83265
103.308
1.2
4.78661
107.629
5-02773
107.664
5
28143
107.696
5.54834
107.725
5-82913
107.751
I-2S
4.78127
112.053
5.02265
112.096
5
27662
112.134
5-54375
112.169
5.82476
112. 2OO
i-3
4-77503
116.488
5.01672
"6.537
5
27096
116.581
5-53837
116.620
5.81964
116.656
i-35
4.76802
120.936
5.01005
120.989
5
26461
121.038
5-53233
121.081
5-81389
121. 121
14
4.76042
125-397
5.00281
125-454
5
25772
125.506
5.52578
125.553
5-80765
125.595
145
4.75240
129.873
4.99518
129.932
5
25046
129.986
5-51887
130-035
5.80108
130.079
-5
4-744I5
134-364
4-98734
134.424
5
23728
134-479
5-5"77
134.529
5-79434
134-573
•55
4-73592
138.870
4-9795°
138.930
5
23553
138.984
5.50468
I39.034
5-78758
139.078
.6
4.72784
143-392
4.97183
I4345I
5
22825
143-503
549775
I43.55I
5.78099
143-593
.65
4.72017
147.930
4-96454
147.984
S
22131
148.034
549"5
148.078
5-77472
I48.II9
•7
4.71308
152.482
4.95780
152-531
5
21490
152.576
548505
152.617
5-76982
152.653
•75
4-70675
157.046
4.95I77
157.090
5
20917
157.129
5.47961
157-165
5.76375
I57-I97
.8
4-70I34
161.622
4.94662
161.659
5
20428
161.691
547495
161.721
5-75932
161.748
•85
4.69697
166.208
4.94249
166.236
5
20035
166.261
5.47122
166.284
5.75576
166.305
•9
4.69377
170.801
4-93944
170.820
5
I974S
170.837
5.46847
170.853
5-753i6
170.867
1-95
4.69182
175.399
4-93759
175.409
5
19569
175.418
5.46679
175.426
5-75*56
175433
2.O
4.69117
180.000
4.93696
180.000
5
19510
180.000
546623
180.000
5-75I03
iSo.OOO
Example. sinh (2.40 + * 0.4) = 549775 /36°-449 = 5-49775 /36°. 26^.56'.
[99]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
x = 2.5
= 2.55
* = 2.6
x = 2.65
x = 2.7
o
0.05
O.I
0.15
O.2
6.05020
6.05071
6.05223
6.05471
6.05808
o.ooo
4.561
9.120
13.676
18.228
6.36451
6.36499
6.36644
6.36879
6.37201
o.ooo
4-555
9.109
13.660
18.206
6.69473
6.69518
6.69656
6.69880
6.70185
o.ooo
4-550
9.098
13.644
18.187
7.04169
7.04213
7-04343
7-04556
7.04847
0.000
4-545
9.089
13-631
18.169
7.40626
7.40668
7.40791
7.40994
7.41271
o.ooo
4-540
9.080
13.618
18.153
O.25
o-3
0-35
0.4
0-45
6.06229
6.06722
6.07273
6.07869
6.08496
22.774
27-314
3!-845
36.368
40.882
6.37601
6.38068
6.38592
6.39160
6-39755
22.748
27.284
31.812
36.333
40.845
6.70565
6.71011
6.71509
6.72048
6.72616
22.725
27.257
31.782
36.301
40.813
7.05208
7.05631
7.06105
7.06618
7.07158
22.703
27.232
3I-756
36.273
40.783
7.41614
7.42017
7-42467
742955
7.43468
22.684
27.210
36.246
40.756
o-S
0-55
0.6
0.65
0.7
6.09139
6.09781
6.10406
6.10999
6.11547
45-386
49.881
54-366
58-843
63.3"
6.40367
6.40977
6.41572
6.42137
6.42658
45-350
49.845
54-332
58.810
63.282
6.73197
6.73778
6-74343
6.74881
6.75376
45-316
49.812
54-300
58.781
63.255
7.07711
7.08264
7.08802
7-09313
7.09784
45.286
49.782
54.272
58.754
63.230
7-43994
7-445I9
745032
7-455I8
7.45966
45-259
49-755
54.246
58.730
63.209
o-75
0.8
0.85
0.9
0-95
6.12034
6.12450
6.12785
6.13030
6.13179
67.772
72.226
76.674
81.119
85.560
6.43121
6.43518
6.43836
6.44069
6.44212
67.746
72.204
76.658
81.107
85-554
6.75818
6.76195
6.76499
6.76720
6.76855
67.722
72.185
76.643
81.097
85.549
7.10204
7.10563
7.10851
7.11062
7.11191
67.701
72.167
76.629
81.088
85-544
7.46366
7.46708
7.46982
7-47I83
7.47306
67.682
72.152
76.617
81.080
85-540
I.OO
1.05
I.I
1-15
1.2
6.13229
6.13179
6.13030
6.12785
6.12450
90.000
94.440
98.881
103.326
107.774
6.44259
6.44212
6.44069
6.43836
6.43518
90.000
94-446
98-893
103.342
107.796
6.76901
6.76855
6.76720
6.76499
6.76195
90.000
94.451
98.903
103.357
107.815
7.11234
7.11191
7.11062
7.10851
7.10563
90.000
94.456
98.912
103.371
107.833
7-47347
7.47306
7-47i83
7.46982
7.46708
90.000
94.460
98.920
103-383
107.848
1-25
i-35
1.4
1-45
6.12034
6.11547
6.10999
6.10406
6.09781
112.228
116.689
121.157
125.634
130.119
6.43121
6.42658
6.42137
6.41572
6.40977
112.254
116.718
121.190
125.668
130.155
6.75818
6.75376
6.74881
6-74343
6-73778
112.278
116.745
121.219
125.700
130.188
7.10204
7.09784
7-093I3
7.08802
7.08264
112.299
116.770
121.246
125.728
130.218
7.46366
7.45966
745032
7-445I9
112.318
116.791
121.270
125.754
130.245
i-55
1.6
1-65
6.09139
6.08496
6.07869
6.07273
6.06722
134.614
139.118
143.632
148.155
152.686
6.40367
6-39755
6.39160
6.38592
6.38068
134-651
i39-I55
143.667
148.188
152.716
6.73197
6.72616
6.72048
6.71509
6.71011
134.684
139.187
143.699
148.218
7.07711
7.07158
7.06618
7.06105
7.05631
I34-7I3
139.217
I43-727
148.244
152.768
7-43994
7.43468
7-42955
7.42467
7.42017
I34.74I
139.244
143-754
148.269
152.790
i-75
1.8
1-85
1.9
1-95
6.06229
6.05808
6.05471
6.05223
6.05071
157.226
161.772
166.324
170.880
175-439
6.37601
6.37201
6.36879
6.36644
6.36499
157-252
161.794
166.340
170.891
175-445
6.70565
6.70185
6.69880
6.69656
6.69518
157-275
161.813
166.356
170.902
175-450
7.05208
7.04847
7-04566
7-04343
7.04213
157.297
161.831
166.369
170.911
175-455
7.41614
7.41271
7.40994
7.40791
7.40668
157.316
161.847
166.382
170.920
175.460
2.0
6.05020
180.000
6.36451
180.000
6.69473
180.000
7.04169
180.000
7.40626
180.000
Example.
sinh (2.6
+ io.6)
= 6.74343
/54°-300 = 6.74343 /54°-i8'oo".
[ 100 ]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
x = 2.75
= 2.85
* = 2.9
x = 2.95
0
7.78935
o.ooo
8.19192
o.ooo
8.61497
o.ooo
9-05956
o.ooo
9.52681
o.ooo
o.os
7.78975
4-537
8.19230
4-533
8.61532
4.530
9.05990
4-527
9-52713
4.525
O.I
7.79092
9-073
8.19341
9.066
8.61639
9.060
9.06091
9-054
9-52809
9.049
0.15
7.79285
13.607
8.19524
I3-596
8.61813
13-587
9.06257
13.579
9.52967
I3.57I
0.2
7-79547
18.138
8.19774
18.125
8.62051
18.113
9.06483
18.102
9.53182
18.093
0.25
7-79875
22.666
8.20085
22.650
8.62347
22.636
9.06764
22.623
9-53450
22.611
7.80257
27.190
8.20449
27.172
8.62691
27-155
9.07093
27.141
27.127
0-35
7.80685
31.709
8.20857
31.689
8.63080
31.671
9.07461
31.655
9-54"3
31.640
0.4
7.81149
36.223
8.21298
36.202
8.63500
36.183
9.07861
36.165
9-54492
36-150
0-45
7-81637
40.731
8.21762
40.710
8.63941
40.690
9.08281
40.671
9.54892
40.655
o-5
7.82138
45-234
8.22238
45.212
8.64395
45.192
9.08711
45.173
9-55301
45-157
0.55
7.82638
49-731
8.22713
49.709
8.64846
49.689
9.09142
49.671
9-557"
49.655
0.6
7-83125
54-222
8.23177
54-201
8.65287
54.182
9.09561
54.165
9.56109
54-149
0.65
7-83588
58.708
8.23617
58.688
8.65706
58.670
9.09959
58.654
9.56488
58.640
0.7
7-84015
63.189
8.24024
63.171
8.66092
63-155
9.10327
63.140
9-56838
63.127
0-75
7.84395
67-665
8.24385
67.649
8.66436
67.636
9.10655
67.622
9.57I50
67.611
0.8
7.84720
72.137
8.24694
72.124
8.66731
72.112
9.10934
72.102
9.57416
72.092
0.85
7.84980
76.606
8.24942
76-596
8.66967
76.587
9.11160
76.579
9-57630
76.571
0.9
7.85172
8l.O72
8.25124
81.065
8.67140
81.059
9.11324
81.053
9-57787
81.048
0-95
7.85288
85-536
8-25235
85-533
8.67246
85-530
9.11424
85.527
9.57882
85-524
I.O
7-85328
90.000
8.25273
90.000
8.67281
90.000
9.11458
90.000
9-579I5
90.000
1.05
7.85288
94.464
8-25235
94467
8.67246
94-470
9.11424
94-473
9-57882
94.476
i.i
7.85172
98.928
8.25124
98.935
8.67140
98.941
9.11324
98.947
9-57787
98.952
1.15
7.84980
103.394
8.24942
103.404
8.66967
103.413
9.11160
103.421
9-57630
103.429
1.2
7.84720
107.863
8.24694
107.876
8.66731
107.888
9.10934
107.898
9.57416
107.908
1-25
7.84395
"2-335
8.24385
"2.351
8.66436
112.364
9.10655
112.378
9-57I50
112.389
7.84015
116.811
8.24024
116.829
8.66092
116.845
9.10327
116.860
9.56838
116.873
1-35
7-83588
121.292
8.23617
121.312
8.65706
121.330
9.09959
121.346
9.56488
121.360
7-83125
125.778
8.23177
125.799
8.65287
125.818
9.09561
125-835
9.56109
125-851
1-45
7.82638
130.269
8.22713
130.291
8.64846
130.311
9.09142
130.329
9-557"
I30.345
i-5
7.82138
134.766
8.22238
134.788
8.64395
134.808
9.08711
134.827
9-55301
134-843
1-55
7.81637
139.269
8.21762
139.290
8.63941
139.310
9.08281
139-329
9.54892
139-345
1.6
7.81149
143-777
8.21298
143.798
8.63500
143.817
9.07861
143-835
9.54492
143.850
1.65
7.80685
148.291
8.20857
148.311
8.63080
148.329
9.07461
148-345
9-54"3
148.360
7-80257
152.810
8.20449
152.828
8.62691
152.845
9.07093
152.859
152-873
i-75
7-79875
157-334
8.20085
I57-350
8.62347
157-364
9.06764
157-377
9-53450
I57-389
1.8
7-79547
161.862
8.19774
161.875
8.62051
161.887
9.06483
161.898
9.53182
161.907
1-85
7.79285
166.393
8.19524
166.404
8.61813
166.413
9.06257
166.421
9-52967
166.429
1.9
7.79092
170.927
8.19341
170.934
8.61639
170.940
9.06091
1 70.946
9.52809
170.951
1-95
7-7897S
175-463
8.19230
175467
8.61532
175.470
9.05990
175-473
9-52713
175-475
2.0
7-7893S
180.000
8.19192
180.000
8.61497
180.000
9-05956
180.000
9.52681
180.000
Example.
sinh (2.95
+ i 1.95)
= 9-52713
/i75°.475
i = 9-527I3 /I7S°-28'.30*.
[I0t]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED
* = 3-o
* = 3-05
x = 3.1
* = 3-15
X = 3.2
9
r y
r y
r y
f y
r y
o
0.05
O.I
0.15
O.2
0
10.01787 o.ooo
10.01820 4.522
10.01910 9.044
10.02060 13.565
10.02260 18.084
0
10.53399 o.ooo
10.53430 4.520 -
10.53520 9.040
10.53660 13.559
I0-538so 18.076
o
11.07645 o.ooo
11.07670 4.518
11.07750 9.036
11.07890 13.553
11.08080 18.068
0
11.64661 o.ooo
11.64690 4.516
11.64770 9.033
11.64890 13.548
11.65070 18.062
0
12.24588 o.ooo
12.24610 4.515
12.24690 9.030
12.24810 13.543
12.24980 18.056
0.25
0-35
0.4
0-45
10.02520 22.601
10.02820 27.115
10.03150 31.627
10.03510 36.135
10.03890 40.640
10.54090 22.591
10.54380 27.104
10.54690 31.615
10.55040 36.122
10.55400 40.627
11.08310 22.583
11.08580 27.094
11.08880 31.604
11.09200 36.111
11.09540 40.615
11.65290 22.575
11.65540 27.085
11.65830 31.594
11.66140 36.100
11.66470 40.604
12.25180 22.568
12.25430 27.077
12.25700 31.585
12.26000 36.091
12.26310 40.594
o-S
0-55
0.6
0.65
o-7
10.04280 45.142
10.04670 49.640
10.05050 54.135
10.05410 58.626
10.05743 63.114
10.55770 45-129
10.56140 49.627
10.56500 54-122
10.56840 58.614
10.57160 63.104
11.09900 45.116
11.10250 49.615
11.10600 54.110
11.10920 58.604
II.II22O 63.094
11.66810 45.105
11.67140 49.604
11.67470 54-100
11.67780 58.594
11.68060 63.085
12.26630 45.095
12.26950 49.594
12.27260 54.090
12.27550 58.585
12.27820 63.077
0-75
0.8
0.85
0.9
0-95
10.06040 67.600
10.06292 72.083
10.06500 76.564
10.06645 81.044
10.06737 85.522
10.57440 67.591
10.57680 72.075
10.57880 76.558
10.58020 81.040
10.58110 85.520
II.II49O 67.582
II.II72O 72.068
II.II900 76.553
II.I2O4O 81.036
II.I2I2O 85.518
11.68320 67.574
11.68540 72.062
11.68710 76.548
11.68840 81.032
11.68920 85.516
12.28070 67.567
12.28280 72.056
12.28440 76.543
12.28560 81.029
12.28640 85.515
1.0
1-05
i.i
1.2
10.06766 90.000
10.06737 94.478
10.06645 98.956
10.06500 103.436
10.06292 107.917
10.58135 90.000
10.58110 94.480
10.58020 98.960
10.57880 103.442
10.57680 107.925
II.I2I50 9O.OOO
II.I2I2O 94.482
II.I2O4O 98.964
II.II900 103.447
II.II72O .107.932
11.68946 90.000
11.68920 94.484
11.68840 98.968
11.68710 103.452
11.68540 107.938
12.28665 90.000
12.28640 94.485
12.28560 98.971
12.28440 103.457
12.28280 107.944
1-25
1.4
i-45
10.06040 112.400
10.05743 116.886
10.05410 121.374
10.05050 125.856
10.04670 130.360
10.57440 112.409
10.57160 116.896
10.56840 121.386
10.56500 125.878
10.56140 130.373
II.II49O II2.4I8
II.II22O 116.906
II.I092O 121.396
II.I0600 125.890
11.10250 130.385
11.68320 112.426
11.68060 116.915
11.67780 121.406
11.67470 125.900
11.67140 130.396
12.28070 112.433
12.27820 116.923
12.27550 121.415
12.27260 125.910
12.26950 130.406
i-S
I.63
1-65
10.04280 134.858
10.03890 139.360
10.03510 143.865
10.03150 148.373
10.02820 152.885
10.55770 134.871
10.55400 139.373
10.55040 143.878
10.54690 148.385
10.54380 152.896
II.O99OO 134.884
11.09540 139.385
II.O92OO 143.889
II.08880 148.396
11.08580 152.906
11.66810 134.895
11.66470 139.396
11.66140 143.900
11.65830 148.406
11.65540 152.915
12.26630 134.905
12.26310 139.406
12.26000 143.909
12.25700 148.415
12.25430 152.923
i!s
1.85
1.9
1.95
10.02520 157.399 •
10.02260 161.916
10.02060 166.435
10.01910 170.956
10.01820 175.478
10.54090 157.409
10.53850 161.924
10.53660 166.441
10.53520 170.960
10.53430 175-480
II.083IO 157.417
II.o8o8o 161.931
11.07890 166.447
11.07750 170.964
11.07670 175.482
11.65290 157-425
11.65070 161.938
11.64890 166.452
11.64770 170.967
11.64690 175.484
12.25180 157.432
12.24980 161.944
12.24810 166.457
12.24690 170.970
12.24610 175.485
2.O
10.01787 180.000
Jo.53399 180.000
11.07645 iSo.OOO
11.64661 180.000
12.24588 180.000
Example.
sinh (3.2 + i i.i) =
12.28560 /98°.97i =
12.28560 /98°.58'.i6
t
[102]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /T. CONTINUED
= 3.25
= 3-3
= 3-35
* = 3-4
* = 3-45
o
12.87578 o.ooo
13-53788 o.ooo
14.23382 o.ooo
14.96536 o.ooo
I5-73432 o-ooo
0.05
12.87600 4.514
13.53810 4.512
14.23403 4.511
14.96556 4.510
15-73450 4-509
O.I
12.87670 9.027
13.53878 9.024
14.23470 9.022
14.96616 9.020
15.73510 9.018
0.15
12.87790 13.539
13.53991 13.536
14.23573 13.532
14.96716 13.529
15.73603 13-526
O.2
12.87949 18.051
13.54141 18.046
14.23718 18.041
14.96855 18.038
15.73736 18.034
0.25
12.88146 22.561
13.54322 22.555
14.23897 22.550
14.97023 22.545
15.73898 22.541
o-3
12.88380 27.070
13.54550 27.063
14.24106 27.057
14.97225 27.052
15.74088 27.047
o-3S
12.88637 31.577
13.54796 31.569
14.24339 31.563
14.97448 31.557
15.74300 31.551
0.4
12.88910 36.082
13.55062 36.074
14.24595 36.067
14.97690 36.061
15.74529 36.055
o-45
12.89215 40.585
13.55346 40.577
14.24863 40.570
14.97945 40.563
15.74772 40.557
o-S
12.89518 45.086
13-55633 45.078
14.25137 45.071
14.98205 45.064
15.75020 45.058
0-55
12.89820 49.585
13.55918 49-577
14.25412 49.570
14.98466 49-563
15.75270 49-557
0.6
12.90117 54.082
13.56203 54-074
14.25614 54.067
14.98721 54.061
I5-755IO 54-055
0.65
12.90398 58.577
13.56470 58.569
14.25933 58.563
14.98960 58.557
I5.75740 58.551
0.7
12.90658 63.070
13.56718 63.063
14.26167 63.057
14.99186 63.052
I5-75950 63.047
0-75
12.90888 67.561
13.56936 67.555
14.26377 67.550
14.99385 67.545
15.76144 67.541
0.8
12.91085 72.051
I3-57I23 72-046
14.26556 72.041
14-99555 72.038
15-76303 72-034
0.85
12.91240 76.539
I3-57275 76.535
14.26700 76.532
14.99692 76.529
15-76433 76-526
0.9
12.91360 81.027
13-57387 81.024
14.26805 81.022
14.99790 81.020
15-76530 81.018
o-9S
12.91430 85.514
13-57455 85.512
14.26870 85.511
14.99853 85.510
15-76587 85.509
I.O
12.91456 90.000
I3-57476 90.000
14.26891 90.000
14.99874 90.000
15.76607 90.000
1.05
12.91430 94.486
13-57455 94488
14.26870 94.489
14.99853 94.490
15-76587 94-491
i.i
12.91360 98.973
I3-57387 98-976
14.26805 98.978
14.99790 98.980
15-7653° 98-982
i-iS
12.91240 103.461
I3-5727S 103.465
14.26700 103.468
14.99692 103.471
I5-76433 103.474
1.2
12.91085 107.949
13.57123 107.954
14.26556 107.959
14.99555 107.963
15-76303 107.966
1-25
12.90888 112.439
13-56936 112.445
14.26377 112.450
14.99385 112.455
15.76144 112.459
J-3
12.90658 116.930
13.56718 116.937
14.26167 116.943
14.99186 116.948
I5-75950 "6.953
i-3S
12.90398 121.423
13.56470 121.431
14.25933 121.437
14.98960 121.443
I5-75740 121.449
1.4
12.90117 125.918
13.56203 125.926
14.25614 125.933
14.98721 125.939
15.75510 125.945
i-4S
12.89820 130.415
13.55918 130.423
14.25412 130.430
14.98466 130.437
I5-75270 130.443
J-5
12.89518 134.914
I3-55633 I34-922
14.25137 134.929
14.98205 134.936
15.75020 134-942
i-55
12.89215 139.415
13.55346 139-423
14.24863 139.430
14.97945 139.437
15.74772 139-443
1.6
12.88910 143.918
13.55062 143.926
14.24595 143.933
14.97690 143.939
I5.74529 143-945
i.6S
12.88637 148.423
13.54796 148.431
14.24339 148.437
14.97448 148.443
15.74300 148.449
i-7
12.88380 152.930
I3.54550 152.937
14.24106 152.943
14.97225 152.948
15.74088 152.953
i-75
12.88146 157.439
13.54322 157-445
14.23897 157.450
14.97023 157.455
15.73898 157-459
1.8
12.87949 161.949
13.54141 161.954
14.23718 161.959
14.96855 161.962
I5-73736 161.966
i.8S
12.87790 166.461
I3-5399I 166.464
14.23573 166.468
14.96716 166.471
15-73603 166.474
1.9
12.87670 170.973
13.53878 170.976
14.23470 170.978
14.96616 170.980
15.73510 170.982
i-95
12.87600 175.486
1^.53810 175.488
14.23403 175-489
14.96556 175.490
I5-73450 I75-49I
s.o
12.87578 180.000
13.53788 180.000
14.23382 180.000
14.96536 180.000
I5-73432 180.000
Example. sinh (3.3 + i 1.3) = 13.56718,
= 13.56718 /i i60..s6'.i3*.
[103]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r {y. CONTINUED
= 3-5
= 3-55
x = 3.6
* = 3-65
= 3-7
o
16.54263
0.000
17.39230
o.ooo
18.28546
o.ooo
19.22434
o.ooo
20.21129
o.ooo
0.05
16.54281
4.508
17.39248
4-507
18.28560
4-507
19.22448
4.506
20.21140
4.506
O.I
16.54337
9.016
17.39300
9.015
18.28611
9.013
19.22496
9.012
20.21189
9.011
0.15
16.54428
13.524
17.39386
13-522
18.28694
13-519
19.22577
13.518
20.21268
13-516
0.2
16.54550
18.031
I7-39504
18.028
18.28805
18.025
19.22681
18.023
20.21365
18.021
O.25
16.54702
22.537
17.39650
22.534
18.28942
22.530
19.22813
22.527
20.21490
22.525
o-3
16.54885
27.042
17.39822
27.038
18.29109
27-035
19.22968
27.031
20.21639
27.028
o-3S
16.55087
31-547
17.40015
31-542
18.29292
31.538
19.23142
3I-S3S
20.21804
3I-S31
04
16.55307
36.050
17.40222
36.045
18.29490
36.041
I9-23331
36.037
20.21981
36-033
0-45
16.55538
40-552
17.40441
40.547
18.29699
40.542
I9-23530
40.538
20.22170
40-535
o-S
16.55774
45-052
17.40665
45-047
18.29912
45-043
19.23732
45-039
20.22365
45-035
0-55
16.56010
49-552
17.40900
49-547
18.30126
49.542
19.23940
49.538
20.22560
49-535
0.6
16.56240
54-050
17.41110
54-045
18.30335
54.041
19.24134
54-037
20.22750
54-033
0.65
16.56460
58.547
17.41310
58.542
18.30530
58.538
19.24323
58.535
20.22930
58.531
0.7
16.56660
63.042
17.41510
63.038
18.30715
63-035
19.24496
63-031
20.23090
63.028
0-7S
16.56840
67-537
17.41680
67-533
18.30880
67-530
19.24650
67-527
20.23240
67-525
0.8
16.56996
72.031
17.41830
72.028
18.31020
72.025
19.24790
72.023
20.23365
72.021
0.85
16.57120
76.524
17.41945
76.521
18.31130
76.519
19.24890
76.518
20.23460
76.516
0.9
16.57210
81.016
17.42030
81.015
18.31210
81.013
19.24970
81.012
20.23540
81.011
o-9S
16.57260
85.508
17.42083
85-507
18.31260
85-507
19.25015
85-506
20.23585
85.506
I.O
16.57282
90.000
17.42102
90.000
18.31278
90.000
19-25033
90.000
20.23601
90.000
1.05
16.57260
94.492
17.42083
94-493
18.31260
94-493
19.25015
94-494
20.23585
94.494
i.i
16.57210
98.984
17.42030
98.985
18.31210
98.987
19.24970
98.988
20.23540
98.989
"S
16.57120
103.476
I7-4I945
103.479
18.31130
103.480
19.24890
103.482
20.23460
103.484
1.2
16.56996
107.969
17.41830
107.972
18.31020
107.975
19.24790
107.977
20.23365
107.979
1-25
16.56840
112.463
17.41680
112.467
18.30880
112.470
19.24650
112.473
20.23240
"2.475
i-3
16.56660
116.958
17.41510
116.962
18.30715
116.965
19.24496
116.969
20.23090
116.972
1-35
16.56460
121.453
17.41310
121.458
18.30530
121.462
19.24323
121.465
20.22930
121.469
1.4
16.56240
125.950
17.41110
125-955
18.30335
125-959
19.24134
125.963
20.22750
125.967
i-4S
16.56010
130.448
17.40900
130.453
18.30126
130.458
19.23940
130.462
20.22560
130.465
i-5
16.55774
134.948
17.40665
134-953
18.29912
134-957
19.23732
134.961
20.22365
I34-965
i-SS
16.55538
139.448
17.40441
139-453
18.29699
I39-458
19.23530
139.462
20.22170
139-465
1.6
16.55307
I43-950
17.40222
143-955
18.29490
143-959
i9-2333i
143-963
20.21981
143.967
1.65
16.55087
148.453
17.40015
148.458
18.29292
148.462
19.23142
148.465
20.21804
148.469
i-7
16.54885
152.958
17.39822
152.962
18.29109
152.965
19.22968
152.969
20.21639
152.972
i-7S
16.54702
IS7-463
17.39650
157.466
18.28942
I57-470
19.22813
157-473
20.21490
157-475
1.8
16.54550
161.969
17.38504
161.972
18.28805
161.975
19.22681
161.977
20.21365
161.979
1.85
16.54428
166.476
17.39386
166.478
18.28694
166.481
19.22577
166.482
20.21268
166.484
1.9
16.54337
170.984
17.39300
170.985
18.28611
170.987
19.22496
170.988
20.21189
170.989
1-95
16.54281
175.492
17.39248
175-493
18.28560
175-493
19.2244^
175-494
20.21140
175-494
2.O
16.54263
180.000
17.39230
180.000
18.28546
180.000
19.22434
180.000
20.21129
180.000
Example. sinh (3.65 -+- to. 25) = 19.22813 /22°.527 = 19.22813 /22°.3i'.37*
[104]
TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r fy. CONTINUED
X =
3-75
x =
3-8
x =
3.85
x =
3-9
* =
3-95
9
r
7
r
y
r
•y
r
7
r
7
o
o
o
o
o
o
0.05
O.I
0.15
0.2
21
21
21
21
21
.24878
.24891
•24935
.25006
.25102
O.OOO
4.505
9.010
13-514
18.019
22.33941
22.33952
22-33995
22.34061
22.34153
O.OOO
4-504
9.009
I3-5I3
I8.0I7
23.48589
23.48601
23.48640
23.48704
23.48791
O.OOO
4.504
9.008
I8.0I5
24.69110
24.69120
24.69159
24.69223
24.69302
O.OOO
4.504
9.007
13-511
18.014
25.95806
25.95820
25-95854
25-95911
25-95991
O.OOO
4.503
9.007
18.013
0.25
0-3
0-35
0.4
0-45
21
21
21
21
21
.25221
•25362
.25520
.25685
.25869
22.522
27.026
31-528
36.030
40.531
22.34270
22.34401
22.34550
22.34712
22.34883
22.52O
27.023
3I-526
36.027
40.528
23.48900
23.49028
23-49H5
23.49322
23.49486
22.518
27.O2I
3I-523
36-025
40.526
24.69406
24.69528
24.69662
24.69809
24.69964
22.517
27.019
31-521
36.022
40.523
25.96090
25.96205
2S-96333
25.96471
25.96618
22.515
27.017
31.519
36.020
40.521
o-S
0-55
0.6
0.65
0.7
21
21
21
21
21
.26052
.26236
.26415
.26586
.26745
45-032
49-531
54-030
58.528
63.026
22.35060
22.35230
22-35403
22.35565
22.35716
45.029
49.528
54-027
58-526
63.023
23.49652
23.49820
23.49980
23.50136
23-50277
45.026
49.526
54-025
58.523
63.021
24.70121
24.70280
24.70440
24.70580
24.70720
45.023
49-523
54-022
58.521
63.019
25.96770
25.96920
25.97066
25.97206
25-97337
45-021
49-521
54-020
58-519
63.017
o-75
0.8
0.85
0.9
o-95
21
21
21
21
21
.26885
.27000
.27100
.27170
.27212
67-522
72.019
76.514
81.010
85.505
22.35850
22.35962
22.36055
22.36122
22.36163
67.520
72.017
76-513
81.009
85-504
23.50404
23-50512
23.50600
23.50664
23.50702
67.518
72.015
76.512
8I.008
85.504
24.70840
24.70940
24.71024
24.71090
24.71120
72.014
76.511
81.007
85.504
25.97450
25-97550
25-97630
25.97680
25.97720
67-5I5
72-013
76.510
81.007
85-503
I.O
1.05
i.i
1.2
21
21
21
21
21
.27230
.27212
.27170
.27100
.27000
90.000
94-495
98.990
103.486
107.981
22.36178
22.36163
22.36122
22.36055
22.35962
90.000
94.496
98.991
103.487
107.983
23-50717
23.50702
23.50664
23.50600
23-50512
90.000
94.496
98.992
103.488
107.985
24-71135
24.71120
24.71090
24.71024
24.70940
90.000
94.496
98.993
103.489
107.986
25.9773I
25.97720
25.97680
25.97630
25-97550
90.000
94-497
98.993
103.490
107.987
1.25
1.4
1-45
21
21
21
21
21
.26885
•26745
.26586
.26415
.26236
112.478
116.974
121.472
125.970
130.469
22.35850
22.35716
22.35565
22.35403
22.35230
112.480
116.977
121.474
125-973
130.472
23.50404
23.50277
23.50136
23.49980
23.49820
112.482
II6.979
121.477
125-975
130.474
24.70840
24.70720
24.70580
24.70440
24.70280
112.483
116.981
121.479
125.978
130.477
25.97450
25-97337
25.97206
25.97066
25.96920
112.485
116.983
121.481
125.980
130.479
i-SS
1.6
-65
-7
21
21
21
21
21
.26052
.25869
.25685
.25520
.25362
134.968
139.469
143.970
148.472
152.974
22.35060
22.34883
22.34712
22.34550
22.34401
I34.97I
139.472
143-973
148.474
152.977
23.49652
23.49486
23.49322
23-49" 5
23.49028
134-974
139-474
143-975
148.477
152.979
24.70121
24.69964
24.69809
24.69662
24.69528
134.976
139-477
143.978
148.479
152.981
25.96770
25.96618
25.96471
25.96205
134-979
139-479
143.980
148.481
152.983
•75
.8
•85
•9
1-95
21
21
21
21
21
.25221
.25102
.25006
•24935
.24891
I57-478
161.981
166.486
170.990
175-495
22.34270
22.34153
22.34061
22-33995
22.33952
157.480
161.983
166.487
170.991
175.496
23.48900
23.48791
23.48704
23.48640
23.48601
157.482
161.985
166.488
I7O.992
175.496
24.69406
24.69302
24.69223
24.69159
24.69120
157.483
161.986
166.489
170.993
175.496
25.96090
25-9599I
25-959"
25-95854
25.95820
I57.485
161.987
166.490
170.993
175-497
2.O
21.24878
180.000
22.33941
180.000
23.48589
iSo.OOO
24.69110
180.000
25.95806
180.000
Example.
sinh (3.90 + * 1.90) =
24.69159
/I70°.993 = 24-69159 /I70°.59'
35'.
[105]
TABLE XI. HYPERBOLIC COSINES, cosh (* + ig) = r /y
x = 0.05
x, = 0.15
o
0.05
O.I
0.15
O.2
I.OOOOO O.OO
0.99692 o.oo
0.98769 o.oo
0.97237 o.oo
0.95106 o.oo
1.00125 o-00
0.99817 0.225
0.98895 0.453
0.97366 0.687
0.95237 0.930
1.00500 o.oo
1.00194 0.450
0.99275 0.905
0.97752 1.371
0.95632 1.855
1.01127 o.oo
1.00822 0.671
0.99909 1.351
0.98396 2.047
0.96290 2.769
1.02007 o.oo
1.01704 0.890
1.00800 1.791
0.99300 2.713
0-97213 3-669
O.25
0-3
0-35
0.4
0-45
0.92388 o.oo
0.89101 o.oo
0.85264 o.oo
0.80902 o.oo
0.76041 o.oo
0.92523 1.186
0.89237 1.458
0.85407 1.754
0.81053 2.079
0.76202 2.443
0.92930 2.364
0.89662 2.907
0.85851 3.495
0.81520 4.142
0.76698 4.866
0.93607 3.529
0.90364 4.338
0.86583 5.213
0.82291 6.174
0.77527 7.247
0.94556 4-674
0.91347 5-743
0.87609 6.897
0.83370 8.161
0.78661 9-569
o-5
0.55
0.6
0.65
0.7
0.70711 o.oo
0.64945 o.oo
0.58779 o.oo
0.52250 o.oo
0-45399 0-0°
0.70803 2.860
0-65135 3-348
0.58989 3.934
0.52487 4.661
0.45672 5.600
0.71417 5.692
0.65713 6.656
0.59626 7.811
0.53201 9.238
0.46491 1 1. 068
0.72296 8.468
0.66667 9-889
0.60676 11.581
0.54376 13.656
0.47831 16.289
0.73521 11.165
0.67994 13.012
0.62131 15.198
0-55995 17-853
0.49663 21.175
o-7S
0.8
0.85
0.9
0-95
0.38268 o.oo
0.30902 o.oo
0.23345 o.oo
0.15643 o.oo
0.07846 o.oo
0.38594 6.877
0.31304 8.741
0.23874 11.755
0.16424 17.507
0.09305 32.407
0.39558 13-529
0.32485 17-053
0.25403 22.545
0.18576 32.181
0.12724 51.700
0.41124 19.771
0.34375 24.618
0.27779 3^805
0.21712 43.229
0.16978 62.139
0.43242 25.478
0.36882 31.277
0.30827 39.424
0.25497 5I-254
0.21608 68.261
I.O
1.05
i.i
1.2
o.oo 180
0.07846 180
0.15643 180
0.23345 180
0.30902 1 80
0.05002 90.000
0.09305 147-593
0.16424 162.493
0.23874 168.245
0.31304 171.259
0.10017 90.000
0.12724 128.300
0.18576 147.819
0.25403 157-455
0.32485 162.947
0.15056 90.000
0.16978 117.861
0.21712 136.771
0.27779 148.195
0.34375 155.382
0.20134 90.000
0.21608 111.739
0.25497 128.746
0.30827 140.576
0.36882 148.723
1-25
i-35
1.4
MS
0.38268 180
0-45399 180
0.52250 180
0.58779 180
0.64945 1 80
0.38594 173-123
0.45672 174.400
0.52487 175-339
0.58989 176.066
0.65135 176.652
0.39558 166.471
0.46491 168.932
0.53201 170.762
0.59626 172.189
0.65713 173-344
0.41124 160.229
0.47831 163.711
0.54376 166.344
0.60676 168.419
0.66667 170.111
0.43242 154.522
0.49663 158.825
0-55995 162.147
0.62131 164.802
0.67994 166.988
i-SS
1.6
1.65
0.70711 180
0.76041 1 80
0.80902 1 80
0.85264 1 80
0.89101 180
0.70803 177-139
0.76202 177.557
0.81053 177.921
0.85407 178.246
0.89237 178.542
0.71417 174.308
0.76698 175.134
0.81520 175.858
0.85851 176.505
0.89662 177.093
0.72296 171.532
0.77527 172.753
0.82291 173.826
0.86583 174.787
0.90364 175.662
0.73521 168.835
0.78661 170.431
0.83370 171-839
0.87609 173.103
0.91347 174-257
i-75
1.8
1.85
1.9
i-95
0.92388 180
0.95106 180
0.97237 180
0.98769 180
0.99692 180
0.92523 178.814
0.95237 179.070
0.97366 179-313
0.98895 179.547
0.99817 179-775
0.92930 177.636
0.95632 178.145
0.97752 178.629
0.99275 179.095
1.00194 179.550
0.93607 176.471
0.96290 177.231
0.98396 177.953
0.99909 178.649
1.00822 179.329
0.94556 175-326
0.97213 176.331
0.99300 177.287
1.00800 178.209
1.01704 179.110
2.O
i.ooooo 180
1.00125 180.000
1.00500 180.000
1.01127 180.000
1.02007 180.000
Example.
cosh (o.io + 10.55) = 0.65713 /6°.6^6 = 0.65713 /6°.39
'.22*.
[106]
TABLE XI. HYPERBOLIC COSINES, cosh (x + ig) = r Ay. CONTINUED
X =
0.25
x =
o-3
x =
0-35
x =
0.4
x —
0.45
9
r
7
r
y
r
y
r
y
r
y
0
o
o
o
o
0
1.03141
o.ooo
1-04534
o.ooo
i. 06 1 88
o.ooo
1.08107
O.OOO
1.10297
O.OOO
0.05
1.02843
1.104
1.04239
1-313
1.05898
1.516
1.07822
I.7I3
1.10017
I.9O2
O.I
1.01949
2.221
1-03357
2.642
1.05029
3-050
1.06970
3-444
1.09182
3-823
0.15
1.00464
3-365
1.01894
4.001
1.03590
4.617
1-05557
5-212
1.07798
5-784
O.2
0.98403
4-550
0.99862
5-407
1.01592
6.238
1-03597
7.038
1.05880
7.806
0.25
0-95779
5-793
0.97277
6.881
0.99052
7-932
1.01107
8.944
1.03446
9-9I3
°-3
0.92612
7-ii3
0.94161
8.443
0.95994
9.726
0.98113
10.957
1.00521
12.132
o-35
0.88927
8-536
0-90539
IO.I22
0.92444
11.647
0.94642
13.107
0.97136
14.496
0.4
0.84754
10.090
0.86443
11.951
0.88436
13-733
0.90732
I5-432
0-93330
17.042
0-45
0.80127
11.815
0.81912
I3-972
0.84012
16.029
0.86426
17.979
0.89149
I9.8l6
0-5
0.75088
13.762
0.76989
16.241
0.79220
18.592
0.81775
20.804
0.84469
22.875
o-55
0.69685
16.001
0.71730
18.834
0.74119
21-497
0.76844
23.982
0.79895
26.288
0.6
0.63977
18.629
0.66199
2 1 .849
0.68781
24.843
0.71708
27.607
0.74969
30.144
0.65
0.58036
21.785
0.60476
25-425
0.63292
28.763
0.66462
31.800
0.69968
34-546
0.7
0.51954
25-673
0.54666
29-758
0.57766
33-!57
0.61223
36.712
0.65012
39.626
0-75
0-45854
30.595
0.48906
35-II8
0.52348
39-079
0.56140
42.529
0.60249
45-527
0.8
o-399J3
37.008
0.43385
41-878
0.47231
45-992
0.51401
49.464
0.55860
52-399
0.85
0.34396
45-057
0.38370
50.507
0.42671
54-484
0.47246
57-712
0.52062
60.358
0.9
0.29782
57.110
0-34235
61.467
0.38994
64.786
0-43953
67-37I
0.49093
69.424
0-95
0.26452
72.186
0.31447
74.882
0.36571
76.831
0.41818
78.297
0.47191
79-433
I.O
0.25261
90.000
0.30452
90.000
0.35719
90.000
0.41075
90.000
0.46534
90.000
1.05
0.26452
107.814
0.31447
I05.II8
0.36571
103.169
0.41818
101.703
0.47191
100.567
i.i
0.29782
122.890
0.34235
"8.533
0.38994
115.214
0-43953
112.629
0.49093
110.576
US
0.34396
134-943
0.38370
129.493
0.42671
125.516
0.47246
122.288
0.52062
119.642
1.2
0.39913
142.992
0-43385
138.122
0.47231
134.008
0.51401
130.536
0.55860
127.601
1-25
0.45854
149.405
0.48906
144.882
0.52348
140.921
0.56140
I37.47I
0.60249
134-473
i-3
0.51954
I54-327
0.54666
150.242
0.57766
146.843
0.61223
143-288
0.65012
140.374
i-3S
0.58036
158.215
0.60476
154-575
0.63292
151-237
0.66462
148.200
0.69968
145-454
1.4
0.63977
161.371
0.66199
I58.I5I
0.68781
I55.I57
0.71708
152.393
0.74969
149-856
i-45
0.69685
163.999
0.71730
l6l.l66
0.74119
158.503
0.76844
156.018
0.79895
I53.7I2
i-5
0.75088
166.238
0.76989
163.759
0.79220
161.408
0.81775
159.196
0.84649
157-125
i-5S
0.80127
168.185
0.81912
166.028
0.84012
163.971
0.86426
162.021
0.89149
160.184
1.6
0.84754
169.910
0.86443
168.049
0.88436
166.267
0.90732
164.568
0.93330
162.958
1.65
0.88927
171.464
0-90539
169.878
0.92444
168.353
0.94642
166.893
0.97136
165.504
i-7
0.92612
172.887
0.94161
I7L557
0.95994
170.274
0.98113
169.043
1.00521
167.868
1-75
0-95779
174.207
0.97277
I73-H9
0.99052
172.068
1.01107
171.056
1.03446
170.087
1.8
0.98403
I75-450
0.99862
174-593
1.01592
173.762
1-03597
172.962
1.05880
172.194
1.85
i .00464
176.635
1.01894
175-999
1.03590
175-383
1-05557
174-788
1.07798
174.216
1.9
1.01949
177.779
1-03357
177.358
1.05029
176.950
1.06970
176.556
1.09182
176.177
i-9S
1.02843
178.896
1.04239
178.687
1.05898
178.484
1.07822
178.287
1.10017
178.098
2.0
1.03141
180.000
1-04534
180.000
1.06188
180.000
1.08107
180.000
1.10297
180.000
Example. cosh (0.40 + i 0.5) = 0.81775
= 0.81775
[107]
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED
* =
o.S
X =
0-55
x =
0.6
x =
0.65
x =
0.7
q
r
7
r
7
r
7
r
7
r
7
o
0
o
o
0
o
1.12763
o.ooo
i.iSSio
o.ooo
1.18547
o.ooo
1.21879
O.OOO
1-25517
o.ooo
0.05
1.12489
2.083
1.15244
2.256
1.18287
2.420
1.21626
2.576
1.25271
2.723
O.I
1.11672
4.186
1.14446
4-533
1.17510
4.879
1.20871
5-J74
1-24538
5.468
0.15
1.10320
6.331
1.13127
6.852
1.16226
7-347
1.19623
7-815
1.23327
8.256
O.2
1.08446
8.539
1.11300
9-237
1.14448
9.898
1.17892
10.523
1.21653
II. IIO
0.25
1.06070
10.836
1.08987
11.713
I.I22OO
12.541
I-I57I5
13.322
1.19541
14.054
o-3
1.03220
13.249
1.06214
14-307
1.09509
15-304
1.13108
16.240
1.17019
17.116
0-35
0.99927
15.811
1.03004
17.052
I.064II
18.216
I.IOIII
19.306
1.14125
20.323
0.4
0.96232
18.559
0-99437
19.984
1.02948
21-315
1.06769
22.555
1.10904
23.706
0-45
0.92182
21.538
0.95524
23.146
0.99174
24.640
1-03134
26.024
1.07409
27.302
o-S
0.87837
24.803
0.91338
26.589
0.95149
28.238
0.99270
29-755
1.03704
31.148
o-SS
0.83266
28.416
0.86951
30.372
0.90946
32.162
0.95249
33-796
0.99861
35.284
0.6
0.78552
32-458
0.82447
34-563
0.86650
36.471
0.91156
38.197
0.95966
39-755
0.65
0-73793
37.020
0.77927
39-241
0.82361
41.231
0.87090
43-on
0.92112
44.603
0.7
0.69112
42.207
0.73510
44.489
0.78194
46.506
0.83160
48.290
0.88406
49.867
0-75
0.64652
48.129
0.69333
50.390
O.74282
52.358
0.79492
54-074
0.84965
55-574
0.8
0.60583
54.889
0.65555
57.010
0.70769
58.826
0.76220
60.387
0.81911
61.737
0.85
0.57100
62.547
0.62350
64-375
0.67810
65.914
0.73482
67.219
0.79369
68-335
0.9
0.54407
71.082
0.59894
72.441
0.65559
73-568
0.71409
74.5I4
0-77455
75.3I5
o.QS
0.52697
80.335
0.58345
81.064
0.64147
81.663
0.70115
82.161
0.76263
82.581
I.O
0.52110
90.000
0.57815
90.000
0.63665
90.000
0.69675
90.000
0.75858
90.000
1.05
0.52697
99.665
0.58345
98-936
0.64147
98.337
0.70115
97.839
0.76263
97.419
i.i
0.54407
108.918
0.59894
107-559
0.65559
106.432
0.71409
105.486
0-77455
104.685
i-iS
0.57100
117.453
0.62350
115.625
0.67810
114.086
0.73482
112.781
0.79369
111.665
1.2
0.60583
125.111
0.65555
122.990
0.70769
121.174
0.76220
119.613
0.81911
118.263
I.2S
0.64652
131.871
0.69333
129.610
0.74282
127.642
0.79492
125.926
0.84965
124.426
i-3
0.69112
137.793
o.735io
I35-51!
0.78194
133-494
0.83160
131.710
0.88406
130.133
i-35
0-73793
142.980
0.77927
140.759
0.82361
138.769
0.87090
136.989
0.92112
135-397
1.4
0.78552
147.542
0.82447
145-437
0.86650
I43-529
0.91156
141.803
0.95966
140.245
i. 45
0.83266
151-584
0.86951
149.628
0.90946
147.838
0.95249
146.204
0.99861
144.716
1.50
0.87837
155.198
0.91338
153-4"
0.95149
151.762
0.99270
150.245
1.03704
148.853
i-SS
0.92182
158.462
0.95524
156.854
0.99174
155-360
1-03134
I53.976
1.07409
152.698
i. 60
0.96232
161.441
0-99437
160.016
I.O2948
158.685
1.06769
157-445
1.10904
156-294
1.65
0.99927
164.189
1.03004
162.948
1.06411
161.784
I.IOIII
160.694
1.14125
159.677
1.70
1.03220
166.751
1.06214
165.693
1.09509
164.696
1.13108
163.760
1.17019
162.884
i-75
1.06070
169.164
1.08987
168.287
.I22OO
167.459
1.15715
166.678
1.19541
165.946
1.8
i .08446
171.461
1.11300
170.763
.14448
170.102
1.17892
169.477
1.21653
168.890
1.85
1.10320
173.669
1.13127
173.148
.16226
172.653
1.19623
172.185
1.23327
171.744
i. 9
1.11672
175.814
1.14446
175467
.17510
175.121
1.20871
174.826
1-24538
I74-532
i-9S
1.12489
177.917
1.15244
177-744
.18287
177.580
1.21626
177.424
1.25271
177.277
2.0
1.12763
180.000
1.15510
180.000
I.I8547
180.000
1.21879
180.000
1-25517
180.000
Example. cosh (0.65 + i i.o) = 0.69675 /go0.
[108]
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r fy. CONTINUED
x = 0.75
= 0.85
x = o.g
x = 0.95
o
1.29468
0.000
1 -33 743
o.ooo
I-38353
o.ooo
1.43309
o.ooo
1.48623
o.ooo
0.05
1.29230
2.862
I-335I3
2.992
1.38130
3-"3
1.43094
3.227
1.48415
3-332
O.I
1.28520
5-745
1.32825
6.004
1.37466
6.246
1.42452
6.473
1.47797
6.684
0.15
1.27346
8.670
1.31691
9-058
1.36369
9.420
I-4I395
9-758
1.46778
10.071
O.2
1.25726
11.661
1.30124
12.175
1-34858
12.656
1-39937
13.102
1.45371
13-516
0.2S
1.23689
14.740
1.28152
15-379
I-32955
15-974
1.38105
16.526
1.43611
17.037
o-3
1.21248
17-933
1.25802
18.693
1.30693
19.398
I-35927
20.051
1.41519
20.653'
0.35
1.18457
21.267
1-23115
22.143
1.28107
22.952
1-33444
23.699
1.39136
24-387
0.4
1-15356
24.772
1.20135
25-755
1.25246
26.601
1.30700
27.493
1-36505
28.257
0-45
I.I200I
28.478
1.16917
29-559
1.22163
30-550
1.27748
31.457
1.33682
32.286
o-S
1.08453
32-442
1.13522
33-586
1.18918
34.647
1.24649
35-614
1.30723
36-493
0-55
1.04785
36-637
1.10024
37.865
1-15583
38.978
1.21471
39.986
1.27697
40.898
0.6
I.OI079
41.160
1.06500
42.426
1.12234
43-567
1.18289
44-593
1.24674
45-517
0.65
0.97427
46.026
1.03041
47.298
1.08957
48.435
1.15184
49-453
1.21732
50.363
0.7
0.93932
51-263
0.99742
52-500
1.05843
53-599
1.12243
54-574
1.19227
55-443
0-7S
O.9O7OO
56.889
0.96705
58.045
1.02985
59.062
1-09553
59.961
1.16418
6o.755
0.8
0.87846
62.907
0.94033
63.927
1.00481
64.819
1.07202
65.601
1.14209
66.288
0.85
0.85481
69.294
0.91828
70.123
0.98420
70.842
1.05273
71.471
1.12400
72.020
0.9
0.83 706
75-998
0.90178
76.585
0.96883
77.091
1-03837
77-532
1.11056
77-9I5
0-95
0.82605
82.936
0.89157
83.241
0-95933
83-503
1.02951
83-730
1.10227
83.927
I.O
O.82232
90.000
0.88811
90.000
0.95612
90.000
1.02652
90 ooo
1.09948
90.000
1.05
0.82605
97.064
0.89157
96.759
0-95933
96.497
1.02951
96.270
1.10227
96.073
i.i
0.83706
104.002
0.90178
103.415
0.96883
102.909
1-03837
102.468
1.11056
102.085
I.IS
0.85481
110.706
0.91828
109.877
0.98420
109.158
1.05273
108.529
I 12400
107.980
1.2
0.87846
117.093
0.94033
116.073
1.00481
115.181
1.07202
114.399
1.14209
113.712
1. 25
0.90700
123.111
0.96705
121.955
1.02985
120.938
1-09553
1 20.039
1.16418
119.245
i-3
0.93932
128.737
0.99742
127.500
1.05843
126.401
1.12243
125.426
1.19227
124-557
i-35
0.97427
133-974
1.03041
132.702
1.08957
i3!-565
1.15184
130.547
1.21732
129.637
1.4
I.OI079
138.840
1.06500
137-574
1.12234
136.433
1.18289
135.407
1.24674
134-483
1-45
1.04785
I43-363
1.10024
142-135
1-15583
141.022
1.21471
140.014
1.27697
139.102
i-5
1.08453
I47-578
1.13522
146.414
1.18918
145-353
1.24649
144.386
1.30723
I43-507
i-SS
I.I2OOI
151-522
1.16917
150.441
1.22163
149.450
1.27748
148.543
1.33682
147.714
1.6
I-I5356
155.228
1.20135
I54-24S
1.25246
153-399
1.30700
152.507
1-36505
I5I-743
1.65
1.18457
158.733
i.23"5
I57-857
1.28107
157.048
1.33444
156.301
1.39136
I55-6I3
i-7
I.2I248
162.067
1.25802
161.307
1.30693
160.602
I-35927
159-949
1.41519
159-347
i-75
1.23689
165.260
1.28152
164.621
I-32955.
164.026
1.38105
163.474
1.43611
162.963
1.8
1.25726
168.339
1.30124
167.825
1-34858
167.344
1-39937
166.898
1.45371
166.484
1.85
1.27346
171-330
1.31691
170.942
1.36369
170.580
I-4I395
170.242
1.46778
169.929
1.9
I.2852O
174-255
1.32825
173.996
1.37466
173-754
1.42452
173-527
1.47797
I73-3I6
i -95
1.29230
177.138
I-335I3
177.008
1.38130
176.887
1.43094
176.773
1.48415
176.668
2.O
1.29468
180.000
1-33743
180.000
I-38353
180.000
1-43309
180.000
148623
180.000
Example.
cosh (0.90 + * 0.5)
= 1.24649 /35°-6i4
= 1.24649 /35°-36'.5o'.
[109]
TABLE XI. HYPERBOLIC COSINES, cosh (at + iq) = r /y. CONTINUED
x = 1.05
x = 1.15
o
0.05
O.I
0.15
O.2
1.54308 o.ooo
1.54108 3.276
1.53513 6.878
1.52532 10.362
1.51182 13.899
1.60379 o.ooo
1.60187 3.521
1.59614 7.059
1.58671 10.631
1-57374 14-253
1.66852 o.ooo
1.66667 3-605
1.66117 7.226
1.65211 10.879
1.63965 14.579
1.73741 o.ooo
1.73564 3-683
1-73036 7-380
1.72166 11.107
1.70971 14.880
i. 81066 o.ooo
1.80895 3.754
1.80389 7.522
1-79555 11-318
1.78409 15.156
0.25
0-3
0-35
0.4
0-45
1.49487 17.509
1.47479 21.209
1.45193 25.019
1.42675 28.957
1.39976 33-043
1-55747 17-944
1.53820 21.720
1.51630 25.599
1.49220 29.597
1.46641 33-732
1.62404 18.344
1.60558 22.189
1.58460 26.130
1.56156 30.182
1.53694 34-360
1.69472 18.713
1.67705 22.620
1.65699 26.616
1-63497 30.716
1.61147 34.931
1.76975 19.050
1.75282 23.014
1-73363 27.061
1.71260 31.203
1.69018 35-451
o-5
0-55
0.6
0.65
0.7
1.37153 37.293
1.34272 41.724
1.31400 46.349
1.28612 51.179
1.25984 56.216
1.43950 38.019
1.41207 42.470
1.38479 47-098
I-35837 5i-910
I-3335I 56-907
1.51128 38.677
1.48517 43-H5
1.45926 47-773
1.43421 52-565'
1.41070 57-523
1.58701 39.275
1.56217 43-755
1.53756 48.380
1.51381 53-J53
i-49I55 58-074
1.66688 39.816
1.64325 44-307
1.61987 48.927
1-59733 53-681
1.57626 58.567
0-75
0.8
0.85
0.9
o-95
1.23594 61.459
1.21515 66.895
1.19816 72.504
1.18552 78.252
1.17782 84.100
1.31095 62.085
1.29137 67.432
1.27540 72.929
1.26358 78.547
1.25631 84.252
1.38939 62.641
1.37093 67.908
1-35590 73-306
1.34478 78.808
1-33795 84.385
1.47141 63.'i36
1.45399 68.330
1-43983 73.639
1.42936 79-039
1.42294 84.503
1.55721 63.579
1.54077 68.706
1.52741 73.935
I.5I755 79-243
1.51150 84.607
I.O
1.05
i.i
i-iS
1.2
1.17520 90.000
1.17782 95.900
1.18552 101.748
1.19816 107.496
1.21515 113.105
1.25386 90.000
1.25631 95.748
1.26358 101.453
1.27540 107.071
1.29137 112.568
1 -33565 90-0°o
1-33795 95-6i5
1.34478 101.192
1.35590 106.694
1.37093 112.092
1.42078 90.000
1.42294 95.497
1.42936 100.961
1.43983 106.361
1.45399 111-670
1.50946 90.000
1.51150 95.393
1.51755 100.757
1.52741 106.065
1.54077 111.294
1-25
i-3
i-35
1.4
i-45
1.23594 118.541
1.25984 123.784
1.28612 128.821
1.31400 133.651
1.34272 138.276
1.31095 117.915
1.33351 123.093
1.35837 128.090
1.38479 132.902
1.41207 137.530
1.38939 H7-359
1.41070 122.477
1.43421 127.435
1.45926 132.227
1.48517 136.855
1.47141 116.864
1.49155 121.926
1.51381 126.847
J-53756 131.620
1.56217 136.245
1.55721 116.421
1.57626 121.433
1.59733 126.319
1.61987 131.073
1.64325 135.693
i-5
i-SS
1.6
1.65
i-7
1.37153 142.707
1.39976 146.957
1.42675 151.043
1.45193 154.981
1.47479 I58.791'
1.43950 141.981
1.46641 146.268
1.49220 150.403
1.51630 154-401
1.53820 158.280
1.51128 141-323
1.53694 145.640
1.56156 149.818
1.58460 1.53.870
1.60558 157.811
1.58701 140.725
1.61147 145.069
1.63497 149.284
1.65699 153-384
1.67705 157-380
1.66688 140.184
1.69018 ^44-549
1.71260 148.797
I.73363 152.939
1.75282 156.986
i-7S
1.8
1.85
1.9
I-9S
1.49487 162.491
1.51182 166.101
1.52532 169.638
i-SSS^ 173-122
1.54108 176.724
1.55747 162.056
1.57374 165.747
1.58671 169.369
1.59614 172.941
1.60187 176.479
1.62404 161.656
1.63965 165.421
1.65211 169.121
1.66117 172.774
1.66667 176.395
1.69472 161.287
1.70971 165.120
1.72166 168.893
1.73036 172.620
1.73564 176.317
1.76975 160.950
1.78409 164.844
1-79555 168.682
1.80389 172.478
1.80895 176.246
2.0
1.54308 180.000
1.60379 180.000
1.66852 180.000
1.73741 180.000
i. 81066 180.000
Example. cosh (1.20 + io) = i. 81066 /o°.
[no]
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED
x = 1.25
x = 1.3
= i-3S
= 1.4
= 1-45
o
1.88842
o.ooo
1.97091
o.ooo
2.05833
o.ooo
2.15090
O.OOO
2.24884
0.000
0.05
1.88679
3.820
1.96935
3.880
2.05684
3-935
2.14947
3.986
2.24747
4.032
O.I
I.88I93
7.652
1.96470
7.772
2.05238
7.882
2.14520
7.982
2-24334
8.075
0.15
1.87394
11.511
1.95704
11.689
2.04505
11.851
2.13819
I2.OOO
2.23669
12.136
O.2
1.86297
15.410
1.94654
15.642
2.03500
15-855
2.12858
16.049
2.22751
16.226
0.25
1.84924
19.360
1.93341
19.644
2.02245
19.903
2.11658
20.139
2.21604
20.355
°-3
1.83304
23-375
1.91792
23.705
2.00764
24.006
2.10244
24.281
2.20254
24.531
o-35
I.8I4/O
27.467
1 .90040
27.837
1.99091
28.175
2.08647
28.482
2.18731
28.762
0.4
1.79462
31.646
1.88123
32.050
1.972612
32-417
2.06903
32.751
2.17067
33.055
0-45
I-77323
35-923
1.86084
26.352
1-95319
36.742
2.05051
37-095
2.15302
37.416
o-S
I.75I04
40.308
1.83970
40.752
1.93306
4i-i55
2.03135
41.520
2.13478
41.851
0-55
1.72856
44.805
1.81832
45.255
1.91273
45.662
2.OI2OO
46.030
2.11637
46.362
0.6
I-70635
49.420
1.79722
49.865
1.89268
50.265
1.99295
50.627
2.09828
50.953
0.65
1.68497
54-155
1.77694
54.582
1.87343
54.96S
1.97468
55-3"
2.08093
55.621
0.7
I.6650I
59.009
1.75802
59.405
1.85549
59.760
1.95767
60.079
2.06479
60.366
0-75
1.64699
63.974
1.74096
64.327
1.83934
64.644
I.94237
64.927
2.05030
65.182
0.8
I-63I45
69.042
1.72627
69.341
1.82544
69.608
I.9292I
69.847
2.03784
70.061
0.85
1.61884
74-197
1-71435
74-432
1.81417
74.641
1.91856
74.828
2.02775
74-995
0.9
1.60954
79.424
1-70557
79-585
1.80588
79.729
I.9I072
79.857
2.02034
79.972
0-95
1.60384
84.699
1.70019
84.782
1 .80080
84-855
1.90592
84.920
2.01580
84.979
I.O
1.60192
90.000
1.69838
90.000
1.79909
90.000
1.90430
90.000
2.01427
90.000
1.05
1.60384
95-301
1.70019
95.218
1 .80080
95-145
1.90592
95.080
2.01580
95.021
i.i
1.60954
100.576
1-70557
100.415
1.80588
100.271
I.9IO72
100.143
2.02034
100.028
i-iS
1.61884
105.803
1-71435
105.568
1.81417
105-359
1.91856
105.172
2.02775
105.005
1.2
I-63I45
110.958
1.72627
110.659
1.82544
110.392
I.9292I
110.153
2.03784
109.939
I-2S
1.64699
116.026
1.74096
115-673
1.83934
115-356
1.94237
115-073
2.05030
114.818
i-3
1.66501
120.991
1.75802
120.595
1.85549
120.240
1.95767
119.921
2.06479
119.634
i-3S
1.68497
125.845
1.77694
125.418
1.87343
125-035
1.97468
124.689
2.08093
124.379
1.4
1.70635
130.580
1.79722
130.135
1.89268
129-735
1.99295
129.373
2.098^8
129.047
i -45
1.72856
135-195
1.81832
134.745
1.91273
I34-338
2.OI2OO
I33.970
2.11637
133-638
i-5
I.75I04
139-693
1.83970
139.248
1.93306
138.845
2.03135
138.480
2.13478
138.150
i-SS
1.77323
144.077
1.86084
143.648
1.95319
143-258
2.05051
142.905
2.15302
142.584
1.6
1.79462
148.354
1.88123
147.950
1.97262
147-583
2.06903
147.249
2.17067
146.945
1.65
I.8I470
152.533
1 .90040
152.163
1.99091
151-825
2.08647
151-518
2.18731
151-238
i-7
1.83304
156.625
1.91792
156.295
2.00764
155-994
2.10244
I55.7I9
2.20254
155469
i-75
1.84924
160.640
1.93341
160.356
2.02245
160.097
2.II658
159.861
2.21604
I59.645
1.8
1.86297
164.590
1.94654
164.358
2.03500
164.145
2.12858
163.951
2.22751
163.774
1.85
1.87394
168.489
1.95704
168.311
2.04505
168.149
2.13819
168.000
2.23669
167.864
1.9
1.88193
172.348
1.96470
172.228
2.05238
172.118
2.14520
172.018
2-24334
171.925
I-9S
1.88679
176.180
1.96935
176.120
2.05684
176.065
2.14947
176.014
2.24747
175-968
2.0
1.88842
180.000
1.97091
180.000
2.05833
180.000
2.15090
180.000
2.24884
180.000
Example. cosh (1.35 -f i 1.30) = 1.85549 / 120°. 240 = 1.85549 / 120°. 14'. 24*.
[in]
TABLE XI. HYPERBOLIC COSINES, cosh (x + ig) = r /y. CONTINUED
x = 1.5
= 1-55
x = 1.6
* = 1.65
x = 1.7
o
2.35241 o.ooo
2.46186 o.ooo
2.57746 o.ooo
2.69952 o.ooo
2.82832 o.ooo
0.05
2.35110 4.075
2.46061 4.113
2.57627 4.149
2.69839 4.181
2.82723 4.210
o.r
2.34720 8.158
2.45688 8.235
2.57272 8.305
2.69499 8.369
2.82399 8.427
0.15
2.34080 12.260
2.45076 12.374
2.56687 12.477
2.68941 12.571
2.81867 12.657
O.2
2.33202 16.389
2.44238 16.536
2.55887 16.671
2.68178 16.794
2.81138 16.906
0.25
2.32107 20.552
2.43193 20.732
2.54890 20.895
2.67226 21.044
2.80230 21.179
o-3
2.30819 24.759
2.41976 24.967
2.53717 25.155
2.66108 25.327
2.79164 25.483
o-35
2.29365 29.016
2.40577 29.248
2.52395 29.458
2.64847 29.649
2.77963 29.822
0.4
2-27779 33-33°
2.39066 33.580
2-50955 33-808
2.63476 34.014
2.76657 34.201
0-45
2.26098 37.706
2.37465 37-970
2.49430 38.209
2.62024 38.426
2.75274 38.622
o-5
2.24362 42.150
2.35812 42.421
2.47857 42.666
2.60527 42.887
2.73850 43.089
0-55
2.22612 46.663
2.34148 46.934
2.46274 47.180
2.59021 47.401
2.72418 47.602
0.6
2.20892 51-247
2.32513 51.512
2.44721 51-752
2.57544 51.968
2.71014 52.163
0.65
2.19245 55-901
2.30949 56.154
2.43235 56.381
2.56133 56.585
2.69673 56.771
0.7
2.17714 60.624
2.29496 60.856
2.41856 61.065
2.54824 61.253
2.68430 61.422
o-7S
2.16340 65.410
2.28193 65.615
2.40619 65.800
2.53650 65.966
2.67316 66.116
0.8
2.15159 70-254
2.27074 70.426
2-39558 70.581
2.52644 70.720
2.66361 70.845
0.85
2.14204 75-x45
2.26169 75.279
2.38701 75.400
2.51831 75.508
2-65591 75-605
0.9
2.13502 80.075
2.25504 80.167
2.38071 80.249
2.51234 80.323
2.65025 80.390
o-9S
2.13073 85.031
2.25098 85.077
2.37686 85.119
. 2.50869 85.157
2.64679 85.191
I.O
2.12928 90.000
2.24961 90.000
2-37557 90-000
2.50747 90.000
2.64563 90.000
1.05
2.13073 94-969
2.25098 94.923
2.37686 94.881
2.50869 94-843
2.64679 94.809
i.i
2.13502 99.925
2.25504 99.833
2.38071 99.751
2.51234 99.677
2.65025 99.610
i-iS
2.14204 104.855
2.26169 104.721
2.38701 104.600
2.51831 104.492
2.65591 104.395
1.2
2.15159 109.746
2.27074 109.574
2.39558 109.419
2.52644 109.280
2.66361 109.155
I.2S
2.16340 114-59°
2.28193 114.385
2.40619 114.200
2.53650 114.034
2.67316 113.884
i~3
2.17714 119.376
2.29496 119.144
2.41856 118.935
2.54824 118.747
2.68430 118.578
•35
2.19245 124.099
2.30949 123.846
2.43235 123.619
2.56133 123.415
2.69673 123.229
•4
2.20892 128.753
2.32513 128.488
2.44721 128.248
2.57544 128.032
2.71014 127.837
•45
2.22612 133.337
2.34148 133.066
2.46274 132.820
2.59021 132.599
2.72418 132.398
•5
2.24362 137-850
2.35812 137.580
2-47857 137-334
2.60527 137.113
2.73850 136.911
•55
2.26098 142.294
2.37465 142.030
2.49430 141-791
2.62024 141.574
2.75274 141.378
.6
2.27779 146.670
2.39066 146.420
2.50955 146.192
2.63476 145.986
2.76657 145.799
•65
2.29365 150.984
2.40577 150.752
2.52395 150.542
2.64847 150.351
2.77963 150.178
•7
2.30819 155-241
2.41976 155.033
2.537I7 I54.845
2.66108 154.673
2.79164 154.517
•75
2.32107 159.448
2.43193 159.268
2.54890 159.105
2.67226 158.956
2.80230 158.821
.8
2.33202 163.611
2.44238 163.464
2.55887 163.329
2.68178 163.206
2.81138 163.094
•85
2.34080 167.740
2.45076 167.626
2.56687 167.523
2.68941 167.429
2.81867 167.343
•9
2.34720 171.842
2.45688 171.765
2.57272 171.695
2.69499 171.631
2.82399 171.573
•95
2.35110 175-925
2.46061 175.887
2.57627 175.851
2.69839 175.819
2.82723 175.790
2.0
2.35241 180.000
2.46186 180.000
2.57746 180.000
2.69952 180.000
2.82832 180.000
Example. cosh (1.6 + i 1.6) = 2.50955 /I46°.i92 = 2.50955 /i46°.n'.3iy.
[112]
TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r /y. CONTINUED
x =
1-75
x = 1.8
x =
1.85
* =
1.9
X =
i-95
9
r
y
r
Y
r
7
r
7
r
Y
0
0
0
o
o
0
2.96419
o.ooo
3.10747
O.OOO
3.
25853
o.ooo
3.4I773
o.ooo
3-58548
o.ooo
0.05
2.96315
4.237
3.10648
4.262
3'
25759
4.284
3.41683
4-304
3.58462
4.322
O.I
2.96006
8.480
3-10353
8.529
3'
25477
8.572
3-41415
8.612
3.58207
8.648
0.15
2.95498
12-735
3.09869
12.806
3.
25015
12.871
3-40975
12.929
3-57788
12.983
0.2
2.94804
17.008
3.09206
17.100
3'
24384
17.184
3-40373
17.260
3-572I3
17.330
0.25
2.93938
21.302
3.08382
21.414
3'
23598
21.516
3-39624
21.608
3-56500
21.692
O.3
2.92921
25-625
3-074I3
25-754
3'
22675
25.871
3-38745
25-977
3-55663
26.073
o-3S
2.91777
29.980
3-06323
30.123
3'
21636
30.252
3-37756
30.369
3-54721
30.476
0.4
2.90532
34-370
3-05138
34-524
3'
20507
34.663
3.36681
34.789
3-53698
34.904
0-45
2.89217
38.800
3-03885
38.961
3'
i93J5
39.107
3-35546
39-239
3.52617
39.358
o-5
2.87861
43.270
3-02595
43-435
3
,18088
43.584
3-34378
43-719
3-51507
43.840
0-55
2.86499
47.784
3.01300
47-947
3'
,16856
48.096
3-33207
48.230
3-50393
48.351
0.6
2.85165
52-339
3.00031
52.499
3'
15650
52.643
3.32060
52.773
3-49302
52.890
0.65
2.83891
56.937
2.98821
57.088
3-
14500
57.224
3-30967
57.346
3-48263
57-457
0.7
2.82710
6i.575
2.97699
61.713
3.
13434
61.837
3-29955
61.949
3-47301
62.050
0-75
2.81653
66.250
2-96695
66.371
3'
,12481
66.480
3.29049
66.579
3.46441
66.668
0.8
2.80747
70.958
2-95835
7L059
3'
,11665
71.150
3.28274
71.233
3-45702
71-307
0-85
2.80016
75.693
2.95142
75-772
3'
,11006
75.842
3.27649
75.906
3-45II2
75-964
0.9
2.79479
80.449
2-94633
80.503
3
10523
80.552
3.27191
80.595
3-44675
80.635
o-95
2.79151
85.221
2.94322
85.248
3.
,10228
85-273
3.26911
85-295
3.44410
85-3I5
I.O
2.79041
90.000
2.94217
90.000
3-
10129
90.000
3.26816
90.000
3-44321
90.000
1.05
2.79151
94-779
2.94322
94-752
3
,10228
94.727
3.26911
94-705
3.44410
94.685
i.i
2-79479
99-551
2.94633
99-497
3,
10523
99.448
3.27191
99.405
3-44675
99-365
1.15
2.80016
104.307
2.95142
104.228
3'
,11006
104.158
3-27649
104.094
3.45112
104.036
1.2
2.80747
109.042
2-95835
108.941
3'
,11665
108.850
3-28274
108.767
3-45702
108.693
1.25
2.81653
II3-750
2.96695
113.629
3
,12481
113.520
3.29049
113.421
3.46441
II3-332
2.82710
118.425
2.97699
118.287
3'
13434
118.163
3-29955
118.051
3-47301
117.950
i-35
2.83891
123.063
2.98821
122.912
3-
,14500
122.776
3.30967
122.654
3.48263
122.543
2.85165
127.661
3.00031
127.501
3'
15650
127.357
3.32060
127.227
3-49302
127.110
1-45
2.86499
132.216
3.01300
132-053
3'
16856
131.904
3-33207
131.770
3-50393
131.649
1-5
2.87861
136.730
3-02595
136-565
3'
,18088
136.416
3-34378
136.281
3-5I507
136.160
i-SS
2.89217
141.200
3-03885
141.039
3'
I93I5
140.893
3-35546
140.761
3-52617
140.642
1.6
2.90532
145.630
3-05138
I45-476
3.
20507
145.337
3.36681
145.211
3-53698
145.096
1-65
2.91777
150.020
3-06323
149.877
3-
21636
149.748
3-37756
149.631
3-54721
149.524
2.92921
154-375
3-07413
154.246
3'
22675
154.129
3-38745
154.023
3-55663
I53-927
1-75
2.93938
158.698
3.08382
158-586
3'
23598
158.484
3-39624
158.392
3-56500
158.308
1.8
2.94804
162.992
3.09206
162.900
3,
24384
162.816
3.40373
162.740
3-572I3
162.670
1.85
2.95498
167.265
3.09869
167.194
3.
25015
167.129
3-40975
167.071
3-57788
167.017
1.9
2.96006
171.520
3-10353
171.471
3'
25477
171.428
3-41415
171.388
3-58207
171-352
i-95
2.96315
I75-763
3.10648
I75-738
3'
•25759
175.716
3.41683
175-696
3.58462
175-678
2.O
2.96419
180.000
3.10747
180.000
3
•25853
180.000
3-4I773
180.000
3.58548
180.000
Example.
cosh (1.95
+ »' 1.25)
= 3.4644I /ii3°.33i
! = 3-46441 /H3°.I9'.
s£
[113]
TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r /T. CONTINUED
x = 2.05
a; = 2.15
o
0.05
O.I
0.15
0.2
3.76220 o.ooo
. 3.76137 4-339
3.75894 8.68!
3-75495 13-031
3.74948 17-392
3.94832 o.ooo
3.94753 4.354
3.94521 8.711
3.94142 13.075
3.93620 17.449
4.14431 o.ooo
4-14357 4-368
4.14136 8.738
4-13774 13-115
4.13278 17.501
4.35067 o.ooo
4.34996 4.380
4.34785 8.763
4.34440 13.151
4.33968 17.548
4.56791 o.ooo
4.56723 4-391
4.56523 8.785
4.56167 13.184
4.55744 I7-591
0.25
0-35
0.4
0-45
3.74268 21.768
3.73470 26.160
3-72574 30-573
3.71600 35.008
3-70572 39467
3.92972 21.836
3.92213 26.239
3-9r359 30-660
3.90432 35.102
3-89453 39-565
4.12661 21.899
4.11938 26.311
4.11125 30.740
4.10243 35.187
4.09311 39.653
4.33381 21.956
4.32693 26.376
4.31918 30.812
4.31079 35-264
4.30193 39.734
4.55184 22.007
4.54529 26.435
4-53793 30-877
4.52993 35-334
4.52150 39.807
0-5
0-55
0.6
0.65
o-7
3.69515 43-951
3.68455 48.461
3.67418 52.996
3.66430 57-557
3.65517 62.142
3.88448 44-051
3.87440 48.560
3-86454 53-092
s-Sss^ 57-647
3.84648 62.224
4.08355 44.141
4.07396 48.649
4.06459 53.179
4.05566 57.729
4.04740 62.299
4.29282 44.223
4.28371 48.730
4.27479 53.257
4.26630 57.803
4.25846 62.366
4.51285 44.297
4.50417 48.804
4-49570 53-329
4.48763 57-870
4.48017 62.427
0-75
0.8
0.85
0.9
0-95
• 3.64699 66.748
3.64000 71.374
3-63437 76-016
3.63023 80.670
3.62771 85.333
3.83870 66.820
3.83206 71-434
3.82671 76.062
3.82279 80.702
3.82039 85.349
4.04002 66.886
4.03371 71.489
4.02863 76.105
4.02490 80.731
4.02263 85.364
4.25145 66.945
4.24544 71.538
4.24061 76-143
4.23707 80.757
4.23491 85.377
4.47350 66.998
4.46780 71.582
4.46322 76.177
4.45985 80.780
4-45779 85.389
I.O
1.05
i.i
1.2
3.62686 90.000
3.62771 94.667
3.63023 99.330
3.63437 103.984
3.64000 108.626
3.81958 90.000
3.82039 94-651
3.82279 99.298
3.82671 103.938
3.83206 108.566
4.02186 90.000
4.02263 94-636
4.02490 99.269
4.02863 103.895
4.03371 108.511
4.23419 90.000
4.23491 94.623
4.23707 99.243
4.24061 103.857
4.24544 108.462
4.45711 90.000
4-45779 94-6ii
4.45985 99.220
4.46322 103.823
4.46780 108.418
1.25
1.4
i-45
3.64699 113.252
3.65517 117-858
3.66430 122.443
3.67418 127.004
3.68455 I3I-539
3.83870 113.180
3.84648 117.776
3.85515 122.353
3.86454 126.908
3.87440 131.440 «
4.04002 113.114
4.04740 117.701
4.05566 122.271
4.06459 126.821
4.07396 131.351
4.25145 113.055
4.25846 117-634
4.26630 122.197
4.27479 126.743
4.28371 131.270
4.47350 113.002
4.48017 117.573
4.48763 122.130
4.49570 126.671
4.50417 131.196
i-SS
1.6
1-65
3.69515 136.049
3.70572 140.533
3.71600 I44-992
3.72574 149-427
3.73470 153.840
3.88448 135-950
3.89453 140.435
3.90432 144.898
3-9I359 149-340
3.92213 153.761
4.08355 135.859
4.09311 140.347
4.10243 144.813
4.11125 149.260
4.11938 153.689
4.29282 135.778
4.30193 140.266
4.31079 144-736
4.31918 149.188
4.32693 153.624
4.51285 135.703
4.52150 140.193
4.52993 144.666
4-53793 149-123
4.54529 153.565
1.8
1.85
1.9
1-95
3.74268 158.232
3.74948 162.608
3.75495 166.969
3.75894 171-319
3.76137 i75-66i
3.92972 158.164
3.93620 162.551
3.94142 166.925
3.94521 171.289
3-94753 I75-646
4.12661 158.101
4.13278 162.499
4.13774 166.885
4.14136 171.262
4.14357 175.632
4.33381 158.044
4.33968 162.452
4.34440 166.849
4.34785 171.237
4.34996 175-620
4.55184 157-993
4.55744 162.409
4.56167 166.816
4.56523 171.215
4.56723 175.609
2.0
3.76220 180.000
3.94832 180.000
4.14431 180.000
4-35067 180.000
4.56791 180.000
Example, cosh (2.0 + i 0.5) = 3.69515
= 3-695*5 /43°-57'.04g-
C"4]
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = 7/7. CONTINUED
X =
2.25
x =
2-3
x =
2-35
x =
2.4
T ST
2-45
q
r
y
r
y
r
y
r
y
r
y
o
o
0
0
o
0
4-79657
O.OOO
5.03722
O.OOO
5-
29047
O.OOO
5-55695
O.OOO
S-83732
0.000
0.05
4-79593
4.401
5.03661
4411
5-
28989
4419
5-55639
4.427
5.83680
4-434
O.I
4.79402
8.805
5A3479
8.824
5-
28816
8.840
5-55474
8.856
5-83522
8.869
0.15
4.79088
13.214
5.03181
13.241
5.
28531
I3-265
5-55204
13.288
5-83265
13-308
O.2
4.78661
17.629
5-02773
17.664
5-
28143
17.696
5-54834
17-725
5.82913
I7.75I
0.25
4.78127
22.053
5.02265
22.096
5'
,27662
22.134
5-54375
22.169
5.82476
22.200
°-3
4-77503
26.488
5.01672
26-537
5'
,27096
26.581
5-53837
26.620
5.81964
26.656
o-3S
4.76802
30.936
5.01005
30.989
5'
,26461
3I-038
5-53233
31.081
5-81389
31.121
0.4
4.76042
35-397
5.00281
35-454
5-
25772
35-506
5-52578
35-553
5.80765
35-595
0-45
4.75240
39-873
4.99518
39-932
5-
25046
39.986
5-51887
40-035
5.80108
40.079
o-5
4-744I5
44-364
4-98734
44.424
5'
23728
44-479
5-5II77
44-529
5-79434
44-573
o-55
4-73592
48.870
4-97950
48.930
5'
23553
48.984
5.50468
49.034
5.78758
49.078
0.6
4.72784
53-392
4.97183
53-451
5
,22825
53-503
5-49775
53-551
5-78099
53-593
0.65
4.72017
57-930
4.96454
57-984
5'
22131
58-034
5-49II5
58.078
5-77472
58.119
0.7
4.71308
62.482
4.95780
62.531
5'
,21490
62.576
5-48505
62.617
5.76892
62.653
0-75
4.70675
67.046
4-95I77
67.090
5
.20917
67.129
5.47961
67.165
5-76375
67.197
0.8
4.70134
71.622
4.94662
7I-659
5
.20428
71.691
5-47495
71.721
5-75932
71.748
0.85
4.69697
76.208
4.94249
76-236
5
.20035
76.261
5.47122
76.284
5^5576
76-305
0.9
4-69377
80.801
4-93944
80.820
5
.19745
80.837
5.46847
80.853
5-753i6
80.867
o-9S
4.69182
85-399
4-93759
85.409
5
.19569
85.418
5.46679
85.426
5.75I56
85433
I.O
4.69117
90.000
4.93696
90.000
5
.19510
90.000
5.46623
90.000
5-75103
90.000
1.05
4.69182
94.601
4-93759
94.591
5
.19569
94.582
5.46679
94-574
5.75I56
94-567
i.i
4-69377
99.199
4-93944
99.180
5
.19745
99.163
5.46847
99.147
5.753i6
99-133
i-iS
4.69697
103.792
4.94249
103.764
5
-20035
103-739
5.47122
103.716
5^5576
103.695
1.2
4-70I34
108.378
4.94662
108.341
5
.20428
108.309
5-47495
108.279
5-75932
108.252
1-25
4.70675
112.954
4-95I77
112.910
5
.20917
112.871
5.47961
112.835
5.76375
112.803
J-3
4.71308
117.518
4-9578o
117.469
5
.21490
117.424
5-48505
117-383
5.76892
"7-347
I-3S
4.72017
122.070
4.96454
122.016
5
.22131
121.966
5-49"5
121.922
5-77472
121.881
1.4
4.72784
126.608
4.97183
126.549
5
.22825
126.497
5-49775
126.449
5.78099
126.407
i-4S
4-73592
131.130
4-97950
131.070
5
•23553
131.016
5.50468
130.966
5.78758
130.922
i-5
4-744IS
I35-636
4-98734
I35-576
5
.23728
I35-52I
5-5II77
I35-47I
5-79434
135427
i-SS
4.75240
140.127
4.99518
140.068
5
.25046
140.014
5-51887
I39-965
5.80108
139.921
1.6
4.76042
144.603
5.00281
144.546
5
.25772
144.494
5-52578
144.447
5-80765
144.405
1-65
4.76802
149.064
5.01005
149.011
5
.26461
148.962
5-53233
148.919
5-81389
148.879
i-7
4-77503
I53-5I2
5.01672
I53-463
5
.27096
I53-4I9
5-53837
i53-38o
5.81964
153-344
i-7S
4.78127
157-947
5.02265
157.904
5
.27662
157-866
5-54375
157-831
5.82476
157.800
1.8
4.78661
162.371
5-02773
162.336
5
.28143
162.304
5-54834
162.275
5-82913
162.249
1.85
4.79088
166.786
5.03181
166.759
5
.28531
166.735
5-55204
166.712
5-83265
166.692
1.9
4.79402
I7I-I95
5-03479
171.176
5
.28816
171.160
5-55474
171.144
5-83522
171.131
i-95
4-79593
175-599
5.03661
I75-589
5
.28989
I75-58I
5-55639
175-573
5-83680
I75.566
2.0
4-79657
180.000
5.03722
180.000
5
.29047
180.000
5.55695
180.000
5.83732
180.000
Example. cosh (2.40 + i 2.0) = 5.55695 /i8o° = 5.55695 \i8o°.
["Si
TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED
* = 2.5
a = 2.55
* = 2.6
x = 2.65
* = 2.7
9
r 7
r V
r 7
r 7
r V
0
o
o
0
e
o
6.13229 o.ooo
6.44259 o.ooo
6.76901 o.ooo
7.11234 o.ooo
7-47347 o.ooo
0.05
6.13179 4-440
6.44212 4.446
6.76855 4.451
7.11191 4.456
7.47306 4.460
O.I
6.13030 8.881
6.44069 8.893
6.76720 8.903
7.11062 8.912
7.47183 8.920
0.15
6.12785 13.326
6.43836 13.342
6.76499 13.357
7.10851 13.371
7.46982 13.383
O.2
6.12450 17-774
6.43518 17-796
6.76195 17.815
7.10563 17.833
7.46708 17.848
0.2S
6.12034 22.228
6.43121 22.254
6.75818 22.278
7.10204 22.299
7.46366 22.318
°-3
6.11547 26.689
6.42658 26.718
6.75376 26.745
7.09784 26.770
7.45966 26.791
o-35
6.10999 31.157
6.42137 31.190
6.74881 31.219
7.09313 31.246
7-455i8 31.270
0.4
6.10406 35-634
6.41572 35.668
6-74343 35-700
7.08802 35.728
7.45032 35.754
0-45
6.09781 40.119
6.40977 40.155
6.73778 40.188
7.08264 40.218
7.44519 40.245
o-5
6.09139 44.614
6.40367 44.651
6.73197 44-684
7.07711 44-713
7-43994 44-741
0-55
6.08496 49.118
6.39755 49-155
6.72616 49.187
7.07158 49-217
7.43468 49.244
0.6
6.07869 53.632
6.39160 53-667
6.72048 53-699
7.06618 53.727
742955 53-754
0.65
6.07273 58.155
6.38592 58.188
6.71509 58.218
7.06105 58.244
7.42467 58.269
0.7
6.06722 62.686
6.38068 62.716
6.71011 62.743
7.05631 62.768
7.42017 62.790
0-75
6.06229 67.226
6.37601 67.252
6.70565 67.275
7.05208 67.297
7.41614 67.316
0.8
6.05808 71.772
6.37201 71-794
6.70185 71.813
7.04847 71.831
7.41271 71.847
0.85
6.05471 76-324
6.36879 76.340
6.69880 76.356
7.04556 76.369
7.40994 76.382
0.9
6.05223 80.880
6.36644 80.891
6.69656 80.902
7.04343 80.911
7.40791 80.920
o-95
6.05071 85.439
6.36499 85.445
6.69518 85.450
7.04213 85.455
7.40668 85.460
I.O
6.05020 90.000
6.36451 90.000
6.69473 90.000
7.04169 90.000
7.40626 90.000
1.05
6.05071 94-561
6.36499 94-555
6.69518 94-55°
7.04213 94.545
7.40668 94.540
i.i
6.05223 99.120
6.36644 99.109
6.69656 99.098
7.04343 99.089
7.40791 99.080
i-iS
6.05471 103.676
6.36879 103.660
6.69880 103.644
7.04556 103.631
7.40994 103.618
1.2
6.05808 108.228
6.37201 108.206
6.70185 108.187
7.04847 108.169
7.41271 108.153
1.25
6.06229 112.774
6.37601 112.748
6.70565 112.725
7.05208 112.703
7.41614 112.684
i-3
6.06722 117.314
6.38068 117.284
6.71011 117.257
7.05631 117.232
7.42017 117.210
i-3S
6.07273 121.845
6.38592 121.812
6.71509 121.782
7.06105 121.756
7.42467 121.731
1.4
6.07869 126.368
6.39160 126.333
6.72048 126.301
7.06618 126.273
7.42955 126.246
i-45
6.08496 130.882
6-39755 130-845
6.72616 130.813
7.07158 130.783
7.43468 130.756
i-5
6.09139 135.386
6.40367 135-35°
6-73J97 i35-3i6
7.07711 135.286
7-43994 I35-259
i-55
6.09781 139.881
6.40977 139-845
6.73778 139.812
7.08264 139.782
7-445I9 139-755
1X3
6.10406 144.366
6.41572 144-332
6.74343 144.300
7.08802 144.272
7.45032 144.246
1.65
6.10999 148.843
6.42137 148.810
6.74881 148.781
7.09313 148.754
7.45518 148.730
i-7
6.11547 153.3"
6.42658 153.282
6.75376 I53-255
7.09784 153.230
7.45966 153.209
i-75
6.12034 157.772
6.43121 157-746
6.75818 157-722
7.10204 157.701
7.46366 157.682
1.8
6.12450 162.226
6.43518 162.204
6.76195 162.185
7.10563 162.167
7.46708 162.152
1.85
6.12785 166.674
6.43836 166.658
6.76499 166.643
7.10851 166.629
7.46982 166.617
i. g
6.13030 171.119
6.44069 171.107
6.76720 171.097
7.11062 171.088
7.47183 171.080
1-95
6.13179 175-560
6.44212 175.554
6.76855 175-549
7.11191 175.544
7.47306 175.540
2.0
6.13229 180.000
6.44259 180.000
6.76901 180.000
7.11234 180.000
7-47347 180.000
Example.
cosh (2.65 + i 0.75)
= 7.05208 /67°.297
= 7.05208 /67°.i7'
.49'.
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED
x = 2.75
x =.2.85
* = 2.9
x = 2.95
9
r
7
r
7
r
7
r
7
r
7
o
o
o
o
o
o
7-85328
O.OOO
8.25273
O.OOO
8.67281
O.OOO
9.11458
O.OOO
9.57915
O.OOO
0.05
7.85288
4.464
8-25235
4.467
8.67246
4.470
9.11424
4-473
9-57882
4.476
O.I
7.85172
8.928
8.25124
8-935
8.67140
8.941
9.11324
8-947
9.57787
8.952
0.15
7.84980
13-394
8.24942
13.404
8.66967
13.413
9.11160
13.421
9-57630
13-429
0.2
7.84720
17.863
8.24694
17.876
8.66731
17.888
9.10934
17.898
9.57416
17.908
0.25
7-84395
22-335
8.24385
22.351
8.66436
22.364
9.10655
22.378
9-57150
22.389
o-3
7.84015
26.8II
8.24024
26.829
8.66092
26.845
9.10327
26.860
9.56838
26.873
035
7.83588
31.292
8.23617
3I.3I2
8.65706
3J.330
9.09959
3I-346
9.56488
31-360
0.4
7-83125
35-778
8.23177
35-799
8.65287
35-8i8
9.09561
35-83S
9.56109
35-851
o-4S
7.82638
40.269
8.22713
40.291
8.64846
40.311
9.09142
40-329
9-557II
40.345
o-S
7.82138
44.766
8.22238
44-788
8.64395
44.808
9.08711
44.827
9-55301
44.843
o-55
7.81637
49.269
8.21762
49.290
8.63941
49.310
9.08281
49-329
9.54892
49-345
0.6
7.81149
53-777
8.21298
53-798
8.63500
53-817
9.07861
53-835
9.54492
53-850
0.65
7.80685
58.291
8.20857
58-311
8.63080
58-329
9.07461
58.345
9-54113
58-360
0.7
7.80257
62.810
8.20449
62.828
8.62691
62.845
9.07093
62.859
9.53762
62.873
0-75
7-79875
67-334
8.20085
67-350
8.62347
67.364
9.06764
67-377
9-53450
' 67-389
0.8
7-79547
71.862
8.19774
7I-875
8.62051
71.887
9.06483
71.898
9-53182
71.907
0.85
7-79285
76.393
8.19524
76.404
8.61813
76.413
9-06257
76.421
9.52967
76.429
0.9
7.79092
80.927
8.19341
80.934
8.61639
80.940
9.06091
80.946
9.52809
80.951
o-95
7-78975
85-463
8.19230
85.467
8.61532
85-470
9.05990
85-473
9-52713
85475
I.O
7-78935
90.000
8.19192
90.000
8.61497
90.000
9.05956
90.000
9.52681
90.000
1.05
7-78975
94-537
8.19230
94-533
8.61532
94-530
9.05990
94.527
9-52713
94.525
i.i
7.79092
99.073
8.19341
99.066
8.61639
99.060
9.06091
99.054
9.52809
99.049
i-iS
7.79285
103.607
8.19524
103.596
8.61813
103.587
9.06257
103.579
9.52967
103.571
1.2
7-79547
108.138
8.19774
108.125
8.62051
108.113
9.06483
108.102
9-53182
108.093
".«S
7-79875
112.666
8.20085
112.650
8.62347
112.636
9.06764
112.623
9-53450
112.611
i-3
7.80257
117.190
8.20449
117.172
8.62691
"7-155
9.07093
117.141
9-53762
117.127
•35
7.80685
121.709
8.20857
121.689
8.63080
121.671
9.07461
121.655
9.54ii3
121.640
•4
7.81149
126.223
8.21298
126.202
8.63500
126.183
9.07861
126.165
9-54492
126.150
•45
7.81637
I30-73I
8.21762
130.710
8.63941
130.690
9.08281
130.671
9.54892
130.655
•5
7-82138
I35-234
8.22238
135.212
8.64395
I35-I92
9.08711
I35.I73
9-55301
135.157
•55
7.82638
I39-73I
8.22713
139.709
8.64846
139.689
9.09142
139.671
9-557II
I39.655
.6
7-83125
144.222
8-23177
144.201
8.65287
144.182
9.09561
144.165
9.56109
144.149
•6S
7-83588
148.708
8.23617
148.688
8.65706
148.670
9.09959
148.654
9-56488
148.640
•7
7.84015
153-189
8.24024
i53-I7i
8.66092
I53-I55
9.10327
153-140
9.56838
I53""?
•75
7.84395
I57-665
8.24385
157.649
8.66436
I57-636
9.10655
157.622
9-57I50
157-611
.8
7.84720
162.137
8.24694
162.124
8.66731
162.112
9.10934
162.102
9.57416
162.092
•85
7.84980
166.606
8.24942
166.596
8.66967
166.587
9.11160
166.579
9-57630
166.571
•9
7.85172
171.072
8.25124
171.065
8.67140
171.059
9.11324
171-053
9.57787
171.048
•95
7.85288
I75-536
8-25235
175.533
8.67246
175-530
9.11424
I75.527
9-57882
I75-524
2.O
7-85328
180.000
8.25273
180.000
8.67281
180.000
9.11458
180.000
9.579I5
180.000
Example.
cosh (2.90 + i 0.9)
= 9.06091 /8o°.946
= 0.96091 /8o°.<;6'.46'.
[117]
TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED
x =
3-o
x = 3.05
* = 3-i
x = 3.15 x =
3-2
<7
r
T
r
7
r
y
r
7
r
•y
0
0
0
o
o
0
10.06766
o.ooo
10.58135
O.OOO
11.12150
O.OOO
11.68946
o.ooo
12.28665
O.OOO
0.05
10.06737
4.478
10.58110
4.480
II.I2I2O
4.482
11.68920
4.484
12.28640
4-485
O.I
10.06645
8.956
10.58020
8.960
II.I2O4O
8.964
11.68840
8.968
12.28560
8.971
0.15
10.06500
13-436
10.57880
13.442
II.II9OO
13-447
11.68710
13-452
12.28440
13.457
O.2
10.06292
17.917
10.57680
I7-925
II.II72O
17.932
11.68540
17.938
12.28280
17.944
0.25
10.06040
22.400
10.57440
22.409
II.II49O
22.418
11.68320
22.426
12.28070
22.433
o-3
10.05743
26.886
10.57160
26.896
II.II22O
26.906
11.68060
26.915
12.27820
26.923 .
0-35
10.05410
3J-374
10.56840
3I-386
II.IO92O
31.396
11.67780
31.406
12.27550
3!-4i5
0.4
10.05050
35-865
10.56500
35.878
II.IO6OO
35.890
11.67470
35-900
12.27260
35.910
0-45
10.04670
40.360
10.56140
40-373
II.IO25O
40.385
11.67140
40.396
12.26950
40.406
o-S
10.04280
44.858
10-55770
44.871
II.O99OO
44.884
11.66810
44.895
12.26630
44.905
10.03890
49.360
10.55400
49-373
11.09540
49.385
11.66470
49.396
12.26310
49.406
0.6
10.03510
53-865
10.55040
53-878
II.0920O
53-889
11.66140
53.900
12.26000
53-909
0.65
10.03150
58.373
10.54690
58-385
II.08880
58.396
11.65830
58.406
12.25700
58-415
0.7
10.02820
62.885
10.54380
62.896
11.08580
62.906
11.65540
62.915
12.25430
62.923
0-75
10.62520
67-399
10.54090
67.409
II.O83IO
67.417
11.65290
67-425
12.25180
67.432
0.8
10.02260
71.916
10.53850
71.924
II.o8o8o
71.932
11.65070
71-938
12.24980
71.944
0.85
10.02060
76.435
10.53660
76.441
11.07890
76.447
11.64890
76.452
12.24810
76.457
0.9
10.01910
80.956
10.53520
80.960
11.07750
80.964
11.64770
80.967
12.24690
80.970
o-95
10.01820
85.478
10.53430
85.480
II.O767O
85.482
11.64690
85.484
12.24610
85-485
I.O
10.01787
90.000
10-53399
90.000
11.07645
9O.OOO
11.64661
90.000
12.24588
90.000
1.05
10.01820
94.522
10.53430
94.520
II.O767O
94.518
11.64690
94.516
12.24610
94.5I5
i.i
10.01910
99.044
10.53520
99.040
11.07750
99.036
11.64770
99-033
12.24690
99.030
I-I5
10.02060
103-565
10.53660
103.559
11.07890
103-553
11.64890
103.548
12.24810
103-543
1.2
10.02260
108.084
10.53850
108.076
II.o8o8o
108.068
11.65070
108.062
12.24980
108,056
!.25
10.02520
112.601
10.54090
112.591
II.O83IO
112.583
11.65290
112.575
12.25180
112.568
1-3
10.02820
117.115
10.54380
117.104
11.08580
117.094
11.65540
117.085
12.25430
117.077
i-35
10.03150
121.627
10.54690
121.615
II.08880
I2I.6O4
11.65830
121.594
12.25700
121.585
1.4
10.03510
126.135
10.55040
126.122
II.O92OO
I26.III
11.66140
126.100
12.26000
126.091
i-45
10.03890
130.640
10.55400
130.627
11.09540
130.615
11.66470
130.604
12.26310
130.594
i-5
10.04280
I35.I42
10.55770
I35-I29
II.O9900
I35.II6
11.66810
135-105
12.26630
I35.095
i-55
10.04670
139.640
10.56140
139.627
II.IO250
139.615
11.67140
139.604
12.26950
139-594
1.6
10.05050
I44.I35
10.56500
144.122
II.IO6OO
I44.IIO
11.67470
144.100
12.27260
144.090
1-65
10.05410
148.626
10.56840
148.614
11.10920
148.604
11.67780
148.594
12.27550
148.585
10.05743
I53-II4
10.57160
I53-I04
II.II22O
I53-094
11.68060
153-085
12.27820
I53-077
i-75
10.06040
157.600
10.57440
I57-59I
II.II490
I57-582
11.68320
157-574
12.28070
I57-567
1.8
10.06292
162.083
10.57680
162.075
II.II72O
162.068
11.68540
162.062
12.28280
162.056
1-85
10.06500
166.564
10.57880
166.558
n.II90O
166.553
11.68710
166.548
12.28440
166.543
1.9
10.06645
171.044
10.58020
171.040
11.12040
171.036
11.68840
171.032
12.28560
171.029
I-9S
10.06737
I75-522
10.58110
I75-520
II.I2I20
I75-5I8
11.68920
175-516
12.28640
I75-5I5
2.0
10.06766
180.000
10.58135
180.000
II.I2I50
iSo.OOO
11.68946
180.000
12.28665
iSo.OOO
Example.
cosh (3.15
+ 2Q.I5)
= 11.68710
/i3°.452 = 11.68710 /i3°.27'.07*.
TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r £y. CONTINUED
= 3-25
*' = 3-3
= 3-35
x = 3.4
= 3-45
9
r
7
r
V
r
7
r
7
r
y
o
o
o
o
0
o
12.91456
0.000
I3-57476
O.OOO
14.26891
O.OOO
14.99874
o.ooo
15.76607
O.OOO
0-05
12.91430
4.486
13-57455
4.488
14.26870
4.489
4.490
15-76587
4491
O.I
12.91360
8-973
I3-57387
8.976
14.26805
8.978
14.99790
8.980
15-76530
8.982
0.15
12.91240
13.461
I3-57275
13-465
14.26700
13.468
14.99692
13-471
iS-76433
13-474
O.2
12.91085
17.949
i3-57!23
17.954
14.26556
17-959
14-99555
17.963
15-76303
17.966
0.25
12.90888
22.439
13-56936
22.445
14.26377
22.450
14.99385
22.455
15.76144
22.459
o-3
12.90658
26.930
13.56718
26.937
14.26167
26.943
14.99186
26.948
I5-75950
26.953
o-35
12.90398
31.423
13.56470
31.431
14.25933
31-437
14.98960
3L443
I5-75740
31-449
0.4
12.90117
35-9x8
13.56203
35-926
14.25614
35-933
14.98721
35-939
i5-755io
35-945
o-45
12.89820
40.415
I3-559I8
40.423
14.25412
40.430
14.98466
40-437
15.75270
40.443
0-5
12.89518
44.914
I3-55633
44.922
14-25137
44-929
14.98205
44.936
15.75020
44.942
0-55
12.89215
49-4I5
I3-55346
49.423
14.24863
49.430
14-97945
49-437
I5-74772
49-443
0.6
12.88910
53.918
13.55062
53.926
14.24595
53-933
14.97690
53-939
I5-74529
53-945
0.65
12.88637
58-423
13.54796
58.431
14.24339
58.437
14.97448
58.443
15.74300
58.449
o-7
12.88380
62.930
I3-54550
62.937
14.24106
62.943
14.97225
62.948
15.74088
62.953
0-75
12.88146
67439
I3-54322
67.445
14.23897
67450
14.97023
67455
15-73898
67459
0.8
12.87949
71.949
I3-54I4I
71-954
14.23718
71-959
14.96855
71.963
I5-73736
71.966
0.85
12.87790
76.461
I3-5399I
76.465
I4-23573
76.468
14.96716
76.471
15-73603
76.474
0.9
12.87670
80.973
13-53878
80.976
14.23470
80.978
14.96616
80.980
i5-735io
80.982
0-95
12.87600
85.486
13-53810
85.488
14.23403
85-489
14.96556
85.490
I5-73450
85.491
I.O
12.87578
90.000
13.53788
90.000
14.23382
90.000
14.96536
90.000
I5-73432
90.000
1.05
12.87600
94-5H
13-53810
94.512
14.23403
94-5"
14.96556
94.510
I5-73450
94.509
i.i
12.87670
99.027
I3-53878
99.024
14.23470
99.022
14.96616
99.020
i5-735io
99.018
1.15
12.87790
103.539
I3-5399I
103.535
I4-23573
103.532
14.96716
103.529
15-73603
103.526
1.2
12.87949
108.051
I3-54I4I
108.046
14.23718
108.041
14.96855
108.037
I5-73736
108.034
T.25
12.88146
112.561
I3-54322
"2.555
14.23897
"2.550
14.97023
"2-545
15-73898
112.541
1.3
12.88380
117.070
I3.54550
117.063
14.24106
117.057
14.97225
117.052
15.74088
117.047
1.35
12.88637
121.577
I3-54796
121.569
14.24339
121.563
14.97448
121-557
15.74300
121-551
1.4
12.88910
126.082
13.55062
126.074
14.24595
126.067
14.97690
126.061
I5-74529
126.055
i-45
12.89215
130-585
I3-55346
130.577
14.24863
130.570
14.97945
130.563
I5-74772
130.557
i-S
12.89518
135.086
I3-55633
135-078
14-25137
I35-07I
14.98205
135.064
15.75020
I35-058
i-55
12.89820
139-585
I3-559I8
139-577
14.25412
I39-570
14.98466
139-563
15-75270
139-557
1.6
12.90117
144.082
13.56203
144.074
14.25614
144.067
14.98721
144.061
i5-755io
I44.055
1.65
12.90398
148.577
13.56470
148.569
14-25933
148.563
14.98960
148.557
I5-75740
148.551
12.90658
I53-070
13.56718
153-063
14.26167
I53.057
14.99186
153-052
I5-75950
I53-047
1-75
12.90888
I57-56I
13-56936
157-555
14.26377
IS7.550
14-99385
157-545
15.76144
I57.54I
1.8
12.91085
162.051
13-57123
162.046
14.26556
162.041
14-99555
162.037
15-76303
162.034
1.85
12.91240
166.539
I3-57275
166.535
14.26700
166.532
14.99692
166.529
I5-76433
166.526
1.9
12.91360
171.027
I3-57387
171.024
14.26805
171.022
14.99790
171.020
15-76530
171.018
i-95
12.91430
I75-5I4
13-57455
175-512
14.26870
175-5"
14.99853
i75-5io
15-76587
I75.509
2.0
12.91456
180.000
I3-57476
180.000
14.26891
180.000
14.99874
180.000
15.76607
180.000
Example. cosh (3.4 + to. 75) = 14.97023
= 14-97023 /67°.27'.i8».
["9]
TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED
x = 3-5
x = 3-55
x = 3.6
x = 3-65
* = 3-7
9
r y
r y
r y
r y
r y
o
o
o
0
0
o
16.57282 o.ooo
17.42102 o.ooo
18.31278 o.ooo
19.25033 o.ooo
20.23601 o.ooo
0.05
16.57260 4.492
17.42083 4.493
18.31260 4.493
19.25015 4.494
20.23585 4.495
O.I
16.57210 8.984
17.42030 8.985
18.31210 8.987
19.24970 8.988
20.23540 8.989
0.15
16.57120 13.476
17.41945 13.479
18.31130 13.481
19.24890 13.483
20.23460 13.484
O.2
16.56996 17.969
17.41830 17-972
18.31020 17.975
19.24790 17.977
20.23365 17.979
O.25
16.56840 22.463
17.41680 22.467
18.30880 22.470
19.24650 22.473
20.23240 22.475
O-3
16.56660 26.958
17.41510 26.962
18.30715 26.965
19.24496 26.969
20.23090 26.972
0-35
16.56460 31.454
17.41310 31.458
18.30530 31.462
19.24323 31.466
20.22930 31.469
0.4
16.56240 35.950
17.41110 35.955
18-30335 35-959
19.24134 35.963
20.22750 35.967
0.45
16.56010 40.448
17.40900 40.453
18.30126 40.458
19.23940 40.462
20.22560 40.465
o-5
16.55770 44-948
17.40665 44-953
18.29910 44-957
19.23732 44.961
20.22365 44-965
0-55
16.55538 49-448
17.40441 49-453
18.29699 49.458
19.23530 49.462
20.22170 49.465
0.6
16.55307 53-950
17.40222 53.955
18.29490 53-959
19-23331 53-963
20.21981 53.967
0.65
16.55087 58.454
17.40015 58-458
18.29292 58.462
19.23142 58.465
20.21804 58.469
0.7
16.54885 62.958
17.39822 62.962
18.29109 62.965
19.22968 62.969
20.21639 62.972
0-75
16.54702 67.463
17.39650 67.466
18.28942 67.470
19.22813 67.473
20.21490 67.475
0.8
16.54550 71-969
17.39504 71-972
18.28805 71-975
19.22681 71.977
20.21365 71-979
0.85
16.54428 76.476
17.39386 76.479
18.28694 76.481
19.22577 76.483
20.21268 76.484
0.9
16.54337 80.984
17.39300 80.985
18.28611 80.987
19.22496 80.988
20.21189 80.989
0-95
16.54281 85.492
17.39248 85.493
18.28560 85.493
19.22448 85.494
20.21140 85.495
I.O
16.54263 90.000
17.39230 90.000
18.28546 90.000
19.22434 90.000
20.21129 90.000
1-05
16.54281 94.508
17.39248 94.507
18.28560 94.507
19.22448 94.506
20.21140 94.506
i.i
16.54337 99.016
17.39300 99.015
18.28611 99.013
19.22496 99.012
20.21189 99.011
1.15
16.54428 103.524
17.39386 103.522
18.28694 103.519
19.22577 103.518
20.21268 103.516
1.2
16.54550 108.031
I7-395O4 108.028
18.28805 108.025
19.22681 108.023
20.21365 108.021
•25
16.54702 112.537
17.39650 112.534
18.28942 112.530
19.22813 112.527
20.21490 112.525
•3
16.54885 117.042
17.39822 117.038
18.29109 117-035
19.22968 117.031
20.21639 117.028
•35
16.55087 121.547
17.40015 121.542
18.29292 121.538
19.23142 121.535
20.21804 121.531
•4
16.55307 126.050
17.40222 126.045
18.29490 126.041
19.23331 126.037
20.21981 126.033
•45
16.55538 130.552
17.40441 130.547
18.29699 130.542
19.23530 130.538
20.22170 130.535
•5
16.55770 135.052
17.40665 135.047
18.29910 135.043
19.23732 135.039
20.22365 135.035
•55
16.56010 139.552
17.40900 139.547
18.30126 139.542
19.23940 139.538
20.22560 139.535
.6
16.56240 144.050
17.41110 144-045
18.30335 144.041
19.24134 144.037
20.22750 144.033
•65
16.56460 148.546
17.41310 148.542
18.30530 148.538
19.24323 148.534
20.22930 148.531
•7
16.56660 153.042
17.41510 153.038
18.30715 153.035
19.24496 153.031
20.23090 153.028
•75
16.56840 157-537'
17.41680 157-533
18.30880 157.530
19.24650 157-527
20.23240 157.525
.8
16.56996 162.031
17.41830 162.028
18.31020 162.025
19.24790 162.023
20.23365 162.021
1.85
16.57120 166.524
17.41945 166.521
18.31130 166.519
19.24890 166.517
20.23460 166.516
1.9
16.57210 171.016
17.42030 171.015
18.31210 171.013
19.24970 171.012
20.23540 171.011
i-95
16.57260 175.508
17.42083 175-507
18.31260 175-507
19.25015 175.506
20.23585 175-505
2.O
16.57282 180.000
17.42102 180.000
18.31278 180.000
19-25033 180.000
20.23601 180.000
Example. cosh (3.65 + i 0.05) = 19.25015 /4°.494 = 19.25015 /4°.29/.38*.
[120]
TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /r CONTINUED
x = 3-75
* = 3-
= 3-8S
= 3-95
0
21.27230 o.ooo
22.36178 o.ooo
23-50717 o.ooo
24.71135 o.ooo
25-9773I o.ooo
0.05
21.27212 4.495
22.36163 4.495
23.50702 4.496
24.71120 4.496
25-97720 4.497
O.I
2I.27I7O 8.990
22.36122 8.991
23.50664 8.992
24.71090 8.993
25.97680 8.993
0.15
2I.27IOO 13.486
22.36055 13.487
23.50600 13.488
24.71024 13.489
25-97630 13.490
0.2
2I.270OO 17.981
22.35962 17-983
23.50512 17.985
24.70940 17.986
25-97550 I7-988
0.2S
21.26885 22.478
22.35850 22.480
23.50404 22.482
24.70840 22.483
25-97450 22.485
o-3
21.26745 26.974
22.35716 26.977
23.50277 26.979
24.70720 26.981
. 25.97337 26.983
o-3S
21.26586 31.472
22.35565 31-474
23-50136 31-477
24.70580 31.479
25.97206 31.481
0.4
21.26415 35.970
22.35403 35-973
23.49980 35.975
24.70440 35.978
25.97066 35.980
o-4S
21.26236 40.469
22.35230 40.472
23.49820 40.474
24.70280 40.477
25.96920 40.479
o-5
2I.26O52 44.968
22.35060 44-971
23.49650 44-974
24.70120 44.977
25.96770 44-979
o-55
21.25869 49.469
22.34883 49-472
23.49486 49.474
24.69964 49.477
25.96618 49.479
0.6
21.25685 53.970
22.34712 53.973
23.49322 53.975
24.69809 53.978
25.96471 53.980
0.65
21.25520 58.472
22.34550 58.474
23-49II5 58.477
24.69662 58.479
25-96333 58-481
0.7
21.25362 62.974
22.34401 62.977
23.49028 62.979
24.69528 62.981
25.96205 62.983
0-75
21.25221 67.478
22.34270 67.480
23.48900 67.482
24.69406 67.483
25.96090 67.485
0.8
21.25102 71.981
22.34153 71-983
23.48791 71.985
24.69302 71.986
25.95991 71.988
0.85
2I.250O6 76.486
22.34061 76.487
23.48704 76.488
24.69223 76.489
25.95911 76.490
0.9
21.24935 80.990
22.33995 80.991
23.48640 80.992
24.69159 80.993
25-95854 80.993
o-95
21.24891 85.495
22.33952 85.496
23.48601 85.496
24.69120 85.496
25.95820 85.497
I.O
21.24878 90.000
22.33941 90.000
23.48589 90.000
24.69110 90.000
25.95806 90.000
1.05
21.24891 94-50S
22.33952 94-504
23.48601 94.504
24.69120 94.504
25.95820 94-503
i.i
21.24935 99-010
22-33995 99-009
23.48640 99.008
24.69159 99.007
25-95854 99-007
i-iS
21.25006 103.514
22.34061 103.513
23.48704 103.512
24.69223 103.511
25.95911 103.510
1.2
2I.25IO2 IO8.OI9
22.34153 108.017
23.48791 108.015
24.69302 108.014
25-9599I 108.013
I-2S
21.25221 112.522
22.34270 112.520
23.48900 112.518
24.69406 112.517
25.96000 112.515
i-3
21.25362 II7.O26
22.34401 117.023
23.49028 117.021
24.69528 117.019
25.96205 117.017
i-3S
2I.2552O 121.528
22.34550 121.526
23.49115 121.523
24.69662 121.521
25-96333 121.519
1.4
21.25685 I26.O3O
22.34712 126.027
23.49322 126.025
24.69809 126.022
25.96471 126.020
i. 45
21.25869 130.531
22.34883 130.528
23.49486 130.525
24.69964 130.523
25.96618 130.521
i-S
21.26052 135.032
22.35060 135.029
23.49650 135.026
24.70120 135.023
25.96770 135.021
i-55
21.26236 I39-531
22.35230 139.528
23.49820 139.526
24.70280 139.523
25.96920 139.521
1.6
21.26415 144.030
22.35403 144.027
23.49980 144-025
24.70440 144.022
25.97066 144.020
1.65
21.26586 148.528
22.35565 148.526
23.50136 148.523
24.70580 148.521
25.97206 148.519
i-7
21.26745 153.026
22.35716 153.023
23-50277 153-021
24.70720 153.019
25-97337 I53-OI7
i-75
21.26885 I57-522
22.35850 157.520
23.50404 157-518
24.70840 157.517
25-9745° I57-5I5
1.8
21.27000 I62.OI9
22.35962 162.017
23.50512 162.015
24.70940 162.014
25.97550 162.012
1.85
2I.27IOO 166.514
22.36055 166.513
23.50600 166.512
24.71024 166.511
25.97630 166.510
1.9
2I.27I7O I7I.OIO
22.36122 171.009
23.50664 171.008
24.71090 171.007
25.97680 171.007
i-9S
21.27212 I75-505
22.36163 175-505
23.50702 175-504
24.71120 i75-5°4
25.97720 175.503
2.0
2I.2723O iSo.OOO
22.36178 180.000
23.50717 180.000
24.71135 180.000
25-9773I 180.000
Example. cosh (3.85 + i 1.05) = 23.48601
= 23.48601 /94°.3o/.i4>.
[121]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r
* =
0
X =
0.05
* =
O.I
* =
3.15
X =
O.2
9
r
7
r
7
f
7
r
7
r
7
0
o
o
0
0
0
0.00
90
0.04996
o.oo
0.09967
O.OO
0.14889
O.OO
0.19738
O.OO
0.05
0.07870
9P
0.09322
57.368
0.12699
37.846
0.16840
27.190
0.21246
20.849
O.I
0.15838
90
0.16607
72.040
O.I87II
56.914
0.21732
45.420
0.25294
36.955
0.15
0.24008
90
0.24520
77.558
0.25987
66.084
0.28232
56.148
0.31045
47.863
0.2
0.32492
90
0.32870
80.329
0.33969
71.092
0-35699
62.612
0-37939
55.054
0.25
0.41421
90
0.41727
81.937
0.42568
74.107
0-43932
66.700
0.45731
59.848
0-3
0.50953
9°
0.51180
82.942
0.51851
76.025
0.52931
69.373
0.54368
63.082
o-3S
0.61280
90
0.61455
83.585
0.61970
77.267
0.62802
71.131
0.63915
65.250
0.4
0.72654
90
0.72778
83.987
o.73J43
78.047
0-73734
72.245
0.74525
66.641
0-45
0.85408
90
0.85476
84.209
0.85678
78.478
0.86004
72.864
0.86439
67.419
0-5
I.OOOOO
90
I.OOOOO
84.279
I.OOOOO
78.616
I.OOOOO
73.064
I.OOOOO
67.670
0-5S
1.17085
90
1.16991
84.209
1.16717
78.478
1.16274
72.864
1.15688
67.419
0.6
1.37638
90
1.37404
83.987
1.36718
78.047
1.35623
72.245
1-34183
66.641
0.65
1.63185
90
1.62722
83.585
1.61369
77.267
1.59231
71.131
1.56459
65.250
0.7
1.96261
90
1.95388
82.942
1.92859
76.025
1.88925
69.373
I.83933
63.082
0-75
2.41421
90
2.39735
81.937
2.34919
74.107
2.27623
66.700
2.18669
59.848
0.8
3.07768
90
3-04234
80.329
2.94391
71.092
2.80120
62.612
2.63581
55-054
0.85
4.16530
90
4.07824
77.558
3.84810
66.084
3.54212
56.148
3-22115
47-863
0.9
6.31375
90
6.02149
72.040
5.34442
56.914
4.60155
45.420
3-95347
36.955
o-9S
12.70620
90
10.72750
57.368
7.87464
37.846
5-93842
27.190
4.70673
20.849
I.O
00
90
20.01667
G.OO
10.03331
O.OO
6.71659
O.OO
5.06649
O.OO
1.05
12.70620
90
10.72750
57-368
7.87464
37.846
5-93842
27.190
4.70673
20.849
.1
6.31375
90
6.02149
72.O4O
5.34442
36.914
4.60155
45.420
3-95347
36.955
•IS
4.16530
90
4.07824
77.558
3.84810
66.084
3-54212
56.148
3-22115
47.863
.2
3-07768
90
3-04234
80.329
2.94391
71.092
2.80120
62.612
2.63581
55.054
•25
2.41421
90
2-39735
81.937
2.34919
74.107
2.27623
66.700
2.18669
59.848
•3
1.96261
90
1.95388
82.942
1.92859
76.025
1.88925
69.373
I-83933
63.082
•35
1.63185
90
1.62722
83.585
1.61369
77.267
1-59231
71.131
1.56459
65.250
1.4
1-37638
90
1.37404
83.987
1.36718
78.047
1.35623
72.245
1-34183
66.641
1-45
1.17085
90
1.16991
84.209
1.16717
78.478
1.16274
72.864
1.15688
67.419
•5
I.OOOOO
90
I.OOOOO
84.279
I.OOOOO
78.616
I.OOOOO
73.064
1.09000
67.670
•55
0.85408
90
0.85476
84.209
0.85678
78.478
0.86004
72.864
0.86439
67.419
.6
0.72654
90
0.72778
83-987
0.73*43
78.047
0.73734
72.245
0.74525
66.641
•65
0.61280
90
0.61455
83.585
0.61970
77.267
0.62802
71.131
0.63915
65.250
•7
0.50953
90
0.51180
82.942
0.51851
76.023
0.52931
69.373
0.54368
63.082
•75
0.41421
90
0.41727
81.937
0.42568
74.107
0.43932
66.700
0.45731
59.848
.8
0.32492
90
0.32870
80.329
0.33969
71.092
0.35699
62.612
0-37939
55.054
•85
0.24008
90
0.24520
77.558
0.25987
66.084
0.28232
56.148
0.31045
47.863
•9
0.15838
9P
0.16607
72.O4O
0.18711
56.914
0.21732
45.420
0.25294
36.953
•95
0.07870
90
0.09322
57.368
0.12699
37.846
0.16840
27.190
0.21246
20.849
2.0 o.oo 90 0.04996 o.oo 0.09967 o.oo 0.14889 o.oo 0.19738 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (o.i + i 0.25) = 0.42568 /74°. 107 = 0.42568 /74°.o6'.2$".m
tanh (o.i + i 1.2) = 2.94391 \7i°.o92 = 2.94391 \7i°.o5'.3i*.
[122]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r
* = 0.25
x = 0.3
= o-3S
x = 0.4
CONTINUED
= 0.45
q
r
V
r
7
r
7
f
7
r
7
o
0
o
o
o
o
0.24492
o.oo
0.29131
O.OO
0.33638
o.oo
0.37995
o.oo
0.42190
0.00
0.05
0.25721
16.710
0.30168
13-805
0-34534
11.652
0.38784
9.990
0.42894
8.665
O.I
0.29145
30.669
0.33123
25.891
0.37127
22.164
0.41000
19.185
0.44965
16.753
0.15
0.34237
41.063
0.37657
35-492
0.41192
30.900
0.44758
27.076
0.48295
23.858
O.2
0.40561
48.442
0-43445
42.715
0.46491
37-770
0.49617
33-498
0.52758
29.796
0.2S
0.47875
53-612
0.50275
48.001
0.52849
42.989
0.55525
38.527
0.58242
34-560
0-3
0.56098
57-214
0.58056
51-799
0.60177
46.843
0.62401
42.331
0.64675
38.242
0-35
0.65262
59-679
0.66796
54-453
0.68466
49-590
0.70225
45-093
0.72031
40.958
0.4
0.75486
61.281
0.76580
56.200
0.77774
51-423
0.79033
46.961
0.80327
42.815
0-45
0.86968
62.184
0.87570
57-194
0.88225
52.474
0.88914
48.039
0.89620
43.896
0-5
I.OOOOO
62.476
I.OOOOO
57.518
I.OOOOO
52.817
I.OOOOO
48.392
I.OOOOO
44.250
o-55
1.14985
62.184
1.14195
57-194
1-13347
52.474
1.12469
48.039
1.11583
43.896
0.6
1.32476
61.281
1.30582
56.200
1.28577
51-423
1.26529
46.961
1.24492
42.815
0.65
1.53228
59-679
1.49710
54-453
1.46059
49.590
1.42400
45.093
1.38830
40.958
0.7
1.78259
57-214
1.72246
51-799
1.66176
46.843
1.60255
42.331
1.54620
38.242
0-75
2.08878
53-612
1.98907
48.001
1.89219
42.989
1.80100
38.527
1.71698
34-560
0.8
2.46545
48.442
2.30177
42.715
2.15096
37-770
2.01545
33-498
1.89545
29.796
0.85
2.92081
41.063
2-65553
35-492
2.42764
30.900
2.23422
27.076
2.07060
23.858
0.9
3-43113
30.669
3.01903
25.891
2.69344
22.164
2.43370
19.185
2.22397
16.753
0-95
3.88795
16.710
3.31480
13-805
2.89571
11.652
2.57838
9.990
2.33132
8.665
I.O
4.08299
0.00
3-43274
o.oo
2.97287
o.oo
2.63193
o.oo
2.37024
o.oo
1.05
3.88795
16.710
3.31480
13.805
2.89571
11.652
2.57838
9.990
2.33132
8.665
i.i
3-43113
30.669
3.01903
25.891
2.69344
22.164
2.43370
19-185
2.22397
16.753
i-iS
2.92081
41.063
2.65553
35.492
2.42764
30.900
2.23422
27.076
2.07060
23.858
1.2
2.46545
48.442
2.30177
42.715
2.15096
37.770
2.01545
33-498
1.89545
29.796
1.25
2.08878
53.612
1.98907
48.001
1.89219
42.989
1.80100
38.527
1.71698
34.560
i-3
1.78259
57.214
1.72246
51-799
1.66176
46.843
1.60255
42.331
1.54620
38.242
i-3S
1.53228
59.679
1.49710
54-453
1.46059
49.590
1.42400
45.093
1.38830
40.958
1.4
1.32476
61.281
1.30582
56.200
1.28577
51-423
1.26529
46.961
1.24492
42.815
i-4S
1.14985
62.184
1.14195
57-194
1-13347
52.474
1.12469
48.039
1.11583
43.896
i-5
I.OOOOO
62.476
I.OOOOO
57.518
I.OOOOO
52.817
I.OOOOO
48.392
I.OOOOO
44.250
i-55
0.86968
62.184
0.87570
57-194
0.88225
52.474
0.88914
48.039
0.89620
43.896
1.6
0.75486
61.281
0.76580
56.200
0.77774
51-423
0.79033
46.961
0.80327
42.815
1.65
0.65262
59-679
0.66796
54-453
0.68466
49.590
0.70225
45.093
0.72031
40.958
i-7
0.56098
57-214
0.58056
51-799
0.60177
46.843
0.62401
42.331
0.64675
38.242
i-7S
0.47875
53-612
0.50275
48.001
0.52849
42.989
0.55525
38.527
0.58242
34.560
1.8
0.40561
48.442
0.43445
42.715
0.46491
37.770
0.49617
33.498
0.52758
29.796
1.85
0.34237
41.063
0.37657
35-492
0.41192
30.900
0.44758
27.076
0.48295
23.858
1.9
0.29145
30.669
0.33123
25.891
0.37127
22.164
0.41090
19.185
0.44965
16.753
i. 95
0.25721
16.710
0.30168
13-805
0.34534
11.652
0.38784
9.990
0.42894
8.665
2.0 0.24492 o.oo 0.29131 o.oo 0.33638 o.oo 0.37995 o.oo 0.42190 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (0.4 + i o) = 0.37995 /o°
tanh (0.45 + * i.i) = 2.22397 \i6°.753 = 2.22397 \i6°.45'.ii*.
[123]
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + *?) = r /y. CONTINUED
* = o.s
* = 0.55
* = 0.6
x = 0.65
* = 0.7
q
r y
r 7
r 7
r 7
r 7
0
o
o
o
0
o
0.46211 o.oo
0.50052 o.oo
0.53704 o.oo
0.57167 o.oo
0.60437 o.oo
0.05
0.46846 7.582
0.50628 6.680
0.54230 5.917
0.57648 5.263
0.60878 4.696
O.I
0.48720 14-732
0.52334 13.027
0.55790 11.570
0.59079 10.312
0.62194 9.217
0.15
0.51758 21.122
0.55115 18.773
0.58344 16.739
0.61428 14.966
0.64357 13.409
0.2
0.55865 26.572
0.58900 23.753
0.61835 21.276
0.64650 19.090
0.67331 17.153
0.2S
0.60952 31.035
0.63616 27.897
0.66205 25.101
"0.68696 22.604
0.71076 20.372
0-3
0.66956 34-544
0.69209 31.204
0.71405 28.190
0.73523 25.471
0.75548 23.017
0-35
0.73847 37.I69
0.75645 33.707
0.77399 30.553
0.79092 27.683
0.80711 25.074
0.4
0.8l62* 38.983
0.82914 35.453
0.84168 32.214
0.85377 29.248
0.86531 26.539
0-45
0.90328 40.046
0.91025 36.482
0.91703 33.J98
0.92354 3°.l8o
0.92973 27.414
0-5
1. 00000 40.395
i.ooooo 36.822
i.ooooo 33.524
i.ooooo 30.489
i.ooooo 27.705
0-55
1.10708 40.046
1.09860 36.482
1.09048 33.198
1.08279 30.180
1.07558 27.414
0.6
1.22508 38.983
1.20607 35-453
1. 18810 32.214
1.17128 29.248
1.15566 26.539
0.65
1-35414 37.169
1-32197 33.707
1.29201 30.553
1.26434 27.683
1.23898 25.074
0.7
1-49352 34-544
1.44490 31.204
1.40047 28.190
1.36012 25.471
1.32366 23.017
0-75
1.64064 31.035
1.57193 27.897
1.51047 25.101
1.45568 22.604
1.40695 20.372
0.8
1.79004 26.572
1.69780 23.753
1.61722 21.276
1.54680 19.090
1.48519 17.153
0.85
1.93206 21.122
1.81438 18.773
1.71398 16.739
1.62793 14.966
1.55384 13.409
0.9
2.05254 14.732
1.91081 13.027
1.79243 11.570
1.69266 10.312
1.60788 9.217
o-9S
2.13465 7.582
1.97520 6.680
1.84400 5.917
1.73467 5.263
1.64262 4.696
1.0
2.16395 o.oo
1.99792 o.oo
1.86202 o.oo
1.74926 o.oo
1.65462 o.oo
1.05
2.13465 7.582
1.97520 6.680
1.84400 5.917
1.73467 5.263
1.64262 4.696
i.i
2.05254 14-732
1.91081 13.027
1.79243 11.570
1.69266 10.312
1.60788 9.217
i.iS
1.93206 21.122
1.81438 18.773
1.71398 16.739
1.62793 14.966
1.55384 13.409
1.2
1.79004 26.572
1.69780 23.753
1.61722 21.276
1.54680 19.090
1.48519 17.153
I-2S
1.64064 31.035
1.57193 27-897
1.51047 25.101
1.45568 22.604
1.40695 20.372
i-3
1.49352 34-544
1.44490 31.204
1.40047 28.190
1.36012 25.471
1.32366 23.017
i-3S
1.35414 37-169
1.32197 33-707
1.29201 30.553
1.26434 27.683
1.23898 25.074
1.4
1.22508 38.983
1.20607 35-453
I. 18810 32.214
1.17128 29.248
1.15566 26.539
1-45
1.10708 40.046
1.09860 36.482
1.09048 33.198
1.08279 30.180
1.07558 27.414
i-S
i.ooooo 40.395
i.ooooo 36.822
i.ooooo 33.524
i.ooooo 30.489
i.ooooo 27.705
i-SS
0.90328 40.046
0.91025 36.482
0.91703 33-198
0.92354 30.180
0.92973 27.414
1.6
0.81628 38.983
0.82914 35.453
0.84168 32.214
0.85377 29.248
0.86531 26.539
1.65
0.73847 37-i69
0.75645 33.707
0.77399 30.553
0.79092 27.683
0.80711 25.074
i-7
0.66956 34-544
0.69209 31.204
0.71405 28.190
0.73523 25.471
0.75548 23.017
i-7S
0.60952 31-035
0.63616 27.897
0.66205 25.101
0.68696 22.604
0.71076 20.372
1.8
0.55865 26.572
0.58900 23.753
0.61835 21.276
0.64650 19.090
0.67331 17-153
1.85
0.51758 21.122
0.55115 18.773
0.58344 16.739
0.61428 14.966
0.64357 13.409
1.9
0.48720 14.732
0.52334 13.027
0.55790 11.570
0.59079 10.312
0.62194 9.217
1-95
0.46846 7.582
0.50628 6.680
0.54230 5.917
0.57648 5.263
0.60878 4.696
2.0 0.46211 o.oo 0.50052 o.oo 0.53704 o.oo 0.57167 o.oo 0.60437 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (0.7 + i 0.7) = 1.32366 /23°.oi7 = 1.32366 /23°.oi>.oi*'.
tanh (0.6 + » 1.5) = i.oooo \33°.524 = i.oooo \33°.3i'.26"'.
[124]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + ig) = r Ay. CONTINUED
x = 0.75
x = 0.85
x = o.g
x = 0.95
0
0.63515
o.oo
0.66403
0.00
0.69107
o.oo
0.71629
o.oo
0.73978
o.oo
0.05
0.63921
4.202
0.66777
3.767
0.69451
3.384
0.71947
3-043
0.74269
2.741
O.I
0.65131
8.257
0.67892
7.411
0.70478
6.662
0.72892
5-995
0.75141
5.401
0.15
0.67125
12.036
0.69730
10.819
0.72172
9-737
0-74453
8.771
0.76578
7-909
O.2
0.69871
15-432
0.72264
13.898
0.74509
12.526
0.76607
11.297
0.78561
10.196
O.2S
0-73333
18.371
0.75461
16.576
0-77459
14.964
0.79326
I3-5I3
0.81065
12.208
0-3
0.77471
20.804
0.79285
18.807
0.80986
17.004
0.82576
15-375
0.84054
13-904
0.35
0.82247
22.707
0.83695
20.559
0.85051
18.613
0.86316
16.848
0.87492
15-250
0.4
0.87623
24.068
0.88650
21.819
0.89611
19-773
0.90504
17.914
0.91333
16.226
0.45
0-93557
24.885
0.94104
22.576
0.94614
20.472
0.95086
18.557
0.95523
16.816
0.5
I.OOOOO
25-157
.00000
22.828
I.OOOOO
20.706
I.OOOOO
18.772
I.OOOOO
17.013
0.55
1.06887
24.885
.06265
22.576
1.05693
20.472
1.05168
18.557
1.04687
16.816
0.6
1.14125
24.068
.12803
21.819
1.11594
19-773
1.10492
17.914
1.09490
16.226
0.65
1.21585
22.707
.19482
20.559
1.17576
18.613
1.15853
16.848
1.14297
15-250
0.7
1.29081
20.804
.26128
18.807
1.23478
17.004
I.2IIOI
15-375
1.18971
13.904
0-75
1.36365
18.371
1-32519
16.576
1.29101
14.964
I.26O62
I3-5I3
1.23358
12.208
0.8
1.43121
15-432
1.38382
13.898
1.34212
12.526
1.30536
11.297
1.27289
10.196
0.85
1.48976
12.036
1.43411
10.819
L38559
9-737
I-343I3
8.771
1.30586
7.909
0.9
1-53537
8.257
1.47293
7.411
1.41889
6.662
I.37I89
5-995
1.33083
5.401
o-9S
1.56444
4.202
I-4975I
3-767
1.43986
3-384
1.38992
3-043
1.34645
2.741
I.O
1-57443
o.oo
I.50594
o.oo
1.44703
o.oo
1.39606
o.oo
1.35175
o.oo
1.05
1.56444
4.202
J-49751
3.767
1.43986
3.384
1.38992
3.043
1.34645
2.741
i.i
1-53537
8.257
1.47293
7.411
1.41889
6.662
I.37I89
5-995
1.33083
5.401
i-i5
1.48976
12.036
1.43411
10.819
I.38559
9-737
L343I3
8.771
1.30586
7.909
1.2
1.43121
15.432
1.38382
13.898
1.34212
12.526
1.30536
11.297
1.27289
10.196
1-25
1-36365
18.371
1-325*9
16.576
.29101
14.964
1.26062
I3.5U
1-23358
12.208
i-3
1.29081
20.804
1.26128
18.807
•23478
17.004
I.2IIOI
15-375
1.18971
13.904
1-35
1-21585
22.707
1.19482
20.559
•17576
18.613
I.I5853
16.848
1.14297
15.250
1.4
1.14125
24.068
1.12803
21.819
-II594
19-773
I.I0492
17.914
1.09490
16.226
i -45
1.06887
24.885
1.06265
22.576
•05693
20.472
I.05I68
18.557
1.04687
16.816
i-5
I.OOOOO
25.157
I.OOOOO
22.828
I.OOOOO
20.706
I.OOOOO
18.772
I.OOOOO
17.013
1-55
0-93557
24.885
0.94104
22.576
0.94614
20.472
0.95086
iS-557
0.95523
16.816
1.6
0.87623
24.068
0.88650
21.819
0.89611
19-773
0.90504
17.914
0.91333
16.226
1.65
0.82247
22.707
0.83695
20.559
0.85051
18.613
0.86316
16.848
0.87492
15-250
i.7
0.77471
20.804
0.79285
18.807
0.80986
17.004
0.82576
15-375
0.84054
13.904
i-75
0-73333
18.371
0.75461
16.576
0-77459
14.964
0.79326
13-513
0.81065
12.208
1.8
0.69871
15.432
0.72264
13.898
0-74509
12.526
0.76607
11.297
0.78561
10.196
1.85
0.67125
12.036
0.69730
10.819
0.72172
9-737
0.74453
8.771
0.76578
7.909
1.9
0.65131
8.257
0.67892
7.411
0.70478
6.662
0.72892
5-995
0.75141
5.401
i-95
0.63921
4.202
0.66777
3.767
0.69451
3-384
0.71947
3.043
0.74269
2.741
2.0 0.63515 o.oo 0.66403 o.oo 0.69107 o.oo 0.71629 o.oo 0.73978 o.oo
Note.
Negative quantities are in heavy type.
Examples, tanh (0.9 + i i.o) = 1.39606 /o°.
tanh (0.95 + i 1.55) = 0.95523 \i6°.8i6 = 0.95523 \i6°.48'.s8'.
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) =
x = 1.05
* = 1.15
1
0.76159 o.oo
0.78181 o.oo
0.80050 o.oo
0.81775 °-oo
0.83365 o.oo
5.05
0.76428 2.470
0.78427 2.228
0.80277 2.OIO
0.81984 1.815
0.83557 1-639
3.1
0.77229 4.870
0.79164 4.394
0.80954 3.966
0.82605 3-582
0.84127 3.235
'•IS
0.78552 7.134
0.80380 6.440
0.82071 5.815
0.83630 5.254
0.85067 4.747
X2
0.80376 9.206
0.82058 8.315
0.83611 7.513
0.85043 6.790
0.86361 6.138
5.2S
0.82678 11.032
0.84172 9.971
0.85551 9.015
0.86822 8.151
0.87991 7.371
5-3
0.85425 12.575
0.86693 11.373
0.87862 10.288
0.88939 9.306
0.89927 8.419
5-35
0.88580 13.802
0.89585 12.492
0.90509 11.305
0.91359 10.231
0.92138 9.258
3.4
0.92098 14.694
0.92802 13.305
0.93449 12.045
0.94042 10.904
0.94585 9.870
3-45
0.95925 15.233
0.96294 13.798
0.96632 12.495
0.96941 11.313
0.97223 10.242
3-5
i.ooooo 15.414
i.ooooo 13.963
i.ooooo 12.646
i.ooooo 11.451
i.ooooo 10.368
^•55
1.04248 15.233
1.03849 13.798
1.03486 12.495
1.03155 11.313
1.02854 10.242
3.6
1.08581 14.694
1.07756 13.305
1.07010 12.045
1.06335 10.904
1.05725 9.870
x6S
1.12892 13.802
1.11626 12.492
1.10486 11.305
1.09458 10.231
1.08533 9-258
3-7
1.17061 12.575
1.15349 11.373
1.13815 10.288
1.12437 9.306
I.II20I 8.419
3-75
1.20951 11.032
1.18804 9.971
1.16889 9-OI5
1.15179 8.151
1.13649 7.371
D.8
1.24415 9.206
1.21866 8.315
1.19602 7.513
1.17587 6.790
1.15793 6.138
3.85
1-27305 7-134
1.24409 6.440
1.21846 5.815
1.19574 5.254
I-I7555 4-747
D.9
1.29485 4.870
1.26320 4-394
i-23527 3-966
1.21058 3.582
1.18868 3.235
3-95
1.30843 2.470
1.27506 2.228
1.24569 2.OIO
1.21976 1.815
1.19680 1.639
i.o
1.31304 o.oo
1.27008 o.oo
1.24922 o.oo
1.22286 o.oo
1.19954 o.oo
i-°S
1.30843 2.470
1.27506 2.228
1.24569 2.0IO
1.21976 1.815
1.19680 1.639
i.i
1.29485 4.870
1.26320 4.394
1.23527 3.966
1.21058 3.582
1.18868 3.235
•IS
1-27305 7.134
1.24409 6.440
1.21846 5.815
1.19574 5.254
I-I7555 4-747
.2
1.24415 9.206
1.21866 8.315
1.19602 7.513
1.17587 6.790
1.15793 6.138
•25
1.20951 11.032
1.18804 9.971
1.16889 9'015
1.15179 8.151
1.13649 7.371
•3
1.17061 12.575
1.15349 11.373
1.13815 10.288
1.12437 9.306
I.II20I 8.419
•35
1.12892 13.802
1.11626 12.492
1.10486 11.305
1.09458 10.231
I-08533 9.258
•4
1.08581 14.694
1.07756 13.305
1.07010 12.045
1.06335 10.904
I.O5725 9.870
•45
1.04248 15.233
1.03849 13.798
1.03486 12.495
1.03155 11.313
I.O2854 IO.242
•5
i.ooooo 15.414
i.ooooo 13.963
i.ooooo 12.646
i.ooooo 11.451
i.ooooo 10.368
•55
0.95925 15.233
0.96294 13.798
0.96632 12.495
0.96941 11.313
0.97223 10.242
.6
0.92098 14.694
0.92802 13.305
0.93449 12.045
0.94042 10.904
0.94585 9.870
•65
0.88580 13.802
0.89585 12.492
0.90509 11.305
0.91359 10.231
0.92138 9.258
•7
0.85425 12.575
0.86693 "-373
0.87862 10.288
0.88939 9.306
0.89927 8.419
•75
0.82678 11.032
0.84172 9.971
0.85551 9.015
0.86822 8.151
0.87991 7.371
.8
0.80376 9.206
0.82058 8.315
0.83611 7.513
0.85043 6.790
0.86361 6.138
•85
0.78552 7.134
0.80380 6.440
0.82071 5.815
0.83630 5.254
0.85067 4.747
•0
0.77229 4.870
0.79164 4.394
0.80954 3.966
0.82605 3.582
0.84127 3.235
1-95
0.76428 2.470
0.78427 2.228
O.80277 2.OIO
0.81984 1.815
0.83557 »-°39
2.0 0.76159 o.oo 0.78181 o.oo 0.80050 o.oo 0.81775 o.oo 0.83365 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (i.i + i 0.7) = 1.13815 /io°.288 = 1.13815 /io°.i7/.i7*.
tatth (1.2 + f
0.89927 \8°.4i9 = 0.89927 \8°.2s'.o8*.
[126]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r
CONTINUED
* = 1.25
* = 1.3
* = i-35
x = 1.4
x = 1-45
9
f t
r y
f y
f 7
r 7
o
o
o
o
0
0
0.84828 o.oo
0.86172 o.oo
0.87405 o.oo
0.88535 o.oo
0.89569 o.oo
o.os
0.85004 1.481
0.86333 1.338
0.87552 1. 210
0.88669 I-O94
0.89692 0.989
O.I
0.85526 2.924
0.86811 2.643
0.87990 2.389
0.89069 2.161
0.90057 1.953
0.15
0.86387 4.291
0-87599 3-879
O.887II 3.508
0.89728 3.172
0.90659 2.869
O.2
0.87573 5-549
0.88684 5-017
0.89702 4.538
0.90634 4.104
0.91485 3.712
0.25
0.89063 6.666
0.90047 6.029
0.90947 5.454
0.91769 4.934
0.92521 4.463
0-3
0.90833 7.616
0.91663 6.890
O.9242I 6.234
0.93114 5.640
0.93746 5.103
0-35
0.92852 8.378
0.93504 7.581
0.94099 6.86O
0.94642 6.207
0-95*37 5-6i7
0.4
0.95082 8.933
0.95534 8.085
0-95947 7-3I7
0.96323 6.622
0.96665 5.993
0-45
0.97481 9.272
0.97715 8.393
0.97928 7.596
0.98122 6.875
0.98298 6.222
o-S
i.ooooo 9.385
i.ooooo 8.496
i.ooooo 7.689
i.ooooo 6.960
i.ooooo 6.299
o-SS
1.02584 9.272
1.02338 8.393
1.02116 7.596
1.01914 6.875
I.OI733 6.222
0.6
1.05173 8.933
1.04674 8.085
1.04224 7.317
1.03817 6.622
1.03450 5.993
0.65
1.07699 8.378
1.06948 7.581
1.06271 6.860
1.05661 6.207
1.05112 5.617
0.7
1.10092 7.616
1.09096 6.890
1.08200 6.234
1.07395 5-640
1.06671 5.103
0-75
1.12280 6.666
1.11054 6.029
1.09955 5-454
1.08963 4.934
1.08084 4-463
0.8
1.14192 5.549
1.12760 5.017
1.11483 4.538
1.10335 4.104
1.09308 3.712
0.85
1.15758 4-291
1.14156 3.879
1.12726 3.508
1.11448 3.172
1.10304 2.869
0.9
1.16924 2.924
1.15193 2.643
1.13650 2.389
1.12272 2.161
1.11041 1.953
0-95
1.17642 1.481
1.15831 1.338
I.I42I8 1. 210
1.12779 1-094
1.11493 0.989
I.O
.17885 o.oo
1.16047 o.oo
1.14410 o.oo
1.12950 o.oo
.11646 o.oo
1.05
.17642 1.481
1.15831 1.338
I.I42I8 I.2IO
1.12779 1.094
.11493 0.989
.1
.16924 2.924
1.15193 2.643
I.I3650 2.389
1.12272 2.161
.11041 1.953
•IS
.15758 4.291
1.14156 3.879
I.I2726 3.508
1.11448 3.172
.10304 2.869
.2
.14192 5.549
1.12760 5.017
I.II483 4.538
1.10335 4.104
.09308 3.712
•25
.12280 6.666
1.11054 6.029
1.09955 5-454
1.08963 4.934
1.08084 4.463
•3
.10092 7.616
1.09096 6.890
1.08200 6.234
1.07395 5.640
1.06671 5.103
•35
.07699 8.378
1.06948 7.581
I.O627I 6.86O
1.05661 6.207
1.05112 5.617
1.4
1.05173 8.933
1.04674 8.085
1.04224 7.317
1.03817 6.622
1.03450 5.993
i-45
1.02584 9.272
1.02338 8.393
1.02116 7.596
1.01914 6.875
I.OI733 6.222
-5
i.ooooo 9.385
i.ooooo 8.496
i.ooooo 7.689
i.ooooo 6.960
i.ooooo 6.299
•55
0.97481 9.272
0.97715 8.393
0.97928 7.596
0.98122 6.875
0.98298 6.222
.6
0.95082 8.933
0-95534 8.085
0.95947 7.317
0.96323 6.622
0.96665 5-993
•65
0.92852 8.378
0.93504 7-58i
0.94099 6.860
0.94642 6.207
0.95137 5-617
-7
0.90833 7-616
0.91663 6.890
0.92421 6.234
0.93114 5.640
0.93746 5.103
-75
0.89063 6.666
0.90047 6.029
0.90947 5-454
0.91769 4.934
0.92521 4-463
.8
0-87573 5-549
0.88684 5.017
0.89702 4.538
0.90634 4.104
0.91485 3.712
•85
0.86387 4.291
0-87599 3.879
0.88711 3.508
0.89728 3.172
0.90659 2.869
•9
0.85526 2.924
0.868 1 r 2.643
0.87990 2.389
0.89069 2.161
0.90057 1.953
•95
0.85004 1.481
0.86333 1.338
0.87552 I.2IO
0.88669 1.094
0.89692 0.989
2.0
0.84828 o.oo
0.86172 o.oo
0.87405 o.oo
0.88535 o-oo
0.89569 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh
(1.25 + to. 25)
= 0.89063 /6°.666
= 0.89063 /6°.3g
'•58'.
tanh (1.25 + i 1.25) = 1.12280 \6°.666 = 1. 12280 \6°.39'.58*.
[127]
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r /y. CONTINUED
*= i-S
x = i-5S
x — 1.6
x = 1.65
x = 1.7
q
r 7
r y
r 7
r 7
r 7
o
o
o
o
0
o
0.90515 o.oo
0.91379 o.oo
0.92167 o.oo
0.92886 o.oo
0.93541 o.oo
0.05
0.90627 0.894
0.91481 0.809
0.92260 0.732
0.92970 O.662
0.93618 0.599
O.I
0.90960 1.767
0.91785 1.598
0.92537 1.446
0.93223 1.308
0.93848 1.183
0.15
0.91509 2.595
0.92285 2.347
0.92993 2.123
0.93638 I.92I
0.94226 1.738
0.2
0.92263 3.357
0.92972 3.038
0.93619 2.748
0.94207 2.486
0.94744 2.249
0.2S
0.93207 4-038
0.93832 3-653
0.94401 3.305
O.9492O 2.990
0.95392 2.705
o-3
0.94323 4.617
0.94847 4.177
0.95325 3-780
0.95760 3.420
0.96155 3.095
o-35
0.95588 5.083
0.95998 4.599
0.96371 4.161
0.96710 3.766
0.97018 3.407
0.4
0.96976 5.423
0.97259 4.908
0.97516 4.440
0.97749 4.019
0.97961 3.636
0-45
0.98458 5.631
0.98603 5.096
0-98735 4-6n
0.98854 4.173
0.98962 3.776
o-S
i.ooooo 5.700
i.ooooo 5.159
i.ooooo 4.668
i.ooooo 4.225
i.ooooo 3.822
o-5S
1.01566 5.631
1.01417 5.096
1.01281 4.611
1.01159 4-173
1.01049 3-776
0.6
1.03118 5.423
1.02818 4.908
1.02548 4.440
1.02303 4.019
1.02082 3.636
0.65
1.04616 5.083
1.04169 4.599
1.03766 4.161
1.03402 3.766
1.03074 3.407
0.7
1.06019 4.617
1-05433 4-177
1.04904 3.780
1.04428 3.420
1-03999 3-095
0-75
1.07289 4.038
1.06574 3.653
1-05931 3-305
1-05353 2.990
1.04831 2.705
0.8
1.08386 3.357
1-07559 3-038
1.06817 2.748
1.06149 2.486
1.05548 2.249
0.85
1.09279 2.595
1.08360 2.347
1-07535 2.123
1.06795 1-921
1.06128 1.738
0.9
1.09938 1.767
1.08951 1.598
1.08065 1.446
1.07270 1.308
1.06556 1.183
o-95
1.10343 0.894
1.09313 0.809
1.08390 0.732
1.07562 0.662
1.06817 0.599
I.O
1.10479 o.oo
1.09436 o.oo
1.08500 o.oo
1.07659 o.oo
1.06906 o.oo
1.05
1.10343 0.894
1.09313 0.809
1.08390 0.732
1.07562 0.662
1.06817 0.599
i.i
1.09938 1.767
1.08951 1.598
1.08065 1.446
1.07270 1.308
1.06556 1.183
i-iS
1.09279 2.595
1.08360 2.347
1-07535 2.123
1.06795 1-921
1.06128 1.738
1.2
1.08386 3.357
1-07559 3.038
1.06817 2.748
1.06149 2.486
1.05548 2.249
1.25
1.07289 4.038
1.06574 3-653
1.05931 3.305
1.05353 2.990
1.04831 2.705
i-3
1.06019 4.617
1.05433 4.177
1.04904 3.780
1.04428 3.420
1.03999 3-095
1-35
1.04616 5.083
1.04169 4.599
1.03766 4.161
1.03402 3.766
1.03074 3.407
1.4
1.03118 5.423
1.02818 4.908
1.02548 4.440
1.02303 4.019
1.02082 3.636
I-4S
1.01566 5.631
1.01417 5.096
1.01281 4.611
1.01159 4.173
1.01049 3.776
i-S
i.ooooo 5.700
i.ooooo 5.159
i.ooooo 4.668
i.ooooo 4.225
i.ooooo 3.822
1-55
0.98458 5.631
0.98603 5.096
0.98735 4.611
0.98854 4.173
0.98962 3.776
1.6
0.96976 5-423
0.97259 4.908
0.97516 4.440
0.97749 4.019
0.97961 3.636
1.65
0.95588 5.083
0.95998 4.599
0.96371 4.161
0.96710 3.766
0.97018 3.407
i-7
0.94323 4.617
0.94847 4.177
0.95325 3.780
0.95760 3.420
0.96155 3.095
i-75
0.93207 4.038
0.93832 3.653
0.94401 3.305
0.94920 2.990
0.95392 2.705
1.8
0.92263 3-357
0.92972 3.038
0.93619 2.748
0.94207 2.486
0.94744 2.249
1.85
0.91509 2.595
0.92285 2.347
0.92993 2.123
0.93638 1.921
0.94226 1.738
1.9
0.90960 1.767
0.91785 1.598
0.92537 1.446
0.93223 1.308
0.93848 1.183
1-95
0.90627 0.894
0.91481 0.809
0.92260 0.732
0.92970 0.662
0.93618 0.599
2.0 0.90515 o.oo 0.91379 o.oo 0.92167 o.oo 0.92886 o.oo 0.93541 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (1.7 + *' Q-7) = 1-03999 /3°-°95 = 1-03999 /3°-°5>-42y.
tanh (1.6 + » u) = 0.95325 \3°.78o = 0.95325 \3°46'.48*.
[128]
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r fa. CONTINUED
= 1.75
x = 1.8
* = 1.85
* = 1.9
x = 1.95
9
r y
r y
r y
r y
r 7
o
o
o
o
o
o
0.94138 o.oo
0.94681 o.oo
0.95175 o.oo
0.95624 o.oo
0.9603 2 O.OO
0.05
0.94208 0.542
0.94745 0.490
0.95232 0.443
0.95677 0.401
0.96080 0.363
O.I
0.94417 1.070
0.94935 0.968
0.95406 0.876
0.95834 0.793
O.96222 O.7I7
0.15
0.94761 1.572
0.95247 1.422
0.95690 1.287
0.96092 1.165
0.96457 1.053
O.2
0.95232 2.035
0.95676 1.841
0.96079 1.666
0.96445 1.507
0.96778 1.364
0.25
0.95821 2.448
0.96211 2.215
0.96565 2.004
0.96886 1.813
0.97178 1.640
°-3
0.96514 2.800
0.96840 2.533
0.97136 2.292
0.97405 2.074
0.97649 1.877
o-3S
0.97297 3.083
0.97551 2.789
0.97781 2.524
0.97990 2.285
0.98179 2.067
0.4
0.98153 3.291
0.98327 2.977
0.98485 2.694
0.98628 2.438
0.98757 2.2O6
o-4S
0.99061 3.417
0.99149 3.092
0.99230 2.798
0.99303 2.531
0.99369 2.291
o-S
i. ooooo 3.459
i.ooooo 3.130
i.ooooo 2.833
i.ooooo 2.562
i.ooooo 2.319
o-5S
1.00948 3.417
1.00858 3.092
1.00776 2.798
1.00702 2.531
1.00635 2.291
0.6
1.01882 3.291
1.01702 2.977
1.01539 2-694
1.01392 2.438
1.01258 2.206
0.65
1.02778 3.083
1.02511 2.789
1.02269 2.524
1.02051 2.285
1.01854 2.067
0.7
1.03612 2.800
1.03263 2.533
1.02948 2.292
1.02664 2.074
1.02408 1.877
o.7S
1.04362 2.448
1.03939 2.215
1.03558 2.004
1.03214 1.813
1.02904 1.640
0.8
1.05007 2.035
1.04520 1.841
1.04081 1.666
1.03686 1.507
1.03329 1.364
0.85
1.05529 1.572
1.04990 1.422
1.04504 1.287
1.04067 1.165
1.03673 1.053
0.9
1.05913 1.070
1.05336 0.968
1.04816 0.876
1.04347 0.793
1.03926 0.717
0-95
1.06148 0.542
1.05547 0.490
1.05006 0.443
1.04519 0.401
1.04080 0.363
I.O
1.06228 o.oo
1.05619 o.oo
1.05070 o.oo
1.04576 o.oo
1.04131 o.oo
1.05
1.06148 0.542
1.05547 0.490
1.05006 0.443
1.04519 0.401
1.04080 0.363
i.i
1.05913 1.070
1.05336 0.968
1.04816 0.876
1.04347 0.793
1.03926 0.717
i-iS
1.05529 1.572
1.04990 1.422
1.04504 1.287
1.04067 1.165
1.03673 1.053
1.2
1.05007 2.035
1.04520 1.841
1.04081 1.666
1.03686 1.507
1.03329 1.364
1.25
1.04362 2.448
I-°3939 2.215
1.03558 2.004
1.03214 1.813
1.02904 1.640
1-3
1.03612 2.800
1.03263 2.533
1.02948 2.292
1.02664 2.074
1.02408 1.877
i-3S
1.02778 3.083
1.02511 2.789
1.02269 2.524
1.02051 2.285
1.01854 2.067
1.4
1.01882 3.291
1.01702 2.977
1.01539 2.694
1.01392 2.438
1.01258 2.206
1-45
1.00948 3.417
1.00858 3.092
1.00776 2.798
1.00702 2.531
1.00635 3-29i
i-5
i.ooooo 3.459
i.ooooo 3.130
i.ooooo 2.833
i.ooooo 2.562
i.ooooo 2.319
i-S5
0.99061 3.417
0.99149 3-092
0.99230 2.798
0.99303 2.531
0.99369 2.291
1.6
0.98153 3.291
0.98327 2.977
0.98485 2.694
0.98628 2.438
0.98757 2.206
1.65
0.97297 3.083
0.97551 2.789
0.97781 2.524
0.97990 2.285
0.98179 2.067
1-7
0.96514 2.800
0.96840 2.533
0.97136 2.292
0.97405 2.074
0.97649 1.877
J-75
0.95821 2.448
0.96211 2.215
0.96565 2.004
0.96886 1.813
0.97178 1.640
1.8
0.95232 2.035
0.95676 1.841
0.96079 1.666
0.96445 1.507
0.96778 1.364
1.85
0.94761 1.572
0.95247 1.422
0.95690 1.287
0.96092 1.165
0.96457 1.053
1.9
0.94417 1.070
0.94935 0.968
0.95406 0.876
0.95834 0.793
0.96222 0.717
1-95
0.94208 0.542
0.94745 0.490
0.95232 0.443
0.95677 0.401
0.96080 0.363
2.O
0.94138 o.oo
0.94681 o.oo
o-9S 1 75 °-°°
0.95624 o.oo
0.96032 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh
(1.9 + i 0.05) =
0.95677 /o°.40i
= 0.95677 /o°.24'
,04'.
tanh (i 95 + i .
i.ooo\2°.i9'.o8*.
[129]
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) - f
CONTINUED
x = 2.05
* = 2.15
0
o.os
O.I
0.15
O.2
0.96403 o.oo
0.96446 0.3 28
0.96576 0.649
0.96789 0.953
0.97080 1.234
0.96740 o.oo
0.96779 0.297
0.96897 0.587
0.97090 0.862
0.97354 1.117
0.97045 o.oo
0.97081 0.268
0.97188 0.531
0.97363 0.780
0.97603 i.oio
0.97323 o.oo
0.97355 0.243
0.97452 0.481
0.97611 0.706
0.97829 0.914
0.97574 o.oo
0.97604 O.22O
0.97692 0.435
0.97842 0.639
0.98033 0.827
0.25
0-3
0-35
0.4
0-45
0.97443 1.484
0.97870 1.698
0.98351 1.870
0.98875 1.996
0.99429 2.072
0.97684 1.343
0.98071 1.537
0.98507 1.692
0.98981 1. 806
0.99483 1.876
0.97902 1.215
0.98253 1.390
0.98648 1.531
0.99078 1.634
0.99532 1.698
0.98100 1.099
0.98418 1.258
0.98776 1.385
0.99165 1.479
0.99577 1-536
0.98279 0.995
0.98567 1.138
0.98892 1.253
0.99244 1.337
0.99617 1.389
0-5
0-55
0.6
0.65
0.7
i.ooooo 2.098
1.00574 2.072
1.01138 1.996
1.01676 1.870
1.02176 1.698
i.ooooo 1.899
1.00520 1.876
1.01029 1. 806
1.01516 1.692
1.01967 1.537
i.ooooo 1.718
1.00470 1.698
1.00931 1.634
1.01371 1.531
1.01778 1.390
i.ooooo 1.555
1.00425 1.536
1.00842 1.479
1.01240 1.385
1.01608 1.258
i.ooooo 1.406
1.00385 1.389
1.00762 1.337
I.OII2I 1.253
1.01454 1.138
0-75
0.8
0.85
0.9
0-95
1.02624 1.484
1.03008 1.234
1.03318 0.953
1.03545 0.649
1.03685 0.328
1.02371 1.343
1.02718 1.117
1.02998 0.862
1.03203 0.587
1.03328 0.297
1.02143 1-215
1.02456 i.oio
1.02708 0.780
1.02894 0.531
1.03007 0.268
1.01937 1.099
I.O222O 0.914
1.02447 0.706
1.02614 0.481
1.02717 0.243
1.01751 0.995
1.02006 0.827
1.02206 0.639
1.02363 0.435
I.O2455 O.22O
I.O
1.05
i.i
1.2
1.03731 o.oo
1.03685 0.328
1.03545 0.649
1.03318 0.953
1.03008 1.234
1.03370 o.oo
1.03328 0.297
1.03202 0.587
1.02998 0.862
1.02718 1.117
1.03045 o.oo
1.03007 0.268
1.02894 0.531
1.02708 0.780
1.02456 i.oio
1.02751 o.oo
1.02717 0.243
1.02614 0.481
1.02447 0.706
I.O222O O.9I4
1.02486 o.oo
1.02455 0.220
1.02363 0.435
1.02206 0.639
1.02006 0.827
1.25
1-35
1.4
1-45
1.02624 1.484
1.02176 1.698
1.01676 1.870
1.01138 1.996
1.00574 2.072
1.02371 1.343
1.01967 1.537
1.01516 1.692
1.01029 1-806
1.00520 1.876
1.02143 1-215
1.01778 1.390
1.01371 1.531
1.00931 1.634
1.00470 1.698
1.01937 1.099
1.01608 1.258
1.01240 1.385
1.00842 1.479
1.00425 1.536
1.01751 0.995
1.01454 1.138
I.OII2I 1.253
1.00762 1.337
1.00385 1.389
i-S
i-SS
1.6
1.65
i-7
i.ooooo 2.098
0.99429 2.072
0.98875 1.996
0.98351 1.870
0.97870 1.698
i.ooooo 1.899
0.99483 1.876
0.98981 1. 806
0.98507 1.692
0.98071 1.537
i.ooooo 1.718
0.99532 1.698
0.99078 1.634
0.98648 1.531
0.98253 1.390
i.ooooo 1.555
0.99577 1-536
0.99165 1.479
0.98776 1.385
0.98418 1.258
i.ooooo 1.406
0.99617 1.389
0.99244 1.337
0.98892 1.253
0.98567 1.138
i.'s
1.85
1.9
1-95
0.97443 1.484
0.97080 1.234
0.96789 0.953
0.96576 0.649
0.96446 0.328
0.97684 1.343
0.97354 1. 117
0.97090 0.862
0.96897 0.587
0.96779 0.297
0.97902 1.215
0.97603 i.oio
0.97363 0.780
0.97188 0.531
0.97081 0.268
0.98100 1.099
0.97829 0.914
0.97611 0.706
0.97452 0.481
0.97355 0.243
0.98279 0.995
0.98033 0.827
0.97842 0.639
0.97692 0.435
0.97604 O.22O
2.0
0.96403 o.oo
0.96740 o.oo
0.97045 o.oo
0.97323 o.oo
0.97574 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (2.2 + 40) =0.97574 /o°.
tanh (2.2 + i 1.95) = 0.97604 \o°.22o = 0.97604 \o°.i3'.i2*.
[130]
TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r /y. CONTINUED
x = 2.25
x = 2.3
X = 2.35
x = 2.4
x = 2.45
1
r y
* y
r y
r y
r y
o
o
o
o
0
o
0.97803 o.oo
0.98010 o.oo
0.98197 o.oo
0.98367 o.oo
0.98522 o.oo
0.05
0.97829 0.199
0.98034 0.180
0.98219 0.163
0.98388 0.147
0.98540 0.134
O.I
0.97909 0.394
0.98106 0.356
0.98285 O.322
0.98447 0.291
0.98594 0.264
0.15
0.98040 0.578
0.98225 0.523
0.98393 0.473
0.98544 0.428
0.98682 0.388
0.2
0.98219 0.748
0.98387 0.677
0.98539 0.6l3
0.98677 0.554
0.98802 0.501
0.25
0.98441 0.900
0.98589 0.814
0.98722 0-737
0.98843 0.666
0.98953 0.603
0-3
0.98703 1.030
0.98825 0.932
0.98937 0.843
0.99037 0.763
0.99129 0.690
o-3S
0.98997 1.134
0.99092 1.027
0.99178 0.929
0.99256 0.841
0.99326 0.760
0-4
0.99316 1. 211
0.99381 1.095
0.99440 0.991
0.99493 0.896
0.99541 0.812
o.4S
0.99653 1.257
0.99686 1.138
0.99716 I.O3O
0.99743 0.931
0.99767 0.843
0-5
i.ooooo 1.273
i.ooooo 1.152
i.ooooo 1.042
i.ooooo 0.942
i.ooooo 0.853
o-5S
1.00348 1.257
1.00315 1.138
1.00285 1-030
1.00258 0.931
1.00233 0-843
0.6
1.00689 I-2II
1.00623 I-°95
1.00564 0.991
1.00510 0.896
1.00461 0.812
0.65
I.OIOI4 I.I34
1.00917 1.027
1.00829 0.929
1.00750 0.841
1.00678 0.760
0.7
I.OI3I4 1.030
1.01189 0.932
1.01075 0.843
1.00972 0.763
1.00879 0.690
0-75
1.01583 O.gOO
1.01431 0.814
1.01295 0.737
1.01171 0.666
1.01058 0.603
0.8
I.Ol8l4 0.748
1.01640 0.677
1.01482 0.613
1.01340 0.554
I.OI2I2 O.5OI
0.85
I.OI999 0.578
1.01807 0.523
1.01634 0.473
1.01477 0.428
1.01336 0.388
0.9
I.02I36 0.394
1.01930 0.356
1.01745 °-322
1.01578 0.291
I.OI426 0.264
0-95
I.O22I9 0.199
1.02005 0.180
1.01813 0.163
1.01639 0.147
I.OI482 0.134
I.O
I.O2247 O-°O
1.02031 o.oo
1.01836 o.oo
1.01659 o.oo
1.01500 o.oo
1.05
I.O22I9 O.IQ9
1.02005 0.180
1.01813 0.163
1.01639 0.147
1.01482 0.134
i.i
I.O2I36 O.394
1.01930 0.356
1.01745 0.322
1.01578 0.291
1.01426 0.264
i-iS
1.01999 O.578
1.01807 0.523
1.01634 0.473
1.01477 0.428
1.01336 0.388
1.2
I.Ol8l4 0.748
1.01640 0.677
1.01482 0.613
1.01340 0.554
1 .01 21 2 O.5OZ
•25
1.01583 O.90O
1.01431 0.814
1.01295 0.737
1.01171 0.666
I.OIO58 O.6O3
•3
I.OI3I4 I.03O
1.01189 0.932
1.01075 0.843
1.00972 0.763
1.00879 0.60X)
•35
I.OIOI4 I-I34
1.00917 .027
1.00829 0.929
1.00750 0.841
1.00678 0.760
•4
1.00689 I -2 1 1
1.00623 .095
1.00564 0.991
1.00510 0.896
1.00461 0.812
•45
J.00348 1.257
1.00315 .138
1.00285 1.030
1.00258 0.931
1.00233 0.843
•5
i.ooooo 1.273
i.ooooo .152
i.ooooo 1.042
i.ooooo 0.942
i.ooooo 0.853
•55
0.99653 1.257
0.99686 .138
0.99716 1.030
0.09743 °-93i
0.99767 0.843
1.6
0.99316 Z.2II
0.99381 .095
0.99440 0.991
0.99493 0.896
0.99541 0.812
1.65
0.98997 1. 134
0.99092 1.027
0.99178 0.929
0.99256 0.841
0.99326 0.760
i-7
0.98703 1.030
0.98825 0.932
0.98937 0.843
0.99037 0.763
0.99129 0.690
1-75
0.98441 O.9OO
0.98589 0.814
0.98722 0.737
0.98843 0.666
0.98953 0.603
1.8
0.98219 0.748
0.98387 0.677
0.98539 0.613
0.98677 0.554
0.98802 0.501
1.85
0.98040 O.578
0.98225 0.523
0.98393 0.473
0.98544 0.428
0.98682 0.388
1.9
0.97909 0.394
0.98106 0.356
0.98285 0.322
0.98447 0.291
0.98594 0.164
1-95
0.97829 0.199
0.98034 0.180
0.98219 0.163
0.98388 0.147
0.98540 0.134
*.o 0.97803 o.oo 0.98010 o.oo 0.98197 o.oo 0.98367 o.oo 0.98522 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (2.45 + » oj) = 1.00879 /o°-69Q = 1.00879 /o°.4if-24'-
tanh (2.45 + * rj) = 0.99129 \o°.6oo = 0.99129 \o°.4i'-24r.
[131]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r fa. CONTINUED
* = 2.5
x = 2.55
X = 2.6
* = 2.65
x = 2.7
9
r y
f 7
r 7
r 7
f Y
0
o
0
o
0
o
0.98661 o.oo
0.98788 o.oo
0.98903 o.oo
0.99007 o.oo
0.99101 o.oo
0.05
0.98678 O.I2I
0.98803 0.109
0.98916 0.099
0.99019 0.089
0.99112 0.081
O.I
0.98727 0.239
0.98847 0.216
0.98956 0.195
0.99055 0.177
0.99145 0.160
0.15
0.98806 0.350
0.98919 0.317
0.99022 0.287
0.99114 0.260
0.99198 0.235
O.2
0.98916 0.454
0.99018 0.411
0.99111 0.372
0.99196 0.336
0.99272 0.304
0.2$
0.99052 0.546
0.99142 0.494
0.99223 0.447
0.99297 0.405
0-99363 0-366
0-3
O.992II O.625
0.99286 0.565
0.99354 0.512
0.99415 0.463
0.99471 0.419
0-35
0.99390 0.688
0.99448 0.623
0.99500 0.563
0.99548 0.510
0.99591 0.461
0.4
0.99584 0.734
0.99624 0.665
0.99660 0.601
0.99692 0.544
0.99721 0.492
o-4S
0.99789 0.763
0.99809 0.690
0.99828 0.624
0.99844 0.565
0.99859 0.511
o.S
i.ooooo 0.772
i.ooooo 0.699
i.ooooo 0.632
i.ooooo 0.573
i.ooooo 0.518
0-55
I.OO2II 0.763
1.00191 0.690
1.00173 0.624
1.00156 0.565
1.00141 0.511
0.6
I.004I7 0.734
1.00377 0.665
1.00342 0.601
1.00309 0.544
1.00280 0.492
0.65
1.00614 0.688
1.00555 0-623
1.00502 0.563
1.00454 0.510
1.00411 0.461
0.7
1.00795 0.625
1.00719 0.565
1.00651 0.512
1.00589 0.463
1.00532 0.419
0-75
1.00958 0.546
1.00866 0.494
1.00783 0.447
1.00708 0.405
1.00641 0.366
0.8
1.01096 0.454
1.00991 0.411
1.00897 0.372
1.00811 0.336
1.00733 0.304
0.85
1.01208 0.350
1.01092 0.317
1.00988 0.287
1.00894 0.260
1.00808 0.235
0.9
1.01290 0.239
1.01166 0.216
1.01055 0.195
1.00954 0.177
1.00863 o.i 60
o-9S
1.01340 o.i 21
I.OI2I2 O.IO9
1.01096 0.099
1.00991 0.089
1.00896 0.081
I.O
1.01357 0.00
1.01227 o.oo
I. OHIO O.OO
1.01003 O'O°
1.00907 o.oo
i. 05
I.OI340 O.I2I
I.OI2I2 O.IO9
1.01096 0.099
1.00991 0.089
1.00896 0.081
i.i
1.01290 0.239
1.01166 0.216
1.01055 0-195
1.00954 0.177
1.00863 o.i 60
i.iS
1.01208 0.350
I.OIO92 O.3I7
1.00988 0.287
1.00894 0.260
1.00808 0.235
1.2
1.01096 0.454
I.OO99I O.4II
1.00897 0.372
1. 0081 1 0.336
1.00733 0.304
I.2S
1.00958 0.546
I.OO866 0.494
1.00783 0.447
1.00708 0.405
1.00641 0.366
1-3
1.00795 0.625
I.007I9 0.565
1.00651 0.512
1.00589 0.463
1.00532 0.419
1-35
1.00614 0.688
!-oo555 0.623
1.00502 0.563
1.00454 0.510
1.00411 0.461
1.4
1.00417 0.734
1.00377 0.665
1.00342 0.601
1.00309 0.544
1.00280 0.492
i-45
I.OO2II 0.763
1.00191 0.690
1.00173 0.624
1.00156 0.565
1.00141 0.511
i-5
i.ooooo 0.772
i.ooooo 0.699
i.ooooo 0.632
i.ooooo 0.573
i.ooooo 0.518
i-SS
0.99789 0.763
0.99809 0.690
0.99828 0.624
0.99844 0.565
0.99859 0.511
1.6
0.99584 0.734
0.99624 0.665
0.99660 0.601
0.99692 0.544
0.99721 0.492
1.65
0.99390 0.688
0.99448 0.623
0.99500 0.563
0.99548 0.510
0.99591 0.461
1-7
0.99211 0.625
0.99286 0.565
0.99354 0.512
0.99415 0.463
0.99471 0.419
1-75
0.99052 0.546
0.99142 0.494
0.99223 0.447
0.99297 0.405
0.99363 0.366
1.8
0.98916 0.454
0.99018 0.411
0.99111 0.372
0.99196 0.336
0.99272 0.304
1.85
0.98806 0.350
0.98919 0.317
0.99022 0.287
0.99114 0.260
0.99198 0.235
1.9
0.98727 0.239
0.98847 0.216
0.98956 0.195
0.99055 0.177
0.99145 0.160
i-95
0.98678 O.I2I
0.98803 0.109
0.98916 0.099
0.99019 0.089
0.99112 0.081
2.0 0.98661 o.oo 0.98788 o.oo 0.98903 o.oo 0.99007 o.oo 0.99101 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (2.5 + * 0.25) = 0^99052 /o°.546 = 0.99052 / o° .
tanh (2.5 + * 1.75) = 0.99052 \o°.546 = 0.99052 \o°.32'.46*.
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) - r^y. CONTINUED
= 2.75
= 2.8
* = 2.85
x = 2.9
x = 2.95
o
0.05
O.I
0.15
O.2
0.99186 o.oo
0.99196 0.073
0.99226 0.145
0.99275 0.213
0.99341 0.275
0.99263 o.oo
0.99272 0.066
0.99299 0.131
0.99343 0.192.
0.99404 0.249
Q-99333 o.oo
0.99341 0.060
0.99366 0.119
0.99406 0.174
0.99460 0.226
0.99396 o.oo
0.99404 0.054
0.99426 0.107
0.99462 0.158
0.99511 0.204
0.99454 o.oo
0.99460 0.049
0.99480 0.097
0.99513 0.143
0-99558 0.184
O.2S
°-3
0.35
0.4
0.45
0.99424 0.331
0.99521 0.379
0.99630 0.417
0.99748 0.446
0.99872 0.463
0.99478 0.300
0.99566 0.343
0.99665 0.378
0.99772 0.403
0.99884 0.419
0.99528 0.271
0.99607 0.310
0.99697 0.342
0.99793 0.365
0.99895 0.379
0-99573 0.245
0.99645 0.281
0.99726 0.309
0.99813 0.330
0.99905 0.343
0.99613 O.222
0.99679 0.254
0.99752 O.28O
0.99831 0.299
0.99914 0.310
o-5
o.SS
0.6
0.65
0.7
i .00000 0.468
1.00128 0.463
1.00253 0-446
1.00372 0.417
1.00482 0.379
i .00000 0.424
1.00116 0.419
1.00229 °-4°3
1.00336 0.378
1.00436 0.343
i .00000 0.383
1.00105 0.379
1.00207 0.365
1.00304 0.342
1.00394 0.310
i .00000 0.347
1.00095 0.343
1.00187 0.330
1.00275 0.309
1.00356 0.281
i. ooooo 0.313
1.00086 0.310
1.00169 0.209
1.00249 0.280
1.00323 0.254
0-75
0.8
0.85
0.9
o-95
1.00580 0.331
1.00664 0.275
1.00731 0.213
1.00780 0.145
1.00810 0.073
1.00524 0.300
1.00600 0.249
i .0066 1 0.192
1.00706 0.131
1.00733 0.066
1.00474 0.271
1.00543 0.226
1.00598 0.174
1.00638 0.119
1.00663 0.060
1.00429 0.245
1.00491 0.204
1.00541 0.158
1.00578 0.107
1.00600 0.054
1.00388 O.222
1.00444 0.184
1.00489 O.I43
I.OO522 O.O97
1.00543 0.049
I.O
1.05
I.I
1-15
1.2
1.00821 o.oo
1.00810 0.073
1.00780 0.145
1.00731 0.213
1.00664 0.275
1.00742 o.oo
1.00733 0.066
1.00706 0.131
1.00661 0.192
1.00600 0.249
1.00671 o.oo
1.00663 0.060
1.00638 0.119
1.00598 0.174
1.00543 0.226
1.00607 o.oo
1.00600 0.054
1.00578 0.107
1.00541 0.158
1.00491 0.204
1.00549 o.oo
1.00543 0.049
I.OO522 O.O97
1.00489 O.I43
I.OO444 0.184
1.25
i-3
1-35
1.4
1-45
1.00580 0.331
1.00482 0.379
1.00372 0.417
1.00253 0-446
1.00128 0.463
1.00524 0.300
1.00436 0.343
1.00336 0.378
1.00229 0.403
1.00116 0.419
1.00474 0.271
1.00394 0.310
1.00304 0.342
1.00207 0.365
1.00105 0.379
1.00429 0.245
1.00356 0.281
1.00275 0.309
1.00187 0.330
1.00095 0.343
1.00388 O.222
1.00323 0.254
I.OO249 O.28O
I.OOl69 O.299
1.00086 0.310
•5
•55
.6
-65
•7
1 .00000 0.468
0.99872 0.463
0.99748 0.446
0.99630 0.417
0.99521 0.379
i .00000 0.424
0.99884 0.419
0.99772 0.403
0.99665 0.378
0.99566 0.343
1 .00000 0.383
0.99895 0.379
0.99793 0.365
0.99697 0.342
0.99607 0.310
1 .00000 0.347
0.99905 0.343
0.99813 0.330
0.99726 0.309
0.99645 0.281
i .ooooo 0.313
0.99914 0.310
0.99831 0.299
0.99752 0.280
0.99679 0.254
•75
.8
.85
•9
•95
0.99424 0.331
0.99341 0.275
0.99275 0.213
0.99226 0.145
0.99196 0.073
0.99478 0.300
0.99404 0.249
0.99343 0.192
0.99299 0.131
0.99272 0.066
0.99528 0.271
0.99460 0.226
0.99406 0.174
0.99366 0.119
0.99341 0.060
0-99573 0.245
0.99511 0.204
0.99462 0.158
0.99426 0.107
0.99404 0.054
0.99613 0.222
0.99558 0.184
0.99513 0.143
0.99480 O.O97
0.99460 O.O49
2.0
0.99186 o.oo
0.99263 o.oo
0-99333 o-°o
0.99396 o.oo
0.99454 0.00
Note. Negative quantities are in heavy type.
Examples, tanh (2.9 + i 0.5) =
i.oooo/o°.347 =
i.oooo /o°.26'.49ir.
tanh (2.95 + * 1.75) = 0.99613 \o°.222 = 0.99613 \o°.i3'.i9'.
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r
CONTINUED
X = .
5.0
x = 3
•05
* = .
J-i
* = 3
•15
* = .
5-2
9
r
7
r
y
r
7
r
7
f
7
0
o
o
o
o
0
0-9950S
o.oo
0.99552
O.OO
0-99595
O.OO
0.99633
O.OO
0.99668
O.OO
0.05
0.99511
0.044
0.99558
O.O4O
0.99600
0.037
0.99638
0.033
0.99672
0.030
O.I
0.99530
0.088
0-99575
0.079
0.99615
O.O72
0.99651
0.065
0.99685
0.059
0.15
0-99559
0.129
0.99601
O.II7
0.99639
0.106
0.99673
0.096
0.99704
0.086
O.2
0.99600
0.167
0.99638
O.ISI
0.99672
0.137
0.99703
0.124
0.99732
0.112
0.25
0.99650
O.2OI
0.99683
0.182
0.99714
0.165
0.99741
0.149
0.99765
0.135
o-3
0.99709
0.229
0-99737
O.2O8
0.99762
o.i 88
0.99784
0.170
0.99805
0.154
o-3S
0-99775
0.253
0.99797
O.229
0.99816
0.207
0.99833
o.i 88
0.99849
0.170
0.4
0.99847
O.27O
0.99862
0.244
0.99875
O.22I
0.99887
O.2OO
0.99897
0.181
0-45
o.99923
0.281
0.99930
0.254
0-99937
0.230
0-99943
O.2O8
0.99948
o.i 88
o-5
I.OOOOO
0.284
I.OOOOO
0.257
I.OOOOO
0.233
I.OOOOO
O.2II
I.OOOOO
0.191
0-55
1.00078
0.28l
1.00070
0.254
1.00064
0.230
1.00058
O.2O8
1.00052
o.i 88
0.6
1.00153
0.27O
1.00139
0.244
1.00125
0.221
1.00114
O.2OO
1.00103
0.181
0.65
1.00225
0.253
1.00204
O.229
1.00185
O.2O7
1.00167
o.i 88
1.00151
0.170
0.7
1.00292
O.229
I 00264
0.208
1.00239
o.i 88
1.00216
0.170
1.00196
0.154
0-75
1.00351
O.2OI
1.00318
0.182
1.00287
0.165
1.00260
0.149
1.00235
0.135
0.8
1.00402
0.167
1.00363
O.I5I
1.00329
0.137
1.00297
0.124
1.00269
O.I 1 2
0.85
1.00443
O.I29
1.00400
O.II7
1.00360
o.i 06
1.00328
0.096
1.00297
0.086
0.9
1.00473
0.088
1.00427
0.079
1.00387
0.072
1.00350
0.065
1.00316
0.059
o-9S
1.00491
O.O44
1.00444
O.O4O
1.00399
0.037
1.00363
0.033
1.00329
0.030
.0
1.00497
O.OO
1.00450
O.OO
1.00407
O.OO
1.00368
O.OO
1.00333
O.OO
•05
1.00491
0.044
1 .00444
0.040
1.00399
0.037
1.00363
0.033
1.00329
0.030
.1
1.00473
0.088
1.00427
0.079
1.00387
0.072
1.00350
0.065
1.00316
0.059
•15
1.00443
0.129
1.00400
0.117
1.00360
o.i 06
1.00328
0.096
1.00297
0.086
.2
1.00402
0.167
1.00363
0.151
1.00329
0.137
1.00297
0.124
1.00269
O.II2
•25
1.00351
0.201
1.00318
0.182
1.00287
0.165
1.00260
0.149
1.00235
0.135
•3
1.00292
O.229
1.00264
0.208
1.00239
0.188
1.00216
0.170
1.00196
0.154
•35
1.00225
0.253
1.00204
0.229
1.00185
0.207
1.00167
0.188
1.00151
O.I7O
•4
1.00153
O.27O
1.00139
0.244
1.00125
O.22I
1.00114
O.2OO
1.00103
O.lSl
1-45
1.00078
0.281
1.00070
0.254
1.00064
O.23O
1.00058
O.2O8
1.00052
o.i 88
i-5
I.OOOOO
0.284
I.OOOOO
0.257
I.OOOOO
0.233
I.OOOOO
O.2II
I.OOOOO
0.191
i-55
0.99923
0.28l
0.99930
0.254
0-99937
0.230
0.99943
O.2OS
0.99948
0.188
1.6
0.99847
O.27O
0.99862
0.244
0.99875
O.22I
0.99887
O.200
0.99897
0.181
1.65
0.99775
0.253
0.99797
0.229
0.99816
O.2O7
0.99833
0.188
0.99849
0.170
i-7
0.99709
O.220
0.99737
0.208
0.99762
0.188
0.99784
0.170
0.99805
o.i54
1-75
0.99650
O.2OI
0.99683
0.182
0.99714
0.165
0.99741
0.149
0.99765
0-I3S
1.8
0.99600
O.l67
0.99638
0.151
0.99672
0.137
0.99703
0.124
0.99732
O.II2
1.85
0.99559
O.I29
0.99601
0.117
0.99639
0.106
0.99673
0.096
0.99704
0.086
1.9
0.99530
0.088
0-99575
0.079
0.99615
0.072
0.99651
0.065
0.99685
0.059
1-95
0.99511
0.044
0.99558
0.040
0.99600
0.037
0.99638
0.033
0.99672
O.O3O
2.0
0.99505
O.OO
0.99552
O.OO
0-99595
O.OO
0.99633
O.OO
0.99668
O.OO
Note. Negative quantities are in heavy type.
Examples, tanh (3.2 + 10) = 0.99668 /o°.
tanh (3.2 + i 1.05) = 1.00329 \o°.o3o = 1.00329 \o°.i'.48".
[134]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r % CONTINUED
= 3-25
= 3-35
* = 3-4
* = 3-45
o
0.99700 o.oo
0.99728 o.oo
0.99754 o.oo
0.99777 o.oo
0.99799 o.oo
0.05
0.99703 0.027
0.99732 0.024
0.99757 0.022
0.99780 O.O2O
0.99801 0.018
O.I
0.99714 0.053
0.99741 0.048
0.99766 0.044
0.99788 0.039
0.99809 0.036
0.15
0.99732 0.078
0.99758 0.071
0.99781 0.064
0.99802 0.058
0.99821 0.052
O.2
0-99757 O.IOI
0.99780 0.092
0.99801 0.083
0.99820 0.075
0.99837 0.068
0.25
0.99788 0.122
0.99807 o.iio
0.99826 0.100
0.99843 0.090
0.99858 0.082
0-3
0.99823 0.139
0.99840 0.126
0.99855 0.114
0.99869 0.103
0.99882 0.094
0-35
0.99864 0.153
0.98877 0.139
0.99888 0.126
0.99899 0.114
0.99909 0.103
0.4
0.99907 0.164
0.99916 0.148
0.99924 0.134
0.99931 O.I 21
0.99938 o.iio
0-45
0-99953 0.170
0.99958 0.154
0.99962 0.139
0.99965 0.126
0.99968 0.114
0-5
i.ooooo 0.172
i.ooooo 0.156
i.ooooo 0.141
i.ooooo 0.128
i.ooooo 0.116
0-55
1.00047 0.170
1.00042 0.154
1.00039 0.139
1.00035 0.126
1.00032 0.114
0.6
1.00093 0.164
1.00084 0.148
1.00076 0.134
1.00069 O.I 21
1.00062 o.iio
0.65
1.00137 0.153
1.00124 °-I39
I.OOII2 0.126
i.ooioi 0.114
1.00092 0.103
0.7
1.00177 0.139
1.00160 0.126
1.00145 0.114
1.00131 0.103
1.00118 0.094
0-75
I.OO2I3 O.I22
1.00193 o.iio
1.00174 0.100
1.00158 0.090
1.00143 0.082
0.8
1.00244 o.ioi
I.OO22O 0.092
1.00199 0.083
1.00180 0.075
1.00163 0.068
0.85
1.00268 0.078
1.00243 0.071
I.OO22O 0.064
1.00199 0.058
1.00180 0.052
0.9
I.O0286 0.053
1.00259 0.048
I.OO234 O.O44
I.OO2I2 0.039
1.00192 0.036
0-95
I.O0297 O.O27
1.00269 0.024
I.OO243 O.O22
I.OO22O O.O2O
1.00199 0.018
I.O
1.00301 o.oo
I.OO273 °'O°
1.00246 o.oo
1.00223 o.oo
1.00202 O.OO
1.05
I.OO297 0.027
1.00269 0.024
I.OO243 O.O22
I.OO22O O.O2O
1.00199 0.018
.1
1.00286 0.053
1.00259 0.048
I.OO234 O.O44
I.OO2I2 0.039
1.00192 0.036
•15
I.OO268 O.O78
1.00243 0.071
I.OO22O O.OO4
I.OOI99 O.O58
1.00180 0.052
.2
1.00244 o.ioi
1.00220 O.O92
I.OOI99 0.083
I.OOlSo O.O75
1.00163 .0.068
•25
I.OO2I3 O.I22
1.00193 o.iio
1.00174 o.ioo
I.OOI58 O.O9O
1.00143 0.082
•3
I.OOI77 O.I39
1.00160 0.126
I.OOI45 O.II4
I.OOI3I O.IO3
1.00118 0.094
•35
I.OOI37 0.153
1.00124 0.139
I.OOII2 O.I26
i.opioi 0.114
1.00092 0.103
1.4
I.OOO93 0.164
1.00084 0.148
1.00076 O.I34
I.OOO69 O.I2I
1.00062 o.iio
i-45
I.OOO47 O.I7O
1.00042 0.154
1.00039 °-I39
I.OOO35 O.I26
1.00032 0.114
•5
i.ooooo 0.172
i.ooooo 0.156
i.ooooo 0.141
i.ooooo 0.128
i.ooooo 0.116
•55
0.99953 0.170
0.99958 0.154
0.99962 0.139
0.99965 O.I26
0.99968 0.114
.6
0.99907 0.164
0.99916 0.148
0.99924 0.134
0.99931 O.I2I
0.99938 o.iio
.65
0.99864 0.153
0.99877 0.139
0.99888 0.126
0.99899 O.II4
0.99909 0.103
•7
0.99823 0.139
0.99840 0.126
0.99855 0.114
0.99869 0.103
0.99882 0.094
•75
0.99788 O.I22
0.99807 o.iio
0.99826 o.ioo
0.99843 0.090
0.99858 0.082
.8
0.99757 o.ioi
0.99780 0.092
0.99801 0.083
0.99820 O.O75
0.99837 0.068
.85
0.99732 O.O78
0.99758 0.071
0.99781 0.064
0.99802 O.O58
0.99821 0.052
•9
0.99714 0.053
0.99741 0.048
0.99766 0.044
0.99788 O.O39
0.99809 0.036
•95
0.99703 0.027
0.99732 0.024
0-99757 0-022
0.99780 0.020
0.99801 0.018
2.0
0.99700 o.oo
0.99728 o.oo
0.99754 o.oo
0.99777 o.oo
0.99799 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (3.4 + * 0.7) =
1.00131 /o°.io3 =
1.00131 /o0.o6'.ii*.
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r /r CONTINUED
= 3-5
x = 3-55
* = 3-6
x = 3-6S
= 3-7
o
0.99818 o.oo
0.99835 o.oo
0.99851 o.oo
0.99865 o.oo
0.99878 o.oo
0.05
0.99820 0.016
0.99837 0.015
0.99853 0.013
0.99867 O.OI2
0.99879 o.on
O.I
0.99827 0.032
0.99843 0.029
0.99858 0.026
0.99872 0.024
0.99884 O.O22
0.15
0.99838 0.047
0.99853 0.043
0.99867 0.039
0.99880 0.035
0.99891 0.032
O.2
0.99853 0.06 1
0.99867 0.055
0.99879 0.050
0.99891 0.046
0.99901 0.041
0.25
0.99871 0.074
0.99883 0.067
0.98894 0.061
0.99905 0.055
0.99914 0.050
0-3
0.99893 0.084
0.99903 0.076
0.99912 0.069
0.99921 0.063
0.99928 0.057
0-35
0.99917 0.093
0.99925 0.084
0.99932 0.076
0.99939 0.069
0.99945 0.063
0.4
0.99944 0.099
0.99949 0.090
0.99954 0.081
0.99958 0.074
0.99962 0.067
0-45
0.99972 0.103
0.99974 0.093
0.99977 0.085
0.99979 0.076
0.99981 0.069
0-S
i.ooooo 0.104
i.ooooo 0.095
i.ooooo 0.086
i.ooooo 0.077
i.ooooo 0.070
0-55
1.00028 0.103
1.00026 0.093
1.00023 0.085
I.OOO2I 0.076
1.00019 0.069
0.6
1.00056 0.099
1.00051 0.090
1.00046 0.081
1.00042 0.074
1.00038 0.067
0.65
1.00083 o'°93
1.00075 0.084
.00068 0.076
1. 0006 1 0.069
1.00056 0.063
0.7
1.00107 0.084
1.00097 0.076
.00088 0.069
1.00079 0.063
1.00072 0.057
0-75
1.00129 0.074
1.00117 0.067
.00106 0.061
1.00096 0.055
1.00087 0.050
0.8
1.00148 0.061
1.00134 0.056
.OOI 2 1 O.O5O
1.00109 0.046
1.00099 0.041
0.85
1.00163 0.048
1.00147 0.043
.00133 0.039
I.OOI2O O.035
1.00109 0.032
0.9
1.00174 0.032
1.00157 0.029
.00142 0.026
1.00129 0.024
I.OOII7 O.O22
o-9S
1.00180 0.016
1.00163 0.015
1.00148 0.013
I.OOI34 O.OI2
I.OOI2I O.OII
I.O
1.00183 o'°o
1.00165 °'°o
1.00149 o.oo
1.00135 o.oo
I.OOI22 O.OO
1.05
1.00180 0.016
1.00163 0.015
1.00148 0.013
I.OOI34 O.OI2
I.OOI2I O.OII
i.i
1.00174 0.032
1.00157 0.029
1.00142 0.026
1.00129 0.024
I.OOII7 O.O22
i-iS
1.00163 0.048
1.00147 0.043
1.00133 °-°30
I.OOI2O O.O35
I.OOI09 O.O32
1.2
1.00148 0.061
1.00134 0.056
I.OOI2I O.O5O
1.00109 0.046
I.OOO99 O.O4I
1-25
1.00129 0.074
1.00117 0.067
1.00106 0.061
1.00096 0.055
I.OOO87 O.O5O
i-3
1.00107 0.084
1.00097 0.076
1.00088 0.069
1.00079 0.063
I.OOO72 O.O57
i-35
1.00083 0.093
1.00075 0.084
1.00068 0.076
1. 0006 1 0.069
1.00056 O.O63
1.4
1.00056 0.099
1.00051 0.090
1.00046 0.08 1
1.00042 0.074
1.00038 O.067
I- 45
1.00028 0.103
1.00026 0.093
1.00023 0.085
I.OOO2I 0.076
I.OOOI9 O.069
i-5
i.ooooo 0.104
i.ooooo 0.095
i.ooooo 0.086
i.ooooo 0.077
i.ooooo 0.070
i-55
0.99972 0.103
0.99974 0.093
0.99977 0.085
0.99979 0.076
0.99981 0.069
1.6
0.99944 0.099
0.99949 0.090
0.99954 0.081
0.99958 0.074
0.99962 0.067
1-65
0.99917 0.093
0.99925 0.084
0.99932 0.076
0.99939 0.069
0.99945 0.063
i-7
0.99893 0.084
0.99903 0.076
0.99912 0.069
0.99921 0.063
0.99928 0.057
i-7S
0.99871 0.074
0.99883 0.067
0.99894 0.061
0.99905 0.055
0.99914 0.050
1.8
°-99853 0.061
0.99867 0.055
0.99879 0.050
0.99891 0.046
0.99901 0.041
1.85
0.99838 0.047
0.99853 0.043
0.99867 0.039
0.99880 0.035
0.99891 0.032
1.9
0.99827 0.032
0.99843 0.029
0.99858 0.026
0.99872 0.024
0.99884 O.O22
i-9S
0.99820 0.016
0.99837 0.015
0.99853 0.013
0.99867 0.012
0.99879 o.oii
2.0 0.99818 o.oo 0.99835 o.oo 0.99851 o.oo 0.99865 o.oo 0.99878 o.oo
Note. Negative quantities are in heavy type.
Examples, tanh (3.6 -f io) = 0.99851 /o°
tanh (3.7 + i 1.7) = 0.99928 \o°.o57 = 0.99928 \o°.o3'.25".
[136]
TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r /y. CONTINUED
x = .
5-75
x =
3-8
x = ;
[-85
x =
3-9
* = :
•95
?
r
7
r
7
r
7
r
7
r
7
o
o
o
0
o
0
0.99889
o.oo
0.99900
O.OO
0.99909
O.OO
0.99918
0.00
0.99926
O.OO
0.05
0.99891
O.OIO
0.99901
0.009
0.99911
0.008
0.99919
0.007
0.99927
0.007
O.I
0.99895
O.O2O
0.99905
0.018
0.99914
0.016
0.99922
0.015
0-99930
0.013
0.15
0.99902
0.029
0.99911
0.026
0.99919
0.024
0.99927
O.O2I
0-99934
0.019
0.2
0.99911
0.037
0.99919
0.034
0.99927
0.031
0-99934
O.O28
0.99940
0.025
0.25
0.99922
0.045
0.99929
0.041
0.99936
0.037
0.99942
0.033
0.99948
0.030
o-3
0-99935
0.051
0.99941
0.046
0-99947
0.042
0.99952
0.038
0.99956
0.034
0.35
0.99950
0.056
0-99955
0.051
0-99959
0.046
0.99963
O.O42
0.99966
0.038
0.4
0.99966
0.060
0.99969
0.055
0.99972
0.049
0-99975
0.045
0.99977
0.040
0.45
0.99983
0.063
0.99984
0.057
0.99986
0.051
0.99987
0.046
0.99988
0.042
o-5
I.OOOOO
0.063
I.OOOOO
0.057
I.OOOOO
0.052
I.OOOOO
0.047
I.OOOOO
0.043
0-55
1.00017
0.063
1.00016
0.057
1.00014
0.051
1.00013
0.046
I.OOOI2
0.042
0.6
1.00034
0.060
1.00031
0.055
1.00028
0.049
1.00025
0.045
1.00023
0.040
0.65
1.00050
0.056
1.00045
0.051
1.00041
0.046
1.00037
O.042
1.00034
0.038
0.7
1.00065
0.051
1.00059
0.046
1.00053
0.042
1.00048
0.038
1.00044
0.034
o.7S
1.00078
0.045
1.00071
0.041
1.00064
0.037
1.00058
0.033
1.00052
0.030
0.8
1.00089
0.037
1.00081
0.034
1.00073
0.031
1.00066
O.O28
1.00060
0.025
0.85
1.00099
0.029
1.00089
0.026
1.00081
0.024
1.00073
O.O2I
1.00066
0.019
0.9
1.00105
O.O2O
1.00095
0.018
1.00086
0.016
1.00078
O.OI5
1.00071
0.013
0-95
1.00109
O.OIO
1.00099
0.009
1.00089
0.008
1.00081
O.OO7
1.00073
0.007
I.O
I.OOIII
O.OO
I.OOIOO
O.OO
1.00090
O.OO
.00082
0.00
1.00074
0.00
1.05
1.00109
O.OIO
1.00099
0.009
1.00089
0.008
.00081
0.007
1.00073
0.007
i.i
1.00105
O.O2O
1.00095
0.018
i. '00086
0.016
.00078
0.015
1.00071
0.013
i.'S
1.00099
O.029
1.00089
0.026
1.00081
0.024
.00073
O.O2I
1.00066
0.019
1.2
1.00089
0.037
1.00081
0.034
1.00073
0.031
.00066
O.O28
1.00060
0.025
1-25
1.00078
0.045
1.00071
0.041
1.00064
0.037
1.00058
0-033
.00052
0.030
1-3
1.00065
O.O5I
1.00059
0.046
1.00053
0.042
1.00048
0.038
.00044
0.034
1-35
1.00050
O.O56
1.00045
0.051
1.00041
0.046
1.00037
O.042
.00034
0.038
1.4
1.00034
O.06O
1.00031
0.055
1.00028
0.049
1.00025
0.045
.00023
0.040
i -45
1.00017
0.063
1.00016
0.057
1.00014
0.051
1.00013
0.046
.00012
0.042
i-5
I.OOOOO
0.063
I.OOOOO
0.057
I.OOOOO
0.052
I.OOOOO
0.047
I.OOOOO
0.043
i-SS
0.99983
O.O63
0.99984
0.057
0.99986
0.051
0.99987
0.046
0.99988
0.042
1.6
0.99966
O.06O
0.99969
0.055
0.99972
0.049
0-99975
0.045
0-99977
0.040
1.65
0.99950
0.056
0-99955
0.051
0-99959
0.046
0.99963
O.O42
0.99966
0.038
i-7
0-99935
0.051
0.99941
0.646
0-99947
0.042
0.99952
O.O38
0.99956
0.034
1-75
0.99922
0.045
0.99929
0.041
0.99936
0.037
0.99942
0.033
0.99948
0.030
1.8
0.99911
0.037
0.99919
0.034
0.99927
0.031
0-99934
0.028
0.99940
0.025
1.85
0.99902
O.O29
0.99911
0.026
0.99919
0.024
0.99927
O.O2I
0-99934
0.019
1.9
0.99895
O.O2O
0.99905
0.018
0.99914
0.016
0.99922
O.OI5
0.99930
0.013
i -95
0.99891
O.OIO
0.99901
0.009
0.99911
0.008
0.99919
O.OO7
0.99927
0.007
2.O
0.99889
O.OO
0.99900
0.00
0.99909
O.OO
0.99918
O.OO
0.99926
O.OO
Note. Negative quantities are in heavy type.
Examples, tanh (3.95 + £0.9) = 1.00071 /o°.oij
tanh (3.95 + i 1.9)
TABLE XIII. FUNCTIONS OF 4 + ig. f (4 + »?) = « +
sinh
cosh
tanh
9
u v
U V
u v
3
27.28992 o.oo
27.30823 o.oo
0.99933 o.oo
3.05
27.20579 2.14258
27.22405 2.14114
0.99934 o.oooio
3.1
26.95392 4-27195
26.97202 4.26908
0.99936 O.OOO2I
3-IS
26.53588 6.37498
26.55370 6.37071
0.99940 0.00030
D.2
25-95425 8.43871
25.97166 8.43305
0.99946 0.00039
3.2S
25.21260 10.45041
25.22951 10.44340
0.99953 0.00047
3-3
24.31551 12.39768
24.33181 12.38935
0.99961 0.00054
5-35
23.26848 14.26851
23.28410 14.25895
0.99970 0.00060
3.4
22.07800 16.05138
22.09282 16.04061
0.99979 0.00064
5-45
20.75141 17-73528
20.76534 17.72339
0.99989 0.00066
3-5
19.29688 19.30983
19.30983 19.29688
i.ooooo 0.00067
3-55
I7-72339 20.76534
17.73528 20.75141
i.ooon 0.00066
3.6
16.04061 22.09282
16.05138 22.07800
I.OOO2I O.00064
3.65
14.25895 23.28410
14.26851 23.26848
i .0003 o o . 00060
3-7
12.38935 24.33181
12.39768 24.31551
1.00039 0.00054
3.75
10.44340 25.22951
10.45041 25.21260
1.00047 0.00047
3.8
8.43305 25.97166
8.43871 25.95425
1.00054 0.00039
3.85
6.37071 26.55370
6.37498 26.53588
1.00060 0.00030
3.9
4.26908 26.97202
4.27195 26.95392
I.OOO64 O.OOO2I
3-95
2.14114 27.22405
2.14258 27.20579
1.00066 o.ooon
I.O
o.oo 27.30823
o.oo 27.28992
1.00067 o.oo
•05
2.14114 27.22405
2.14258 27.20579
1.00066 o.ooon
.1
4.26908 26.97202
4.27195 26.95392
I.OOO64 O.0002I
•15
6.37071 26.55370
6.37498 26.53588
1.00060 0.00030
.2
8.43305 25.97166
8.43871 25.95425
1.00054 0.00039
•25
10.44340 25.22951
10.45041 25.21260
1.00047 0.00047
•3
12.38935 24.33181
12.39768 24.31551
1.00039 0.00054
•35
14.25805 23.28410
14.26851 23.26848
1.00030 0.00060
•4
16.04061 22.09282
16.05138 22.07800
I.OOO2I O.OOO64
•45
17.72339 20.76534
17.73528 20.75141
i.ooon 0.00066
•5
19.29688 19.30983
19.30983 19.29688
i.ooooo 0.00067
•55
20.75141 17.73528
20.76534 17-72339
0.99989 0.00066
.6
22.07800 16.05138
22.09282 16.04061
0.99979 0.00064
•65
23.26848 14.26851
23.28410 14.25895
0.99970 0.00060
•7
24.31551 12.39768
24.33181 12.38935
0.99961 0.00054
•75
25.21260 10.45041
25.22951 10.44340
0-99953 0.00047
.8
25-95425 8.43871
25.97166 8.43305
0.99946 0.00039
.85
26.53588 6.37498
26.55370 6.37071
0.99940 0.00030
•9
26.95392 4-27195
26.97202 4.26908
0.99936 O.O002I
•95
27.20579 2.14258
27.22405 2.14114
0.99934 o.oooio
2.0 27.28992 0.00
27.30823 o.oo
0-99933 o.oo
Note.
Examples.
Negative quantities are in heavy type.
sinh (4 + i 0.7) =
cosh (4 + *' i. 25)
12.38935 + * 24.33181.
* —10.45041 + i 25.21260.
[138]
TABLE XIII. FUNCTIONS OF 4 + iq. / (4 + iq) = r
sin'
l
cos
h
tan)
1
9
f
7
r
7
r
T
o
o
o
o
27.28992
o.oo
27.30823
o.oo
0-99933
o.oo
0.05
27.29002
4-503
27.30810
4-497
0.99934
0.006
O.I
27.29036
9.006
27.30780
8.994
0.99936
O.OI2
0.15
27.29090
13-509
27.30723
I3-49I
0.99940
O.OlS
O.2
27.29166
18.011
27.30650
17.989
0.99946
0.023
0.25
27.29260
22.514
27-3055°
22.486
0-99953
O.027
°-3
27.29370
27.016
27.30445
26.984
0.99961
0.031
0-35
27.29492
31-517
27.30324
31-483
0.99970
0.034
0.4
27.29624
36.018
27.30190
35-982
0.99979
0.037
0-4S
27.29764
40.519
27.30050
40.481
0.99990
0.038
o-5
27.29908
45.019
27.29908
44.981
I.OOOOO
0.038
0-55
27.30050
49-519
27.29764
49.481
I.OOOIO
0.038
0.6
27.30190
54.018
27.29624
53-982
I.0002I
0-037
0.65
27.30324
58.517
27.29492
58-483
1.00030
0.034
0.7
27.30445
63.016
27.29370
62.984
1.00039
0.031
o.7S
27-30550
67.514
27.29260
67.486
1.00047
O.O27
0.8
27.30650
72.011
27.29166
71.989
1.00054
0.023
0.85
27.30723
76.509
27.29090
76.491
1.00060
O.OlS
0.9
27.30780
81.006
27.29036
80.994
1.00064
O.OI2
o.QS
27.30810
85-503
27.29002
85-497
1.00066
O.OO6
1.0
27.30823
90
27.28992
90
1.00067
0.00
1.05
27.30810
94-497
27.29002
94-503
1.00066
0.006
i.i
27.30780
98.994
27.29036
99.006
1.00064
0.013
i-i5
27.30723
103.491
27.29090
103.509
1.00060
O.OlS
1.2
27.30650
107.989
27.29166
108.011
1.00054
O.O23
I.2S
27-30550
112.486
27.29260
112.514
1.00047
O.027
i-3
27-30445
116.984
27.29370
117.016
1.00039
0.031
I-3S
27.30324
121.483
27.29492
121.517
1.00030
0.034
1.4
27.30190
125.982
27.29624
126.018
I.O002I
0.037
1-45
27-30050
130.481
27.29764
130-519
I.OOOIO
0.038
•5
27.29908
134.981
27.29908
I35-019
I.OOOOO
0.038
•55
27.29764
139.481
27.30050
139-519
0.99990
0.038
.6
27.29624
143.982
27.30190
144.018
0.99979
0.037
.65
27.29492
148.483
27.30324
148.517
0.99970
0.034
•7
27.29370
152.984
27.30445
153.016
0.99961
O.O3I
•75
27.29260
157.486
27-30550
I57-5I4
0-99953
O.O27
.8
27.29166
161.989
27.30650
162.011
0.99946
O.O23
1.85
27.29000
166.491
27.30723
166.509
0.99940
O.OlS
1.9
27.29036
170.994
27.30780
171.006
0.99936
O.OI2
1-95
27.29002
175-497
27.30810
I75-503
0-99934
O.006
2.0 27.28992 180
27.30823 180
0-99933 °-°°
Note. Negative quantities are in heavy type.
Examples, sinh (4 + * i.o) = 27.30823 /go0.
tanh (4 + * 1.5) = i.oooo \o°.038 = i.oooo \o°.o2'.i7*
T'ABLE XIV. SEMI-EXPONENTIALS, and logw
e* . e*
x - log™ -
e* . ex
x - log.o -
e* . e*
x - loglo -
4.00 27.299 1.4361479
4.01 27.573 1.4404909
4.02 27.851 1.4448338
4.03 28.130 1.4491768
4.04 28.413 1.4S3SI97
4.50 45-009 1.6532952
4.51 45.461 1.6576381
4.52 45.918 1.6619811
4-53 46.379 1.6663240
4.54 46.845 1.6706669
5.00 74-207 1.8704424
5.01 74.952 1.8747854
5.02 75.706 1.8791283
5.03 76.467 1.8834712
5.04 77.235 1.8878142
4.05 28.699 1.4578627
4.06 28.987 1.4622056
4.07 29.278 1.4665485
4.08 29.573 1.4708915
4.09 29.870 1.4752344
4-55 47-3 1(5 1.6750099
4.56 47-792 1.6793528
4.57 48.272 1.6836958
4.58 48.757 1.6880387
4.59 49.247 1.6923817
5.05 78.011 1.8921571
5.06 78.795 1.8965001
5-07 79-587 1.9008430
5.08 80.387 1.9051860
5.09 81.195 1.9095289
4.10 30.170 1.4795774
4.11 30.473 1.4839203
4.12 30.780 1.4882633
4.13 31.089 1.4926062
4.14 31.401 1.4969492
4.60 49.742 1.6967246
4.61 50.242 1.7010676
4.62 50.747 1.7054105
4.63 51.257 1.7097535
4.64 51.772 1.7140964
5.10 82.011 1.9138719
5.11 82.835 1.9182148
5.12 83.668 1.9225577
5.13 84.509 1.9269007
5.14 85.358 1.9312436
4.15 31.717 1.5012921
4.16 32.036 1.5056350
4.17 32.358 1.5099780
4.18 32.683 1.5143209
4.19 33.011 1.5186639
4.65 52.292 1.7184393
4.66 52.818 1.7227823
4-67 53-349 1.7271252
4.68 53.885 1.7314682
4.69 54-427 1.7358m
5.15 86.216 1.9355866
5.16 87.082 1.9399295
5.17 87.957 1.9442725
5.18 88.841 1.9486154
5.19 89.734 1.9529584
4-20 33.343 1-5230068
4.21 33.678 1.5273498
4.22 34.017 1.5316927
4-23 34-359 1-5360357
4.24 34.704 1.5403786
4.70 54-974 I-740IS4I
4.71 55-526 1.7444970
4.72 56.084 1.7488400
4.73 56.648 1.7531829
4.74 57.217 1.7575258
5.20 90.636 1.9573013
5.21 91.547 1.9616443
5.22 92.467 1.9659872
5-23 93.396 1.9703301
5.24 94.335 I-974673I
4-25 35-053 I.54472I5
4.26 35.405 1.5490645
4-27 35.76i 1.5534074
4.28 36.120 1.5577504
4.29 36.483 1.5620933
4-75 57-792 1.7618688
4-76 58.373 1.7662117
4.77 58.960 1.7705547
4.78 59.552 1.7748976
4.79 60.151 1.7792406
5.25 95.283 1.9790160
5.26 96.241 1.9833590
5.27 97.208 1.9877019
5.28 98.185 1.9920449
5.29 99.172 1.9963878
4.30 36.850 1.5664363
4.31 37.220 1.5707792
4-32 37-594 1.5751222
4-33 37-972 1.5794651
4-34 38.354 1.5838081
4.80 60.755 1-7835835
4.81 61.366 1.7879265
4.82 61.983 1.7922694
4.83 62.605 1.7966123
4.84 63.235 1.8009553
5.30 100.168 2.0007308
5.31 101.175 2.0050737
5.32' 102.192 2.0094166
5.33 103.219 2.0137596
5.34 104.256 2.0181025
4-35 38.739 1.588151°
4.36 39.129 1.5924939
4-37 39-522 1.5968369
4-38 39-919 1.6011798
4-39 40-320 1.6055228
4.85 63.870 1.8052982
4.86 64.512 1.8096412
4.87 65.160 1.8139841
4.88 65.815 1.8183271
4.89 66.477 1-8226700
5-35 105.304 2.0224455
5.36 106.362 2.0267884
5.37 107.431 2.0311314
5.38 108.511 2.0354743
5.39 109.602 2.0398173
4.40 40.725 1.6098657
4.41 41.135 1.6142087
4.42 41-548 1.6185516
4.43 41.966 1.6228946
4-44 42.387 1.6272375
4.90 67.145 1.8270130
4.91 67.820 1.8313559
4.92 68.501 1.8356989
4.93 69.190 1.8400418
4.94 69.885 1.8443847
5.40 110.703 2.0441602
5.41 111.816 2.0485031
5.42 112.940 2.0528461
5.43 114-075 2.0571890
5.44 115.221 2.0615320
4.45 42.813 1.6315804
4.46 43.244 1.6359234
4.47 43-678 1.6402663
4.48 44.117 1.6446093
4.49 44-561 1.6489522
4.95 70.587 1.8487277
4.96 71.297 1.8530706
4.97 72.013 1.8574136
4.98 72.737 1.8617565
4.99 73.468 1.8660995
5.45 116.379 2.0658749
5.46 117-549 2.0702179
5.47 118.730 2.0745608
5.48 119.923 2.0789038
5.49 121.129 2.0832467
4.50 45-009 1.6532952
5.00 74.207 1.8704424
5.50 122.346 2.0875897
e4'*0 /c4'20\
Example. — = 33.343 logio \—) = 1.5230068.
[140]
TABLE XIV. SEMI-EXPONENTIALS. - and loglo ( - ). CONTINUED
* 7 log"> 7
ex . e*
x Iog10 -
2 2
e* . ex
x - loglo -
5.50 122.346 2.0875897
5.51 123.576 2.0919326
5.52 124.818 2.0962755
5.53 126.072 2.1006185
5-54 127.339 2.1049614
6.00 201.714 2.3047369
6.01 203.742 2.3090785
6.02 205.789 2.3134222
6.03 207.858 2.3177667
6.04 209.947 2.3221098
6-5° 332-57I 2.5218844
6-51 335.9I3 2.5262268
6.52 339-289 2.5305699
6.53 342.699 2.5349128
6.54 346.143 2.5392556
5.55 128.619 2.1093044
5.56 129.911 2.1136473
5.57 131.217 2.1179903
5.58 132.536 2.1223332
5.59 133.868 2.1266762
6.05 212.057 2.3264527
6.06 214.188 2.3307951
6.07 216.340 2.3351368
6.08 218.514 2.3394793
6.09 220.711 2.3438220
6-55 349-622 2.5435988
6-56 353.135 2.5479408
6-57 356-685 2.5522849
6.58 360.270 2.5566281
6-59 363-890 2.5609701
5.60 135-213 2.1310191
5.61 136.572 2.1353620
5.62 137-945 2.1397050
5.63 139.331 2.1440479
5.64 140.731 2.1483909
6.10 222.929 2.3481666
6. it 225.169 2.3525086
6.12 227.432 2.3568516
6.13 229.718 2.3611951
6.14 232.027 2.3655386
6.60 367.547 2.5653129
6.61 371.241 2.5696560
6.62 374.973 2.5740000
6.63 378.741 2.5783424
6.64 382.547 2.5826849
5.65 142.146 2.1527338
5.66 143-574 2.1570768
5.67 i45-OI7 2.1614197
5.68 146.475 2.1657627
5.69 147.947 2.1701056
6.15 234.359 2.3698817
6.16 236.714 2.3742240
6.17 239.093 2.3785669
6.18 241.496 2.3829100
6.19 243.923 2.3872527
6.65 386.391 2.5870270
6.66 390.275 2.5913708
6.67 394.197 2.5957134
6.68 398.160 2.6000576
6.69 402.161 2.6044000
5.70 149-434 2.1744485
5.71 150.936 2.1787915
5.72 152.452 2.1831344
5-73 I53-985 2.1874774
5-74 I55-532 2.1918203
6.20 246.375 2.3915966
6.21 248.851 2.3959394
6.22 251.352 2.4OO2824
6.23 253.877 2.4046234
6.24 256.429 2.4089672
6.70 406.202 2.6087420
6.71 410.285 2.6130856
6.72 414.409 2.6174292
6.73 418.574 2.6217721
6.74 422.780 2.6261144
5.75 157-095 2.1961633
5.76 158.674 2.2005062
5.77 160.269 2.2048492
5.78 161.880 2.2091921
5-79 163.507 2.2135351
6.25 259.006 2.4133099
6.26 261.609 2.4176526
6.27 264.239 2.4219970
6.28 266.894 2.4263388
6.2Q 269.576 2.4306813
6.75 427.030 2.6304584
6.76 431.321 2.6348006
6.77 435-656 2.6391436
6.78 440.034 2.6434862
6.79 444.457 2.6478298
5.80 165.150 2.2178780
5.81 166.810 2.2222209
5.82 168.486 2.2265639
5.83 170.179 2.2309068
5.84 171.890 2.2352498
6.30 272.285 2.4350237
6.31 275.022 2.4393675
6.32 277.786 2.4437104
6.33 280.578 2.4480536
6.34 283.398 2.4523967
6.80 448.923 2.6521719
6.81 453-435 2.6565151
6.82 457-993 .2.6608589
6.83 462.595 2.6652009
6.84 467-244 2.6995437
5.85 173-617 2.2395927
5.86 175-362 2.2439357
5.87 177.124 2.2482786
5.88 178.905 2.2526216
5.89 180.703 2.2569645
6.35 286.246 2.4567394
6.36 289.123 2.4610826
6.37 292.029 2.4654260
6.38 294.964 2.4697690
6.39 297.928 2.474III4
6.85 471.940 2.6738868
6.86 476.683 2.6782296
6.87 481.474 2.6825728
6.88 486.312 2.6869150
6.89 491.200 2.6912584
5.90 182.519 2.2613074
5.91 184.353 2.2656504
5.92 186.206 2.2699933
5.93 188.077 2.2743363
5.94 189.967 2.2786792
6.40 300.922 2.4784540
6.41 303.947 2.4827979
6.42 3O7.OO2 2.4871412
6.43 310.087 2.4914836
6.44 313.203 2.4958260
6.90 496.137 2.6956016
6.91 501.123 2.6999443
6.92 506.160 2.7042878
6.93 511.246 2.7086299
6.94 516.386 2.7129744
5.95 191.877 2.2830222
5.96 193.805 2.2873651
5.97 195.753 2.2917081
5.98 197.720 2.2960510
5.99 199.707 2.3003939
6.45 316.352 2.5001705
6.46 319.530 2.5045II6
6.47 322.742 2.5088555
6.48 325.985 2.5I3I977
6.49 329.262 2.5I754I6
6.95 521.575 2.7173168
6.96 526.816 2.7216589
6-97 532.112 2.7260030
6.98 537-459 2.7303454
6.99 542.860 2.7346878
6.00 201.714 2.3047369
6-50 332.571 2.5218844
7.00 548.317 2.7390317
gb.60
Example. - = 135-213
2
>-
1310191.
[141]
FABLE XIV. SEMI-EXPONENTIALS. ^ and Iogi0 (-\ CONTINUE
ez ex
x log 10 —
2 2
ex e*
x log 10 —
2 2
e' e'
x - loglo -
7.00 548.317 2.7390317
7.01 553.827 2.7433741
7.02 559-393 2.7477170
7.03 565.015 2.7520600
7.04 570.694 2.7564033
7.50 904.021 2.9561785
7.51 913.107 2.9605217
7.52 922.284 2.9648647
7-53 931-553 2.9692076
7.54 940.915 2.9735504
8.00 1490.479 3.1733259
8.01 1505.457 3.1776683
8.02 1520.589 3.1820118
8-03 1535.870 3.1863546
8.04 1551.306 3.1906976
7.05 576.429 2.7607458
7.06 582.223 2.7650893
7.07 588.074 2.7694320
7.08 593-984 2.7737747
7.09 599.954 2.7781180
7-55 950.371 2.9778932
7.56 959.923 2.9822364
7-57 969-570 2.9865792
7.58 979-3I4 2.9909220
7.59 989.157 2.9952653
8.05 1566.895 3-1950399
8.06 1582.645 3.1993836
8.07 1598.552 3.2037268
8.08 1614.617 3.2080694
8.09 1630.841 3.2124116
7.10 605.984 2.7824612
7.11 612.074 2.7868039
7.12 618.225 2.7911467
7.13 624.439 2.7954901
7.14 630.714 2.7998325
7.60 999.098 2.9996080
7.61 1009.139 3.0039510
7.62 1019.281 3.0082939
7.63 1029.525 3.0126369
7.64 1039.872 3.0169799
8.10 1647.234 3.2167554
8.ii 1663.789 3.2210981
8.12 1680.510 3.2254412
8.13 1697.400 3.2297842
8.14 1714.458 3.2341270
7.15 637.053 2.8041755
7.16 643.456 2.8085189
7.17 649.922 2.8128612
7.18 656.454 2.8172043
7.19 663.052 2.8215476
7.65 1050.323 3.0213229
7.66 1060.879 3.0256659
7.67 1071.541 3.0300088
7.58 1082.310 3.0343517
7.69 1093.187 3.0386944
8.15 1731-690 3.2384703
8.16 1749.092 3.2428130
8.17 1766.672 3.2471560
8.18 1784.427 3.2514988
8.19 1802.364 3-2558425
7.20 669.715 2.8258901
7.21 676.446 2.8302331
7.22 683.245 2.8345765
7.23 690.111 2.8389189
7.24 697.047 2.8432620
7.70 1104.174 3.0430376
7.71 1115.271 3.0473806
7.72 1126.480 3.0517234
7-73 ii37-8oi 3.0560663
7.74 1149.236 3.0604092
8.20 1820.476 3.2601848
8.21 1838.774 3.2645284
8.22 1857.251 3.2688706
8.23 1875.914 3.2732129
8.24 1894.770 3-2775566
7.25 704.052 2.8476047
7.26 711.128 2.8519478
7.27 718.275 2.8562908
7.28 725-494 2.8606338
7.29 732.785 2.8649766
7.75 1160.786 3.0647523
7.76 1172.452 3.0690950
7.77 1184.236 3.0734383
7.78 1196.137 3.0777810
7.79 1208.159 3.0821242
8.25 1913.812 3.2188994
8.26 1933.047 3.2862424
8.27 1952.473 3.2905850
8.28 1972.098 3.2949284
8.29 1991.913 3.2992704
7.30 740.150 2.8693197
7.31 747.589 2.8^36629
7.32 755.102 2.8780056
7.33 762.691 2.8823487
7-34 770.356 2.8866915
7.80 1220.301 3.0864670
7.81 1232.565 3.0908098
7.82 1244.953 3.0951531
7.83 1257.465 3.0994961
7.84 1270.102 3.1038386
8.30 2011.936 3.3036142
8.31 2032.158 3-3079575
8.32 2052.580 3.3123000
8.33 2073.206 3.3166425
8.34 2094.045 3.3209860
7-35 778-098 2.8910343
7.36 785.918 2.8953772
7-37 793-8i7 2.8997205
7.38 801.795 2.9040633
7.39 809.853 2.9084062
7.85 1282.867 3.1081818
7.86 1295.760 3.1125246
7.87 1308.783 3.1168677
7.88 1321.936 3.1212105
7.89 1335.222 3-1255535
8.35 2115.092 3-3253293
8.36 2136.347 3.3296718
8.37 2157.819 3.3340150
8.38 2179.505 3-3383578
8.39 2201.409 3.3427008
7.40 817.992 2.9127491
7.41 826.213 2.9170920
7.42 834.517 2.9214351
7.43 842.904 2.9257782
7.44 851.375 2.9301209
7.90 1348.641 3.1298964
7.91 1362.195 3.1342394
7.92 1375-886 3.1385826
7.93 1389.713 3.1429254
7.94 1403.680 3.1472680
8.40 2223.533 3-3470436
8.41 2245.881 3-35I3868
8-42 2268.452 3.3557296
8.43 2291.250 3.3600725
8-44 2314.277 3.3644154
7.45 859.932 2.9344641
7.46 868.574 2.9388068
7-47 877.303 2.9431496
7.48 886. 120 2.9474925
7.49 895.026 2.9518356
7.95 1417-787 3-1516110
7.96 1432-036 3-1559539
7.97 1446.429 3.1602971
7.98 1460.966 3.1646402
7.99 1475.648 3.1689827
8-45 2337.536 3-3687583
8.46 2361.030 3.3731014
8.47 2384.752 3-3774433
8.48 2408.725 3.3817872
8.49 2432.926 3.3861290
7.50 904.021 2.9561785
8.00 1490-479 3-I733259
8.50 2457.383 3-3904730
e7-10
Example.
2
/C7.1(K
= 605.984 log 10 (^ ,
) = 2.7824612.
[142]
TABLE XIV. SEMI-EXPONENTIALS. - and Iog10 ( ^
CONTINUED
X
e*
i e*
log 10 —
X
e*
log 10 -
X
e?
g*
log 10 -
2
2
2
2
2
2
8.50
2457.383
3.3904730
9-00
4051.543
3.6076204
9-50
6679.863
3.8247676
8.51
2482.082
3.3948162
9-OI
4092.263
3.6119636
9-51
6746.988
3.8291101
8.52
2507.027
3.3991590
9-02
4I33-388
3.6163062
9-52
6814.805
3-8334534
8-53
2532.221
3.4035016
9-°3
4174.929
3.620649!
9-53
6883.295
3.8377964
8-54
2557.672
3.4078448
9.04
4216.889
3.6249922
9-54
6952475
3.8421394
8-55
2583-380
34I2I882
9-°S
4259.264
3.6293345
9-55
7022.345
3.8464822
8.56
2609.341
3.4165308
9.06
4302.076
3.6336780
9-56
7092.923
3.8508252
8-57
2635.562
3.4208732
9.07
4345.302
3.6380200
9-57
7164.203
3.8551679
8.58
2662.052
3.4252166
9.08
4388.982
3.6423638
9-58
7236.210
3.8595112
8-59
2688.810
3.4295601
9.09
4433-098
3.6467073
9-59
7308.929
3.8638537
8.60
2715.830
3.4339026
9.10
4477.646
3.6510498
9.60
7382.390
3.8681970
8.61
2743.126
3.4382458
9.11
4522.647
3.6553927
9.61
7456.583
3.8725398
8.62
2770.693
3.4425884
9.12
4568.100
3-6597356
9.62
753I-526
3.8768830
8.63
2798.535
3.4469308
9-13
4614.016
3.6640791
9-63
7607.221
3.8812260
8.64
2826.665
3.4512744
9.14
4660.383
3.6684216
9.64
7683.672
3.8855688
8.65
2855.070
3.4556167
9-iS
4707.211
3.6727637
9.65
7760.882
3.8899111
8.66
2883.767
3.4599602
9.16
4754.528
3.6771074
9.66
7838.890
3.8942546
8.67
2912.745
3.4643025
9.17
4802.308
3.6814500
9.67
7917.680
3.8985980
8.68
2942.023
3.4686462
9.18
4850.577
3.6857934
9.68
7997.247
3.9029406
8.69
2971.592
3.4729891
9.19
4899.328
3.6901365
9.69
8077.622
3.9072835
8.70
3001.456
34773320
9.20
4948.563
3.6944792
9.70
8158.802
3.9116264
8.71
3031.621
3.4816749
9.21
4998.284
3.6988209
9.71
8240.792
3.9159690
8.72
3062.088
3.4860178
9.22
5048.532
3.7031652
9.72
8323.623
3.9203124
8-73
3092.852
3.4903592
9-23
5099.272
3.7075082
9-73
8407.262
3.9246546
8.74
3123.948
3.4947038
9.24
5150.519
3.7II85IO
9-74
8491.770
3.9289982
8-75
3*55-337
3.4990458
9-25
5202.272
3.7161930
9-75
8577.112
3-93334II
8.76
3187.054
3.5033896
9.26
5254.569
3.7205370
9.76
8663.316
3.9376842
8-77
3219.085
3-5077325
9.27
5307-367
3.7248791
9-77
8750.384
3.9420270
8.78
3251.440
3.5120756
9.28
5360.716
3.7292228
9.78
8838.326
3.9463700
8.79
3284.114
3.5164182
9.29
54I4-587
3-7335654
9-79
8927.154
3.9507131
8.80
3317.122
3.5207614
9-30
5469.009
3.7379086
9.80
9016.875
3.9550560
8.81
3350.460
3.5251044
9-31
5523-975
3.7422517
9.81
9107.481
3.9593983
8.82
3384-I33
3-5294474
9-32
5579-491
3.7465946
9.82
9199.026
3.9637418
8.83
3418.141
3.5337900
9-33
5635-563
3-7509373
9-83
9291.480
3.9680850
8.84
3452.496
3-538I332
9-34
5692.203
3.7552804
9.84
9384.860
3.9724278
8.85
3487.197
3.5424766
9-35
5749405
3.7596229
9-85
9479.163
3.9767701
8.86
3522.243
3.5468192
5807.194
3.7639664
9.86
9574-444
3.9811136
8.87
3557-63I
3.5511609
9-37
5865.555
3.7683091
9.87
9670.678
3.9854569
8.88
3593-395
3-5555050
9-38
5924.507
3.7726522
9.88
9767.860
3.9897994
8.89
3629.512
3.5598482
9-39
5984.054
3.7769956
9.89
9866.020
3.9941420
8.90
3665-986
3.5641908
9.40
6044.19!
3.7813382
9.90
9965.186
3.9984854
8.91
3702.820
3.5685326
9.41
6104.922
3.7856801
9.91
10065.350
4.0028289
8.92
3740.045
3.5728768
9.42
6l66.29O
3.7900240
9.92
10166.494
4.007I7I2
8-93
3777-635
3.5772201
9-43
6228.269
3.7943674
9-93
10268.667
4.0II5I4I
8.94
38I5-597
3.5815626
9-44
6290.860
3.7987100
9-94
10371.873
4.0158572
8-95
3853-937
3.5859044
9-45
6354.080
3.8030526
9-95
10476.107
4.0201999
8.96
3892.678
3.5902486
9.46
64I7-943
3.8073958
9.96
10581.397
4.0245430
8-97
3931-795
3.5945909
9-47
6482.450
3.8II7392
9-97
10687.745
4.0288860
8.98
3.5989344
9.48
6547.591
3.8l6o8l6
9.98
10795.160
4.0332290
8.99
4011.228
3.6032773
9-49
6613.388
3.8204240
9-99
10903.652
4.0375721
9.00
4051.543
3.6076204
9-50
6679.863
3.8247676
IO.OO
11013.233
4.0419148
gS-M
Example.
i
= 3665.986
log 10 (
) = 3
.5641908.
[143]
TABLE XV
REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + to
Sinh0
Cosh0 Tanh0 Cothe Sech0 Cosech 6 0
o.oo
o.oo
I.OO
o.oo
00
I.OO
00
0.00
O.OI
O.OIOOOO
1.000050
O.OIOOO
IOO.
0.9999
IOO.
O.OI
0.02
O.O20OOI
I.OOO2OO
O.O2OOO
50.
0.9998
50.
O.O2
0.03
0.030005
I.OOO45O
0.02999
33-34
0.9995
33-333
0.03
0.04
0.04001 1
1.000800
0.03998
25.013
0.9992
24.99
O.O4
O.OS
0.050021
I.OOI25O
0.04996
20.016
0.9987
19.992
0.05
0.06
0.060036
1.001801
0.05993
16.686
0.9982
16.657
O.O6
O.O7
0.070057
1.002451
0.06989
14.308
0-9975
14.274
O.O7
0.08
0.080085
1.003202
0.07983
12.527
0.9968
12.487
0.08
O.OQ
0.090122
1.004053
0.08976
11.141
0-9959
11.097
0.09
O.IO
0.100167
1.005004
0.09967
10.033
0.9950
9-983
O.IO
O.II
O.IIO222
1.006056
0.10956
9.128
0.9940
9-073
O.II
O.I 2
0.120288
1.007209
0.11943
8.373
0.9928
8.314
O.I 2
0.13
0.130366
1.008462
0.12927
7-735
0.9916
7.669
0.13
0.14
0.140458
1.009816
0.13909
7.189
0.9902
7.120
O.I4
O.IS
0.150563
1.011271
0.14888
6.716
0.9888
6.642
0.15
0.16
0.160684
1.012827
0.15865
6.303
0.9873
6.223
0.16
0.17
O.I7O82O
1.014485
0.16838
5-939
0.9857
5-854
0.17
0.18
0.180974
1.016244
0.17808
5-6i5
0.9840
5-525
0.18
0.19
O.I9II45
1.018104
0.18775
5-325
0.9822
S-232
0.19
O.2O
0.201336
1.020067
0.19737
5.067
0.9803
4.967
O.2O
O.2I
0.2II547
1.022131
0.20696
4.832
0.9784
4.726
O.2I
O.22
O.22I779
1.024298
0.21652
4.618
0.9763
4-509
O.22
0.23
0.232033
1.026567
0.22603
4-425
0.9742
4-3io
0.23
O.24
O.2423II
1.028939
0.23549
4.246
0.9719
4.127
O.24
0.25
O.2526I2
1.031413
0.24492
4.083
0.9695
3-959
0.25
O.26
0.262939
1.033991
0.25430
3-932
0.9671
3-803
0.26
O.27
0.273292
1.036672
0.26363
3-793
0.9646
3-659
O.27
0.28
0.283673
1.039457
0.27290
3.664
0.9620
3-52S
0.28
0.29
0.294082
1.042346
0.28214
3-544
0.9591
3.400
O.29
0.30
0.304520
1-045339
0.29131
3-433
0.9566
3-284
0.30
0.31
0.314989
1.048436
0.30043
3-328
0-9537
3-r75
0.31
0.32
0.325489
1.051638
0.30951
3-231
0.9511
3-072
0.32
0-33
0.336022
1.054946
0.31852
3.140
0.9479
2.976
0-33
0-34
0.346589
1-058359
0.32748
3-053
Q-9447
2.885
o-34
0-35
0.357190
1.061878
0.33637
2-973
0.9416
2.800
o-35
0.36
0.367827
1.065503
0.34522
2.897
0-9385
2,719
0.36
0-37
0.378500
1.069234
0.35399
2.825
0-9353
2.642
0-37
0.38
0.389212
1-073073
0.36271
2-757
0.9319
2.569
0.38
0-39
0.399962
1.077019
0.37136
2.693
0.9285
2.500
o-39
0.40
0.410752
1.081072
0.37995
2.632
0.9250
2-434
0.40
O.4I
0.421584
1.085234
0.38847
2-574
0.9215
2.372
0.41
O.42
0-432457
1.089504
0.39693
2-519
0.9178
2.312
0.42
0-43
0-443374
1.093883
0.40532
2.467
0.9141
2.256
0-43
0.44
0-454335
1.098372
0.41365
2.417
0.9103
2.2OI
0.44
0-45
0.465342
1.102970
0.42190
2-370
0.9066
2.149
0-45
0.46
0.476395
1.107679
0.43009
2-325
0.0925
2.099
0.46
0.47
0.487496
1.112498
0.43820
2.282
0.8988
2.051
0.47
0.48
0.498646
1.117429
0.44624
2.241
0.8949
2.O06
0.48
0.49
0.509845
1.122471
0.45421
2.2O2
0.8909
I.96l
0-49
Example. sinh 0.25 = 0.252612.
TABLE XV
REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + io. CONTINUED
e
Sinhfl
Coshtf
Tanh0
Cothfl
Sechfl
Cosech &
e
0.50
0.521095
1.127626
0.46211
2.164
0.8868
1.919
0.50
0.51
0-532398
1.132893
0.46995
2.128
0.8827
1.878
0.51
0.52
0-543754
1.138274
0.47769
2.093
0.8785
1.839
0.52
o-S3
o-555i64
1.143769
0.48538
2.060
0.8743
1.801
o-53
0-54
0.566629
1.149378
0.49299
2.028
0.8700
1-765
0-54
0-55
0.578152
1.155101
0.50052
1.998
0.8658
1-730
0-55
0.56
0-589732
1.160941
0.50797
1.969
0.8614
1.696
0.56
0-57
0.601371
1.166896
0-51536
1.940
0.8570
1.663
0-57
0.58
0.613070
1.172968
0.52266
i-9i3
0.8525
1.631
0.58
o-59
0.624831
1.179158
0.52990
1.887
0.8480
1.601
0-59
0.60
0.636654
1.185465
0.53704
1.862
0.8435
I-57I
0.60
0.61
0.648540
1.191891
0.54413
1-838
0.8390
1.542
0.61
0.62
0.660492
1.198436
0.55112
1.814
0.8344
i.SH
0.62
0.63
0.672509
1.205101
0.55805
1.792
0.8298
1.487
0.63
0.64
0.684504
1.211887
0.56490
1.770
0.8251
1.461
0.64
0.65
0.696748
1.218793
0.57166
1.749
0.8205
1-435
0.65
6.66
0.708970
1.225822
0.57836
1.729
0.8158
1.410
0.66
0.67
0.721264
1.232973
0.58498
1.709
0.8110
1-387
0.67
0.68
0.733630
1.240247
0.59152
1.690
0.8065
1-363
0.68
0.69
0.746070
1.247646
0.59798
1.672
0.8015
1.340
0.69
0.70
0.758584
1.255169
0.60437
1-655
0.7967
1.318
0.70
0.71
0.771174
1.262818
0.61067
1.637
0.7919
1.297
0.71
0.72
0.783840
1.270593
0.61691
1.621
0.7870
1.276
0.72
o.73
0.796586
1.278495
0.62306
1.605
0.7821
1.255
0-73
0.74
0.809411
1.286525
0.62914
1.590
0-7773
1.235
0.74
o.7S
0.822317
1.294683
0.63516
1-5744
0.7724
i. 216
0-75
0.76
0.835305
1.302971
0.64108
1-5599
0.7675
1.1972
0.76
0.77
0.848377
1.311390
0.64693
1-5457
0.7625
1.1787
0.77
0.78
0.861533
1.319939
0.65271
1-5320
0.7576
1.1607
0.78
0.79
0.874776
1.328621
0.65842
1.5188
0.7527
1.1431
0.79
0.80
0.888106
1-337435
0.66403
I.5059
0.7477
1.1259
0.80
0.81
0.901525
1-346383
0.66959
1-4934
0.7427
1.1092
0.81
0.82
0.915034
1.355466
0.67507
1.4813
0-7377
1.0928
0.82
0.83
0.928635
1.364684
0.68047
1.4696
0.7327
1.0768
0.83
0.84
0.942328
I-374039
0.68580
1.4582
0.7278.
1.0612
0.84
0.85
0.956116
1-383531
0.69107
1.4470
0.7228
1.0459
0.85
0.86
0.969999
1.393161
0.69626
1-4362
0.7178
1.0309
0.86
0.87
0.983980
1.402931
0.70137
1.4258
0.7128
1.0163
0.87
0.88
0.998058
1.412841
0.70642
1.4156
0.7078
I.OO2O
0.88
0.89
1.012237
1.422893
0.71139
1-4057
0.7028
0.9881
0.89
0.90
1.026517
1433086
0.71629
1.3961
0.6978
0-9737
0.90
0.91
1.040899
1.443423
0.72114
1.3867
0.6928
0.9607
0.91
0.92
1.055386
I-453905
0.72591
1.3776
0.6878
0-9475
0.92
0-93
1.069978
1.464531
0.73060
1-3687
0.6828
0.9346
o-93
0.94
1.084677
1.475305
0.73522
1.3600
0.6778
0.9219
0.94
o-95
1.099484
1.486225
0-73979
i.35i8
0.6728
0.9095
0-95
0.96
1.114402
I-497295
0.74427
1.3436
0.6678
0.8973
0.96
0.97
1.129431
1.508514
0.74870
1.3356
0.6629
0.8854
0.97
0.98
I-I44573
1.519884
0.75306
1.3279
0.6579
0.8737
0.98
0.99
1.159829
1.531406
0.75736
1.3204
0.6529
0.8621
0-99
Example. cosh 0.55 = 1.155101.
[i45]
TABLE XV
REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + io. CONTINUED
e
Sinh0
Coshfl
Tanh0
Cothfl
Sech0
Cosech 6
e
1. 00
1.175201
1.543081
0.76159
1-3130
0.6480
0.8509
I.OO
I.OI
1.190691
1.554910
0.76576
1-3059
0.6431
0.8395
I.OI
1.02
1.206300
1.566895
0.76987
1.2989
0.6382
0.8290
1.02
1.03
1.222029
I-579036
0.77391
1.2921
0.6333
0.8183
1-03
1.04
1.237881
i-59I336
0.77789
1.2855
0.6284
0.8078
I.O4
1.05
1.253857
1.603794
0.78181
1.2791
0.6235
0-7975
1.05
1. 06
1.269958
1.616413
0.78566
1.2728
0.6 1 86
0.7874
1. 06
1.07
1.286185
1.629194
0.78946
1.2666
0.6138
0.7777
1.07
1. 08
1.302542
1.642138
0.79320
1.2607
0.6090
0.7677
1. 08
1.09
1.319029
1-655245
0.79688
1-2549
0.6042
0.7581
I.O9
I.IO
I-33S647
1.668519
0.80050
1.2492
0-5993
0.7487
I.IO
I. II
1.352400
1.681959
0.80406
1-2437
0-5945
0-7393
I. II
I.I2
1.369287
1-695567
0.80757
1.2382
0.5898
0.7302
1. 12
I-I3
1.386312
1.709345
0.81102
1.2330
0.5850
0.7215
I-!3
I.I4
I-403475
1.723294
0.81441
1.2279
0.5803
0.7125
1.14
I-IS
1.420778
I.7374I5
0.81775
1.2229
0-5755
0.7038
1-15
1.16
1.438224
1.751710
0.82104
i. 2180
0.5708
0.6953
1.16
1.17
1-455813
1.766180
0.82427
1.2132
0.5662
0.6869
1.17
1.18
1.473548
1.780826
0.82745
1.2085
0.5616
0.6786
1.18
1.19
1.491430
I.79565I
0.83058
1.2040
0-5569
0.6705
1.19
1.20
1.509461
1.810656
0.83365
I-I995
0-5523
0.6625
i. 20
I.2I
1.527644
1.825841
0.83668
1.1952
0-5477
0.6546
1. 21
1.22
1-545979
1.841209
0.83965
1.1910
o.543i
0.6468
1.22
1.23
1.564468
1.856761
0.84258
1.1868
0.5385
0.6392
1.23
1.24
i.583"5
1.872499
0.84546
1.1828
0-5340
0.6317
1.24
I.2S
1.601919
1.888424
0.84828
1.1789
0.5296
0.6242
1-25
1.26
1.620884
1.904538
0.85106
1.1750
0.5251
0.6170
1.26
1.27
1.640010
1.920842
0.85380
1.1712
0.5206
0.6098
1.27
1.28
1.659301
1-937339
0.85648
1.1675
0.5162
0.6026
1.28
I.2Q
1.678758
1.954029
0.85913
1.1640
0.5118
0-5957
1.29
1.30
1.698382
1.970914
0.86172
1.1605
0.5074
0.5888
1.30
I-3I
1.718177
1.987997
0.86428
1.1570
0.5030
0.5820
I-3I
1.32
1.738143
2.005278
0.86678
i-i537
0.4987
0-5753
1.32
1-33
1.758283
2.022760
0.86925
1.1504
0.4944
0.5687
i-33
1-34
I-778599
2.040445
0.87167
1.1472
0.4901
0.5623
i-34
I-3S
1.799093
2-058333
0.87405
1.1441
0.4858
0-5559
i-35
1.36
1.819766
2.076427
0.87639
1.1410
0.4816
0-5495
1-36
1-37
1.840622
2.094729
0.87869
1.1380
0-4773
0-5433
i-37
1.38
1.861662
2.113240
0.88095
L^Si
0.4732
0.5372
1-38
1-39
1.882887
2.131963
0.88317
1-1323
0.4690
0.53H
1-39
1.40
1.904302
2.150898
0.88535
1.1295
0.4649
0.5252
1.40
1.41
1.925906
2.170049
0.88749
1.1268
0.4608
0.5192
1.41
1.42
L947703
2.189417
0.88960
1.1241
0.4568
o.5i34
1.42
1-43
1.969695
2.209004
0.89167
1.1215
0.4527
0.5077
i-43
1.44
1.991884
2.228812
0.89370
1.1189
0.4486
0.5020
1.44
i. 45
2.014272
2.248842
0.89569
1.1165
0.4446
0.4964
1-45
1.46
2.036862
2.269098
0.89765
1.1140
0.4407
0.4909
1.46
1.47
2-059655
2.289580
0.89958
1.1116
0.4367
0.4855
1-47
1.48
2.082654
2.310292
0.90147
1.1093
0.4329
0.4802
1.48
1.49
2.105861
2-331234
0.90332
1.1070
0.4290
0.4749
1.49
Example. tanh 1.25 = 0.84828.
[146]
TABLE XV
REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + to. CONTINUED
e
Sinh0
Cosh0 Tanh0 Coth 0 Sech 6 Cosechfl 6
1.50
2.129279
2.352410
0.90515
1.1048
0.4251
0.4697
1.50
1-51
2.152910
2.373820
0.90694
1.1026
0.4212
0.4645
i-Si
1-52
2.176757
2.395469
0.90870
1.1005
0.4174
0.4594
1.52
1-53
2.200821
2.417356
0.91042
1.0984
0.4137
0-4543
1-53
1-54
2.225105
2.439486
0.91212
1.0963
0.4099
0.4494
1-54
i-SS
2.249611
2.461859
0.91379
1.0944
0.4062
0.4444
1-55
1.56
2-274343
2.484479
0.91542
1.0924
0.4025
0.4398
1.56
1-57
2.299302
2.507347
0.91703
1.0905
0.3988
0.4350
1-57
1.58
2.324490
2-530465
0.91860
1.0886
0-3952
0.4302
1.58
1-59
2.349912
2.553837
0.92015
1.0868
0.3916
0.4255
1-59
i. 60
2-37SS68
2.577464
0.92167
1.0850
0.3879
0.4209
1.60
1.61
2.401462
2.601349
0.92316
1.0832
0.3844
0.4164
1.61
1.62
2.427596
2.625495
0.92462
1.0815
0.3809
0.4119
1.62
1.63
2-453973
2.649902
0.92606
1.0798
0-3774
0.4075
1.63
1.64
2.480595
2.674575
0.92747
1.0782
0-3739
0.4031
1.64
1-65
2-507465
2-699515
0.92886
1.0766
0.3704
0.3988
1.65
1.66
2.534586
2.724725
0.93022
1.0750
0.3670
0-3945
1.66
1.67
2.561960
2.750207
0.93155
1.0735
0.3636
0.3903
1.67
1.68
2-589591
2-775965
0.93286
, 1.0719
0.3602
0.3862
1.68
1.69
2.617481
2.802000
0.93415
1.0704
0-3569
0.3820
1.69
1.70
2.645632
2.828315
0-93541
1.0691
0.3536
0.3780
1.70
1.71
2.674048
2.854914
0.93665
1.0676
0-3503
0.3740
1.71
1.72
2.702731
2.881797
0.93786
1.0662
0.3470
0.3700
1.72
1-73
2.731685
2.908969
0.93906
1.0649
0.3438
0.3661
1-73
1.74
2.760912
2.936432
0.94023
1.0636
0.3405
0.3622
1.74
i-75
2.790414
2.964188
0.94138
1.0623
0-3373
0.3584
i-75
1.76
2.820196
2.992241
0.94250
1.0610
0.3342
0.3546
1.76
1.77
2.850260
3.020593
0.94361
1-0597
0.3310
0.3508
1.77
1.78
2.880609
3.049247
0.94470
1-0585
0.3279
o.347i
1.78
1.79
2.911246
3.078206
0.94576
1-0573
0.3248
0-3435
1.79
1.80
2.942174
3-107473
0.94681
1.0562
0.3218
0-3399
1.80
1.81
2-973397
3-137051
0.94783
1-0550
0.3187
0.3363
1.81
1.82
3.004916
3.166942
0.94884
1.0539
0.3158
0.3328
1.82
1-83
3-036737
3-r97i5o
0.94983
1.0528
0.3128
0.3293
1.83
1.84
3.068860
3.227678
0.95080
1.0517
0.3098
0.3258
1.84
1.85
3.101291
3-258528
0.95175
1.0507
0.3069
0.3224
1.85
1.86
3.134032
3.289705
0.95268
1.0497
0.3040
0.3191
1.86
1.87
3.167086
3.321210
0-95359
1.0487
0.3011
0.3157
1.87
1.88
3.200457
3-353047
0.95449
1.0477
0.2982
0-3125
1.88
1.89
3.234148
3.385220
0-95537
1.0467
0.2954
0.3092
1.89
1.90
3.268163
3-4I7732
0.95624
1.0458
0.2926
0.3059
1.90
1.91
3.302504
3-450585
0.95709
1.0448
0.2897
0.3028
1.91
1.92
3-337I76
3-483783
0.95792
1.0439
0.2870
0.2997
1.92
i-93
3.372181
3-5I7329
0.95873
1.0430
0.2843
0.2965
1-93
1.94
3.407524
3-55I227
0-95953
1.0422
0.2816
0.2935
1.94
I- 95
3.443207
3-58548i
0.96032
1.0413
0.2789
0.2904
i-95
1.96
3.479234
3.620093
0.96109
1.0405
0.2762
0.2874
1.96
1.97
3.515610
3-655067
0.96185
1.0397
0.2736
0.2844
1.97
1.98
3-552337
3.690406
0.96259
1.0389
0.2710
0.2815
1.98
1.99
3-5894I9
3-726115
0.96331
1.0380
0.2684
0.2786
1.99
Example. coth 1.70 = 1.0691.
[147]
TABLE XV
REAL HYPERBOLIC FUNCTIONS, f (x + io) = u + io. CONTINUED
e
Sinh0
Cosh0
Tanhfl
Cothfl
Sechfl
Cosech 0
0
2.OO
3.626860
3.762196
0.96403
J-0373
0.2658
0.2757
2.00
2.OI
3.66466
3-79865
0.96473
1-0365
0.2632
0.2729
2.OI
2.O2
3.70283
3-83549
0.96541
1.0358
0.2607
0.2701
2.O2
2.03
3-74I38
3.87271
0.96608
1-0351
0.2582
0.2673
2.03
2.04
3.78029
3.91032
0.96675
1.0344
0.2557
0.2645
2.04
2.05
3-81958
3-94832
0.96740
1-0337
0-2533
0.2618
2.05
2.O6
3.85926
3.98671
0.96803
1.0330
0.2508
0.2596
2.06
2.07
4.02550
0.96865
1.0323
0.2484
0.2565
2.07
2.08
3-93977
4.06470
0.96926
1.0317
0.2460
0.2538
2.08
2.O9
3.98061
4.10430
0.96986
1.0310
0.2436
0.2512
2.09
2.IO
4.02 1£6
4.14431
0.97045
1.0305
0.2413
0.2486
2.IO
2. II
4.06350
4.18474
0.97103
1.0298
0.2389
0.2461
2. II
2.12
4-10555
4-22558
0.97159
1.0293
0.2366
0.2436
2.12
2.13
4.14801
4.26685
0.97215
1.0286
0.2344
0.2411
2.13
2.14
4.19089
4-30855
0.97269
1.0280
0.2321
0.2386
2.14
2.15
4.23419
4-35067
0.97323
1.0275
0.2298
0.2362
2-15
2.l6
4.27791
4.39323
0-97375
1.0269
0.2276
0.2338
2.l6
2.17
4.32205
4-43623
0.97426
1.0264
0.2254
0.2314
2.17
2.18
4.36663
447967
0.97477
1.0259
0.2232
0.2290
2.18
2.19
4.41165
4.52356
0.97526
1.0254
O.22II
0.2267
2.19
2.20
4-457"
4.56791
0-97574
1.0249
0.2189
0.2244
2. 2O
2.21
4.50301
4.61271
0.97622
1.0243
0.2168
0.2221
2.21
2.22
4-54936
4-65797
0.97668
1.0239
0.2147
0.2198
2.22
2.23
4.59617
4.70370
0.97714
1.0234
O.2I26
O.2I76
2.23
2.24
4.64344
4.74989
0.97758
1.0229
0.2105
0.2154
2.24
2.25
4.69117
4.79657
0.97803
1.0225
0.2085
O.2I32
2.25
2.26
4-73937
4.84372
0.97847
I.022O
0.2064
O.2IIO
2.26
2.27
4.78804
4.89136
0.97888
I. O2 1 6
0.2044
0.2089
2.27
2.28
4.83720
4.93948
0.97929
I.O2II
O.2024
O.2067
2.28
2.29
4.88683
4.98810
0.97970
I.O2O7
0.2OO5
O.2047
2.29
2.30
4.93696
5.03722
0.98010
1.0203
0.1985
O.2O26
2.30
2.31
4.98758
5.08684
0.98049
I.OI99
0.1966
0.2005
2.31
2.32
5.03870
5^3697
0.98087
I.OI95
0.1947
0.1985
2.32
2-33
5.09032
5.18762
0.98124
I.Oigi
0.1928
0.1965
2-33
2-34
5^4245
5-23879
0.98161
I.OI87
0.1909
0.1945
2-34
2-35
5.19510
5.29047
0.98198
I.Ol84
0.1890
0.1925
2-35
2.36
5-24827
5-34269
0.98233
I.OlSo
0.1872
0.1905
2.36
2-37
5.30196
5-39544
0.98268
I.OI77
0.1854
0.1886
2-37
2.38
5.35618
5-44873
0.98302
I.OI73
0.1835
0.1867
2.38
2-39
5-41093
5-50256
0.98335
1.0169
O.l8l7
0.1848
2-39
2.40
5.46623
5-55695
0.98368
1.0166
0.1800
0.1829
2.40
2.41
5.52207
5.611-89
0.98399
1.0163
0.1782
0.1811
2.41
2.42
5-57847
5-66739
0.98431
1.0159
0.1765
0.1793
2.42
2-43
5-63542
5-72346
0.98462
1.0156
0.1747
0.1775
2-43
2-44
5.69294
5.78010
0.98492
1-0153
0.1730
0.1757
2-44
2-45
5-7SI03
5.83732
0.98522
1.0150
O.I7I3
0.1739
2-45
2.46
5.80969
5.89512
0.98551
1.0147
0.1696
0.1721
2.46
2-47
5.86893
5-95352
0.98579
1.0144
o.i 680
0.1704
2-47
2.48
5.92876
6.01250
0.98607
1.0141
0.1663
0.1687
2.48
2.49
5-98918
6.07209
0.98635
1.0138
0.1647
0.1670
2.49
Example.
sech 2.00 = 0.2658.
[148]
TABLE XV
REAL HYPERBOLIC FUNCTIONS. / (x + to) = u + io. CONTINTJED
0
Sinhfl
Coshfl
Tanh0
Cothfl
Sech0
Cosech 8
6
*-5
6.05020
6.13229
0.98661
1.0136
0.1631
0.1653
2-5
2.6
6.69473
6.76901
0.98903
I.OIII
0.1477
0.1494
2.6
2.7
7.40626
7-47347
0.99101
1.0091
0.1338
0.1350
2.7
2.8
8.19192
8.25273
0.99263
1.0074
O.I2I2
O.I22I
2.8
2.9
9-05956
9.11458
0.99396
1. 0061
0.1097
0.1104
2.9
3-o
10.01787
10.06766
0-99505
1.0050
0.0937
0.09982
3-o
3-i
11.07645
11.12150
0-99595
1.0041
0.0899
0.0003
3-i
3-2
12.24588
12.28665
0.99668
1.0033
0.0814
0.08 1 6
3-2
3-3
13.53788
I3-57476
0.99728
1.0027
0.0736
0.0739
3-3
3-4
14.96536
14.99874
0.99778
1.0022
0.0667
0.0668
3-4
3-5
16.54263
16.57282
0.99818
1.0018
0.0604
0.0604
3-S
3-6
18.28546
18.31278
0.99851
1.0015
0.0546
0.0547
3.6
3-7
20.21129
20.23601
0.99878
I.OOI2
0.0494
0.0495
3-7
3-8
22.33941
22.36178
0.99900
I.OOIO
0.0447
0.0448
3-8
3-9
24.69110
24.7"3S
0.99918
1.0008
0.0405
0.0405
3-9
4.0
27.28992
27.30823
0-99933
1.0007
0.0366
0.0366
4.0
4.1
30.16186
30.17843
0-99945
1. 0006
0.0331
0.0332
4-1
4-2
33-33567
33-35066
0-99955
1.0005
0.0300
0.0300
4.2
4-3
36.84311
36.85668
0.99963
1.0004
0.0271
0.0271
4-3
4.4
40.71930
40.73157
0.99970
1.0003
0.0245
0.0245
4-4
4-5
45.00301
45.01412
0-99975
1.0003
O.O222
O.O222
4-5
4.6
49-737I3
49.74718
0.99980
I.OOO2
O.O2OI
O.O2OI
4.6
4-7
54.96904
54.97813
0.99983
I.OO02
0.0182
0.0182
4-7
4-8
60.75109
60.75932
0.99986
I.OOOI
0.0165
0.0165
4.8
4-9
67.14117
67.14861
0.99989
1. 0001
0.0149
0.0149
4-9
S-°
74.20321
74.20995
0.99991
I.OOOI
0.0135
0.0135
5-o
5-1
82.0079
82.0140
0-99993
1.00007
0.01219
0.01219
5-i
5-2
90.6334
90.6389
0.99993
1.00007
0.01103
0.01103
S-*
5-3
100.1659
100.1709
0-99994
1.00006
0.00998
0.00998
5-3
5-4
110.7009
110.7055
0.99995
1.00005
0.00903
0.00903
5-4
5-5
122.3439
122.3480
0.99996
1.00004
0.00818
0.00818
5-5
5-6
135.2114
135.2150
0-99997
1.00003
0.00740
0.00740
5-6
5-7
149.4320
J49-4354
0.99998
I.OOO02
0.00669
0.00669
5-7
5-8
165.1483
165.1513
0.99998
I.OOOO2
0.00606
0.00606
5-8
5-9
182.5174
182.5201
0.99998
1.00002
0.00548
0.00548
5-9
6.0
201.7132
201.7156
0-99999
I.OOOOI
0.00496
0.00496
6.0
6.1
222.9278
222.9300
I.OOO
I.OOO
0.00449
0.00449
6.1
6.2
246.3735
246.3755
I.OOO
I.OOO
0.00406
0.00406
6.2
6-3
272.2850
272.2869
I.OOO
I.OOO
0.00367
0.00367
6-3
6.4
300.9217
300.9233
I.OOO
I.OOO
0.00332
0.00332
6.4
6-5
332.5701
332.5716
I.OOO
I.OOO
0.00301
0.00301
6-5
6.6
367.5469
367-5483
I.OOO
I.OOO
0.00272
0.00272
6.6
6-7
406.2023
406.2035
I.OOO
I.OOO
0.00246
0.00246
6.7
6.8
448.9231
448.9242
I.OOO
I.OOO
0.00223
0.00223
6.8
6.9
496.1369
496.1379
I.OOO
I.OOO
O.OO2O2
0.00202
6.9
7.0
548.3161
548.3170
I.OOO
I.OOO
0.00182
0.00182
7-o
7-i
605.9831
605.9839
I.OOO
I.OOO
0.00165
0.00165
7-i
7-2
669.7150
669.7158
I.OOO
I.OOO
0.00149
0.00149
7.2
7-3
740.1496
740.1503
I.OOO
I.OOO
0.00135
0.00135
7-3
7-4
817.9919
817.9925
I.OOO
I.OOO
0.00122
O.OOI22
7-4
7-5
904.0209
904.0215
I.OOO
I.OOO
O.OOIII
O.OOI 1 1
7-S
Example. cosech 2.50 = 0.1653.
[149]
TABLE XVI. SUBDIVISIONS OF A DEGREE — AUXILIARY TABLE
O.OI
00.36
O.4I
24.36
0.81
48.36
OI
0.0167 •
4i
0.6833
21
0.0058
O.O2
OI.I2
0.42
25.12
0.82
49.12
02
0.0333
42
O.7OOO
22
0.006 1
0.03
01.48
0-43
25.48
0.83
49.48
03
0.0500
43
0.7167
23
0.0064
O.O4
02.24
0.44
26.24
0.84
50.24
04
0.0667
44
0.7333
24
0.0067
O.OS
03.00
o-4S
27.00
0.85
51.00
05
0.0833
45
0.7500
25
0.0069
O.O6
03-36
0.46
27-36
0.86
51.36
06
O.IOOO
46
0.7667
26
0.0072
O.O7
O4.I2
0.47
28.12
0.87
52.12
07
0.1167
47
0.7833
27
0.0075
0.08
04.48
0.48
28.48
0.88
52.48
08
0.1333
48
O.SOOO
28
0.0078
O.O9
05.24
0.49
29.24
0.89
53-24
09
0.1500
49
0.8167
29
0.0081
O.IO
O6.OO
0.50
30.00
0.90
54-oo
IO
0.1667
So
0.8333
30
0.0083
O.I I
06.36
0.51
30-36
0.91
54.36
II
0.1833
Si
0.8500
31
0.0086
O.I2
07.12
0.52
31.12
0.92
5S-12
12
0.2OOO
52
0.8667
32
0.0089
0.13
07.48
0-53
31.48
o-93
55-48
13
0.2167
53
0.8833
33
0.0092
O.I4
08.24
0-54
32.24
0.94
56-24
14
0.2333
54
0.9OOO
34
0.0094
O.IS
09.00
o-SS
33-00
0-95
57.00
15
0.2500
55
0.9167
35
0.0097
0.16
09.36
0.56
33-36
0.96
57-36
16
0.2667
56
0-9333
36
O.OIOO
0.17
IO.I2
0-57
34-12
0.97
58.12
17
0.2833
57
0.9500
37
0.0103
0.18
10.48
0.58
34-48
0.98
58.48
18
0.3000
58
0.9667
38
0.0106
0.19
11.24
o-S9
35-24
0.99
59-24
19
0.3167
59
0-9833
39
0.0108
0.20
12.00
0.60
36.00
I.OO
60.00
20
0-3333
60
I.OOOO
40
O.OI 1 1
a
0
O.2I
12.36
0.61
36.36
21
O.3SOO
OI
0.0003
41
0.0114
O.22
I-J.I2
0.62
OX/"»JV'
37.12
22
oo^^
0.3667
02
0.0006
42
0.0117
O.23
*O* *
IV48
0.63
o / • *•*
27.48
23
v"O'-'v-' /
0.3833
03
0.0008
43
0.0119
^•"O
O.24
J. J.4J.V
14.24
*•" "O
0.64
O 1 *^**
38.24
*j
24
•J JO
0.4000
o
°4
O.OOII
~O
44
O.OI 2 2
o.2<;
I^.OO
0.6?
O" *^
30.00
2?
0.4167
oc
0.0014
41;
O.OI25
•*•*• j
0.26
* J**"
1C. 76
"•"o
0.66
o y
30.36
~«j
26
O.4333
J
06
0.0017
~j
46
O.OI28
0.27
O O
16.12
0.67
ov o
40.12
• • • •
• • . «
27
V.*TOOO
0.4500
07
0.0019
47
O.OI3I
0.28
16.48
0.68
40.48
28
O.4.66?
08
O.OO22
48
O.O1 11
O.2Q
. vs.tfi_r
17.24
0.60
*T ^
4.1.24.
2O
v.lfwu y
O.4833
oo
O.OO25
•f**
4O
•WAOi>
0.0136
y
O.1O
• / • *"T
iS.OO
' y
O.7O
^* • *^
42.OO
*y
30
^•t^OO
0.5000
vy
IO
O.OO28
ty
CO
O.OI 3O
*~*o
**• f >*
T^
o
/ //
ow
0
-WAoy
0.31
18.36
0.71
42.36
0.001
00.03.6
3i
0.5167
II
0.0031
51
0.0142
0.32
19.12
0.72
43.12
O.O02
00.07.2
32
0-5333
12
0.0033
52
0.0144
o-33
19.48
o-73
43-48
0.003
06.10.8
33
0.5500
13
0.0036
53
0.0147
0-34
20.24
0.74
44.24
0.004
00.14.4
34
0.5667
14
0.0039
54
0.0150
0-35
21.00
0-75
45-00
0.005
00.18
35
0-5833
15
O.OO42
55
0.0153
0.36
21.36
0.76
45-36
0.006
OO.2I.6
36
0.6000
16
O.OO44
56
0.0156
0-37
22.12
0.77
46.12
0.007
00.25.2
37
0.6167
17
O.0047
57
0.0158
0.38
22.48
0.78
46.48
0.008
00.28.8
38
0.6333
18
O.OO5O
58
0.0161
o-39
23.24
0.79
47.24
0.009
00.32.4
39
0.6500
19
0.0053
59
0.0164
0.40
24.00
0.80
48.00
O.OIO
00.36
40
0.6667
20
0.0055
60
0.0167
Examples. o°.4i = o°.24'.j6"
o°.oo5 = o°.oo'.i8*
o°.4i'.oo* = o°.6833.
o°.oo'.46* = o°.oi28.
[150]
EXPLANATORY TEXT
EXPLANATORY TEXT
INTRODUCTION
THE Tables in this book are designed primarily for presenting hyperbolic functions
of a complex variable either in the rectangular coordinate form of that variable (x -f- iy)
or the polar coordinate form (p/8). They are also designed secondarily for presenting
circular functions of a complex variable. A few formulas are added as aids to the
conversion of such functions. The most extensive range offered is in Tables VII to
XIV inclusive, between which, the functions sinh (x + iy), cosh (x + iy), tanh (x -f iy),
expressed in the result either in rectangular coordinates u + iv or in polar coordinate
quantities r /y, may be obtained between the limits of o and ± 10 of x, and between the
limits of o and ± « for y. It is shown, moreover, to be an easy matter to extend the
range of x beyond the offered range of ± 10, should such an extension be required.
The practical need for tabulated values of hyperbolic functions of (x + iy) beyond the
range of x = ±10 appears to be so small that any such extension is left to the reader.
As the author's applications for financial assistance in the computation of the Tables
were unsuccessful, the steps in x and y (0.05 and 0.07854 respectively) are larger than
were originally intended; i.e., for reducing the work of the user to the lowest practic-
able limits. Consequently, interpolation must ordinarily be resorted to, when three or
more significant digits are needed in the results. Such interpolations require an appre-
ciable amount of time to effect in two dimensions; i.e., for both x and y. In order to
render such interpolation unnecessary for ordinary engineering purposes, where three, or
at most four, significant digits may be needed, a separate atlas of 23 large-scale charts,
45 cm. X 45 cm. over ruled areas, has been prepared, and is published as an adjunct
to these Tables. The necessary interpolation can very swiftly be made on the charts,
by inspection.
COMPLEX QUANTITIES
The following brief outline of complex quantities is offered in view of their funda-
mental importance in connection with the Tables, for the assistance of those who have
studied elementary mathematics, but who may not have become familiar with com-
plex numbers. For a more comprehensive discussion of complex quantities, the reader
must be referred to special treatises on the subject.
Ordinary numerical quantities, or the numbers dealt with in ordinary arithmetic,
may be considered to range between zero and either positive or negative infinity, by
indefinitely small gradations. Such numbers may be represented geometrically by
distances, in either direction, from a zero point on an infinite straight line. Thus in
Fig. i, we may consider that the straight line — XOX extends from minus infinity on
the left, to plus infinity on the right, O being the zero point. The point x\. would then
represent + i, and so on. That is, the number + i may be regarded as represented
on the line — XOX either by the position of the point xi with respect to the zero point 0;
[153]
EXPLANATORY TEXT
or, as the vector Ox\; i e., the straight line drawn from the origin O to the point x\ and
forming a part of the reference line — XOX. Under these assumptions, the ordinary
numbers of arithmetic may be represented geometrically as vectors; but such vectors
are confined to a single straight-line direction from O towards .X" for positive numbers,
and from O towards — X for negative numbers.
Complex quantities, or complex numbers, cannot be completely represented by
reference to a single direction, or to vectors along one and the same straight line. They
may, however, be represented geometrically by the position, in an infinite plane, of a
movable point with respect to a fixed point as origin. Thus, in Fig. i, the plane XOY
is the plane of reference, and the fixed point 0 is the origin. Then any point PI in the
plane represents a complex number, and any complex number may be represented by
a point on the plane.
-Y
2 X
FIG. 2. — Plane Vector 2.236 e"-106 or
FIG. i. — Complex quantity i + » a. designated by 2.236 763°. 26'.
A complex number may be specified either in rectangular coordinates, or in polar
coordinates, as may be preferred. Thus, the same vector OP\ is represented in Fig. i
to rectangular coordinates, and in Fig. 2 to polar coordinates. In Fig. i, the X axis
— XOX passing through the origin O is the fundamental reference axis, and the Y axis
— YOY, perpendicular thereto in the reference plane, immediately follows. Then the
point Pi, measuring + i along OX, and -f- 2 along OF, may be defined by the expres-
sion (i + 22), where the symbol i signifies measurement along 'the subordinate axis.
It is shown in mathematical treatises that * = V — i. The vector OP\ of Fig. i may
therefore be expressed as (i + V~ 1-2) and a vector from 0 to any point in the plane
may be represented by x + V~i y — x + *3S where x and y may have any positive or
negative numerical values, including zero.
In pursuance of time-honored terminology, the axis — XOX is sometimes called the
"real" axis, and — YOY the "imaginary," axis; so that the ^-component of a
complex number becomes the " real component," and the ^-component the " imaginary
component." The symbol i still stands for the imaginary component. In mathematics
as applied to electrical engineering, the symbol i commonly designates electric current-
strength, and so, in order to prevent the possibility of confusion, the symbol j is
frequently substituted as the sign of the imaginary. Under such a convention, the
plane-vector, or complex quantity, OP\, would be represented as i -\- j 2. As, however,
in this book we necessarily consider complex quantities from a broader viewpoint than
that offered by electrical engineering, we shall use the symbol i to denote the imaginary
component, perpendicularly rotated with respect to the fundamental X axis.
[154]
EXPLANATORY TEXT
Complex quantities may also be expressed in polar coordinates, as in Fig. 2, where
the fundamental reference axis OX is drawn in the positive direction in the reference
plane, from the origin 0, and the circular angle 5i is measured in the positive or counter-
clockwise direction from OX to OP\. The vector OPi is then specified in polar co-
ordinates by its length pi and by its angle 5i. The length p\ is called the modulus of the
vector, and the angle 81 is called the argument. This argument may be expressed in
circular radians, in degrees-minutes-seconds, quadrants, or any other recognized unit
of circular angle. Thus, in Fig. 2, the vector OPi may be represented to polar co-
ordinates symbolically by pi/8\ or, using numbers, by 2. 236/63°. 26', where 2.236 is
the modulus to the same scale of linear measure as in Fig. i, and 63°. 26' is the argument.
If one and the same complex quantity be expressed both in rectangular and polar
coordinates, as follows:
x + iy = p/8 (i)
it is evident that x = p cos 8, y = p sin 5, ;y/a; = tan 8, and p = V x2 + y2, relations which
enable the coordinates to be changed, at will, from one form to the other. ' Thus in
Figs i and 2, xi = i, yi = 2, p\ — VJ = 2.236, and §1 = tan -1 (2) = 63°.26/.
-X
FIG. 3. — Complex quantity — 2 — » i.
FIG. 4. — Plane-Vector 2.236 e*3-608. or 2.236^206^34'.
Similarly, Figs. 3 and 4 represent the complex quantity or plane vector OPi to rec-
tangular and polar coordinates respectively. Here x% = — 2, yz = — i, pz = Vs = 2.236
and 82 = 206°. 34'.
ADDITION OF COMPLEX QUANTITIES
One vector quantity is added to another, by drawing it in the reference plane from
the extremity of the latter as origin, and then drawing a vector from the origin to its
-X
-Y
FIG. 5. — Addition of two complex quantities
(x + » 2) + (- 2 - * i) = - i + » i.
FIG. 6. — Complex Addition, Polar coordinates.
Pl/Si +pJ/5i! = p3/53
2. 236/63 ".26' + 2.236/2o6°.34' = x .414/135*
Op + pP = OP
free end. The last named vector is the required sum. Thus, in Fig. 5, the complex
quantity OP2 — — 2 — ii of Fig 3 is added to the complex quantity OP\ =1+12
EXPLANATORY TEXT
of Fig i, giving the resultant vector OP = — i + ii. Fig. 6 shows the corresponding
operation with polar coordinate vectors. Here OP2 = 2. 236/206°. 34' of Fig. 4 is added
to OPi = 2. 236/63°. 26' of Fig. 2, to produce OP = 1.414/135° = p3/8z of Fig. 6.
On the drawing-board, the graphic process of adding vectors is as easily effected
when they are expressed in polar as in rectangular coordinates. But the arithmetical
addition is much more easily made with rectangular coordinates. The rule is: find the
vector sum by taking first the sum of the reals, and then the sum of the imaginaries; or
+ xz + . . . -f *„)
(xi + iyj + (xz + iyz) + - . • + (xn + fyB) =
+ i (y* + y* + . . . + ?„) = 2x
In the case of Figs. 5 and 6:
(i + «) + (- 2 - «) = (+ i - 2) + i(2 - i) = - i + ii =
SUBTRACTION OF COMPLEX QUANTITIES
(2)
/i3S°.
Reversing the sign of a rectangular complex quantity means reversing the sign of
both its real and imaginary components. Reversing the sign of a polar complex quan-
tity means changing its argument by 180°.
To subtract one complex quantity A from another B, reverse the sign of A, and then
add it thus reversed to B, by the rules of addition.
Y
o
FIG. 7. — Complex Subtraction
(i + 12) - (- 2 -*i) .. 3 +*3 =OP.
FIG. 8. — Complex Subtraction, Polar Coordinates
pi /Si — p-i/b-i = p3/5a
2.236 763 ".26' - 2.236/206^34' = 4.243/45"
Opt + piP = OP.
In Figs. 7 and 8, the vector Pz of Figs. 3 and 4 is subtracted from the vector P\ of
Figs, i and 2. In Fig. 7, we have
OPi - OP2 = OP.
(i + *2) - (- 2 - *l) = (i + *2) + (2 + *l).
= 3 + *3-
In Fig. 8, pi/3i — p2/52 = psAj
2.236/63°. 26^ — 2. 236/206°. 34' = 2. 236/63°. 26' + 2. 236/26°. 34'.
= 4.243/45°.
Here again the process of complex subtraction, which is only a slight modification of
complex addition, is very easily made on the drawing board by purely geometric proc-
esses, whether the quantities are rectangular or polar. If, however, the process is to
be conducted algebraically, it is much more easily conducted with rectangular coordinates.
[156]
EXPLANATORY TEXT
MULTIPLICATION OF COMPLEX QUANTITIES
Two rectangular complex quantities may be multiplied algebraically by the ordinary
rules of algebra, remembering that i2 = — i. Thus
(*i + fyi) (*2 + iyz) = (xi xz - yi y2) + i(xi y2 + xz yi). (3)
In Fig. 9, the vector OP\ of Figs, i and 2 is multiplied by the vector OP2 of Figs. 3
and 4. The product is the broken line OPz.
For (i + M) X (- 2 - ii) = (- 2 + 2) - i(i + 4) = - *5.
FIG. 9. — Product and Quotient of Complex Quantities
Rectangular Coordinates
(i +.2) X(-2 -ii) = -is =OP3
(-2 +11) -M+i + 12) = -o.8+to.6 = OP4
FIG. 10. — Product of two Complex Quantities, Polar Coordinates
2.236 763°. 26' X 2.236/2o6°.34' = 5 /270° = OPs
If the two quantities to be multiplied are polar; then
Pi /Si X PZ/SZ = Pi Pz/8i +
(4)
Or the rule is form the product of the moduli and add the arguments. Thus in Fig. 10,
OPi = Vj/63°.26' and OP2 = Vs/2o6^3j/ /. OP3 = 5/27o°.oo/.
RECIPROCAL OF A COMPLEX QUANTITY
The reciprocal of a rectangular complex quantity can be reduced to the standard
algebraic form, by multiplying both numerator and denominator by the same complex
quantity with reversed imaginary. Thus:
i i X (x - iy) _ x - iy _ ( x \ _ .f y \ ,.
1 yt.2 / V /L-2 _J /t«2 / \J/
*2 +
x + iy (x + iy) (x - iy)
For example if x + iy = i + 12,
I _ I /I — J2\ _ !_
1+^2 I + 12 \ I — 12 ) I
—12 I 12
= = 0.2 —
+ 4 5 5
The reciprocal of a polar complex quantity is obtained by taking the reciprocal of
its modulus, and reversing its argument. That is
= -I- 8 =
p— p
Ci57]
(6)
EXPLANATORY TEXT
For example:
V5/63°.26' VS
QUOTIENT OF COMPLEX QUANTITIES
To find the quotient of a complex quantity A divided by another B, form the re-
ciprocal of B and then multiply this reciprocal by A .
Thus to find (x\ + iyi)/(x2 + fy2)
/ x
For example:
\ __ — 2 — ii __ — 2 — ii /i — i:
\ i + *2 i -f- 12 \ i — i'.
_ (- 2 - 2) + t(
0 o , • ,
= — o.o + i o.o.
Thus, in Fig. 9,
The quotient of two polar complex quantities is formed by taking the quotient of
their moduli and the difference of their arguments. That is
P2/02 P2 /E ' E /ox
-=7= = — /fc ~ Si. (8)
Pi/5i Pi
Thus in Fig. 1 1 we have the quotient of OP2 of Figs. 3 and 4 divided by OPi of Figs,
i and 2, or
^ = i /I43°.o8/.
* A'
FIG. n. — Quotient of two Complex Quantities,
Polar Coordinates.
8.836 /2o6°.34> -5- 2.236 /63°.26' = I.o/i43°.o8' = O?4
Pz ^83
It is thus evident that in order to find either the sums or the differences of complex
quantities, it is desirable to have them expressed in rectangular coordinates; while,
on the other hand, in order to find products, reciprocals, or quotients, it is preferable
to have them expressed in polar coordinates.
[158]
EXPLANATORY TEXT
POWERS AND ROOTS OF COMPLEX QUANTITIES
It will be evident from the foregoing that
(p/8)n = pn/n8: and \fpjl = y~p/8/n (9)
operations that are readily executed on polar complex quantities.
CIRCULAR AND HYPERBOLIC FUNCTIONS GEOMETRICALLY COMPARED
Since the Tables in this book are adapted for the evaluation of both circular and
hyperbolic functions of a complex variable; that is, either of sin (a; -f- iy), cos (it + iy)
and tan (x + iy); or of sinh (x + iy), cosh (x + iy) and tanh (x + iy), it may be
advisable to consider some propositions in the comparative geometry of the circular
and hyperbolic functions, both real and complex.
REAL CIRCULAR AND HYPERBOLIC FUNCTIONS
The geometry of the real circular functions sin x, cos x and tan x relates, as is well
known, to the motion of a radius vector over a circle. The geometry of the real hyper-
bolic functions sinh x, cosh x and tanh x relates to the motion of a radius vector over a
rectangular hyperbola. In Fig. 12, A b c d E g is a circle x* + y2 = i, assumed to have
unit radius, and center 0. As the radius vector OA rotates in the positive or coun-
terclockwise direction about the center 0, it describes a circular sector such as AOE,
and a circular angle /3, the tangent Ef being always perpendicular to the radius
vector OE. The magnitude of the circular angle /3 may be defined in either of two
ways, namely: —
(1) By the ratio of the circular arc length 5 described during the motion, by the
vector's terminal E, to the constant length p of the radius vector.
(2) By the area of the circular sector AOE swept out by the radius vector during
the motion.
According to definition (i), if the radius vector generates any infinitesimal angle
dfi circular radians, by moving its terminal over an infinitesimally small circular arc ds
then <//3 = — = — circular radians (10)
P i
since the constant radius vector p has been taken equal to unity Consequently, in
passing over any circular arc from distance s\ to distance s%, through a distance s% — s\ = sy
the total circular sector and circular angle generated will of course be: —
/ds
— = ($2 — si) = s circular radians (n)
or the angle /3, as is well known, becomes equal to the length of the circular arc de-
scribed, when expressed in circular radians.
According to definition (2), if in Fig. 12, the radius vector of unit length moves from
the initial position OA to any position such as OE, it will sweep out a circular sector OEA.
[159]
EXPLANATORY TEXT
If the arc AE1 be measured in the negative or clockwise direction equal in length to the
arc AE, then it is well known that the area of the double sector EOE1 shaded in Fig. 12,
is equal to /3 units of area because the area of the whole circle is manifestly TT units,
Consequently, the magnitude of
2/3
and the shaded area is — that of the whole circle.
27T
the angle (3 expressed in circular radians is numerically twice the area of the circular
sector AOE which it covers when the circle has unit radius.
FIG. 12. — Circular Sector and Real
Circular Functions.
FIG. 13. — Hyperbolic Sector and Real
Hyperbolic Functions.
Turning now to the hyperbolic case, let A b c d E Fig. 13, be an arc of a rectangular
hyperbola x2 — y2 = i, assumed to have unit semi-diameter OA, and center 0. As
the radius vector OA rotates in the positive or counterclockwise direction with center O,
it describes a hyperbolic sector AOE*, and also what may conventionally be called
for convenience a " hyperbolic angle " 0.* The tangent Ef to the path of the moving
terminal E always makes a circular angle /3 with the Y axis; or a circular angle of 2/3
with a perpendicular to the radius vector. The magnitude of the hyperbolic angle 6
may be denned in either of two ways; namely: —
(1) By the ratio of the hyperbolic arc length s described during the motion, by the
terminal E, to the integrated mean length of the varying radius vector.
(2) By the area of the hyperbolic sector AOE Fig. 13, swept out by the radius vector
during the motion. f
According to definition (i), if the variable radius vector p generates any infinitesimal
hyperbolic angle dd by moving its terminal over an infinitesimally small hyperbolic arc
ds ; then |
J*
hyperbolic radians (12)
ds
dd = -
P
* It should be pointed out that a " hyperbolic angle " in the sense above denned is not the opening be-
tween two lines intersecting in a plane; but a quantity otherwise analogous to a circular angle, and the
argument x of the functions sinha;, cosh a;, tanh#, etc. The use of the term " hyperbolic angle " can only
be justified by its convenience of analogy.
t GreenhilFs " Differential and Integral Calculus," 1896, p. 108.
I A demonstration of this proposition has been given by the author in " The Application of Hyperbolic
Functions to Electrical Engineering Problems." Appendix L, p. 250. University of London Press, IQII.
[160]
EXPLANATORY TEXT
Consequently, in passing over any hyperbolic arc from distance Si to distance 53 through
a distance s2 — Si = s , the total hyperbolic sector and hyperbolic angle generated will be
6
/•
ds = s_
p~ pi
hyperbolic radians (13)
where p1 is the integrated mean value of p as defined by the last equation. Any in-
finitesimally small angle, whether circular or hyperbolic, is therefore expressed in corre-
ponding radian measure by one and the same term ds/p ; but whereas, in the case of
circular angles, the constancy of the radius vector makes the integral simply s/p, in the
case of hyperbolic angles, the variation of the radius vector makes the integral more
complex. Fig. 14 represents a circular angle of i radian in five sections of 0.2 radian
each; while Fig. 15 represents a hyperbolic angle of i radian correspondingly divided.
The integrated mean radius vector of the full sector A OF intersects the curve in the
point/, the total length of the arc A B C D E F being 1.3167 units.
SINES, COSINES AND TANGENTS OF CIRCULAR AND HYPERBOLIC ANGLES
If, with unit radius, we draw both a circular and a rectangular hyperbolic sector,
as in Figs. 12 and 13, and take OA as the initial line in each; then for any position
FIG. 14. — Circular Sector of i Radian Subdivided into Five
Sectors of 0.2 Radian each.
FIG. 15. — Hyperbolic Sector of i Radian Subdivided into
Five Sectors each of o. 2 Radian.
of the radius vector such as OE, we shall have in either case the following magnitude
conditions: —
The sine will be equal to the length of the perpendicular from the terminal of the
radius vector on to the X axis.
The cosine will be equal to the length of the intercept on the X axis made by the
above-mentioned perpendicular.
The versed sine will be equal to the length XA, Fig. 12, and A X, Fig. 13, between the
intercept on the X axis, and the horizontal unit radius.
[161]
EXPLANATORY TEXT
The tangent will be equal to the length of the perpendicular from the radius vector
(or radius vector produced) on to unit radius point of the X axis. Thus in
Fig. 12, sin/3 = XE.
Fig. 13, sinh 6 = XE.
Fig. 12, cos /3 = OX.
Fig. 13, coshfl = OX.
Fig. 12, tan/3 = At.
Fig. 13, tanh 0 = At.
Whereas the values of sin /3, cos /3 and tan /3 fluctuate periodically in sign as /3 increases
from o to a , the values of sinh 6, cosh 6, and tanh 6 do not change sign, the graphs of
the real hyperbolic functions being indicated in Fig. 16, as far as 6 = 3.0.
i
1
1
I
/
/
/
FIG. 16
Diagram showing the graphs of the
sfne, cosine, tangent, cotangent, secant
and cosecant of a hyperbolic angle.
Ordinates'-- Numerical value of the trigonometrical,
function
Abscissas :- Numerical value of the hyperbolic angle.
/
/
1
/
/
I
/
/
/
I
/
1
/
/
/
/
j
/
1
/
/
[
j
I
1
i
y
\
Jl
* i
I
c \\
^-
'/
: \
6^
r
3
,
\ "
V,
0
j
v
v
/<
~)
0,
i -
-^
^
v
"~
>.y
/\
v—
--^NHT
O.5 -
"7
^
-4
^ft
HI
'PE
RB
OL
C
AN
GLE
Ljr
Q ^
/
1 1
••^•w
'»—
—
—
__^
[162]
EXPLANATORY TEXT
BISECTION or CIRCULAR AND HYPERBOLIC ANGLES
If we take any circular angle BOC Fig. 17, we may of course bisect this angle in either
of two ways: —
FIG. 17. — Bisection of a circular sector in the well-known
manner by a radius vector through the intersection of ter-
minal tangents, or through the midpoint of the chord be-
tween terminal points.
(i) By drawing tangents bb, cc, to the curve at the points B, C, respectively, and
drawing the straight line Od from the center O through the point of intersection d.
FIG. 18. — Bisection of a hyperbolic sector by a radius vector through the intersection of
terminal tangents, or through the midpoint of the chord between terminal points.
[163]
EXPLANATORY TEXT
(2) By drawing the chord BC, and marking the radius OD through the midpoint 8
of this chord.
Similarly, if we take any hyperbolic angle BDC Fig. 18, between the points B and C
of a rectangular hyperbola, we may bisect this angle in either of two ways: —
(1) By drawing tangents bb, cc, to the curve at the points B, C, respectively, and
drawing the straight line Od from the center O through the point of intersection d.
(2) By drawing the chord, BC, and marking the radius OD through the midpoint
5 of this chord.*
COMPARATIVE GEOMETRY OF COMPLEX CIRCULAR AND HYPERBOLIC FUNCTIONS
We have seen that the real circular functions sin x, cos x, may be derived from a
circle diagram, and that the real hyperbolic functions sinh x, cosh x, may be similarly
derived from a rectangular hyperbola diagram. We shall see that both the complex
circular functions sin (x + iy), cos (x + iy), and the complex hyperbolic functions
sinh (x + iy), cosh (x -\- iy), may be derived from a combination circle and hyperbola
diagram.
COMPLEX CIRCULAR FUNCTIONS
CONSTRUCTION FOR sin (x ± iy), AND FOR sin"1 (u ± in)
In Fig. 19, take OA = i along the negative side of the Y axis. From OA as initial
line, mark off the circular angle x = AOB. From OB as initial line, mark off the hyper-
bolic angle y and its sector BOD. Let C be the foot of the perpendicular from D on
OB produced. Drop perpendiculars from C and D on the axis of reals OX, at c and
d respectively. About c as center, rotate cd positively through 90° to cZ. Then will
* This proposition is proved in GreenhilFs " Differential and Integral Calculus," Macmillan & Co., 1896,
page 67, Fig. 16, for the particular case when the angle AOB, in our Fig. 18, is zero. The demonstration of
proposition (i) for the general case of Fig. 18 is not difficult; but that found by the author is rather lengthy.
The demonstration of the general proposition (2) is, however, brief and direct, as follows: —
Let 0i be the hyperbolic angle of the sector AOB.
Let 02 be the hyperbolic angle of the sector AOC.
Then it is required to show that
S£=hA_ = hA =
Of OA i
But from an inspection of the Figure,
Be = sinh 0i, Cg = sinh 02,
Oe = cosh 0i, Og = cosh 02,
eB + gC sinh 0i + sinh 02
so that
2 2
Oe -4- Og cosh 0i + cosh 02
and Of = —
2 2
_„ jft _ sinh 0i -\- sinh 02
Of~ cosh 0i + cosh 02
io
tions," 1909, p. XIV, Formula (49).
(n I rt \
which is a known equivalent expression for tanh — , see Becker and Van Orstrand's "Hyperbolic Func-
[164]
EXPLANATORY TEXT
the complex vector OZ = Oc + icd be the required circular sine of the complex angle
x + iy radians. In the case represented, sin (i + i i) = 1.299 + ^0.635 = 1.446
/26°.o5. As y varies, Z moves along the hyperbola bZ : —
X2 Y*
sin2 x cosz x
and as x varies, Z moves along the ellipse : —
X2 Y2
= i
(14)
cosh2 y ' sinh2 y
Both the hyperbola and the ellipse have as common foci FF', the points X = ± i, F = o.
7
FIG. 19. — Constructions for sin (* + iy) and sin-1 (u =*= I'D).
From the same figure, we have also, if Oc = u and cZ = iv,
sin"1 (u ± iv) = sin"1 Ob ± i cosh"1 OE
/(IT uY + v2- Vd - «)2 + v2 \
2 /
= sn
±i
(16)
^ FZ ~ F'Z A KV FZ + F'Z
since Ob = and OE =
CONSTRUCTIONS FOR cos (x + iy) AND FOR cos"1 (u + iv)
In Fig. 20, take OA as unit distance along the real or X axis, in the positive direction.
From OA as initial line, describe the circular angle x, or the circular sector AOB of area
x/2. On OB as initial line, describe the hyperbolic angle y, or the hyperbolic sector
area BOD of area y/2. Let C be the foot of the perpendicular from D on OB produced.
Drop perpendiculars from C and D on the X axis at c and d respectively. With c as
center, rotate the line cd in the positive direction through 90° into the position cZ; so
that ~cZ = i.~cd. Then the complex quantity OZ = Oc + *'.cd will be the required
circular cosine of the complex angle (x + iy) radians.
[165]
EXPLANATORY TEXT
In the case represented, cos (i + i i) = 0.834 — * 0.989 = i.293\49°.866. As y
varies, Z moves along the hyperbola bZ defined by
X2 Y2
= i
cos^ x sirr x
and as x varies, Z moves along the ellipse ZE, defined by
(18)
cosh2 y sinh2 y
Both the hyperbola and the ellipse have as common foci FF', the points X = ± i , Y = o.
FIG. 20. — Constructions for cos (x =•= iy) and cos"1 (« ± in).
From Fig. 20 we obtain: —
cos"1 OZ = cos"1 (u ± iv) = cos"1 Ob =F cosh"1 OE
E2 1
(19)
snce =
FZ ~ F'Z
and OE =
COMPLEX HYPERBOLIC FUNCTIONS
CONSTRUCTIONS FOR sinh (x ± iy) AND sinh"1 (u ± «'»)
In Fig. 21, take CM as unit length along the real or X axis in the positive direction.
From OA as initial line, describe the circular angle y, or the circular .ector AOB of
area y/z. From OB as initial line describe the hyperbolic angle x, or the hyperbolic
sector BOD of area x/2. Let C be the foot of the perpendicular from D on OB pro-
[166]
EXPLANATORY TEXT
duced. Drop perpendiculars from C and D on the F axis at c and d respectively. With
c as center, rotate the line cd negatively, or clockwise, through 90° to cZ. The
complex quantity OZ = Oc — i.cd will be the required hyperbolic sine of the complex
angle (x + iy) radians.
In the case represented, sinh (i + * i) = 0.635 + * I-2985 = 1.446/63°. 95. As x
varies, Z moves along the hyperbola Zbz: —
F2 X2
= i
sin" y cos" y
and as y varies, Z moves along the ellipse XExy
F2 Y2
7 i - V
(20)
(21)
cosh2 x sinh2 #
The hyperbola and ellipse are confocal at the points F and/ defined by X = o, F = ± i.
FIG. 21. — Constructions for sinh (* * iy) and sinh"1 (u =t in).
From Fig. 21 we also obtain
sinh"1 (« ± w) = sinh~l (cZ ± Oc) = cosh-1 OE ± * sin"1 Ob.
= cosh"1
db i sin"1
(22)
since
[167]
EXPLANATORY TEXT
CONSTRUCTIONS FOR cosh (x -f- iy) AND cosh"1 (u + iv)
In Fig. 22, take OA as unit distance along the real or X axis in the positive direction.
From OA, as initial line, describe the circular angle y, or the circular sector AOB of area
y/z. From OB, as initial line, describe the hyperbolic angle x, or the hyperbolic sector
BOD of area x/2. Let C be the foot of the perpendicular from D on OB produced.
Drop perpendiculars from C and D on the X axis at c and d respectively. About c, as
FIG. 22. — Constructions for cosh (x =•= jy) and cosh ' (u ± MI).
center, rotate the line cd negatively, or clockwise, through 90° to cZ ; so that cZ = — i.cd.
Then the complex quantity OZ = Oc — i.cd will be the required cosine of the complex
angle (x + iy) radians.
In the case represented, cosh (i + i i) = 0.834 + 20.989 = 1.293 749°. 866. As x
varies, Z moves along the hyperbola Zbz
X2
cos2y sin2}/
As y varies, Z moves along the ellipse EZez
F2
= i.
i
= i.
(23)
(24)
cosh2 x sinh2 x
The ellipse and hyperbola are confocal at the points A, a, denned by X = ± i, Y = O.
Li68]
EXPLANATORY TEXT
From the same figure. If Oc — u and cZ = iv
cosh-1 (u ± iv) = cosh"1 (Oc ± i.cZ) = cosh-1 OE ± i cos"1 C
_i_ ~_\
± * cos"1
= cosh"1 V(l + ")2 + ^ + V(l - U)
2
•(25)
CONSTRUCTIONS FOR tan (x ± fy) AND tan-1 (u ± *»)
In Fig. 23, lay off along the X axis a point A distant tan x from 0, and also a point
B such that O.5 = cot x. Draw a circle through A and 5 having its center on OX
at C. The distance OC measures cot 2x and the radius of the circle is cosec 2X. Any
circle thus drawn will intersect the Y axis at two points e and / which are at unit dis-
FIG. 23. — Constructions for tan (* ± iy) and tan~' (u
tances from 0. Then lay off the Y axis two points a and b, distant respectively tanh y
and coth y from O. With center c on the Y axis, draw a circle through a and b. The
distance Oc will be coth 2y, and the radius of the circle will be cosech 2j. Let Z be
the point of intersection of the two circles. Then OZ is the required tangent of (x + iy).
If x is kept constant but y is varied, the point Z moves over the circle AZB. If on the
other hand y is kept constant, but x is varied, Z will move around the circle aZb and
will make one complete revolution for each increase of TT units in x.
[169]
EXPLANATORY TEXT
In the case represented, tan (i + i i) = 0.2718 + i 1.084 = 1.118/75°. 916.
From Fig. 23 it is evident that the angle AeO is equal to x, and angle eAO is thus
the complement of x. Hence half the angle between r\_ and r2 is the complement of x.
Moreover ;y = loge V^/
tan-1 (u ± iv) =
- tan-
Therefore, if OZ = u +
1 ( ^ + tan-1
\± v — i
(26)
CONSTRUCTIONS FOR tanh (x ± iy) AND tanh-1 (w ± n>)
In Fig. 24 mark off on the axis of reals xOX two points T and -X" such that the former
is distant by tanh x and the latter by coth x from the origin O. Find the point C mid-
way between T and X. Incidentally, this point will be distant coth 2X from O. With
FIG. 24. — Constructions for tanh (x =*= »'y) and tanh—1 (w =*= i»)
center C and radius CT = CX = cosech 2x, draw the circle TXZ. Mark off on the
axis of imaginaries yOY, two points t and y such that the former is distant by tan y
and the latter by cot y from the origin O. Find the point c midway between them.
Incidentally, this point will be distant cot 2y from O. With center c and radius ct =
cy = cosec zy, draw the circle By At. This circle will cut the axis of reals at two points
A and B distant each one unit from O. It will also intersect the circle TXZ perpen-
dicularly at Z. Connect OZ. This vector OZ is the required hyperbolic tangent of
the complex angle (x + iy) radians.
[170]
EXPLANATORY TEXT
In the case represented, tanh (i -f- i i) = 1.084 + 20.2718 = 1.118/14°. 084. As x
varies, Z moves along the circle AtB. As y varies, Z m^ves along the circle TZX,
performing one complete revolution for each TT units of increase in y,
From the same Figure, if OZ — u ± iv = op ± ipZ, we have tanh"1 (u ± iv) =
x ± iy.
In this case x = loge Vri/r2
or * = $log.(ri/ra). (27)
and y = - — - where a is the circular angle at Z between the radii vectores r, and r2.
2
Also
Hence
DEGREE OF PRECISION OF TABLES
INTRODUCTION
If a numerical quantity, freed from decimals, is correctly expressed to within say
i part in 1000; i.e., i part in io3, then this degree of precision may conveniently be
described as precision of the third order. In general, therefore, if a numerical quantity
be correctly expressed to within i part in ion, where n is any real positive number, its
precision is of the nth order. The weekly statement of the financial assets of a bank
might be expressed as $186,257,361.26 which, assuming that it is to be taken as being
numerically correct to a single cent, represents 18,625,736,126 cents, an apparent
precision of i in lo10'27, or of the 10.2 7th order. Physical and astronomical precisions
are less ostensibly pretentious, however, and rarely exceed the 6th order. Engineering
computations are commonly satisfied with a precision of the third order; although,
on rare occasions, the order required may be the highest that physics can attain.
The degree of precision corresponding to retaining a specified number of significant
digits correct within unity, in Tables, can only be stated approximately ; since it varies
with the values of the digits. Thus, if we have tables containing entries each of three
significant digits, correct to the last digit, the lowest entry may be 100 and the highest
999. The precision would therefore be i in 100 in the former case, and practically i in
1000 in the latter. That is the order of precision would vary between the second and
the third. The average precision might be stated as of the 2. 5th order. Such tables
of n significant digits lay claim to an average precision of the (n — i/2)th order.
Many tables are, however, employed in which the last digit is stated to be correct to
the nearest digit; that is within half of unity. On that understanding, the precision
of say a three-digit table would vary between 1/2 in 100 to 1/2 in 999 or between the
2. 3rd and the 3. 3rd order, with a mean of the 2.8th order. Consequently, we may say
that such tables, giving n significant digits, lay claim to an average precision of the
(n — o.2)th order.
[171]
EXPLANATORY TEXT
DEGREE OF PRECISION PRESENTED IN THE FOLLOWING TABLES
The tables of complex hyperbolic functions here presented have been prvffciied
with a view to giving five decimal places regularly. This means five significant digits
when the values of the results lie between o.i and unity, six significant digits when they
lie between i and 10, four when they lie between o.i and o.oi and so on. Tables I to
VI inclusive were computed with the aid of five-figure logarithms of real hyperbolic
functions, so that their degree of precision is necessarily limited to, and must on the
average fall below that of such logarithm tables, which, as we have seen, is of the 4. 8th
FIG. 25. — sinh (p/5) expressed in polar coordinates.
order. Exclusive of such mistakes as may exist, they do not claim a degree of pre-
cision beyond the 4. 5th order.
Tables VII to XIII inclusive were, however, computed for the most part from Lig-
owski's gudermannian angles which are tabulated by him for each thousandth of a
hyperbolic radian, to the nearest hundredth of a second of circular arc. The logarithms
of the corresponding real hyperbolic functions were then found in the eight-place
tables of Bauschinger and Peters, which offer such logarithms for each and every second
of circular arc. The results were computed in the formulas to at least six significant
[172]
EXPLANATORY TEXT
digits and the sixth was then frequently discarded to meet the needs 9f the five-decimal
table. Consequently in this group of tables, excluding such errors as may exist, the
precision is on the average of the 4.8th order, and rises to the 5. 8th order, when the
value of the result lies between i and 10. The average precision of the second group
of tables is thus about half an order greater than that of the first group.
PRECISION OF THE CHARTS IN THE ATLAS
The charts of the accompanying Atlas have been prepared with a view to offering
three digits in the deduced quantity, if reasonable care be taken in their use. This
represents an average degree of precision of the 2. 5th order; or about equal to that fur-
FIG. 26. — sinh
in polar coordinates.
nished by an ordinary 25 cm. slide rule. When a higher degree of precision than this
is needed, arithmetical interpolation in the Tables must ordinarily be resorted to; but
even then it is desirable to obtain a preliminary approximate value from the Atlas,
in order to furnish a check against gross error.
[173]
EXPLANATORY TEXT
GRAPHIC REPRESENTATIONS
Figs. 25 and 26 present the results obtained from Table I to true polar coordinates.
Each intersection of the curves corresponds to an entry in the table. Fig. 25 relates
to pages 2, 3, and Fig. 26 to the rest of the table. The curves of constant p intersect
those of constant 5 perpendicularly. That is, each intersection occurs theoretically
at right angles. If, however, an attempt is made to prepare plates corresponding to
Figs. 25 and 26 on a large scale, for a reasonable degree of precision, in rapid interpola-
tion by graphical inspection, difficulties present themselves. Firstly, it has been found
impracticable to procure polar coordinate ruled sheets large enough. Secondly, regular
polar coordinate charts of the type presented in Figs. 25 and 26' necessarily offer very
little graphical interpolation precision at small radial distances from the origin of co-
1 2 3
^•"
s*
i •
\
\
^._
1
Z
i
> 4
FIG. 27. — Polar Coordinate Diagram Regular
Presentation on Circular Sheet.
FIG. 28. — Polar Coordinate Dia-
gram Squared Presentation on
Rectangular Sheet.
ordinates, where the radial lines, sharply converging, crowd the diagram. On the
other hand, they offer relatively great apparent interpolation precision at large dis-
tances from the origin, as the radii diverge. In preparing interpolation charts, therefore,
the author has devised the scheme of using squared paper sheets for presenting such
polar coordinate quantities as appear in Figs. 25. and 26. Fig. 27 represents the regular
polar coordinate r/_y_ diagram, in which the lines of constant r are circles concentric
at 0, and the lines of constant 7 are radii diverging from O. Fig. 28 represents the
corresponding squared polar coordinate diagram, in which the lines of constant r are
the vertical straight lines i-i, 2-2, 3-3, and 4-4. The lines of constant 7 are the parallel
horizontal lines, while the origin point O in Fig. 27 becomes expanded into the original
straight line o-o in Fig. 28. The straight, broken line DU is transformed into the
curved line du, and, in general, orthogonally intersecting curves of one diagram do not
transform into orthogonally intersecting curves on the other.
[i74]
EXPLANATORY TEXT
INTERPOLATION CHARTS
Plates IA, IB, and Ic of the Atlas correspond to Table I and present, to squared polar
coordinates, the results in that table. Each intersection of curves in the plates corre-
sponds to one entry in the table. Plate IA includes the entries on pages 2 and 3 of
the table; while Plates IB and Ic include the entries in the remainder of the table.
The curves of constant p and constant 5 intersect one another at various angles, but the
method of interpolation requires little explanation. The entering quantity will fall
within some particular curvilinear parallelogram. The respective opposite sides may
be subdivided into tenths in any of the three following ways: (i) by direct inspectional
estimate, (2) by graphical subdivision on a sheet of tracing paper laid over the chart,
(3) by means of a radiating decimal scale of lines, prepared in advance, on tracing paper
or thin celluloid. It is not, in general, worth the effort of attempting a closer subdivi-
sion than tenths of the sides of any parallelogram. The point of intersection of lines
parallel to the sides, through the correct decimal points, is then to be marked on the
covering tracing paper, or held with a blunt pointer, such as a knitting needle, on the
chart itself, and the rectangular coordinates of this point read off from the parallel
ruling or background of the plate. That is, the charts are always used with the entering
variable on the curvilinear coordinates, and with the result found on the rectilinear frame-
work in the background; except when inverse functions are sought, and the procedure
consequently reversed.
TABLE I
sinh (p/8) = r£y
POLAR HYPERBOLIC SINES OF A POLAR VARIABLE
Table I, pages 2 to 7, gives the hyperbolic sine of vectors up to 3.0 in modulus, by
steps of o.i, for each degree of argument from 45° to 90°. The results are also expressed
in polar coordinates, as plane vectors, corresponding to the relation: —
sinh (p/5) = r/7 (30)
or p/d_ = sinh-1 (7/7). (31)
The graphs of the results to true polar coordinates appear in Figs. 25 and 26, where
the curves of constant p always intersect orthogonally the curves of constant d ; so that
at any point of intersection the angles of intersection are right angles. In Plates IA,
IB, and Ic of the Atlas, the same graphs are given to squared polar coordinates, the
disadvantages of the distortion being more than outweighed by the advantages in
facility of graphic interpolation. In these charts the curves of constant p do not inter-
sect the curves of constant 8 orthogonally.
[175]
EXPLANATORY TEXT
INTERPOLATION. FIRST CASE. IN MODULUS ONLY
If Table I is entered with a vector quantity of more than one decimal in modulus
and of some exact degree of argument, such as 2.76/70°; then the result will lie nearly
on the line between the results for 2.7/70° and 2.8/70°; namely, between 1.2
and i. 2136/143°. 005. A first approximation may be obtained by proportional parts
between them, thus : —
Required sinh 2.76/70°
by Table, sinh 2.80/70° = 1.2136/143°. 005.
by Table, sinh 2.70/70° = 1.2031/136°. 489.
Difference 0.10/70° = 0.0105 /6°.5i6.
Proportion for 0.06/70° = 0.0063 /3°.Qio.
sinh 2.70/70° = i. 2031/136°. 489.
Result sinh 2.76/70° = 1.2094/140°. 399.
The true value is 1.2086 /140°.366.
INTERPOLATION BY THE USE OF TAYLOR'S THEOREM
When more precise interpolation is required than that by simple intermediate pro-
portion, we may use Taylor's theorem in the following form; since
-cosh*, =sinh0, etc.
d0
sinh (0 + A0) = sinh 0 + A0 cosh 0 + sinh 0 + - cosh 0 + . . (32)
2!
Let 0 = p/5 and A0 = Ap/5.
Then
sinh \(p + Ap)/« j = sinh (p/6) +Ap/5. cosh (p/5) +~ M sinh (p/3) + . . . (33)
The number of correction terms to be retained depends on the interval, and on the
degree of precision desired. It is seldom that more than two correction terms have to
be retained. Thus in the case already considered: —
sinh (2. 76/70°) = sinh (2. 7/70°) -f- 0.06/70°. cosh (2. 7/70°) +0.0018/140°. sinh(2. 7/70°).
By Table II, page 13, cosh (2.7/70°) = 1.3422/153°. 322. Consequently dealing first
with the first correction term only:
sinh (2.76/70°) = i.203i/i36°.489 + 0.06/70° X 1.3422/153°. 322
= i. 2031/136°. 489 + 0.08053/223°. 322
= i. 2031/136°. 489 (i + 0.066937/86°. 833)
= i. 2031/136°. 489 (i + 0.003698 + i 0.06684)
= i 2031/136°. 489 (i. 00592 /3°.8io)
= i. 2102/140°. 299.
[176]
EXPLANATORY TEXT
Taking next the second correction term into account.
sinh (2.76/70°) = i. 2102/140°. 299 -}- 0.0018/140° X i. 203 1/136°. 489
= i. 2102/140°. 299 + 0.002166/276°. 489
= i. 2102/140°. 299 (i -+• 0.001789/136°. 190)
= i. 2102/140°. 299 (i — 0.00129 + i 0.00124)
= i. 2102/140°. 299 (0.99871 -f-io. 00124)
= i. 2102/140°. 299 (0.99871 /o°. 067)
= 1.2086/140°.366.
The correct result is 1.2086/140°.366.
(34)
SECOND AND GENERAL CASE. INTERPOLATION BOTH IN MODULUS AND ARGUMENT
Let the entered quantity be sinh (1.025/80.75°).
We have from Table I the four nearest results as follows: —
sinh 1.0/80° = 0.85125/83^.489. sinh 1.1/80° = 0.90416/84^.286.
sinh 1.0/81° = 0.84940/84°. 156. sinh 1.1/81° = 0.90172/84°. 877.
Difference for i° = — 0.00185 + o°.667.
Proportion for 0.75° = — 0.001388 + 0.500.
sinh i. 0/80°. 75 = 0.84986/83°. 989. sinh i.i/8o
sinh i.i /8o.75 = 0.90233/84°. 729.
Difference for o.i = + 0.05247 /o°.74o.
Proportion for 0.025 = + 0.01312 /o°.i85.
sinh i.Q25/8o°.75 = 0.86298/84°. 174.
The true value = 0.863 72/84°. 166.
Diff. for i° = — 0.00244 + o°-59i-
— 0.00183 + 0.443.
= 0.90233
(35)
DUAL INTERPOLATION BY THE USE or TAYLOR'S THEOREM
Let the nearest tabular function be sinh 0 = sinh (p/5_) and the required function
sinh (0 + A0) = sinh | (p + Ap)/d + A8\.
Then (p + Ap) /S + A3 = p/S + (Ap + *PA5). (36)
where the increment Ap + *pA5 is taken with reference to the vector axis p/5. Refer-
ring this increment to the initial axis of reference,
0 + A0 = (p + Ap)/5+AS = p/5 + V(Ap2 + (PA6)2 /& + tan'1 - (37)
/&
So that A0 = V (Ap)2 + (pAS)2 /8 + tan"1 (pA3/Ap). (38)
When, however, A5 is not very small, the last formula may contain an appreciable
error, and the following method of deducing A0, using rectangular complex quantities,
is to be preferred.
[i77]
EXPLANATORY TEXT
Let 0 = p/_8 = x + iy.
and 0 + A0 = (p -f Ap) /8 + A3 = x + A* + * (y + Ay).
Then A0 = A* + i.
(39)
(40)
= V(Aa)2 + (Ay)2 /tan-1 (Ay/As). (41)
We then have by Taylor's theorem, as before,
sinh (0 + A0) = sinh 0 + A0. cosh 0 + - — - . sinh 0 + - — - . cosh 0 + . . . (42)
2 3.2
a series in which two correcting terms only need ordinarily be retained. Thus, in the
example last considered, 0 = 1.0/80° and 0 + A0 = 1.025/80°. 75. If we form A0 by
the use of (37), we have Ap = 0.025, A5 = o°.75 = 0.01309 radian, pA5 = 0.01309.
A0 = V(o-°25)2 + (o.oi309)2 /8o° + tan"1 (0.01309/0.025)
= 0.02822/80° + 27°. 637
= 0.02822 /107°.637.
If we form A0 by the use of the rigid formula (41)
0 + A0 = 1.025/80°. 75 = 0.164761 + i 1.0116715.
0 = 1.0/80° = 0.173648 + ^0.9848078.
A0 = — 0.008887 + 30.0268637
= 0.028295/108° .306.
Entering now the correction formula (42), we find in the tables:
sinh 1.0/80° = 0.85125/83°. 489, cosh 1.0/80° = 0.57991/14°. 521.
so that
sinh 1.025/80°. 75 = sinh 1.0/80° + 0.028295 /io8°.3o6 X cosh 1.0/80°
6
= 0.85125/83°. 489 -fr- 0.028295 /io8°.3o6 X 0.57991/14°. 521
+ 0.0004003/2 1 6°. 6 1 2 X 0.85125/83°. 489
+ o.ooo ooi /324°.9i8 X 0.57991/14°. 521.
It is evident that in conformity with the precision of the tables, only the first two
correction terms need be included. Taking the first into account, we have:
sinh i. 025/80°. 75 = 0.85125/83^.489 + 0.028295/108°. 306 X 0.57991/1.
= 0.85125/83^.489 + o.oi64o8/i22°.827
= 0.85125/83/^489 (i + o.oi9276/39°.338)
= 0.85125/83^.489 (i + 0.014909 + 30.012219)
= 0.85125/83°. 489 (1.014909 + 30.012219)
= 0.85125/83^.489 X i. 01498/0°. 690
= 0.86400/84°. 1 79.
[178]
EXPLANATORY TEXT
Taking up the second correction term: —
sinh 1.025/80°. 75 = 0.86400 784°. 179 + 0.0004003 /2i6°.6i2 X 0.851 25 783°. 489
= 0.86400/84°. 179 + 0.0003405 /3oo°. IQI
= 0.86400/84°. 1 79 (i + 0.000395/215°. 922)
= 0.86400/84°. 1 79 (i — 0.00032 — 20.000232)
= 0.86400/84°. 1 79 (0.99968 — to. 000232)
= 0.86400/84°. 1 79 X 0.99968X0°. 013
= 0.863727 /84°.166.
The true value is 0.86372 /84°.166.
CONCLUSIONS
In general, dual interpolation by simple proportion, as in (35), will give a result of
the third order of precision. In order to secure precision of the fourth order, inter-
polation by the use of Taylor's theorem as in (42) may be required.
EXTENSION or TABLE BY USE or FORMULA FOR 28
Although Table I is only carried as far as 3.0 in modulus (p — 3) ; yet it may be
used with a little additional calculation in conjunction with Table II, for obtaining
the hyp. sines of plane vector quantities of moduli up to 6.0, by means of the formula:
sinh 29 = 2 sinh 0 cosh 6 (43)
Example: Required sinh 5.0/77°, a quantity outside of Table I. Here 6 = 2.5/77°
is within the limits of the Table; so that
sinh 5-0/77° = 2 X sinh 2.5/77° X cosh 2.5/77° (44)
= 2 X 0.87843 /i 20°. 891 X o.96459/i56°.524
= 2 X 0.87843 X 0.96459/277^.415
= 1.75686 X 0.96459/277^.415
= 1. 69465 /277°.41S.
This method ordinarily calls for interpolation both in sinh 6 and cosh 6. For this
reason, it may be preferable to obtain the required result by the use of either Table
VII or Table X, the limits of which are less restricted.
TABLE II
cosh (p/J>) = r[i_
POLAR HYPERBOLIC COSINES OF A POLAR VARIABLE
Table II gives the value of cosh p/5 between the limits of p = o and p = 3.0 by
steps of o.i, and the limits 5 = 45° and 5 = 90°, by steps of i°. The graphs of these
quantities, to squared polar coordinates, appear in Plates HA and HB of the Atlas.
[179]
EXPLANATORY TEXT
INTERPOLATION BY SIMPLE PROPORTION
In general, as in the case of Table I, a very fair degree of precision in interpolation
can be obtained by taking first simple proportional parts in argument, and then simple
proportional parts in modulus.
Example: Required cosh (0.931 05 1
We have from Table II: -
cosh 0.9/57° = 0.88922/23°. 140.
cosh 0.9/58° = 0.87602/23°. 003.
Difference for i° = — 0.01320 /o°.i37.
Diff. for 0.518° = - 0.00685 /o°.o7i.
cosho. 9/57°. 518 = 0.88237 /23°.o69.
cosh 1.0/57° =
cosh 1.0/58° =
Difference for i° = — 0.01626 /o°.o94.
Diff. for 0.518° = — 0.00844 /o°.Q49.
cosh i.o/57°.5i8 = 0.87132 /28°.868.
cosh 0.9 /57-5i8 = 0.88237 /23.o69.
Difference for o.i = —0.01105 /5°-799-
Diff. for 0.3105 = — 0.00343 /i°.8oo.
cosh 0.93105/57°. 518 = 0.87894 /24°.869.
The correct value is 0.87837 /24.803.
INTERPOLATION OF TAYLOR'S THEOREM
When a higher degree of precision is required than can be expected from simple
proportional parts, we may use Taylor's Theorem in the following form: —
cosh (0 + A0) = cosh 0 + A0 sinh 0 +
cosh 0 + *=f- sinh 0 + . . . (45)
"; 3-
Example: Required cosh 0.93105/57°. 518
having given in Table II cosh 0.9/57° = 0.88922/23°. 140
and in Table I sinh 0.9/57° = 0.85414/64^.^8.
0.93105/57.518 = 0.500 + * 0.785398.
0.900 /57° = 0.49018 + 20.754804.
A0 = 0.00982 + 20.030594
= 0.03214/72°. 196.
cosh 0.93105/57°. 518 = cosh 0.9/57° + 0.03214/72°. 196 X 0.85414/64°. 218
+ o. 00052 /i44°.392 X 0.88922/23°. 140.
[180]
EXPLANATORY TEXT
It is evident that for the Tables here considered only two correction terms need be
included. Taking up the first correction term,
cosh o.93io5/57°.5i8 = 0.88922 / 2^.140 + 0.03214/22^10)6 X o.854i4./64°.2i8
= 0.88922/23°. 140 + 0.02745/136°. 414
= 0.88922/23°. 140 (i + 0.03087 /i 13°. 274)
= 0.88922/23°. 140 (i — 0.01220 + * 0.02835)
= 0.88922/23°. 140 (0.98780 + i 0.02835)
= 0.88922/23°. 140 X 0.98780 (i + i 0.02870)
= 0.88922/23°. 140 X 0.98780 X i. 00041/1°. 645
= 0.87873 /24°.?85.
Taking up the second correction term: —
cosh 0.93105/57°. 518 = 0.87873/24°. 785 + 0.00052/144°. 392 X 0.88922 /23°.i40
= 0.87873/24°. 785 + o.ooo46/i67°.532
= 0.87873/24°. 785 (i + o.ooo524/i42°.747
= 0.87873/24°. 785 (i — 0.000416 -f- i 0.000317)
= 0.87873 /24°.785 (0.999584 + * 0.0003 1 7)
= 0.87873 X 0.999584 /24°^7_8£ (i + i 0.00032)
= 0.87837 /24°.785° X 1/0.018°
= 0.87837 /24°.8Q3.
The correct value is 0.87837 /24°.8Q3.
GRAPHICAL INTERPOLATION
For rapid but less precise work, interpolation may be made by proportional parts
on Plate HA or Plate HB, without arithmetical computation.
TABLE III
tanh (p/3) = r/y
POLAR HYPERBOLIC TANGENTS OF A POLAR VARIABLE
Table III gives in polar coordinates the value of tanh p/3 between the limits p = o
and p = 3.0 by steps of o.i, and the limits 8 = 45° and 8 = 90°, by steps of i°. The
graphs of these quantities, to squared polar coordinates, appear in Plates HlA and
Ills of the Atlas.
INTERPOLATION BY SIMPLE PROPORTION
In general, as in the cases of Tables I and II, a very fair degree of precision can be
obtained by taking first simple proportional parts in argument and then simple propor-
tional parts in modulus.
Example: Required tanh (0.93105/57°. 518).
[181]
EXPLANATORY TEXT
We have from Table III: —
tanh 0.9/57° = 0.96056 /4i°.o78.
tanh 0.9/58° = 0.97069/42°. in.
Difference for i° = 0.01013 /i°.o33.
Diff. for 0.518° = 0.00525 /o°.535.
tanh 0.9/57°. 518 = 0.96581/41°. 613.
tanh 1.0/57° = 1.06648 /37°.Q3S.
tanh 1.0/58° = 1.08054 /38°.oo4.
Difference for i° = 0.01406 /o°.g6g.
Diff. for 0.518° = 0,00728 /o°.5Q2.
tanh i.o/57°.5i8 = 1.07376 /37°-537-
tanh 0.9/57.518 = 0.96581 /4i°.6i3.
Difference for o.i = 0.10795 / — 4°-Q76.
" for 0.3105 = O.03352/— i°.266.
Inferred value of tanh 0.93105 /57°.5i8 = 0.99933 /40°.347.
Correct value of tanh 0.93105/57^.518 = 1.0000 /40°.395.
INTERPOLATION BY TAYLOR'S THEOREM
For a higher degree of interpolation precision than by simple proportion, we may
use Taylor's theorem in the following form: —
(A0)2
tanh (0
A0) = tanh 6 + A0 sech20 -
(A0)3
2!
2 sech20 tanh0
2 sech20 (sech20 - 2 tanh20)
(46)
Example: Required tanh 0.93105/57°. 518.
having given in Table I sinh 0.9/57° = 0.85414/64.218.
II cosh 0.9/57° = 0.88922/23°. 140.
Ill tanh 0.9/57° = Q.96o56/4i°.o78.
Here A0 = 0.03214/72°. 196, as given by (41). Hence by Taylor's theorem as far as
the second correction term inclusive,
0.03214 2.i6
tanh 0.93105/511518 = tanh 0.9/52! + (0.8892 2)
(o.o32i4)2/i44°.392
- (0.88922)2 /46°.28o
Taking up the first correction term: —
tanh o.93iQ5/57°.5i8 = 0.96056 /4i°.Q78 +
= 0.96056/41°. 078 + 0.04065/25°. 916
= 0.96056 /4i°.o78,(i + 0.04084 — 20.01107)
= 0.96056/41°. 078 (1.04084 — i 0.01107)
= 0.96056 /4I.O78 (1.04090X0°. 609)
= 0.99985 /4o°.469.
[182]
EXPLANATORY TEXT
Taking up next the second correction term: —
tanh o.93io5/57°.5i8 = 0.99985/40^469 — 0.00103 /i44°-3Q2 X 0.96056/41^078
(o.88922)2/46°.28o
= 0.99985/40^.469 + 0.00103 X 0.96056X40°. 810
0.79071
= 0.99985/40°. 469 + 0.00126X40°. 810
= 0.99985/40°. 469 (i + 0.00126X81.279)
= 0.99985/40°. 469 (i + 0.00019 ~~ * 0.00125)
= 0.99985/40°. 469 (1.00019 ~ * 0.00125)
= 0.99985/40°. 469 X i.oooi9\o°.o72
= i.oooo /4o°.-
Correct value = 1.0000 /40°.395.
When more than two correction terms have to be retained, it is often easier to deter-
mine sinh (8 + A0) and cosh (6 + A0) by Taylor's theorem, as already described, and
then to take their ratio for tanh (6 + A0).
TABLE IV
sinh 0
POLAR RATIO — - — FOR POLAR VALUES OF 0
v
Table IV has been prepared by dividing the values of sinh 0 found successively in
Table I by their respective values of 0. The object of the table is to facilitate the
computation of the equivalent T or II of any uniform alternating-current line of known
electrical constants.* That is, the table pertains more particularly to the applications
of hyperbolic functions than to the fundamental properties of those functions. The
sinh (p/5)
table gives the vector value of - - for the range p = o to p = 3 by steps of o.i,
p/5
and for 5 = 45° to 5 = 90° by steps of i°. The graphs of the values contained in the
tables are plotted to squared polar coordinates in Charts IVA and IVs of the Atlas, for
rapid graphic interpolation.
INTERPOLATION BY SIMPLE PROPORTION
A fair degree of precision in interpolation can ordinarily be obtained by first taking
simple proportional parts in argument and then simple proportional parts in modulus.
sinh (1.025/80°. 75)
Example: Required -
*"The Application of Hyperbolic Functions to Electrical Engineering Problems," by A. E. Kennelly,
University of London Press, 1914, Chap. III.
[183]
EXPLANATORY TEXT
We have from Table IV the following values of — - — : —
u
For 1.0/80° = 0.85125 /3°.48g. For 1.1/80° = 0.82196 /4°.286.
1.0/81° = 0.84940 /3°.i56. 1.1/81° = 0.81975 /3°.8y7.
Difference for i° = — 0.00185 /— °°-333- Difference for i° = — 0.00221 /— o°.
0.75° = — O.OOI39/— o°.25o. °°-75 = ~~ o. 00166 /— o°.
For i.i/8o°.75 = 0.82030 /^°.g
0.84986 /3°.239. For i.o/8o°.75 = 0.84986 /3°.2
Difference for o.i = — 0.02956 /o°.74o.
for 0.025 = — 0.00739 /o°.i85.
For i. 025/80°. 75 = 0.84247 /3°.424.
Correct value = 0.84265 /3.°416.
When a higher degree of precision is required than can be expected from proportional
parts, the proper value of sinh (6 + A0) should be obtained by Taylor's theorem as
already explained in connection with Table I, and this value divided by (6 + Ad) ;
because the expansion of — , . directly, by Taylor's theorem, does not lend itself
conveniently for computation.
EXTENSION FOR THE RANGE OF THE TABLE BY THE USE OF FORMULA FOR 26
Although Table IV is only carried as far as 3.0 in modulus (p = 3); yet it may be
used with a little additional calculation, in conjunction with Table II, for obtaining
— — , for vector values of 6 with moduli up to 6.0, by means of the formula: —
a
sinh 26 = 2 sinh 6. cosh 6 (47)
sinh 26 sinh 6
whence — - — = — - — . cosh 6. (48)
20 6
Consequently, to find — - — for the double of any quantity within the range of Table
u
IV, find the value of - - for the quantity, by interpolation directly in Table IV, and
V
multiply the result by the hyperbolic cosine of the quantity as obtained from Table II.
Corresponding steps may be taken with Charts II and IV.
sinh (5.0/77°)
Example: Required -; — 5 — , this being outside of the limits of Table IV;
0 * iii,
but not outside twice the value therein obtainable.
Here — — for 0 = 2.5/77° is by Table IV 0.35137
e
and cosh 6 " " " " " II 0.96459 l
[184]
Hence
sinh (5.0/77°)
5-0/77°
EXPLANATORY TEXT
= 0-35137 /43°-89i X Q.96459/i56°.524.
= 0.33893 /200° .415.
sin 6
This procedure calls for interpolation both in - - and in cosh 6. For this reason
e
it may be preferable to obtain the required result by the use of either Table VII or
Table X, the limits of which are less restricted.
TABLE V
tanh 6
POLAR RATIO — - — FOR POLAR VALUES OF 0
u
Table V, like Table IV, has been prepared for electrical engineering applications of
hyperbolic functions, rather than for developing these functions alone. It gives the
tanh (p/6)
vector value of
p/5
for the range p — o to p = 3.0 in modulus, by steps of o.i,
and for the range 5 = 45° to 8 = 90° in argument, by steps of i°. It was computed
directly from Table III by dividing the resulting values successively by their respec-
tive values of 6. The graphs of the values in Table V are presented to squared polar
coordinates in Chart V, for rapid graphic interpolation.
INTERPOLATION BY SIMPLE PROPORTION
Except where a high degree of precision in interpolation is required, it is prefer-
able to interpolate first by simple proportion in argument, and then by simple pro-
portion in modulus; although this order of operations may be inverted.
Example: Required — -r— for 6 = 0.93105^
9
We have from Table V: —
For 6 — 0.9/57° = 1.06729 \i5°.922.
6 = 0.9/58° = 1.07854 \i5°.889.
For 6 = 1.0/57° = i- 06648 \i^<
Difference for i° = o.oii25\— 0.033.
for 0.518° = o.oo583\— 0.017.
For 0 = 0.9/57°. 518 = J-°7312
-9°5-
6 = 1.0/58° = i.o8o54\i9°.996.
Difference for i° = 0.01406 \o°.o3i.
" for o°.5i8 = 0.00728 \o°.oi6.
For 0 = i.o/57°.5i8 = 1.07376X19°. 981.
6 = o.9/57°-5i8 = 1.07312 \i5-905.
Difference for o.i = 0.00064 \4°-o76.
" for 0.03105 = 0.00020 \i°.266.
For0 = o.93iQ5/57°-5i8 = 1.07332X17^171.
Correct value,
[185] •
1. 07406 \17°.123.
EXPLANATORY TEXT
When a higher degree of precision is needed than simple proportion can give, it is
preferable to find the proper interpolated value for tanh Q from preceding tables and
then to divide by 0; since the function - -. — — r — does not lend itself to expansion
(0 + A0)
by Taylor's theorem in a simple form.
Tables IV and V jointly, with their respective graphs in the Atlas, enable the equiva-
lent T or II of any uniform alternating-current line in the steady state, at a single fre-
quency, to be completely determined, provided 0 does not exceed six radians in modulus
(5 lying between 45° and 90°) ; because although in both tables, 0 is not carried beyond
three radians; yet — — — can be found by extension up to six radians, and in the formu-
0
las for deducing the equivalent T or II, — - — - has only to be carried to half the modulus
0
, sinh 0
of-0-'
The following example may illustrate the use of Tables IV and V either with or
without the aid of the graphic interpolation Charts IV and V of the Atlas. An alter-
nating-current line of uniform electrical constants is 250 km. long and has, at a certain
frequency, a total conductor impedance of 565.711 784°. 777 ohms, associated with a
total distributed insulation admittance of 4.3707 X io~3/9o° mhos. Its hyperbolic
angle is therefore Vs. 65711 X 4.3707 X io~1/i74 .777 = 1.5724 787°. 388 hyperbolic
radians. Interpolating either from the tables or the Charts IV and V, we obtain
sinh0 .tanh () tanh 0.7862/87^88
= 0.638 /
. _ 0.786a/8
If we multiply the conductor impedance by — - — , we have
u
565. 711 784°. 777 X 0.638/2°. 6 = 360.69/87°. 377 ohms, and if we multiply half the
insulation admittance by — ff). g , we have
Iff/
2.1854 X io-3/9o° X i.27\i°.5 = 2.78 X io-3/88°.g mhos = 359-77\88°.5 ohms.
If now we apply an artificial condenser leak of 2.78 X io-y88°.5 mhos to each end of a
localised impedance coil of 3 60. 6 9/87^.3^7 ohms, we obtain the "equivalent II" of the
line at the frequency considered, and such a combination of localised impedance and ad-
mittances would behave exactly like the line, at its terminals, or outside them, so as to
be capable of replacing the line in any electrical system, at that frequency.
[186]
EXPLANATORY TEXT
TABLE VI
POLAR FUNCTIONS OF POLAR SEMI-IMAGINARY QUANTITIES
A semi-imaginary quantity is a complex numerical quantity which, when expressed
in rectangular coordinates, has equal real and imaginary components; or, when ex-
pressed in polar coordinates, has an argument of 45°. That is x/± 45° = a rt i a.
The interest of the table pertains primarily to the application of hyperbolic functions
to uniform alternating-current lines of negligibly small linear inductance and leakance,
a case approximated to by cabled lines at low frequencies. The table was first pub-
lished by the author in the transactions of the International Electrical Congress of
St. Louis (1904). The arguments of the results are given in degrees and minutes, and
not in degrees and decimals like the rest of the tables.
The table gives the hyperbolic sine, cosine, tangent, cosecant, secant, and cotangent
of the vector x/4$° for the range x = o to x '= 20.5, by steps in x of o.i up to x = 6,
and of 0.05 beyond that point. At x = 6, the values of the hyp. sine and cosine so
nearly coincide, that they are taken as equal in the table, thus bringing sech x and
cosech x into equality as well as tanh x = coth x = i. Graphs of the functions are
given in Chart VI as far as x = 4, approximately.
INTERPOLATION BY SIMPLE PROPORTION
In general, interpolation may be quickly effected by simple proportional parts of
modulus since the argument is constant at 45°. This procedure is sufficiently evident
to require no exemplification.
INTERPOLATION OF TAYLOR'S THEOREM
When precise interpolation is necessary, we have the following expansions for
f (0 + A0)
sinh (* + A*) /4S! = sinh (2/45°) + (Ax) /4S°. cosh (*/45°) +
sinh (*/45°) + -=r/ilS°. cosh (*/45°) + ..... (49)
O "
cosh | (* + A*)/45°} = cosh (x/4S° + (A*)/45_°. sinh(*/45°) + ^fl^L<
(Ar^3
cosh (*/45°) + ^fr/ias!- sinh (*^0) + — (5o)
, tanh (* + A*)/4,S! = tanh
2 sech2 (»/4S°). tanh (s/4Q - Zl^.0. 2. sech2 (*/45°).
J sech2 (*/45°) - 2 tanh2 (s/4S°) *+.... (51)
Example: Required cosh (3.1/45°), having given in Table VI
sinh (3.0/45°) = 4.1986 /i 2o°.48/.
cosh (3. 0/45°) = 4-U43 /i22°.i6/.
[187]
EXPLANATORY TEXT
Here cosh (3.1/45!) = 4.i443/i22°.i6' + 0.1/45° X 4-1986 /i2o°.48'
0.01/90° o.ooi ,
+ - - X 4.i443/i22°.i6' H — /i35° X 4.1986 / 120°. 48'
= 4.1443 /I22°.l6' + 4.1986/120°. 48 (0.1/45° + 0.00017/135°)
+ 4. 1443 /I 22°. 1 6' (0.005/90°)
= 4. 1443 /i 2 2°. 1 6' (i + 20.005) + 4.i986/i65°.48' (o.i + 20.00017)
= 4.i443/i22°.i6' ( i. 0000/0°. 1 7') + 4. 1986/165°. 48' (o.i /o°.oi')
= 4.1443 /i 22^33' + o.4i99/i65°.49/
:2°.33/ (i + o. 10132/43°. 16')
= 4.1443 /i 22°.33/ (i + 0.07378 + * 0.06944)
= 4.1443 /I22°.33/ (1.07378 + * 0.06944)
= 4.I443/I22°.33' X 1.0760 /3°.42;
which is in substantial agreement with the tabulated value of cosh 3.1.
Beyond x = 6, the value of either sinh (3/45°) or cosh (#/45!) was computed from
the formula: —
sinh (s/45°) = cosh (s/45°) = J — /T/T radians. (52)
v ^
where e = 2.71828 . . .
X 6 141.14
Thus, with x = 7, — T= = 4.9498, - = - = 70.57 at the argument of 4.9498 cir-
\j 2 ^2
cular radians = 283°. 36'; so that: —
sinh (7/45°) = cosh (7/4.5!) = 7Q.57/283°.36/
which coincides with the tabulated value in Table VI.
TABLE VII
sinh (x + iy) = u + vo
RECTANGULAR HYPERBOLIC SINES OF A RECTANGULAR VARIABLE
Tables I to VI contain certain restrictions in range which limit their general appli-
cation. They are primarily designed to cover particular applications of hyperbolic
functions to electrical engineering. Tables VII to XIV, however, are free from such
restrictions, and are intended to furnish the circular as well as the hyperbolic sine, cosine,
and tangent of a complex angle, and to furnish this result either in the rectangular or
polar form. That is, they furnish: —
sinh (x + iy) \ . , . . . . . , ,
; . ; > in the f orm u + vo ; also in the form r /j
or sin (x + iy) j
cosh (x -\- iy)\ . ^ f • , • ,1 e /
. ( } in the form u + vo\ also in the form r /y
or cos (x + iy) J
tanh (x + iy) \ . . , , ,
. ( > in the form u + tv: also in the form r /y
or tan (x + ly) }
[188]
EXPLANATORY TEXT
between the limits, for the hyperbolic functions, x = o and x = ± 10, by steps of 0.05
and between the limits y = o and y = ± oc , by steps of 0.07854 = TT/^O.
PERIODIC PROPERTIES OF THE RECTANGULAR COMPLEX HYPERBOLIC
SINES AND COSINES
It is well known that sinh \x + i (y -f 2mr)\ = sinh \x + iy\ (53)
and cosh \x + i (y + 2mr) \ = cosh \x + iy\ (54)
where n is any integer.
This means that, keeping x constant, the values of the hyp. sine and hyp. cosine
repeat themselves as iy passes through increments of i.2ir\ or they are periodic functions
of iy, having the period 2ir i.
The matter may be visualised more clearly from geometrical reasoning. Consider-
ing the exponential form of the hyperbolic cosine,
c * + 'V _L e -<* + '»)
cosh (x + iy) =- — — . (55)
e* . e~1
This may written in the form: — . e iy -\ e~iv. If x be kept constant, we
require to study the changes produced in this form of the hyp. cosine by varying y.
q'
$
%sZ»**>*
-a d^~ -A
•
P
FIG. 29. — Geometrical constructions for cosh (x + iy) and sinh (* + iy).
In Fig. 29, OA is an initial line and OP a radius vector of length or modulus €*/2>
multiplied by e '»; that is rotated positively about O, from OA through a circular angle
of y radians. Similarly, op is a vector of length or modulus €~*/2 rotated negatively
through a circular angle of y radians from the initial line Oa. The equation (55) states
that the hyperbolic cosine is the plane vector sum of OP and op, or O'p' in the Figure.
If now we steadily increase the value of y, leaving x constant, we cause OP to rotate
steadily counterclockwise, and also op to rotate steadily clockwise, through Ay cir-
cular radians. When Ay = 2ir, both OP and op will have made one complete revo-
lution and will have returned to their initial positions indicated. Consequently, the
value of cosh \x + i (y + 271-) \ repeats that of cosh \x + iy \.
€ x + iy _ f -(x + iy)
Since sinh (x + iy) = - (56)
the same reasoning applies; but the vector op is added in the negative or reversed di-
rection; so that O'q' is the hyperbolic sine of x + iy.
The above mentioned periodic property of the hyp. sine and cosine has been utilized
for shortening the tables of those functions by reducing the circular angle y of Fig. 29
[189]
EXPLANATORY TEXT
from radians to quadrants. That is, any complex angle x + iy represented by a point
P, and radius vector OP, in the complex plane XY, Fig. 30, is first transferred to a
new complex plane XQ, Fig. 31, at the point p = x, q, by keeping x the same in both
planes, but making the points -, — ,—,—,... etc., on the Y axis of the XY diagram,
2222
become the points i, 2, 3, 4 ... etc., on the Q axis of the XQ diagram. Thus if
x + iy = 2.5 + * 6. 2832,
x + iq = 2.5 + i 4.00
where 4.00 is underscored to indicate quadrant measure, instead of the ordinary radian
measure.
"Y
FIG. .30.
FIG 3r.
Transference of a Complex Quantity from the XY to the XQ Plane.
In the case indicated by Fig. 30, x + iy = 3 + i 9 and x + iq — 3 + * 5.74 in Fig. 31.
Consequently, after a complex angle has been transferred from the complex plane
XY to the complex plane XQ, the values of either sinh (x + iq) or cosh (x + iq) exactly
repeat themselves for each 4 units of increase in q; or with reference to Fig. 31 for each
4 quadrants of increase in the circular angle instead of 6.2832 . . . radians. The
operation of transferring the complex angle from the XY to the XQ plane may there-
fore be described as quadranting y; i.e., changing the expression of y from circular
radian units to circular quadrant units.
All of the Tables VII to XIII inclusive require to be entered in terms of x + iq;
so that the complex entering value has to be quadranted by dividing its imaginary or
y-component by the numeric v/2 = 1.57079 This preliminary step occupies
a certain extra time and effort; but it actually economises the total time and effort
involved. If the tables were computed for x -f- iy, they would have to be repeated in
bulk for each TT radians, or 2 quadrants, increase in y. In electrical engineering appli-
[ 190]
EXPLANATORY TEXT
cations, y frequently rises to 100 radians, and might easily be much greater than 100.
In order to go up to 100 radians, the bulk of the Tables VII to XIII would have to be
increased about thirty fold. Altogether, aside from the greatly increased bulk and expense
of such tables, the extra time and effort consumed in turning over the numerous pages
would be comparable with that saved by eliminating the preliminary step of quadranting
the imaginary component or dividing it by ir/2.
RULES FOR THE USE OF TABLE VII
Express the " angle " whose hyperbolic sine is required in the form of an ordinary
rectangular complex quantity x + iy.
Quadrant the imaginary component y through the process of dividing it by ir/2;
i.e., transfer the quantity from the XY to XQ plane; so that the new expression of the
complex quantity is x + iq; where q = y/i. 57079. . . .
If q is greater than 4.6, divide by 4 and retain«only the remainder. If the remainder
exceeds 2, subtract 2 therefrom, and apply a negative sign to the result found in the
table. A change of 2 quadrants simply reverses the sign of the result. If the remainder
on the other hand does not exceed 2, enter Table VII with it, and take out the result
with unchanged sign.
• Example: Required the hyperbolic sine of 0.65 + i 25.75. Here x = 0.65 and
y = 25.75. That is y is 25.75 circular radians. Reduce this to quadrants through
dividing by 1.57079. . . .
25-75 * log 25.75 = 1.4107772.
1.57079 ... log TT/2 = 0.1961199.
log 16.393 = 1.2146573.
The quadranted yalue x + iq = 0.65 + i 16.393
NOTE. — It is found convenient to underscore quadrantal quantities to distinguish
them from radianal quantities.
Rejecting quadrant multiples of 4, i.e., 16 in this case, we enter Table VII with
x -|- iq = 0.65 + i 0.393. The nearest entry to this is x = 0.65, q = 0.4, for which
the hyperbolic sine is 0.56368 + * 0.71639, an ordinary rectangular complex quantity
on the U V plane. Interpolation should be made in this result to meet the change
from q = 0.40 to q = 0.393, as wiH De explained later.
Second Example: Required sinh (x + iy) = sinh (1.15 + i 10.10).
. / . 10.10 \
Quadranting the imaginary, sinh (x + iq) = sinh ( 1.15 + * = 1
= sinh (1.15 + i 6.430).
Rejecting 4'$ from the imaginary = sinh (1.15 + i 2.430).
Deducting 2 from the residual and chang-
ing the sign = — sinh (1.15 -f *'o-43°)-
* This operation would ordinarily be effected with the slide-rule, when a high degree of precision is not
aimed at.
[191]
EXPLANATORY TEXT
We now enter Table VII with x = 1.15 and q = 0.43. The nearest entry is x = 1.15
q = 0.45, the result for which is 1.08037 + * 1.12836. But we must apply a negative
sign to the whole of this result because of the 2 rejected in the quadrantal residuum.
Hence,
sinh (1.15 + i 10. 10) = — (1.08037 + * 1-12836)
= — 1.08037 — i 1.12836 = u + i v
except for the interpolation from q = 0.45 to q = 0.43. The operation of interpolation
will be discussed later on.
Third Example: Required sinh (x + iy) = sinh (3.60 + i 18.1).
Quadranting the imaginary, sinh (x + iq) = sinh ( 3.60 + i— '—^ }
* \ i-57°<V
= sinh (3.60 + i 11.523).
Rejecting 4'$ from the quadrants = sinh (3.60 + ^'3.523).
Deducting 2 from the residual imaginary
and changing the sign .............. = — sinh (3.60 + i 1.523).
Entering Table VII with x = 3.6 and q = 1.523, the nearest entry is x = 3.6 and
q = 1.5, for which the result is — 12.92978 + * 12.94910. But applying the negative
sign to this result because of 2 deducted from the quadrantal imaginary, and we have
finally: —
sinh (3.60 -+- i 18.1) = — (— 12.92978 + i 12.94910)
= 12.92978 — i 12.94910 = u + i v
except for the interpolated correction from q = 1.500 to q = 1.523, to be considered
later.
RANGE OF THE TABLE
Table VII extends by steps of 0.05 in x up to x = 3.95, and in Table XIII up to
x = 4.0. In y, the range is indefinitely great; because after dividing y by ir/2 so as
to reduce it to quadrant measure, all multiples of 4 are rejected. From o to 2, in the
remainder, the table gives the result directly and from 2 to 4, by change of sign in the
total. Cases of x greater than 4.0 are dealt with in connection with Table XIV.
REPETITIONS IN THE TABLE
If sinh { x + i- — a = u + iv (57)
I \2 / )
it is easy to show that:
sinh I * -M ( - + 0 ) \ = — u + iv. (58)
I \2 / J
It follows that in any column of Table VII, the entry for q = (i — a) is the same as
that for q = i + a except for a change in the sign of u. Consequently, the table might
have been reduced to half its present size, if the responsibility for making this change of
sign had been left to the reader. It was considered, however, that since the reader is al-
ready charged with the duty of applying a negative sign to the total result when the q
[ 192]
EXPLANATORY TEXT
residuum lies between 2 and 4, the retention of the full size of the present table was
warranted, especially as the duplication of the text in each column provides a certain
check upon the numerical work of tabulation.
INTERPOLATION BY SIMPLE PROPORTION
As a first approximation, interpolation may be effected by simple proportion, first
in regard to x and second in regard to q.
Example: Required sinh (0.15 -f 3*0.25), having given:
sinh (0.2 -f 3*0.2) = 0.19148 + 3*0.31522. sinh (0.2 + i 0.3) = 0.17939 + 3*0.46310.
sinh (o.i + 3*0.2) = 0.09526 + 3*0.31056. sinh (o.i + 3*0.3) = 0.08925 + 3*0.45626.
Diff. for o.i x = 0.09622 + 3*0.00466. Diff. for o.ix = 0.09014 -f- 3*0.00684.
Diff. for 0.052; = 0.04811 + 3*0.00233. Diff. for 0.052; = 0.04507 + 3*0.00342.
sinh (0.15 + i 0.3) = 0.13432 + 3*0.45968.
sinh (0.15 + 3*0.2) = 0.14337 H~ * 0.31289. sinh (0.15 + 3*0.2) = 0.14337 -f- 3*0.31289.
Diff. for q o.io = — 0.00905 + 3*0.14679.
Diff. for q 0.05 = — 0.00453 + i 0.07340.
sinh (0.15 + 3*0.25) = 0.13885 + 3*0.38629.
Correct value = 0.13910 + 3*0.38700.
INTERPOLATION BY TAYLOR'S THEOREM
When a higher degree of precision is desired than that which can be expected by
simple proportion, we may use Taylor's theorem in the following form: —
sinh (0 + A0) = sinh 0 + A0 cosh 0 + ^- sinh 0 + z=f- cosh 0 + . . . (59)
sinh { (x + iy) + (Ax + i Ay) \ = sinh (x + iy) -f- (Ax + i Ay) cosh (x + iy)
. (Ax + 3* Ay)2 . , f ... (Ax + iAy)z , , . , ,,.%
+ - — — — - sinh (x + 3y) + - —. — - cosh (x + ly) + • • • (60)
* O *
Quadranting imaginaries on both sides; or transferring to the XQ plane,
sinh | (x + iq) + (Ax + i Aq) \ = sinh (x + iq) + \ Ax -f i (IT/ 2) Aq) \ cosh (x + iq)
!A* + 3*(7r/2)A;y!2 . \Ax + 3 Or/2) Aq\3 , , ., ,
+ L - smh (x + iq) + L A-t- *- cosh (x + iq) + . . . (61)
= sinh (x + iq) + A'0 cosh (x + iq) + - — ~- sinh (x + iq)
+ ^ cosh (* + iq) + . . .
3!
where A'0 = Ax + iAy = Ax + i (IT/ 2) Aq. (62)
[193]
EXPLANATORY TEXT
Example (i): With A^ = o.
Required sinh (0.15 + i 0.2), having given in Table VII and in Table VIII:
sinh (o.i + i 0.2) = 0.09526 + i 0.3 1056.
cosh(o.i + 3*0.2) = 0.95582 + i 0.03095. Then by (60);
sinh (0.15 + * 0.2) = sinh (o.i + i 0.2) + 0.05 cosh (o.i + i 0.2) -\ —
sinh (o.i + 3*0.2) -\ — — - cosh (o.i + 3*0.2)
o
= 0.09526 + 3*0.31056 + 0.05 (0.95582 + 30.03095)
+ O.OOI25 (0.09526 + 3*0.31056) + O.OOO02 (0.95582 +
3* 0.03095)
= I.OOI25 (0.09526 + 3*0.31056) + 0.05002 (0.95582 + 3*0.03095)
= (0.09538 + 30.31095) + (0.04781 + 30.00155)
= 0.14319 + 3* 0.3 1250
which is the correct tabular value of sinh (0.15 + i 0.2) in Table VII.
Example (2): With Ax = o.
Required sinh (o.i + 3*0.25), having given in Table VII and in Table VIII:
sinh (o.i + 3*0.2) = 0.09526 + 3*0.31056.
cosh (o.i + 3*0.2) = 0.95582 + 3*0.03095. Then by (62);
sinh (o.i + 3*0.25) = sinh (o.i + 3*0.2) + 3*0.05 X 1.5708 X cosh (o.i + 3*0.2)
., (0.05 X 1.5708)2 . f .
+ 32- — sinh (i.o + 30.2)
2 1
. ., (0.05 X 1.5708^ , t .
+ 33-^ — - — - cosh (i.o + 30.2)
3!
= 0.09526 + 3*0.31056 + 3* X 0.07854 (0.95582 + 3*0.03095)
0.00617 / <:\
(0.09526 + 0.31056)
2
. O.OO048
(0.95582 + 30.03095)
6
= (0.09526 + 30.31056) (i — 0.00309)
+ 3* (0.95582 + 3*0.03095) (0.07854 — 0.00006)
= 0.99691 (0.09526 + 3*0.31056) + 0.07848 (— 0.03095 + 3*0.95582)
= 0.09497 + 3*0.30960 — O.OO243 + 3'O.075OI
= 0.09254 + 3*0.38461.
The tabular value is 0.09254 + i 0.38460.
Example (3) : Interpolation for both Ax and A^.
Required sinh (0.15 + i 0.25), having given
sinh (o.i + 3*0.2) = 0.09526 + 3*0.31056 by Table VII
and cosh (o.i + 3*0.2) = 0.95582 + 3*0.03095 by Table VIII.
Here A6 in formula (59) = (0.05 + i 0.05)
and A'0 in formula (62) = (0.05 + i(ir/2) X 0.05) = (0.05 + 30.07854).
EXPLANATORY TEXT
Thus: -
sinh (0.15 + ^0.25) = sinh (o.i + 10.2) + A'0 cosh (o.i + io.z
(A'0) 2
+ , • sinh (o.i + io.2) + . . .
A'0 = 0.05 + i 0.07854.
(A'8)2 = + 0.0025 — 0.00617 + i 0.00785
= — 0.00367 + io. 00785.
(A'0)2
= — 0.00184 ~t~ ^
(A'8)3 = (0.05 + i 0.07854)3 = — 0.00080 + *'o.oooio.
(A'8)*
— 7 — = — 0.00013 + * 0.00002.
f (A'0)2 ">
sinh (0.15 -f 10.25) = sinh (o.i + i 0.2) j i -f- - — p + • • . >
+ cosh (o.i + * 0.2) j A'0 + ^^ + . . . |
\ 3 /
= (0.09526 + i 0.31056) (0.99816 + i 0.00393) = 0.09386 + i 0.31036
+ (0.95582 + i 0.03095) (0.04987 + i 0.07856) = 0.04524 + i 0.07664
= 0.13910 + i 0.38700.
The correct tabulated value is = 0.13910 + i 0.38700.
EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY
Table VII expresses the relation
sinh (x + iq) = u + iv. (63)
(a) If x be taken with negative sign, we have
sinh (— x + iq) = — u + iv (64)
so that changing the sign of the real component entering the table changes the sign of
the real component in the result; but leaves the sign of the imaginary component
unchanged.
(b) If q be taken with negative sign, we have
sinh (x — iq) = u — iv (65)
so that changing the sign of the imaginary component in the entering quantity changes
the sign of the imaginary component in the result, leaving the sign of the real com-
ponent unchanged.
•
(c) If both x and q be taken with negative sign, we have
sinh (— x — iq) = sinh \ — (x + iq) \ = — u — iv = — (u + «0 (66)
so that changing the sign of the total entering quantity changes the sign of the total
result.
The facts may be summed up by saying that changes in the sign of the entering
quantity produce corresponding changes of sign in the result.
[195]
EXPLANATORY TEXT
CIRCULAR SINES OF COMPLEX " ANGLES "
Since, as is well known: —
sin 6 = — i sinh (iff) (67)
we have
sin (x + iy) = — i sinh (ix — y) (68)
= * sinh (y — ix). (69)
Consequently, in order to find the circular sine of the complex quantity (x + iy), enter
Table VII for sinh (y + ix), which on being quadranted, becomes sinh \ y + ix/(ir/2) \
and let the result be (u + iv). Then sinh (y — ix) = u — iv and sin (x + iy) = v + iu.
In other words, invert the entering components, and then invert the components of the
result.
Example: Required sin (i + i 2) from Table VII. Here 6 = (i + i 2).
Enter the Table with — sinh (id) = sinh (— id) = sinh (2 — i i).
Quadranting the imaginary, we enter the table with (x — iq) = (2 — i 0.6366).
The nearest entry is (2 — i 0.65), for which the hyp. sine is given as 1.89503 — i 3.20780.
Consequently, sin (i + i 2) = 3.2078 -f- i 1.89503, except in so far as interpolation is
needed to reduce sinh (2 — i 0.6366) from sinh (2 — 3' 0.65). In this way any circular
sine of a complex quantity can always be obtained from the table of hyperbolic sines,
between the limits of o and ± 4 in y, and of o and ± °o in x.
GRAPHIC INTERPOLATION BY MEANS OF CHARTS VIlA, Vila, VIIc
Charts VII-VIII A, B, and c, serve for the evaluation of either sinh (x + iq) or
cosh (x + iq), according to the axis of reference selected. Thus, taking Chart VII-
VIIIs, if this is held with the. line SS as the axis of reference or initial line; then by
comparison with the entries in Table VII, it will be found that sinh (x + iq) can be
read from it directly over the range q = o to q = 4, beyond which the values repeat
themselves indefinitely. On the other hand, if the chart be turned through 90°, so
as to bring the line CC as the axis of reference, it will be found by comparison with the
entries of Table VIII, that cosh (x + iq) can be read from it directly over the range
q = o to q = 4.
Chart VII-VIIlA gives sinh (x + iq) and cosh (x + iq) for values of x up to about
0.9. Chart VII-VIIIs gives the corresponding results for values of x up to about
x = 2. Finally, Chart VII-VIIIc provides for values of x up to x = 4. In all of
these charts, interpolation can be made for both x and q to o.oi, by direct inspection.
The graphs on these charts are undistorted, since they give complex functions as results,
in rectangular coordinates. The curves therefore always intersect orthogonally, and
they represent a confocal system of ellipses and hyperbolas, the common foci being at
two points at unit distances from the center, along one of the reference axes. The curvi-
linear rectangles into which the charts are divided have pairs of sides the ratio of
whose lengths tends to the value ir/2.
If the preliminary process of quadranting the imaginary of the entering quantity
were not adopted; that is, if the graphs were entered in terms of (x + iy), instead of
[196]
EXPLANATORY TEXT
(* + iq) ; then it would be necessary* to have a new chart for each range of 2ir units in
3>; or some 16 sets of Charts A, B, and C, in order to reach y = 100. That is, 48 charts
would have to be computed, prepared, drawn, lithographed, bound, sold and operated
instead of the 3 charts actually presented. Moreover, if y were needed greater than
100, the set of 48 would fail; whereas, working with quadrant imaginaries, the three
charts serve up to indefinitely great values of q and y,
GRAPHIC CHART VII-VIIlA
This chart corresponds to Tables VII and VIII at least as far as x = 0.9, or for
pages 42 to 45, and 58 to 61 of this book. To find hyperbolic sines from the chart,
place it facing the observer with the axis OO vertical. This is the major axis of all the
ellipses shown. Starting from this central axis towards the right hand, the successive
ellipses marked o.i, 0.2, 0.3, etc., represent values of x; while the successively rising
hyperbolas o.i, 0.2, represent values of q. These values of q will be found to extend
over two quadrants. Enter the chart on the curvilinear coordinates for x and q. At
the proper intersection read off the u and v coordinates of the rectilinear ruling, u being
the abscissas and v the ordinates.
Conversely, to find sinh"1 (u + iv) within the limits u = o and « = ±i,»=oto
v = 2.0, enter the chart with the same aspect on the rectilinearly ruled coordinates
and read off at the proper intersection the curvilinear values taking x on the ellipses
and q on the hyperbolas.
To find hyperbolic cosines from the chart, rotate it clockwise 90°; so as to have
the axis OO horizontal. Then enter on the curvilinear coordinates with x on the ellipses
and q on the hyperbolas. The first and fourth quadrants only will be presented to the
observer; but from the symmetry of the diagram, it will be easy to reverse the chart,
so as to present the second and third quadrants. Read off the result on the rectilinear
background using u for abscissas and v for ordinates.
Conversely, to find cosh"1 (u + iv) from the chart with the axis OO horizontal, enter
on the rectilinear background and read off at the proper intersection from the curvilinear
coordinates in x and q, taking the ellipses as parts of the it-system and the hyperbolas
as part of the ^-system.
GRAPHIC CHART VII-VIIIs
This chart gives the graph of the functions sinh (x + iq) and cosh (x + iq) from
x = 0.8 at least as far as x = 2.05 along the ellipses and from q = o to q = °o by virtue
of successive rotations. In this and the following charts, the numerical values of q are
all underscored, an indication which may serve readily to distinguish the imaginaries
q, from the reals, x.
* A single set of charts entered in terms of (x + iy) could be used up to y — 6.2832 in one revolution,
and could be used for all larger values by throwing out multiples of 2*-. This operation of dividing y by 2T
would, however, take as much time as the operation of quadranting, and would also lead to a dissymmetrical
chart in the hyperbolas.
EXPLANATORY TEXT
In all of the Charts VII to IX inclusive, the curvilinear rectangles all tend to have
sides in the ratio TT : 2; that is the long side approximates to being 1.57 times the short
side. In IXA exceptions are found; because extra curvilinear coordinates are supplied.
To find sinh (x + iq) from Chart VII-VIIlB, hold the minor axis 55 horizontal.
Enter on the curvilinear coordinates with x on ellipses and g on hyperbolas. At the
proper intersection read off on the rectilinear background in u and v. Proceed inversely
to find the inverse function sinh"1 (u + iv).
To find cosh (x + iq) from the same chart, hold the major axis CC horizontal. Enter
on the curvilinear coordinates with x on ellipses and q on hyperbolas. Read off on the
rectilinear background.
All four quadrants appear in this and the following chart, so that it is not necessary
to limit the value of q to less than 2 quadrants.
GRAPHIC CHART VII-VIIIc
This Chart gives the graph of sinh (x + iq) and cosh (x + iq) from x = 2.0, at least
as far as x = 3.90. The procedure is precisely the same as that for VII-VIIlB already
described.
TABLE VIII
cosh (x + iq) = « + iv
RECTANGULAR HYPERBOLIC COSINES or A RECTANGULAR VARIABLE
Table VIII may be regarded as an inversion of Table VII; because:
cosh 6 = — i sinh (6 + iir/z) (70)
or in quadrant imaginaries,
cosh 6 = — i sinh (6 + i i). (71)
That is the hyp. cosine of any complex quantity (x + iq) is — i times the hyp. sine
of that quantity with an additional quadrant in the imaginary. Thus
cosh (0.5 + io,6) = — isinh (0.5 + i 1.6)
= — * (— 0.42158 + 20.66280)
= + 0.66280 + * 0.42158.
All of the entries in Table VII thus reproduce themselves by inversion in corresponding
parts of Table VIII, a fact which serves as a numerical check upon both.
In order to find the value of cosh (x + iy), quadrant the imaginary quantity y, by
dividing it with IT/ 2, as in entering Table VII. The complex quantity (x + iy) will
now be expressed as (x -\- iq); or will in effect have been transferred from the XY to
the XQ plane. Next throw out multiples of 4 from q, so as to leave a remainder less
than 4. If this remainder exceeds 2, deduct 2 from it, but change the sign of the total
result thereupon deduced. • If the remainder, however, is not greater than 2, then,
the result is taken directly from the table.
Example: To find cosh (1+25 ) = cosh (x + iy).
Quadranting, we have cosh (i + 13.183) = cosh (x + iq).
Deducting 2 from q, cosh (i + i 1.183).
[198]
EXPLANATORY TEXT
With this we enter Table VIII. The nearest entry is x + iq = i +^1^2, the result
for which is — 0.47684 + i 1.11768. This has to be corrected by interpolation from
q = 1.2 to q = 1.183. Reverse the sign of the result to 0.47684 — i 1.11768 for the
deduction of 2 quadrants.
Example 2: Required cosh (0.25 + 2*30 ) = cosh (x + iy).
Quadranting, this becomes cosh (0.25 + i 19.099) = cosh (x + iq).
Rejecting imaginary quadruples = cosh (0.25 + 2*3.099)
= cosh (x + iq).
Deducting 2 quadrants = cosh (0.25 + i 1.099)
= — cosh (x + iq).
The nearest entry is 0.25 + ii.i for which the result is (— 0.16135 + * 0.24950).
Applying the negative sign on account of the two deducted quadrants, the final result
is, neglecting interpolation,
cosh (0.25 + i 30) = 0.16135 ~~ * o-2495o = u — iv.
INTERPOLATION BY SIMPLE PROPORTION
A first approximation can be obtained by interpolating according to simple pro-
portion.
Example: Required cosh (0.55 + 2*0.55) = cosh (x + iq)
having given
cosh (0.6 + 20.5) = 0.83825 + i 0.45018. cosh (0.6 + i 0.6) = 0.69680 + 10.51506.
cosh (0.5 + 2*0.5) = 0.79735 + i 0.36847. cosh (0.5 + 20.6) = 0.66280 + 2*0.42158.
Diff. for x o.i = 0.04090 + 20.08171. Diff. for x o.i = 0.03400 -f- 20.09348.
Diff. for x 0.05 = 0.02045 ~l~ 20.04086. Diff. for x 0.05 = 0.01700 + 2*0.04674.
cosh (0.55 + i 0.6) = 0.67980 + 20.46832.
cosh (o.55 + 2o.jj) = 0.81780 + 20.40933. cosh (0.55 + 2*0.5) = 0.81780 + 10.40933.
Diff. for q cxi = —0.13800 + 2*0.05899.
Diff. for (70.05 = —0.06900 + 2*0.02950.
cosh (0.55 + 2*0.55) = 0.74880 + 20.43883.
Correct value = 0.75018 + * 0.43963.
INTERPOLATION BY TAYLOR'S THEOREM
For a higher degree of precision than simple proportion affords, reference may be
had to Taylor's theorem in the following form : —
cosh (6 + A0) = cosh 6 + A0 sinh Q + - cosh 6 + sinh 0 + . . .
2! 3!
cosh { (x -f- iy) +• (Ax + 2' Ay) \ = cosh (x + iy) + (Ax + iAy) sinh (x + iy)
+ (^±iM! cosh (, + ^ + (a»-M*y)' sinh (, + iy)+... (7ia)
EXPLANATORY TEXT
Quadranting imaginaries on both sides, or transferring to the XQ plane,
cosh \ (x + iq) + (A* + iAq) | = cosh (x + iq) + (Ax + iAq ir/2) sinh (x + iq)
(Ax + iAq 7T/2)2
+ - * -
u t • \
cosh (* + ^q)
, (Ax + 3*A<7 ir/2)3 . , , . .
+ - f*- - sinh (x + 39) + .
O "
= cosh (x + iq) -f A'0 sinh (x + iq)
cosh (x + iq) + . . .
where A'0 = (Ax + * Aj) = (Ax + * A^ x/a). (7ib)
Example: Required cosh (0.5 + 3*0.55) = cosh (x + iq)
having given cosh (0.5 + ^0.5) = 0.79735 + i 0.36847 in Table VIII
and sinh (0.5 + ^0.5) = 0.36847 + 0.79735 in Table VII.
Here Ax = o, Aq = 10.05, A'0 = 30.05 X 1.5708 = 30.07854.
cosh (0.5 + io^) = cosh (0.5 + io.£ i + + + . .
+ sinh (0.5 + 3'o^ A'0
A'0 = 3*0.07854.
(A'0)2
(A'0)2 = — 0.00617. ~~T~ = ~ 0-00309
2!
(A'0)3
(A'0)3 = — 3*0.00048. — p = — 3*0.00008.
(A'0)4
(A'0)4 = + 0.00004. — i = o.ooooo.
4!
cosh (0.5 + 3*0.55) = (o.79735 + ^ 0.36847) (i - 0.00309)
+ (0.36847 + ^' 0.79735) (30.07854 — 3*0.00008)
= (°-79735 + * 0.36847) 0.99691
+ (0.36847 + i 0.79735) 3*0.07846
= 0.79489 -f- 3*0.36733 + 3*0.02891 — 0.06256
= 0.73233 + 3*0.39624.
The tabulated value = 0.73233 + « 0.39624.
In view of the similarity of the interpolation operations by Taylor's theorem to
those already discussed in relation to Table VII, further examples are probably not
needed.
EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY
Since if cosh ( x + iy) = u + iv (72)
cosh (— x + iy} = u — iv (73)
cosh ( x — iy) = u — iv (74)
cosh (— x — iy) = u + iv (75)
changing the 'sign of either the real or imaginary entering component only changes the
sign of the imaginary component in the result; while changing the sign of the enter-
ing quantity as a whole, has no effect on the sign of the result.
[ 200]
EXPLANATORY TEXT
CIRCULAR COSINES OF COMPLEX " ANGLES "
It is well known that if 6 be any angle, real or complex,
cos 6 = cosh (i6). (76)
Consequently,
cos (x + iy) = cosh (— y -f ix) (77)
or, quadranting the imaginary component,
cos (x + iy) = cosh (— y -f- i 2x/ir) = u + iv. (78)
To find the circular cosine of any complex quantity x + iy, we enter Table VIII
with (— y -f * x/ 1. 5 708). The result is the desired cosine.
Example: Required cos (0.4 + i 1.2).
Thus we require cosh (— 1.2 + i 0.2546).
We now enter Table VIII with x = — 1.2 and q = 0.2546 the nearest entry being
x = — 1.2 and q = 0.25, for which the result is 1.67283 — * 0.57765.
Hence cos (0.4 + * 1.2) = 1.67283 — i 0.57765 neglecting interpolation from q = 0.25
to q = 0.2546.
GRAPHIC CHART INTERPOLATIONS
The use of the Graphic Charts VII-VIIlA, B, c, for hyperbolic cosines has already
been described in connection with sines, on pages 197-198.
TABLE IX
tanh (x + iq) = u + iv
RECTANGULAR HYPERBOLIC TANGENTS OF A RECTANGULAR VARIABLE
Entering Process
Let tanh (x + iy} be the required function. Quadrant the imaginary component,
as described under Tables VII and VIII; that is, divide y by IT/ 2; so that y/(*/2) — q.
The required function is now expressed in the form tanh (x + iq). Throw out mul-
tiples of 2 from q and retain only the remainder as q. Enter Table IX with (x + iq),
and find the result directly as u ± iv. It is a well-known property of tanh (x + iy),
that it is periodic in iy, and that the period is i.ir circular radians; or, in quadrants,
i.2. That is
tanh I x + * (y -f mr) \ = tanh (x + iy). (79)
where n is any integer; or, in quadrant measure of the imaginary,
tanh \ x + i (q + 2n) j = tanh (x + iq). (80)
Example: Required tanh (0.25 + *3o) = tanh (x + iy).
Quadranting, tanh (0.25 + i 19.099) = tanh (x + iq).
Rejecting multiples of 2, tanh (0.25 + i 1.099) = tan^ (x + *'?)•
We now enter Table IX with x = 0.25 and q = 1.099, the nearest entry to which
x = 0.25, q = i.i. The result is 2.95122 — i 1.75011.
[201]
EXPLANATORY TEXT
INTERPOLATION
Interpolation may be approximately effected by simple proportion, first in x and then
in q, as indicated in connection with Tables VII and VIII; or, when a higher degree of
precision is required, recourse may be had to Taylor's theorem in the following form: —
tanh (0 + A0) = tanh 6 + A0 sech2 0 - ?—f- 2 sech2 0 tanh 0
+ - 2 sech2 0 (2 tanh2 0 - sech2 0) + . . . (81)
o *
or
tanh i (* + iy) + (A* + iAy) \ = tanh (x + iy) +
(A* + iAy)' . _ _ •
cosh2 (as + «
and quadranting,
tanh \ (x + iq) + (A* + i&q) \ = tanh (x + iq) +
- .
cosh2 (x + i
so that as far as the second correction term: —
tanh \ (x + iq) + (A* + *Ag) } = tanh (* + iq) + cosh2
tanh (, + {„)+ ... (83)
\IQ
(A'O)* tanh (ag + iq)
cosh2 (* + iq)
where A'0 = (A* + iAy) = \Ax + iAq (ir/z) } (85)
Example: Required tanh (0.5 + 3' 0.5 5) = tanh (a: + ^)
having given cosh (0.5 + ^'0.5) = 0.79735 + * 0.3684 7 by Table VIII
= 0.87837 /24°.8o3 by Table XI
and tanh (0.5 + ^'0.5) = 0.76159 -f- i 0.64805 by Table IX
= i. 0/40°. 395 by Table XII.
Here A'0 = (o + ^'0.05 X 1.5708) = (o + ^'0.07854).
*' 0.07854 , 0-00617 X i /4Q°.
tanh (0.5 + .0^5) = 0.76x59 + .0.64805 + 2^ +
= 0.76159 + i 0.64805 +
0.77153 0.77153
= 0.76159 + i 0.64805 + 0.10180/40°. 394 + o.oo8oo\9°.2ii
= 0.76159 + i 0.64805 + 0.07753 + * 0.06597 ~J~ 0.00790 — i 0.00128
= 0.84702 + i 0.71274.
The correct value is 0.84752 + * 0.71229.
As the third correction term is inconvenient for computation, it is often preferable
to obtain a precise interpolation of tanh (x + iy) in working out the correct inter-
polations of sinh (x + iy) and cosh (x + iy) by the methods already illustrated, and
then to take their ratio.
[202]
EXPLANATORY TEXT
EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY
If tanh ( x + iy) u + iv (86)
then tanh ( x — iy) = u — iv (87)
tanh (— x + iy) = — u + iv (88)
and tanh (— x — iy) = — (u + iv). (89)
Consequently, changes in the sign of the entering quantity produce corresponding changes
of sign in the result.
FIG. 32. — Graphs of x + iy and tanh (* + iy) in the XY and UV planes respectively.
CIRCULAR TANGENTS OF COMPLEX " ANGLES "
Since tan 6 = — ita.nh (id), (go)
It follows that tan (x + iy) = — stanh (— y + ix) (91)
or, quadranting the imaginary: —
tan (x + iy) = — i tanh (— y + i 2x/ir) = — i tanh (— y + iq)
= — i (— u + iv)
= v + iu. (92)
Consequently, to find tan (x — iy) from Table IX, enter it with y as x and with x/(ir/2)
as q. Invert the components of the result and the required function is obtained. Thus
[•203]
EXPLANATORY TEXT
required tan (i+ 2* 2). We enter with x = 2 and q = 0.6366. The nearest entry is
x — 2.0 and q = 0.65 for which u + iv = 1.01623 + i 0.03318. Therefore, inverting,
tan (i + i 2) = 0.03318 + i 1.01623
neglecting the interpolation from q = 0.65 to q = 0.6366.
GRAPHIC INTERPOLATION BY MEANS or CHARTS IXA, IXs, AND IXc
These charts contain all of the entries in Table IX, and also a certain number of
additional results. They present circles intersecting circles orthotomically; i.e., by
rectangular intersection. It is clear that for values of x less than o.io, the curves run
off Chart IXA. In fact the first curve shown of x = o.oi extends as far as u = 100.
By taking x small enough, the corresponding values of u and v may become indefinitely
great. The entire UV plane is covered to infinity once between x = o and x = o°,
q = o and q = ^ It is covered once more for each 2 quadrants increase in q.
When entering for tanh (— x ± iq); or for the inverse operation tanh"1 (— u ± »),
it must be remembered that the confocal conic-section diagrams VII and VIII
are complete for negative as well as for positive values of x and q; but that only half
of the UV plane is presented in Charts IX. The full graph is indicated in Fig. 32,
by the aid of which the functions corresponding to negative real values are readily
apprehended.
TABLE X
sinh (x + iq) = r /y
POLAR HYPERBOLIC SINES or A RECTANGULAR VARIABLE
This table corresponds completely to Table VII, already considered, except that it
offers results in polar instead of rectangular coordinates.
To find sinh (x + iy) expressed in polar coordinates, quadrant the imaginary, and
express the entering variable as (x + iq). Reject multiples of 4 in q, and if the re-
mainder exceeds 2, reject 2 but change the sign of the total result.
INTERPOLATION BY SIMPLE PROPORTION
Required sinh (o. 1 5 + i 0.25) having given
sinh (0.2 + 1*0.2) = o.36882/58°.723. sinh (0.2 + 20.3) = 0.49663/68°. 825.
sinh (o.i + 20.2) = 0.32485 /72°.947. sinh (o.i + ^'0.3) = 0.46491 778°. 932.
Diff. for o.i x = +0.043977 — 14°. 224. Diff. for o.i x = +0.031727 — 10°. 107.
Diff. for 0.05 x = 0.02I99/— 7°. 112. Diff. for 0.05 x = 0.015867— 5°.Q54.
sinh (0.15 + foj) = o.48o77/73°.878.
sinh (0.15 + 1*0.2) = 0.34684 /65°.83S. sinh (0.15 + 1*0.2) = 0.34684/65°. 835.
Diff. for i o.i = 0.133937 8°-°43-
Diff. for 2*0.015 = o. 06697 / 4°. 022.
sinh (0.15 + 2*0.25) = 0.41381/69°. 857.
Correct Value 0.41 124/70°. 229.
[ 204]
EXPLANATORY TEXT
INTERPOLATION BY TAYLOR'S THEOREM
For a higher degree of precision than is obtainable by simple proportion, it is con-
venient to use rectangular coordinates and apply formula (62). Thus, required sinh
(0.15 + 20.25). Here referring to Table VII and to the work on page 195, we find
for the result u + iv = 0.13911 + i 0.38701.
Here ^0.38701 = 1.5877110 yo0.^'.^ = jo°.22g ^0.13911 = 1.1433584
log 0.13911 = 1.1433584 log sec. 7o°.229 = 0.4707400
0.4443526 log 0.41124= 1.6140984
log tan 70°. 13' = 0.4440674 Result 0.41124/70°. 229.
2853 Correct Value 0.41 124 /70°.229.
0-72 3968
INTERPOLATION BY CHARTS X-XlA AND X-XIs
These charts present the polar coordinate results on rectangular coordinate sheets,
so that they are not true graphs, but are merely to be regarded as interpolation dia-
grams.
To find sinh (x + fy), proceed as in the use of the tables and quadrant the imaginary
so as to obtain the entering quantity in the form (x + iq}> Enter with the curvilinear
coordinates, taking the more nearly vertical wavy lines for x and the more nearly hori-
zontal lines for q, starting from the line SS as the zero of q. Read off the result on
the rectangular background to the left-hand scale of ordinates.
When we leave X-XlA and enter X-XlB,it is noticeable that the curves of constant
x approach vertical straight lines and the curves of constant q approach horizontal
straight lines. At and beyond x = 3.0, we may approximate to the modulus r at any
required q, by taking the value of r at q = 0.5 and simple proportional parts between
this and r at q = o or r at q = i.o. The change in modulus r between q = 0.5, and
either of the above limits is very nearly e~z/2- Thus at x = 3.5 and q = 0.5, r by the
tables is 16.55774. At q = o, r = 16.54263, a change of — 0.01511, and at q = i.o
r = 16.57282, a change of + 0.01508. The value of c~3-6/2 will be found to be 0.01510,
and over the entire range of q from o to i.o, the change in r follows in nearly simple
proportion.
Beyond x = 3.2, the limit of Chart X-XIs, the values of r can be obtained
by the above rule applied to Table X and with the aid of Chart VII-VIIIc. The
values of the amplitude y beyond x = 3 closely approximate to q quadrants. That
tx
is sinh (x -f- iq) approximates to — ^£, with q in quadrant measure.
TABLE XI
cosh (x -\- iq) — * /7
POLAR HYPERBOLIC COSINES OF A RECTANGULAR VARIABLE
Table XI corresponds completely with Table VIII, except that it gives results
expressed in polar instead of rectangular coordinates. It is entered with (x -f iq} just
as in Table VIII.
[205]
Interpolation may be effected by simple proportion, as in the case of Table X, or
when a higher degree of precision is required, it may be carried on by Taylor's theorem.
In the latter case, it is more convenient to refer to the corresponding entries in Table
VIII, interpolating according to formula (716). The rectangular coordinates duly
interpolated are then transformed into polar coordinates, as in the last example on
page 225.
INTERPOLATION BY CHARTS X-XlA AND X-XIs
When Charts X-XI are used to find cosh (x + iy) , the imaginary is first quadranted
by dividing with ?r/2, so as to obtain the entering variable in the form cosh (x + iq).
Starting then from q = o at the horizontal line CC, near the middle of the chart, the
underscored figures correspond to q for a little more than the first quadrant. The
manifest repetition of the curves enables the lower half of the sheet, however, to be
used for the second quadrant. The result is read off on the rectangular background
to the right-hand scale of argument.
Beyond x = 3.2, the limit of Chart X-XlB reference may be had to Chart VII-
VIIlB; or the approximate formula may be used:
ex
cosh (x + iq} = — /£. (93)
the argument q of the result being interpreted in quadrant measure and converted into
degrees.
TABLE XII
tanh (x + iq) = r /y
POLAR HYPERBOLIC TANGENTS OF A RECTANGULAR VARIABLE
Table XII corresponds completely with Table IX, except that it gives results
expressed in polar instead of rectangular coordinates.
If we desire to find tanh (x + iy), we must first divide y by IT/ 2 so as to obtain the
entering quantity in the form (x + iq). Multiples of 2 are then rejected in q leaving
a remainder less than 2. With this remainder the table is entered.
Interpolation may be made by simple proportion to a moderate degree of precision.
GRAPHIC INTERPOLATION BY MEANS OF CHARTS XIlA, B, c, D
These charts cover between them the full range of Table XII. To find tanh (x + iq)
from them with q less than 2, find the proper chart, and enter orr'the curvilinear co-
ordinates keeping the underscored number for q. Read off the result on the rectilinear
background.
For tan (x + iy) and also for the effects of changes of sign, see directions in the dis-
cussion on Table IX.
To find tanh""1 (r/r), enter immediately on the rectangular background of r and 7
in the proper chart, and read off at the correct intersection the corresponding values
on the curvilinear coordinates. The result will appear in terms of (x + iq). The
imaginary q must be dequadranted, or multiplied by 7r/2, in order to be expressed in
terms of (x + iy).
[ 206]
EXPLANATORY TEXT
TABLE XIII
/ (4 + iq) = u + iv or r /y
RECTANGULAR AND POLAR FUNCTIONS OF THE RECTANGULAR VARIABLE (4 + iq)
In this table the hyperbolic sine, cosine and tangent of (4 + iq) are collected from
q — o to q = 2.0. The results are expressed both in rectangular coordinates (u + iv),
and in polar coordinates r /y.
It will be seen that the moduli of the tangents vary between 0.99933 and 1.00067,
or differ from unity by two thirds of one per mil, at most. The arguments also differ
from o° by less than 0.04°, or about 2'. 17" of arc.
Beyond x = 4, it is evident that the hyp. sine and cosine differ by so small a per-
centage, that no tabulation of these differences would ordinarily be required.
TABLE XIV
e*/2 and log 10 (ex/2)
SEMI-EXPONENTIALS
This table enables the hyp. sine or cosine of any rectangular variable (x + iq) to
be found for values of x greater than 4 and less than 10. It is shown in the preceding
table that when x reaches 4, the ratio of the sine to the cosine never differs from unity
by more than two-thirds of i per mil. This deviation from unity rapidly diminishes as
x is further increased. Consequently, the sine and cosine may each be computed from
the formula.
sinh (x + iq) = cosh (x -f- iq) = — /q. (94)
Example: Required the value of sinh (8.51 -f- i 25.75). The first step is to quad-
rant the imaginary by dividing with x/2, as on page 191. This gives the required func-
tion in the form sinh (8.51 + i 16.393). Rejecting multiples of 4.0 in q, we may then
write it sinh (8.51 + 3*0.393). Turning to the top of page 143, we find ex/2 = 2482.082
for x = 8.51; so that the result is 2482.082 70.393 quadra-nt. Expressing the argument
in .degrees by multiplying with 90 and we have 0.393 X 90 = 35.37°. Thus
sinh (8.51 + i 16.393) = cosh (8.51 + i 16.393) = 2482.082 /35°.37.
INTERPOLATION IN x
. Since ^ = ?^=<-I+Al+(M! + +... (95)
it follows that when Ax is a small quantity, it suffices to multiply the tabular value of
er/2 by (i -f Ax) in order to arrive at the interpolated result unless (A*)2/ 2! the second
correction term, is of sufficient magnitude to need consideration.
[207]
EXPLANATORY TEXT
Example: To find sinh (8.51 + iq), having given that sinh (8.50 + iq) = 2457.383 /£.
Here Ax = o.oi.
2457-383 X 1 = 1 2457.383
2457-383 X Ax = o.oi 24.574
(A*)2
2457-383 X — -— = 0.00005 -123
2482.080
Result 2482. 080 /q.
Tabulated value 2482.082 [^
TABLE XV
/(* + * o)
REAL HYPERBOLIC FUNCTIONS
This is a short table of real, as distinguished from complex hyperbolic functions for
convenience of reference. It was prepared and published by the author in 1903 in re-
lation to continuous-current electric circuit applications, taking the sines, cosines, and
tangents from Ligowski's tables, and adding the corresponding computed reciprocals
for the cosecants, secants, and cotangents. Much more extensive tables of real hyper-
bolic functions are, however, available. See Bibliography, page 211.
TABLE XVI
SUBDIVISIONS or A DEGREE
This is a short table for convenience in changing the expression of a circular angle
from decimals of a degree to minutes and seconds, or inversely. By its aid, three-
decimal subdivisions of a degree may be converted into minutes and seconds of arc, by
direct inspection; or minutes and seconds may be read off as decimals of a degree ta
three-digit accuracy.
METHODS EMPLOYED IN COMPUTATION
Tables I to V, inclusive, were computed as one group, and Tables VII to XIII,
inclusive, as a separate group.
Tables I to V were computed, at first, by using the formulas: —
sinh (x + iy) = V sinh2 x + sin2 y /tan"1 (tan y / tanh x) = ri /yi. (96)
cosh (x + iy) = Vcosh2# — sin2;y /tan"1 (tan y • tanh x) = r2 /72. (97)
tanh (x + iy) = (ri/r2) /TI ~ 72- (98)
At a later stage of the work, the following formulas, kindly suggested by Professor
Bouton, were substituted: —
sinh (x + iy) = Vcosh 2X • sin z /tan"1 (tan y / tanh x) = ri fji. (99)
cosh (x + iy) = V cosh sx • cos z /tan"1 (tan y • tanh x) = r% /jz. (100)
tanh (x + iy) = tan z /y\ — 72- (101)
[208]
EXPLANATORY TEXT
Where the auxiliary circular angle z is defined by :
cos 2y
- = COS 2Z. (102)
cosh 2X
The arithmetical work was conducted with the aid of five-place logarithms, and was
checked by tabulating successive first and second differences in the tabulated results.
Tables VII to XII were computed by means of the following formulas: —
sinh (x + iy) = sinh x cos y -f- i cosh x sin y. (103)
cosh (x -f- iy) = cosh x cos y + * sinh x sin y. (104)
, f . . sinh 2X sin 2y
tanh (x + ty) = — r- — - -f * — — — ^ (105)
cosh 2# -f cos 2y cosh 2* + cos 2y
A standard schedule was prepared and seven-place logarithms used in the compu-
tation. The value of tanh (x + iy) was arrived at in two ways, first by dividing (103)
by (104), and second by the independent formula (105). If these two methods did not
give identical results for tanh (x -f- iy) to five decimal places, when expressed both in
rectangular and polar coordinates, the steps of the computation were gone over afresh.*
Complete agreement being secured, leads to the inference that the values of sinh, cosh,
and tanh (x + iy) are correct, at least as far as their logarithms.
Finally, all of the tables have been reduced to graphic form in the Atlas, each entry
of the tables being marked off on its proper chart with a sharp needle, and the ruling
pen drawn through the successive punctures. In this process a certain number of
errors were discovered and rectified. The tables were then set up in type from the
MSS. used in making the charts, and were proofread three times. By this procedure
it is hoped that the outstanding errors are neither large nor numerous.
BIBLIOGRAPHY AND APPLICATIONS OF HYPERBOLIC FUNCTIONS
Hyperbolic functions of a real variable are employed extensively in mathematics
generally. In particular, they are used in the solution of cubic equations.
In navigation, real hyperbolic functions enter in connection with Mercator sailing.
In cartography, real hyperbolic functions are used in preparing maps on certain pro-
jections, especially on Mercator's projection, which appears to have been the first
application of hyperbolic functions.
In statics, real hyperbolic functions naturally present themselves in relation to the
properties of the catenary and of the funicular polygon; also in the discussion of the
forms and stresses of elastic bodies.
In dynamics, the same functions present themselves in the theory of vibrations,
and in the motion of bodies through a resisting medium.
* The author desires to express his acknowledgement of the care and painstaking effort of his assistants
engaged in computation, namely,
Miss Ethel Smith, A.B. Radcliffe, 1911.
Miss A. F. Daniell, A.B. Radcliffe, 1911.
Miss Mary M. Devlin, A.B. Radcliffe, 1912.
Miss Hope M. Hearn, A.B. Radcliffe, 1912.
C 209]
EXPLANATORY TEXT
A good summary of the historical development of real hyperbolic functions is given
in Becker and van Orstrand's "Hyperbolic Functions," Smithsonian Mathematical
Tables, 1909, together with a fine compendium of formulas involving these functions.*
In electrical engineering, the earliest published application of real hyperbolic func-
tions is perhaps in T. H. Blakesley's "Alternating Currents of Electricity," London,
1889, which also appends a short table of these (real) functions. The real functions
were also introduced by Sir J. J. Thomson, in "The Electrician," Vol. XXVIII, page
599, 1891. "On the Heat Produced by Eddy Currents in an Iron Plate Exposed to an
Alternating Magnetic Field."
The fundamental differential equation of the alternating potential-current, steady-
state distribution along a uniform conductor, involving hyperbolic functions, nominally
real, seems to have been first published by O. Heaviside in 1893, "Electromagnetic
Theory," Vol. I, page 450.
The first published application of complex hyperbolic functions to the last-named
problem was by the author, "On the Fall of Pressure in Long-Distance Alternating-
Current Conductors," Electrical World, N. Y., Vol. XXIII, page 17, January, 1894
and "The Electrician," London (abstract), Vol. XXXII, page 239, January 5, 1894.
Complex hyperbolic functions also present themselves in the discussion of Hertzian-
wave reflections, and in other branches of electrical engineering. They naturally enter
the subject of confocal ellipses and hyperbolas, such as Captain Weir's Azimuth diagram
of these confocals, for indicating the azimuth of a celestial object in terms of the hour-
angle, latitude and declination. (Godfray's "Astronomy," § 222.)
The mathematical discussion of hyperbolic functions is found in GreenhilTs "Differ-
ential and Integral Calculus," Macmillan and Co., 1896; Ligowski's "Tafeln der Hyper-
belfunctionen und der Kreisfunctionen," Berlin, Ernst & Korn, 1890; McMahon's
"Hyperbolic Functions," Wiley and Sons, N. Y., 1896; Becker and van Orstrand's
"Hyperbolic Functions," Smithsonian Institution, 1909; Vassall's "Nouvelles Tables
des Logarithmes," Paris, Gauthier-Villars, 1872; as well as other text-books.
Works dealing with the applications of hyperbolic functions to electrical engineering
are: "The Application of Hyperbolic Functions to Electrical Engineering Problems,"
by the author, The University of London Press, 1911, and Fleming's "The Propagation
of Electrical Currents in Telephone and Telegraph Conductors," Constable & Co.,
London, 1911.
A three-dimensional complex-angle geometrical model,f from which the hyperbolic
sines and cosines of complex angles can be presented projectively, has 'been constructed
and described.
BRIEF BIBLIOGRAPHY or TABLES or HYPERBOLIC FUNCTIONS
(i) "Tafeln der Hyperbelfunctionen und der Kreisfunctionen," by Dr. W. Ligowski,
Berlin, 1890, Ernst and Korn, 104 pages, giving five-figure logarithms of sinh 6, cosh 6,
and tanh 0 up to 8 = 9, by steps of o.ooi up to 6 = 2 and from 2.0 to 9.0 by steps of
* This compendium has, by permission, been included in this book at its second edition, as Table XXIH.
f "A new geometrical model for the orthogonal projection of the cosines and sines of complex angles" by
A. E. Kennelly, Proc. Am. Ac. of Arts and Sciences, Vol. 54, April, 1919, pp. 371-378.
[210]
EXPLANATORY TEXT
o.oi; also the Gudermannian angle to two or more decimals of a second of arc, and
other tables.
(2) Smithsonian Mathematical Tables, "Hyperbolic Functions," by George F.
Becker, and C. E. van Or strand, Smithsonian Institute, Washington, D.C., 1909, 321
pages, giving five-figure logarithms of sinh 6, cosh 0, and tanh 0, by steps of o.oooi
up to o.i, by steps of o.ooi from o.i to 3.0, and by steps of o.oi from 3.0 to 6.0; also
similar five-figure tables of natural real hyperbolic functions, and various other tables.
(3) "Alternating-Current Phenomena in Parallel Conductors," Vol. I by F. E. Pernot,
John Wiley, New York, 1918, containing a Table of six-decimal logarithms of hyperbolic
functions, up to 2.0 by steps of o.ooi. These present a higher order of precision by
one unit, than have been previously available for real hyperbolic functions.
The following is a list of all the tables of Complex Hyperbolic Functions known
to the present writer, in the order of date of publication: —
(4) ChrystaFs "Algebra," Edinburgh, 1889, briefly discusses the theory of sinh 6,
cosh0, and tanh 6 where 6 is complex; or of the form x + iy. Graphs are given in out-
line for these functions, from which a few numerical values may be read.
(5) The paper on "Resonance in Alternating-Current Lines," by E. J. Houston
and A. E. Kennelly, Transactions A. I. E. E., April, 1895, Vol. XII, pages 133-169,
contains a Plate for the graphical evaluation of sinh 6 and of cosh 0, 0 being a complex
variable x + iq, between the limits of x = o and #=i.25;g = o and q = «, by steps
of 0.05 in x and q. The Plate is 40 cm. X 34 cm. and corresponds to Plates VII —
VIILv of the Atlas prepared from tables in this book, except that it gives the result in
regular polar coordinates instead of regular rectangular coordinates. It was produced,
by a graphical process, for a precision of the 2.5th order.
(6) The first tables of complex hyperbolic functions were a short set published by
Dr. James McMahon in his Chapter IV, entitled "Hyperbolic Functions," of a book
by Merriam and Woodward on "Higher Mathematics," pages 107-168. The tables
gave sinh (x + iy) and cosh (x + iy) from x = o to x = 1.5, by steps of o.i, and also
from y = o to y = 1.5, by steps of o.i, Wiley & Sons, New York, 1896. The chapter
has since been issued as a separate volume by the same publishers.
(7) A table of hyperbolic functions of semi-Lmaginaries or sinh, cosh, tanh, coth,
sech and cosech of x /45°, by steps of o.i in x up to x = 20.5, was published by the
present writer in a paper on "The Alternating-Current Theory of Transmission Speed
over Submarine Telegraph Cables," in the Proceedings of The International Electrical
Congress of St. Louis, Section A, Vol. I, pages 68-105, 1904. This table is reproduced
in Table VI of this volume.
(8) Some short tables of sinh, cosh, tanh, coth, sech, and cosech p/5 by steps of o.i
in p, up to p = 1.5, for five particular values of 5, published by the present writer in an
article on "The Distribution of Pressure and Current over Alternating-Current Cir-
cuits," in the Harvard Engineering Journal, 1905-06.
C"! 3
EXPLANATORY TEXT
(9) Short three-digit tables of sinh and cosh (x + iy) up to x = i, and y = i, by
W. E. Miller, in a paper "Formulae, Constants, and Hyperbolic Functions for Trans-
mission-Line Problems" in the General Electrical Review, Schenectady, N. Y., May,
1910. Supplement.
(10) "Tables of Hyperbolic Functions in Reference to Long Alternating-Current
Transmission Lines," published by the present writer in the Transactions of the American
Institute of Electrical Engineers, December 1911, pages 2495-2506. These give sinh,
cosh, and tanh p /5 from p = o to p = 0.5, by steps of o.i, and from 5 = 60° to 5 = 90°
by steps of i°. These tables are incorporated in Tables I, II, and III of this volume.
(n) "Tables of Sines, Cosines, Tangents, Cosecants, Secants, and Cotangents of
Real and Complex Hyperbolic Angles," published by the present writer in The Har-
vard Engineering Journal" 1912. These gave sinh, cosh, and tanh p /8 from p = o
to p — i by steps of o.i, and from 5 = 45° to 5 = 90° by steps of i°; also corresponding
tables of (sinh 0)/0 and of (tanh 0)/0. These tables are published in separate form by
the Harvard Engineering Journal. They are incorporated in tables I, II, III, IV, and V
of this volume.
NEW TABLES INTRODUCED IN THE SECOND EDITION
Tables I to V in this volume were computed for the range of 45° to 90° in the slope
or argument 5 of the entering vector quantity; because at that time it did not appear
that there would be any need for the range from o° to 45°. Alternating-current lines
used for the transmission or distribution of power have linear hyperbolic angles a, the
slope of which is commonly between 80° and 90°, rarely falling as low as 45°. It has
been shown during recent years, however, that railway-signal engineers employ track-
signaling circuits, formed of the rails. These are metallic circuits of low frequency,
small linear capacitance and large distributed linear leakance. The linear hyperbolic
angles a of such circuits develop slopes lying within the range 5 = o° to 45°. It has
therefore become desirable to cover this range, at least as far as p = i. For that pur-
pose, Tables XVII to XXI have been inserted. They run by steps of 0.05 in p, from
o to i.o, and by steps of 5° in 5, from o° to 45°. This new tabulated material is
available for use in track-signaling and similar computations. It is hoped to incorpo-
rate it graphically into the associated Chart Atlas at the first opportunity.
Table XVII presents sinh p/£ as a polar planevector. It corresponds to and may
be regarded as an extension of Table I. Similarly Tables XVIII, XLX, XX and XXI
correspond respectively to Tables II, III, IV and V. Whereas, however, Tables I to V
are carried to five decimal places in the sizes and three decimal places in the slopes of
the evaluated quantities, the new tables are carried to six decimal places in sizes and
four decimal places in slopes. They thus aim at one higher order of precision.*
Table XXII is similar to Table XI and expands the region covered by the first six
entries of the latter into a correspondingly magnified field. It has been found that in
* The author desires to express his acknowledgment of the painstaking assistance, on these tables, by Miss
Lillian L. Hodgdon, of the Harvard Observatory computing staff.
[212]
EXPLANATORY TEXT
dealing with short lengths of alternating-current line, having negligible linear induc-
tance as well as negligible linear leakance, and therefore having a semi-imaginary linear
angle, there is frequent need for a magnified table of this kind. It may be noted that
whereas Table VI expresses slopes in degrees and minutes, Table XXII expresses them
as degrees, and four-place decimals of a degree.
Table XXIII is a useful collection of 238 formulas, with a few insertions, taken
from Becker and van Orstrand's book of Tables of real hyperbolic functions, referred to
in the footnote on page 210.
TABLE XVII. HYPERBOLIC SINES, sinh (P/5) = r
0.05
0
0.000000
0.0000
0.050021
o.oooo
0.100167
o.oooo
5
o.oooooo
5.0000
0.050020
5.0042
0.100164
5.0167
10
o.oooooo
IO.OOOO
0.050020
10.0081
0.100157
10.0325
IS
o.oooooo
15.0000
0.050019
15.0119
0.100144
15.0478
20
0.000000
20.0000
0.050016
20.0153
0.100128
20.0614
25
o.oooooo
25.0000
0.050014
25.0183
0.100107
25.0730
30
o.oooooo
30.0000
0.050011
30.0208
0.100083
30.0828
35
o.oooooo
35.0000
0.050006
35.0222
0.100057
35-0897
40
o.oooooo
40.0000
0.050003
40.0236
0.100029
40.0942
45
o.oooooo
45.0000
0.05000
45.0236
O.IOOOO
45-0952
0-35
0.40
0-45
0
0-357190
o.oooo
0.410752
o.oooo
0.465342
o.oooo
5
0.357081
5.2014
0.410589
5-2625
0.465109
5-3314
10
0.35675?
10.3969
0.410105
10.5172
0.464420
10.6531
15
0.356228
15.5808
0.409316
15-7569
0.463294
15-9558
20
0-355512
20.7472
0.408245
20.9742
0.461767
21.2303
25
0.354628
25.8908
0.406925
26.1625
0.459886
26.4686
30
0.353606
31.0089
0.405397
31.3161
0.457709
31.6630
35
0.352475
36.0961
0.403709
36-4305
0.455303
36.8086
40
0.351270
41-1503
0.401910
41.5017
0.452741
41.8997
45
0.35003
46.1694
0.40006
46.5278
0.45010
46.9338
0.70
o o
o 0.758584 o.oooo
5 0.757700 5-7875
10 0.755078 H.5533
15 0.750800 17-2755
20 0.745004 22.9344
25 0.737873 28.5103
30 0.729632 33.9861
35 0.720538 39.3467
40 0.710872 44.5800
45 0.700934 49-6767
0-75
o
0.822317 o.oooo
0.821227 5.9000
0.817993 11.7753
0.812719 17.6019
0.805576 23.3567
0.796787 29.0175
0.786636 34-5647
0-775439 39-98ii
0.763544 45-2525
0.75I3I7 50-3678
0.80
o
0.888106 o.oooo
0.886780 6.0192
0.882843 1 2.0108
0.876425 17.9481
0.867734 23.8044
0.857048
0.844710
0.831105
0.816660
0.801819
29-5561
35.1800
40.6567
45-9697
51.1064
Examples, sinh (0.35 /35°) = 0.352475 /36°.O96i.
sinh (0.80 /£} = 0.886780 /6°.oi92.
TABLE XVII. HYPERBOLIC SINES.
O.I5 O.2O
sinh (P /5) = r /?.
0.25
CONTINUED
0.30
0
0
0
0
o
O
0.150563
o.oooo
0.201336
O.OOOO
0.252612
0.0000
0.304520
O.OOOO
5
0.150554
5-0372
0.201316
5.0661
0-252573
5-1033
0.304452
• 5-1483
10
0.150529
10.0733
0.201256
10.1303
0-252455
10.2033
0.304248
10.2922
15
0.150488
15.0172
0.201157
15.1906
0.252263
15.2972
0-303915
15.4275
20
0.150432
20.1380
0.201024
20.2450
0.252002
20.3825
0.303465
20.5500
25
0.150362
25.1644
0.200859
25.2919
0.251680
25-4561
0.302909
25.6558
3°
0.150282
30.1858
0.200669
30.3303
0.251308
30.5158
0.302265
30.7419
35
0.150193
35.2017
0.200458
35-3586
0.250896
35-5600
0.301553
35.8058
40
0.150098
40.2114
. 0.200233
40.3761
0.250457
40.5872
0-300795
40.8456
45
0.15000
45.2146
O.20OOO
45.3817
0.25001
45-5965
0.30001
45.8592
o.
50
0-55
0.60
o,
65
0
0
o
o
o
0
0.521095
O.OOOO
0.578152
O.OOOO
0.636654
O.OOOO
0.696748
O.OOOO
5
0.520776
5.4078
0.577726
5-4919
0.636100
5-5833
0.696042
5.6819
10
0.519829
10.8039
0.576463
10.9697
0.634456
11.1500
0.693947
"•3477
15
0.518282
16.1767
0.574400
16.4197
0.631774
16.6842
0.690530
16.9694
20
0.516184
21.5153
0.571604
21.8286
0.628138
22.1700
0.685898
22.5386
25
0.513601
26.8094
0.568162
27.1844
0.623662
27.5933
0.680199
28.0355
3°
0.510612
32.0503
0.564179
32.4764
0.618485
32.9414
0.673609
33-4450
35
0.507309
37-2305
0-559779
37-6956
0.612769
38.2036
0.666335
38.7539
40
0.503794
42.3439
0.555099
42.8344
0.606689
43-3705
0.658601
43-9525
45
0.50017
47.3872
0.550279
47.8880
0.600432
48.4369
0.650644
49-0330
0.85
0.90
0-95
I
.00
o
o
o
o
0
O
0.956116
O.OOOO
1.026517
O.OOOO
1.099484
O.OOOO
1.175201
O.OOOO
5
0.954520
6.1447
1.024615
6.2767
1.097239
6.4147
1.172573
6.5592
IO
0.949784
12.2592
1.018975
12.5200
1.090583
12.7928
1.164779
13-0775
15
0.942064
18.3131
1.009783
18.6967
1.079736
19.0986
1.152083
19.5180
20
0.931612
24.2775
0-997344
24.7753
1.065062
25.2969
1-134913
25.8417
25
0.918768
30.1255
0.982060
30.7253
1.047043
31-3544
1.113841
32.0122
3°
0.903942
35-8314
0.964429
36.5183
1.026266
37.2400
1.089558
37-9955
35
0.887604
41-3730
0.945011
42.1294
1.003398
42.9253
1.062847
43-7603
40
0.870267
46.7314
0.924414
47-5372
0.979156
48.3864
1.034550
49.2789
45
0.852463
51.8917
0.903276
52.7242
0.954292
53-6033
1.005545
54-5292
Examples, sinh (0.90/20°) = 0.997344/2^
sinh (1.0/0°) = 1.175201/0°.
TABLE XVIII. HYPERBOLIC COSINES. cosh(p/5) = r/7-
0.05
o-3S
0.40
0.70
0-7S
O.IO
0
I.OOOOOO
o.oooo
1.001250
o.oooo
1.005004
o.oooo
• 5
I.OOOOOO
o.oooo
1.001231
0.0125
1.004929
0.0497
10
I.OOOOOO
o.oooo
1.001175
0.0244
1.004703
0.0978
15
I.OOOOOO
o.oooo
1.001083
0.0358
1.004335
0.1428
20
I.OOOOOO
o.oooo
1.000958
0.0461
1.003836
0.1836
25
I.OOOOOO
o.oooo
1.000804
0.0547
1.003221
0.2189
3°
I.OOOOOO
o.oooo
1.000626
0.0619
1.002507
0.2478
35
I.OOOOOO
o.oooo
1.000428
0.0672
1.001718
0.2689
40
I.OOOOOO
o.oooo
1.000218
0.0705
1.000876
0.2820
45
I.OOOOOO
o.oooo
I.OOOOO
0.0716
I.OOOOI
0.2864
0.45
o
1.061878
O.OOOO
1.081072
o.oooo
1.102970
o.oooo
5
1.060965
0.5861
1.079886
0.7567
i. 101477
0-9453
10
1.058252
1.1561
1.076361
1-4933
1.097041
1.8669
15
1.053819
1.6950
1.070598
2.1911
1.089786
2.7411
20
1.047791
2.1872
1.062762
2.8303
1.079919
3-5450
25
1.040347
2.6189
1-053079
3-3936
1.067721
4-2567
30
1.031702
2.9772
1.041829
3-8639
1-053539
4-8550
35
I.O22II2
3.2506
1.029339
4.2264
1.037782
5.3208
40
I.OII859
3-4295
1.015974
4.4678
1.020908
5-6375
45
I.OOI25
3-5071
1.00213
4-5784
1.00341
5.7906
0.80
o
1.255169
o.oooo
1.294683
o.oooo
1-337435
o.oooo
5
1.251669
2.1083
1.290690
2-3744
1.332918
2.6483
10
1.241264
4-1750
1.278820
4.7044
1.319493
5-2500
15
1.224242
6.1581
1.259402
6-9456
1-297535
7-7578
20
1.201069
8.0155
1.232970
9.0520
1.267648
10.1236
25
1.172378
9.7039
1.200241
10.9775
1.230648
12.2978
30
1.138949
11.1803
1.162104
12.6747
1.187536
14.2294
35
1.101690
12.4000
1.119583
14.0942
I-I39459
15.8644
40
1.061615
13-3186
1.073820
15.1850
1.087691
17-1458
45
1.019823
13-8911
1.026048
15.8947
1.033602
18.0136
Examples, cosh (0.10/25°) = 1.003221 /o°.2i89.
cosh (0.75/40°) = i.Q7382o/i5°.i850.
TABLE XVIII. HYPERBOLIC COSINES, cosh (p/3) = r /?. CONTINUED
O.I5 O.2O O.2S 0.30
0
o
o
o
o
o
1.011271
o.oooo
1.020067
o.oooo
1.031413
O.OOOO
1.045339
o.oooo
5
I.OIIIOI
O.I I II
1.019765
0.1964
1.030943
0.3047
1.044665
0.4350
10
1.010595
0.2189
1.018868
0.3872
1.029547
0.6006
1.042663
0.8578
IS
1.009769
0.3203
1.017404
0.5664
1.027265
0.8794
1.039390
1.2567
20
1.008648
0.4119
1.015415
0.7292
1.024167
1.1328
1.034944
1.6200
25
1.007265
0.4914
1.012961
0.8703
1.020343
1-3533
1.029455
1-9375
3°
1.005662
0.5561
1. 010116
0.9858
1.015908
1-5344
1.023084
2.J994
35
1.003887
0.6042
1.006966
1.0719
1.010992
1.6703
1. 01602 1
2-3975
40
1.001995
0.6339
1.003604
1.1258
1.005746
1-7567
1.008476
2-5253
45
1.00005
0.6445
1.00013
1.1458
1.00033
1.7901
1.00068
2-5773
0.50
0.55
0.60
0.65
o
o
0
o
o
O
1.127626
O.OOOO
1.155101
O.OOOO
1.185465
o.oooo
1.218793
o.oooo
5
1.125794
1.1506
1.152899
1.3711
1.182861
1.6053
1.215756
1.8514
IO
1.120349
2.2736
1.146351
2.7108
1.175118
3-1753
1.206728
3.6644
15
1. 111445
3-34"
1.135641
3.9872
1.162452
4-6750
1.191956
5.4000
20
1.099330
4.3264
1.121066
5.1694
1.145211
6.0689
1.171848
7.0194
25
1.084345
5.2028
1.103034
6.2267
1.123873
7.3222
1.146955
8.4836
30
1.066914
5-9453
1.082045
7.1292
1.099026
8.4006
1.117959
9-7533
35
1.047534
6.5297
1.058692
7.8481
1.071363
9.2700
1.085656
10.7894
40
1.026760
6-9350
1-033637
8.3566
1.041655
9.8980
1.050938
"•5539
45
1.00520
7.1424
1.007598
8.6311
1.010745
10.2544
1.014773
12.0094
0.85
0.90
0-95
1. 00
0
o
0
0
o
O
1-383531
o.oooo
1.433086
o.oooo
1.486225
o.oooo
1.543081
0.0000
5
1.378460
2.9289
1.427430
3-2150
1.479951
3.5053
1.536155
3.7989
IO
1-363390
5.8092
1.410623
6-3794
1-461313
6.9589
1.515588
7-5475
15
1.338746
8.5914
1-383145
9.4428
1.430851
10.3086
1.481987
11.1858
20
1.305212
11.2253
1-345771
12.3525
1.389438
13.5005
1.436333
14-6653
25
1.263709
13-6589
•299533
I5-0545
1-338233
16.4789
1.379926
17.9267
3°
1-215358
15-8375
.245685
17.4914
1.278634
19.1844
1.314321
20.9097
35
1.161440
17.7030
-185653
19.6019
1.212219
21-5531
1.241261
23-5483
40
1.103367
19.1936
.120988
21.3192
1.140689
23-5142
1.162611
25.7689
45
1.042645
20.2411
•053338
22.5686
1.065840
24.9872
1.080307
27.4870
Examples, cosh (0.25/30°) = 1.015908/1"
cosh (1.00/40°) = 1.162611/2.5
[2173
TABLE XIX. HYPERBOLIC TANGENTS, tanh (p l&) = r
0.05
o
o.oooooo
o.oooo
0.049958
o.oooo
0.099668
o.oooo
5
o.oooooo
5.0000
0.049959
4.9917
0.099673
4.9670
10
o.oooooo
IO.OOOO
0.049961
9-9837
0.099688
9-9347
IS
o.oooooo
15.0000
0.049965
14.9761
0.099712
14.9050
20
o.oooooo
20.0000
0.049968
19.9692
0.099745
19.8778
25
o.oooooo
25.0000
0.049974
24.9636
0.099786
24.8541
3°
o.oooooo
300000
0.049980
29.9589
0.099833
29.8350
35
O.OOOOOO
35.0000
0.040985
34-9550
0.099886
34.8208
40
o.oooooo
40.0000
0.049992
39-9531
0.099941
39.8122
45
o.oooooo
45.0000
0.05000
44.9520
O.IOOOO
44.8087
o-35
0.40
0.70
o-75
0-45
0
0.336375
o.oooo
0-379949
o.oooo
0.421899
o.oooo
s
0.336562
4-6153
0.380215
4-5058
0.422260
4.3861
10
0.337H9
9.2408
0.381011
9.0239
0.423339
8.7862
15
0.338036
13-8858
0.382325
13.5658
0.425124
13-2147
20
0.339296
18.5600
0-384135
18.1439
0.427594
17.6853
25
0.340875
23.2719
0.386414
22.7689
0.430717
22.2119
3°
0.342740
28.0317
0.389121
27.4522
0.434449
26.8080
35
0.344849
32.8455
0.392202
32.2041
0.438727
31.4878
40
0.347153
37.7208
o.39559i
37-0339
0.443469
36.2622
45
0-34959
42.6623
0.39921
41.9494
0.44857
41.1432
0.80
0
0.604367
o.oooo
0.635150
o.oooo
0.664037
o.oooo
s
0-605352
3.6792
0.636270
3-5256
0.665292
3-3709
10
0.608313
7.3783
0.639647
7.0709
0.669078
6.7608
15
0.613278
11.1174
0.645321
10.6563
0.675454
10.1903
20
0.620284
14.9189
0.653363
14.3047
0.684523
13.6808
25
0.629382
18.8064
0.663856
18.0400
0.696420
17.2583
30
0.640619
22.8058
0.676907
21.8900
0.711313
20.9506
35
0.654029
26.9467
0.692614
25.8869
0.729386
24.7922
40
0.669614
31.2614
0.711054
30.0675
0.750820
28.8239
45
0.687309
35-7856
0.732244
34-4731
0.775752
33-0928
Examples, tanh (0.75/25°) = 0.663856/18^.0400.
tanh (0.40/20°) = 0.384135 /i 8°. 1 439^
O8]
TABLE XIX. HYPERBOLIC TANGENTS, tanh (p /6) = r fa. CONTINUED
0.15
0.25
0.30
0
0.148885
o.oooo
0.197375
o.oooo
0.244919
0.0000
0.291312
0.0000
5
0.148901
4.9261
0.197414
4.8697
0.244992
4.7986
0.291435
4-7133
IO
0.148951
9.8544
0.197529
9-7431
0.245210
9.6027
0.291799
9-4344
IS
0.149032
14.7869
0.197718
14.6242
0.245567
14.4178
0.292398
14.1708
20
0.149142
19.7261
0.197972
19-5158
0.246055
19.2497
0.293219
18.9300
25
0.149277
24.6730
0.198289
24.4216
0.246663
24.1028
0.294242
23-7183
30
0.149436
29.6297
0.198659
29-3445
0.247373
28.9814
0.295445
28.5425
35
0.149611
34-5975
0.199071
34.2867
0.248168
33.8897
0.296798
33-4083
40
0.149799
39-5775
0.199514
39-2503
0.249027
38.8305
0.298267
38-3203
45
0.14999
44.5701
0.19997
44.2359
0.24993
43-8064
0.29981
43.2819
0.50
0.55
0.60
0.65
0
0
e
o
0
O
0.462117
0.0000
0.500521
0.0000
0.537049
o.oooo
0.571669
O.OOOO
5
0.462586
4.2572
0.501107
4.1208
0.537764
3-9780
0.572518
3-8305
IO
0.463988
8.5303
0.502867
8.2589
0.539908
7-9747
0.575065
7-6803
15
0.466314
12.8356
0.505793
12.4325
0.543484
12.0092
0.579325
11.5694
20
0.469545
17.1889
0.509875
16.6592
0.548491
16.1011
0.585313
15.5192
25
0.473651
21.6066
0.515090
20-9577
0.554922
20.2711
0.593047
19-5519
30
0.478587
26.1050
0.521400
2S-3472
0.562757
24.5408
0.602535
23.6917
35
0.484289
30.7008
0.528746
29.8475
0.571953
28.9336
0.613763
27-9645
40
0.490664
35.4089
0.537034
34-4778
0.582428
334725
0.626679
32.3986
45
0-49759
40.2448
0.546130
39-2569
0.594049
38.1825
0.641172
37.0236
o.
85
0.90 0.95
1. 00
o
o
e
o
o
O
0.691070
o.oooo
0.716298
O.OOOO
0.739782
O.OOOO
0.761595
O.OOOO
5
0.692454
3-2158
0.717804
3.0617
0.741402
2.9094
0.763317
2.7603
10
0.696634
6.4500
0.722358
6.1406
0.746304
5.8339
0.768533
5.5300
15
0.703692
9.7217
0.730063
9.2539
0.754611
8.7900
0.777391
8.3322
20
0-713763
13.0522
0.741095
12.4228
0.766541
11.7964
0.790146
11.1764
25
0.727041
16.4666
0.755702
15.6708
0.782407
14.8755
0.807174
14.0855
30
0.743766
19-9939
0.774215
19.0269
0.802627
18.0556
0.828989
17.0858
35
0.764227
23.6700
0.797038
22.5275
0.827737
21.3722
0.856264
20. 2 1 2O
40
0.788737
27-5378
0.824642
26.2180
0.858389
24.8722
0.889851
23-5IOO
45
0.817596
31.6506
0.857537
30.1556
0.895343
28.6161
0.930795
27.O422
Examples, tanh (0.25/25°) = o.246663/24°.io28.
tanh (0.90/30°) = 0.774215 1
1 219 ]
o
5
10
25
30
35
40
45
o
5
10
IS
20
25
30
35
40
45
o
5
10
IS
20
25
30
35
40
45
I.OOOOOO
I.OOOOOO
I.OOOOOO
I.OOOOOO
I.OOOOOO
o.oooo
o.oooo
o.oooo
0.0000
o.oooo
I.OOOOOO O.OOOO
I.OOOOOO 0.0000
I.OOOOOO O.OOOO
I.OOOOOO O.OOOO
I.OOOOOO O.OOOO
0-35
o
1.020543 o.oooo
1.020230 0.2014
1.019305 0.3969
1.017796 0.5808
1.015748 0.7472
1.013224 0.8908
1.010302 1.0089
1.007070 1.0961
1.003630 1.1503
1.00008 1.1694
0.70
o
1.083691 o.oooo
1.082429 0.7875
1.078682 1.5533
.072572 2.2755
.064292 2.9344
.054105 3-5103
.042331 3.9861
.029340 4.3467
1.015532 4.5800
1.001334 4-6767
sinh 0
TING FACTOR.
e '&
0.05
O.IO
o
o
1.000420 o.oooo
1.001670 o.oooo
1.000404 0.0042
1.001638 0.0167
1.000398 0.0081
1.001570 0.0325
1.000373 0.0119
1.001442 0.0478
1.000317 0.0153
1.001279 0.0614
1.000279 0.0183
1.001071 0.0730
1.000216 0.0208
1.000834 0.0828
I.OOOI34 O.O222
1.000571 0.0897
1.000061 0.0236
1.000290 0.0942
i.ooooo 0.0236
i.ooooo 0.0952
0.40
0.45
o
o
.026880 o.oooo
1.034093 o.oooo
.026473 0.2625
1-033577 0.3314
.025264 0.5172
1.032044 0.6531
.023290 0.7569
1.029542 0.9558
.020612 0.9742
1.026148 1.2303
.017312 1.1625
1.021969 1.4686
.013493 1.3161
1.017131 1.6630
1.009272 1.4305
1.011784 I. 8086
1.004775 1-5017
1.006092 1.8997
1.00014 1.5278
1.00023 1.9338
o-7S
0.80
o
o
1.096423 o.oooo
1.110132 o.oooo
1.094969 0.9000
1.108475 1.0192
1.090658 1.7753
1.103554 2.0108
1.083625 2.6019
1.095532 2.9481
1.074102 3.3567
1.084668 3.8044
1.062383 4-0175
1.071311 4-5561
1.048848 4.5647
1.055881 5.1800
1.033919 4.9811
1.038881 5.6567
1.018059 5-2525
1.020825 5.9697
1.001756 5.3678
1.002274 6.1064
Example.
sinh (0.40/25°)
0-40/25°
= 1.017312^
C220]
TABLE XX. CORRECTING
O.IS 0.20
FACTO
D sinh 0
= r£r.
CONTINUED
0.30
e
0.25
0
o
o
o
O
1-003753
o.oooo
1.006680
o.oooo
1.010448
o.oooo
1.015067
o.oooo
5
1.003696
0.0372
1.006579
0.066 1
1.010291
0.1033
1,014839
0.1483
10
1.003527
0.0733
1.006278
0.1303
1.009820
0.2033
1.014161
0.2922
is
1.003253
0.0172
1.005784
0.1906
1.009051
0.2972
1.013051
0.4275
20
1.002878
0.1380
1.005119
0.2450
1.008007
0.3825
1.011549
0.5500
25
1.002412
0.1644
1.004296
0.2919
1.006722
0.4561
1.000696
0.6558
3°
1.001878
0.1858
1.003343
0.3303
1.005233
0.5158
1.007550
0.7419
35
1.001287
0.2017
1.002290
0.3586
1.003584
0.5600
1.005177
0.8058
40
1.000653
0.2114
1.001167
0.3761
1.001830
0.5872
1.002651
0.8456
45
I.OOOOO
0.2146
I.OOOOI
0.3817
1.00003
0.5965
1.00005
0.8592
0.50
o-SS
0.60
0.65
o
o
0
e
0
1.042190
O.OOOO
1.051185
o.oooo
1.061090
o.oooo
1.071920
o.oooo
5
1.041552
0.4078
1.050411
0.4919
1.060167
0.5833
1.070833
0.6819
10
1.039657
0.8039
1.048114
0.9697
1.057426
1.1500
1.0^7611
1-3477
IS
1.036564
1.1767
1.044363
1.4197
1.052956
1.6842
1.062354
1.9694
20
1.032369
1-5153
1.039280
1.8286
1.046896
2.1700
1.055228
2.5386
25
1.027202
1.8094
1.033021
2.1844
1.039436
2-5933
1.046459
3-0355
30
1.021223
2.0503
1.025779
2.4764
1.030808
2.9414
1.036322
3-4450
35
1.014618
2.2305
1.017780
2.6956
1.021281
3.2036
1.025131
3-7539
40
1.007587
2.3439
1.009270
2.8344
1.011148
3-370S
1.013232
3.9525
45 •
1.00035
2.3872
1.000508
2.8880
1.000720
3-4369
1.000991
4-0330
0.85
0.00
0.95
1. 00
o
o
o
o
0
1.124842
o.oooo
1.140574
o.oooo
1-157352
o.oooo
1.175201
O.OOOO
5
1.122964
1.1447
1.138461
1.2767
1.154989
1.4147
1-172573
1-5592
10
I.II7393
2.2592
1.132194
2.5200
1.147982
2.7928
1.164779
3-0775
IS
1.108311
3-3131
1.121981
3.6967
1.136564
4.0986
1.152083
4.5180
20
1.096014
4.2775
I. 108160
4-7753
I.I2III7
5.2969
1.134913
5-8417
25
1.080904
5-1255
1.091177
5-7253
1.102151
6.3544
1.113841
7.0122
3°
1.063461
5.8314
1.071587
6.5183
1.080280
7.2400
1.089558
7-9955
35
1.044241
6.3730
1.050012
7.1294
1.056208
7.9253
1.062847
8.7603
40
1.023843
6.7314
1.027127
7-5372
1.030691
8.3864
1.034550
9.2789
45
1.002897
6.8917
1.003640
7.7242
1.004518
8.6033
1.005545
9.5292
Example.
sinh (0.95/25°) _
0-95/25^
[221]
TABLE XXI. CORRECTING FACTOR.
taphg
e
0.05
0
1. 000000
o.oooo
0.999160
o.oooo
0.996680
o.oooo
S
I.OOOOOO
o.oooo
0.999177
0.0083
0.996732
0.0330
10
I.OOOOOO
o.oooo
0.999223
0.0163
0.996882
0.0653
IS
I.OOOOOO
o.oooo
0.999291
0.0239
0.997119
0.0950
20
I.OOOOOO
o.oooo
0.999359
0.0308
0.997448
0.1222
25
I.OOOOOO
o.oooo
0.999475
0.0364
0.997858
0.1459
3°
I.OOOOOO
o.oooo
0.999591
0.0411
0.998331
0.1650
35
I.OOOOOO
o.oooo
0.999707
0.0450
0.998856
0.1792
40
I.OOOOOO
o.oooo
0.999844
0.0469
0.999414
0.1878
45
I.OOOOOO
o.oooo
I. 00000
0.0480
0.99999
0.1913
0.35
0.40
0.45
o
0.961071
o.oooo
0.949872
o.oooo
0.937553
o.oooo
5
0.961606
0.3847
0.950538
0.4942
0-938355
0.6139
10
0.963197
0.7592
0.952528
0.9761
0.940753
1.2138
i5
0.965817
1.1142
0.955812
1.4342
0.944719
1.7853
20
0.969419
1.4400
0.960338
1.8561
0.950209
2.3147
25
0.973929
1.7281
0.966036
2.2311
0.957150
2.7881
3°
0.979258
1.9683
0.972801
2.5478
0.965442
3.1920
35
0.985284
2.1545
0.980505
2.7959
0.974949
3.5122
40
0.991867
2.2792
0.988977
2.9661
0.985487
3-7378
45
0.99883
2.3377
0.99800
3.0506
0.99684
3.8568.
0.70
0.75
0.80
o
0.863381
o.oooo
0.846867
o.oooo
0.830046
o.oooo
5
0.864789
1.3208
0.848360
1-4744
0.831615
1.6291
10
0.869019
2.6217
0.852862
2.9291
0.836347
3.2392
i5
0.876111
3.8826
0.860428
4-3437
0.844318
4.8097
20
0.886121
5.0811
0.871150
5-6953
0-855654
6.3192
25
0.899117
6.1936
0.885142
6.9600
0.870525
7-7417
30
0.915169
7.1942
0.902542
8.1100
0.889141
9.0494
35
0.934328
8.0533
0.923485
9.1131
0.911732
10.2078
40
0.956592
8.7386
0.948072
9.9325
0.938525
11.1761
45
0.981870
9.2144
0.976325
10.5269
0.969690
11.9072
Note. Negative quantities are in heavy type.
Example.
tanh (0.75/25°)
0.75/25°
[223]
XAXAJV^ ± O.J.TI VJ 1 f\\-, J. V
e rti-
^UNTLNUED
o.i 5
O.2O
0.25
0.30
0
o
0
0
o
O
0.992567 o.oooo
0.986875 o.oooo
0.979676 o.oooo
0.971040 o.oooo
5
0.992676 0.0739
0.987069 0.1303
0.979968 0.2014
0.971449 0.2867
10
0.993006 0.1456
0.987642 0.2569
0.980839 0.3973
0.972665 0.5656
15
0.993547 0.2131
0.988588 0.3758
0.982269 0.5822
0.974660 0.8292
20
0.994280 0.2739
0.989860 0.4842
0.984222 0.7503
0-977396 1.0700
25
0.995182 0.3270
0.991446 0.5784
0.986650 0.8972
0.980807 1.2817
30
0.996237 0.3703
0.993295 0.6555
0.989493 1.0186
0.984817 1.4575
35
0.997410 0.4025
0.995357 0.7133
0.992673 x.i 103
0.989327 1.5917
40
0.998660 0.4225
0.997571 0.7497
0.996106 1.1695
0.994224 1.6797
45
0.99995 0.4299
0.99987 0.7641
0.99971 1.1936
0.99937 1-7181
0.50
0.55
O.6o
0.65
0
0
o
o
o
o
0.924234 o.oooo
0.910038 o.oooo
0.895082 o.oooo
0.879491 o.oooo
5
0.925172 0.7428
0.91 1 104 0.8792
0.896274 I.O22O
0.880796 1.1695
10
0.927976 1.4697
0.914304 1.7411
0.899847 2.0253
0.884716 2.3197
15
0.932627 2.1644
0.919624 2.5675
0.905806 2.9908
0.891269 3.4306
20
0.939089 2.8111
0.927046 3.3408
0.914151 3.8989
0.900482 4.4808
25
0.947302 3-3934
0.936527 4-0423
0.924870 4.2289
0.912380 5.4481
30
0.957174 3-8950
0.948001 4.6528
0.937928 5-4592
0.926976 6.3083
35
0.968577 4.2992
0.961356 5-1525
0.953254 6.0664
0.944250 7-0355
40
0.981327 4.59"
0.976426 5.5222
0.970713 6.5275
0.964122 7.6014
45
0.99516 4-7552
0.992963 5.7431
0.990081 6.8175
0.986419 7.9764
0.85
0.90
0.95
1.00
o
o
0
0
o
O
0.813024 o.oooo
0.795887 o.oooo
0.778718 o.oooo
0.761595 o.oooo
5
0.814651 1.7842
0.797560 1.9383
0.780424 2.0906
0.763317 2.2397
IO
0.819569 3-5500
0.802620 3.8594
0-785583 4-1661
0-768533 4.4700
15
0.827873 5-2783
0.811181 5.7461
0.794327 6.2100
0.777391 6.6678
20
0.839722 6.9478
0.823438 7-5772
0.806886 8.2036
0.790146 8.8236
25
0.855342 8.5334
0.839669 9.3292
0.823586 10.1245
0.807174 10.9145
30
0.875019 10.0061
0.860239 10.9731
0.844870 11.9444
0.828989 12.9142
35
0.899091 11.3300
0.885598 12.4725
0.871302 13.6278
0.856264 14.7880
40
0.927926 12.4622
0.916269 13.7820
0.903567 15.1278
0.889851 16.4900
45
0.961877 13.3494
0.952819 14-8444
0.942466 16.3839
0.930795 17.9578
Note. Negative quantities are in heavy type.
tanh (1.0/10°)
Eiample. : — 5 = 0.768533X4^4700.
[223]
TABLE XXIL FUNCTIONS OF SEMI-IMAGINARIES
COMPLEX VARIABLE 0/45° (slope constant).
9
Sinh0/45°
Cosh 0/45°
tanh 6 /45°
Coseche/45°
hyp.
Size
Slope
Size ^
Slope
Size _
Slope
Size
Slope
rads.
numeric
degrees
numeric
degrees
numeric
degrees
numeric degrees
0.00
o.ooooo
'45.0000
I.OOOOO
o.oooo
o.ooooo
45.0000
oc
45.0000
O.OI
O.OIOOO
45.0009
I.OOOOO
0.0028
O.OIOOO
44.9981
IOO.OOOOO
45.0009
O.O2
0.02OOO
45-0037
I.OOOOO
0.0114
O.O2OOO
44.9923
50.00000
45-0037
0.03
0.03000
45.0084
I.OOOOO
0.0257
0.03000
44.9827
33-33333
45.0084
0.04
0.04000
4S-oi5i
I.OOOOO
0.0458
0.04000
44-9693
25.00000
45.0151
O.OS
0.05000
45-o236
I.OOOOO
0.0716
0.05000
44.9520
20.00000
45-0236
O.O6
0.06000
45-0342
I.OOOOO
0.1031
0.06000
44.9310
16.66667
45-0342
O.O?
0.07000
45-0465
.00000
0.1403
0.07000
44.9062
14.28571
45-0465
0.08
0.08000
45.0608
.00000
0.1833
0.08000
44-8775
12.50000
45.0608
O.OQ
0.09000
45.0770
.00001
0.2320
O.OQOOO
44.8450
n. mil
45-0770
O.IO
O.IOOOO
45-0952
.00001
0.2864
O.IOOOO
44.8087
10.00000
45-0952
O.II
O.IIOOO
45-"52
.00001
0.3466
O.IIOOO
44.7686
9.09091
45.«52
O.I2
O.I2OOO
45-I372
.00002
0.4125
O.I2OOO
44.7247
8-33333
45.1372
0.13
0.13000
45.1611
1.00003
0.4841
0.13000
44.6770
7-69231
45.1611
0.14
0.14000
45.1869
1.00004
0.5614
0.13999
44-6255
7.14286
45-1869
0.15
0.15000
45.2146
1.00005
0.6445
0.14999
44-5701
6.66667
452146
0.16
0.16000
45.2442
1.00006
0-7333
0.15999
44.5109
6.25000
45.2442
0.17
0.17000
45-2757
1.00008
0.8278
0.16999
44-4479
5-88235
45-2757
0.18
0.18000
45-3092
1.00009
0.9281
0.17998
44.3811
5.55556
453092
0.19
0.19000
45-3445
I.OOOII
1.0341
0.18998
44.3104
5-26316
45-3445
O.20
O.20OOO
45-38I7
1.00013
1.1458
0.19997
44-2359
5.00000
45-38I7
O.2I
O.2IOOO
45.4208
1. 00016
1.2632
0.20997
44.1576
4.76190
454208
0.22
O.22OOO
45.4619
I.OOO2O
1.3864
0.21996
44-0755
4-54545
45.4619
0.23
0.23000
45-5048
1.00024
1-5152
0.22995
43.9896
4.34783
45.5048
O.24
0.24000
45-5497
1.00028
1.6498
0.23994
43.8999
4.16667
45-5497
0.25
0.25001
45-5965
1.00033
1.7901
0.24993
43.8064
3.99984
45.5965
O.26
0.26001
45-6453
1.00038
1.9361
0.25991
43.7092
3.84601
45-6453
0.27
0.27001
45-6959
1.00044
2.0878
0.26989
43.6081
3.70357
45-6959
0.28
0.28001
45-7485
1.00051
2.2452
0.27987
43-5032
3-57I30
45.7485
O.29
0.29001
45.8029
1.00060
2.4084
0.28984
43-3945
3.44816
45.8029
0.30
0.30001
45-8592
1.00068
2-5773
0.29981
43.2819
3-33322
45-8592
0.31
0.31002
45-9I74
1.00077
2.7520
0.30978
43-1654
3.22560
45.9174
0.32
0.32002
45-9775
1.00087
2.9323
0.31974
43-0452
3.12481
45-9775
0-33
0.33002
46.0396
1.00098
3.«83
0.32970
42-9213
3.03012
46.0396
0-34
0.34002
46.1036
I.OOIII
3-3099
0.33965
42-7937
2.94100
46.1036
0-35
0.35003
46.1694
1.00125
3-507I
0-34959
42.6623
2.85690
46.1694
0.36
0.36003
46.2372
1.00140
3-7ioo
0-35953
42.5272
2-77755
46.2372
0-37
0.37004
46.3069
1.00156
3-9i85
0.36946
42.3884
2.70241
46.3069
0.38
0.38004
46.3786
1.00174
4-1328
0-37939
42-2458
2.63130
46.3786
0-39
0.39005
46.4522
1.00193
4-3528
0.38930
42.0994
2.56377
46.4522
0.40
0.40006
46-5278
1.00213
4-5784
0.39921
41.9494
2.49963
46.5278
0.41
0.41006
46.6053
1.00235
4.8096
0.40910
41-7957
2.43867
46.6053
0.42
0.42007
46.6846
1.00259
5-0464
0.41899
41.6382
2.38056
46.6846
0.43
0.43008
46-7658
1.00285
5.2889
0.42886
41.4778
2-32515
46.7658
o-44
0.44009
46.8489
1.00312
5-5369
0.43872
41.3120
2.27226
46.8489
0-45
0.45010
46.9338
1.00341
5.7906
0.44857
41.1432
2.22173
46.9338
0.46
0.4601 2
47.0206
1.00372
6.0498
0.45841
40.9708
2-17335
47.0206
0.47
0.47013
47.1093
1.00406
6.3146
0.46823
40.7947
2.12707
47.1093
0.48
0.48014
47.2000
1.00442
6.5850
0.47803
40.6150
2.08273
47.2000
0.49
0.49016
47.2926
1.00480
6.8609
0.48782
40.4317
2.04015
47.2926
0.50
0.50017
47-3872
1.00520
7.1424
0-49759
40.2448
1.99932
47.3873
Note.
Negative quantities are in heavy type.
Examples, sinh 0.13 745° =* 0.13000 745°. 1611.
cosh 0.26/45° = 1.00038 /i.°936i.
[224]
TABLE XXII. FUNCTIONS OF SEMI-IMAGINARIES. CONTINUED
COMPLEX VARIABLE 6 /45° (slope constant).
e
Sech 6 /45°
Coth 0 /45°
Sinh 0 /45° / 9 /45°
tanh B /4S° / 0 /45°
hyp.
Size_
Slope
Size
Slope
Size
Slope
Size
Slope
rads.
numeric
degrees
numeric
degrees
numeric
degrees
numeric
degrees
0.00
I.OOOOO
o.oooo
oc
45.0000
I.OOOOO
o.oooo
I.OOOOO
o.oooo
0.01
I.OOOOO
0.0028
IOO.OOOOO
44.9981
I.OOOOO
0.0009
I.OOOOO
0.0019
O.02
I.OOOOO
0.0114
50.00000
449923
I.OOOOO
0.0037
I.OOOOO
0.0077
0.03
I.OOOOO
0.0257
33-33333
44.9827
I.OOOOO
0.0084
I.OOOOO
0.0173
O.O4
I.OOOOO
0.0458
25.00000
449693
I.OOOOO
0.0151
I.OOOOO
0.0307
0.05
I.OOOOO
0.0716
20.00000
449520
I.OOOOO
0.0236
I.OOOOO
0.0480
O.O6
I.OOOOO
0.1031
16.66667
44.9310
I.OOOOO
0.0342
I.OOOOO
0.0690
O.O7
I.OOOOO
0.1403
14.28571
44.9062
I.OOOOO
0.0465
0.99999
0.0938
0.08
I.OOOOO
0.1833
12.50000
44-8775
I.OOOOO
0.0608
0-99999
0.1225
O.OQ
0.99999
0.2320
ii.iiin
44.8450
I.OOOOO
0.0770
0.99999
0.1550
O.IO
0.99999
0.2864
IO.OOOOO
44-8087
I.OOOOO
0.0952
0-99999
0.1913
O.I I
0.99999
0.3466
9.09091
44.7686
I.OOOOO
0.1152
0.99998
0.2314
0.12
0.99998
0.4125
8-33333
44-7247
I.OOOOO
0.1372
0.99998
0.2753
0.13
0.99997
0.4841
7.69231
44.6770
I.OOOOO
0.1611
0.99997
0.3230
O.I4
0.99996
0.5614
7-14337
446255
I.OOOOO
0.1869
0.99996
0-3745
0.15
0.99995
0.6445
6.66712
445701
I.OOOOO
0.2146
099995
0.4299
O.l6
0.99994
0-7333
6.25039
445109
I.OOOOO
0.2442
0.99994
o 4891
O.I?
0.99992
0.8278
5.88271
444479
I.OOOOI
0.2757
0.99992
0.5521
0.18
0.99991
0.9281
S-SS6i8
44.3811
I.OOOOI
0.3092
0.99991
06189
O.I9
0.99989
•0341
5.26371
44.3104
I.OOOOI
0-3445
0.99989
0.6896
O.2O
0.99987
.1458
5.00050
44-2359
I.OOOOI
0.3817
0.99987
0.7641
0.21
0.99984
.2632
4-76258
441576
I.OOOO2
0.4208
0.99984
0.8424
O.22
0.99980
.3864
4.54628
44-0755
I.OOOO2
0.4619
0.99981
0.9245
0.23
0.99976
.5152
4-34878
43.9896
I.OOOO2
0.5048
0.99978
1.0104
O.24
0.99972
1.6498
4.16771
438999
I.OOOO2
0-5497
0-99975
I.IOOI
0.25
0.99967
1.7901
4.OOI I 2
43.8064
1.00003
0-5965
0.99971
1.1936
0.26
0.99962
1.9361
3.84748
43.7092
1.00003
0-6453
0.99966
2.2908
0.27
0.99956
2.0878
3.70521
43.6081
1.00003
0-6959
0.99960
1.3919
0.28
0.99949
2.2452
3-57309
43.5032
1.00004
07485
0-99953
1.4968
O.29
0.99940
2.4084
3.45018
433945
1.00004
0.8029
0-99945
1.6055
0.30
0.99932
2-5773
3-33544
43 2819
1.00005
0.8592
0-99937
1.7181
0.31
0.99923
2.7520
3.22810
43.1654
1.00005
0.9174
0.99928
1.8346
0.32
0.99913
29323
3-I2754
430452
1.00006
0-9775
0.99919
1.9548
0-33
0.99902
3.1183
3-0330S
42.9213
1.00006
1.0396
0.99908
2.0787
0-34
0.99889
3 3099
2.94421
42-7937
1.00007
1.1036
0.99896
2.2063
0-35
0.99875
3-5071
2.86049
42.6623
1.00008
1.1694
0.99883
2.3377
0.36
0.99860
3.7100
2.78141
42.5272
1.00009
1.2372
0.99869
2.4728
o-37
0.99844
39185
2.70665
42-3884
I.OOOIO
1.3069
0.99854
2.6116
0.38
0.99826
4-1328
2.63580
42.2458
I.OOOII
1-3786
0-99837
3.7543
o-39
0.99807
43528
2.56870
42.0994
1.00013
1.4522
0.09819
2.9006
0.40
0.99788
4-5784
2.50494
42.9494
1.00014
1.5278
0.99800
3.0506
0.41
0.99766
4.8096
2.44438
4i 7957
1.00016
1.6053
0.99780
3.2043
0.42
0.99742
5.0464
2.38668
41.6382
1.00017
1.6846
0-99759
3.3618
o-43
0.99716
5.2889
2.33174
41.4778
1.00019
1.7658
0.99736
3.5230
o-44
0.99689
5.5369
2-27935
41.3120
I.OOO2I
1.8489
0.99711
3.6880
0-45
0.99660
5.7906
2.22930
41 US*
1.00023
I.9338
0.99684
3.8568
0.46
0.99629
6.0498
2.18145
40.9708
1.00025
2.0206
0.99655
4.0292
0.47
0.99596
6.3146
2.I357I
40.7947
1.00027
2.1093
0.99624
4.2053
0.48
0.99560
6.5850
2.09190
40.6150
1.00030
2.2OOO
0.99591
4.3850
0-49
0.99522
6.8609
2.04995
40.4317
1.00032
2.2926
0-99555
4.5683
0.50
0^9483
7.1424
2.00969
40.2448
1.00035
2.3872
0.99516
4-7552
Note.
Negative quantities are in heavy type.
Examples.
tanh 0.39 /45°
o-39
coth 0.50 /45°
TABLE XXIII. HYPERBOLIC FUNCTION FORMULAS
(from Smithsonian Mathematical Tables No. 1871 of 1909, Becker and van Orstrand's
"Hyperbolic Functions," by permission.)
A. RELATIONS BETWEEN HYPERBOLIC AND CIRCULAR FUNCTIONS
1. sinh u = — i sin iu = tan gd u.
2. cosh u = cos iu = sec gd u.
3. tanh u = — i tan iu = sin gd u.
4. tanh \u— tan \ gd u.
5. 0" = (i + sin gd u) -i- cos gd u,
= [i - cos (J TT + gd tt)3 -i- sin (£ TT + gd «),
= tan (JTT + % gdu).
6. sinh iu = i sin u.
7. cosh ra = cos u.
8. tanh *« = i tan w.
8a. sin u = — i sinh iu = tanh (grf"1 u).
8b. cos M = cosh iu = sech (gd""1 u).
8c. tan M = — i tanh i« = sinh (gd~l u).
9. sinh (u ± w>) = ± i sin (z> =F i«),
= sinh u cos v ± i cosh w sin v.
10. cosh (w ± iv) = cos (v T iu),
= cosh M cos v db t sinh w sin w.
loa. sin (w =t *») = ± i sinh (t» ± tw) = sin « cosh v ± i cos « sinh v.
xob. cos (« =b iv) = cosh (v =F iu) = cos M cosh v =F i sin « sinh ».
1 1 . cosh (»w TT) = cos m IT. (m is an integer.)
12. sinh (2 w + i) \ i IT = i sin (2 w + i) I T. (m is an integer.)
B. RELATIONS AMONG THE HYPERBOLIC FUNCTIONS
13. sinh u= \(eu — e~u) = — sinh (— u) = (csch u) ~l,
= 2 tanh \ u -i- (i — tanh2 \ u) = tanh u -f- (i — tanh 2 «)*.
14. cosh « = £ (eu + e~") = cosh (— u) = (sech u)~l,
= (i + tanh2 i u) -5- (i - tanh2 } M) = i -=- (i - tanh2«)*.
15. tanh u = (eu - e~") -T- (eu + «r«) = - tanh (- w),
= (coth u)~l = sinh M -i- cosh M = (i — sech2 M)*.
16. sechtt = sech (— w) = (i — tanh2«)*.
[226]
FORMULAS
17. cschw = — csch (— «) = (coth2w — i)*.
1 8. coth u = - coth (- u) = (csch2 w + i)*.
19. cosh2 u — sinh2 u = i.
20. sinh £ « = V i (cosh w — i).
21. cosh % w = V i (cosh w + i).
22. tanh ^ « = (cosh w — i) -5- sinh u,
= sinh u -T- (i + cosh w) = V (cosh u — i) -r- (cosh M + i).
23. sinh 2« = 2 sinh w cosh u = 2 tanh M -f- (i — tanh2 u).
24. cosh 2« = cosh2 u + sinh2 w = 2 cosh2 M — i,
= 1+2 sinh2« = (i + tanh2«) -f- (i — tanh2w).
25. tanh 2U = 2 tanh M -f- (i + tanh2«).
26. sinh 3« = 3 sinh w + 4 sinh3 «.
27. cosh 3« = 4 cosh3 « — 3 cosh u.
28. tanh 3« = (3 tanh u + tanh3 w) -f- (i + 3 tanh2 «).
28a> m cosh u + n sinh w = J (w + ») eu + \ (m — ri) e~".
a8b. m eu ± w«e~u = (m ± ri) cosh w + (w =F w) sinh u.
29. sinh «M =
n cosh""1 u sinh w + ^-^ -4-J cosh""3 M sinh3 «+....
6
/M ('M — — T )
30. cosh nu = cosh" u -\ — — cosh n~2 u sinh2 u +
2
31. sinh w + sinh i> = 2 sinh \ (u -\- v) cosh § (M — z>).
32. sinh w — sinh v = 2 cosh !(« + ») sinh % (u — v).
33. cosh u -f cosh 0 = 2 cosh £ (« + v) cosh £ (« — v).
34. cosh « — cosh v = 2 sinh %(u + v) sinh HM ~ *)•
35. sinh u + cosh M = (i + tanh £ w) -H (i — tanh i w).
36. (sinh w + cosh u)n = cosh w« + sinh nu.
36a. a sinh u + b cosh u = V#2 ~ ^2 sinh (« + tanh"1 ^). a> b
= V*2- a2 cosh (« + coth"1 1). 6 > a
__^____^^ i
36b. a cosh « ± i sinh u = Va2 — 62 cosh (w ± tanh"1 j).
37. tanh w + tanh z> = sinh (M + v) •+• cosh w cosh v.
38. tanh M — tanh v = sinh (u — v) -f- cosh w cosh ».
39. coth u + coth t> = sinh (« + v) -i- sinh M sinh ».
[227]
FORMULAS
40. coth u — coth v = — sinh (u — v) -f- sinh u sinh v.
41. sinh (u ± v) = sinh w cosh z> =t cosh M sinh t>.
42. cosh (u =t z>) = cosh w cosh z; ± sinh M sinh z>.
43. tanh (u ± z») = (tanh w ± tanh v) -H (i ± tanh w tanh z>).
44. coth (u± v) = (coth M coth z> ± i) -f- (coth z> ± coth M).
45. sinh (u + v) + sinh (w — zi) = 2 sinh u cosh z>.
46. sinh (u + z>) — sinh (« — z>) = 2 cosh w sinh v.
47. cosh (« + »)+ cosh (u — v) = 2 cosh u cosh z>.
48. cosh (u + v) — cosh (M — ») = 2 sinh u sinh v.
49. tanh \ (u + z;) = (sinh M -f sinh z>) -f- (cosh M + cosh v).
50. tanh ^ (w — v) = (sinh M — sinh v) -f- (cosh w + cosh z>).
51. coth J (M + z>) = (sinh w — sinh v) -t- (cosh M — cosh »).
52. coth 5 (« — ») = (sinh M + sinh z>) -f- (cosh « — cosh z>).
tanh « + tanh v _ sinh (w + v)
tanh M — tanh v sinh (« — z;) *
coth u + coth v _ _ sinh (w + »)
coth M — coth z» sinh (w — v)
55. sinh (w + z>) + cosh (« + v) = (cosh w + sinh M) (cosh » + shih v).
56. shih (w + v) sinh (« — ») = sinh2 u — sinh2 »,
= cosh2 u — cosh2 z>.
57. cosh (u + z>) cosh (u — v) = cosh2 u + sinh2 »,
= sinh2 M + cosh2 v.
58. sinh (#w TT) = o. (m is an integer.)
59. cosh (miir) = ( — i)m.
60. tanh (mi TT) = o.
61. sinh (u + miir) = (— i)msinh«.
62. cosh (u + mi IT) = (— i)mcoshw.
63. sinh (2 m + i) £ *' TT = ± i.
64. cosh (2 w + i) 3 i TT = o.
65. sinh ( — ± « ) = i cosh M.
\ 2 /
. cosh ( — ±w) = =t * sinh M.
\2 /
66.
[228]
FORMULAS
66a. sinh { (u + iv) + i f \ = sinh { (u + iv) + i i \ — i cosh (u + iv).
66b. cosh \ (u + iv) + i f | = cosh | (w + «>) + « T; \ = i.sinh (M + iv).
66c. tanh ^ (w + iv) + i f | = tanh | (w + iv) + i i_ \ = coth (u + iv).
66d. sinh | (w + iv) + i TT \ = sinh \ (u + iv) + is_\ = —sinh (u + iv).
66e. cosh \ (u + iv) + i * \ = cosh \ (u + iv) + i 2 \ = —cosh (u + iv).
66f. tanh \ (u + iv) + i T \ = tanh \ (u + iv) + i 2 \ = tanh (u + iv).
67. tanh (u + iir) = tanh u.
67a. If sinh ^ (u + t (i - q) \ = x + iy; then sinh \ u + i (i + q) \ =- x + iy.
670. If cosh \(u + i(i-q)\=x + iy: then cosh \ u + i (i + q) \ =- x + iy.
67C. If tanh \ u + i (i - q) \ = x + iy: then tanh \ u + i (i 4- q) I = x - iy.
C. INVERSE HYPERBOLIC FUNCTIONS
/T
T-TI — u *
^ + i;»
69. cosh-1 u = log (w + Vw2 — i) = sinh -1 V«2 - i = / , 2 _ ^ •
r du
70. tanh"1 w = \ log (i + «) — £ log (i — u) = I 2*
%/ A **
71. coth"1 M = ^ log (i + u) — i log (u — i) = / j _ W2 :
72.
73. csch"1 u =
74. sin"1 w = — « sinh"1 iu =— i log (I'M + V1 — «2)-
75. cos"1 u =— i cosh"1 M = — i log (M + i Vi ~ «2)-
76. tan-1 u=— i tanh"1 ZM = — : log (i + iu) . log (i — iu).
• t 1. 11:
21
77. cot'1 u- i coth-1 iu = — . log (iu — i) — TJ J°g (*M + J)-
2 Z* • **
78. sin"1 iu = i sinh"1 u = i log (w
C 229 .1
FORMULAS
1 7
80. tan"1 iu = i tanh-1 u = - log (i + «) — - log (i — «).
2 2
81. cot-1 iu =— i cotir1 u = -- log (w + i) + — log (u — i).
2 2
82. cosh-1 \(u + -} = sinh-1 \(u - -} =
\ «/ \ «/
tt
i- i ~
= 2 tanh-1 - = log M.
« + i
83. tanh-1 tan M = \gdzu.
84. tan"1 tanh M = \ gd~l 2 «.
85. cosh"1 esc 2 « = — sinh-1 cot 2 u = — tanh-1 cos 2 u = log tan u.
86. tanh-1 tan2 (J TT + i «) = J log esc w.
87. tankr1 tan2 1 « = ^ log sec u.
88. cosh"1 « ± cosh"1 v = cosh"1 [«» ± V (w2 — i) (v2 — i)].
89. sinh"1 « ± sinh"1 z> = sinh-1 \ju V i + v2 ± z; V i + w2].
D. SERIES
M2 7<3 M4
90. e«=I+M + - + - + -+ . .. (M2<oc)
91. log u = (u - i) - *- (u - i)2 + - (w - i)» - . . . (2> w> o)
93.
94. log (i + M) = u -- w2 H -- «3 - — w4 + .
234
95- log ferj) = 2 tt + — W3 + — M6 + — W7 + . . . 1 (M2 <l)
*>e^-,tt+ife>v7fe)<*;'^]
M3 «5 M7
97- smh« = * + — +—+—+... («2<a)
2
[230]
FORMULAS
'' +JL--U
6!
98. cOSh«=I+^-+- -+-f + . . . («2<cr)
(0\ / A \ , » »
4«2\ / 4M-\ / , 4«2\
1 ~»*~J V1 32r2/ V l "*" ?~W ' * * («2 < <*)
99. tanh « = w «3H w8 — w7 + . . . («2 < i T2)
1,1 2
100. u com « = i H «•* w4 H w8 — . fo2 O2)
3 45 945
1 •> , 5 6 1
101 . sech u = i «^ + — «4
2 24 720
102. U CSCh U = I T-U2 + -~ U4 —
108. com"1 « = tanh"1 - = — I — -. -\ r H 7 + . .
M u 3 «* 5 M5 7 w7
I . 2 IM 13** 135
6 360 15120
103. gdu = <t> = u — z~w3H w5 «7 + . . . (M small)
r i sech3 w 13 sech5 u
= sech u ... (u large)
2 23 245
I - . I s . 6l - / T\
104. tt = ga1<£ = <p-t--^<p'JH <p°H ^07+... (0 <-
6 24 5°40 \ 2/
105. sinh"1 « = «---+--— - - - 4 — + . . . («2<i)
23 245 2467
22 M2 244W4 2466 M6
II I3I 1351
106. cosh"1 u = log 2 w -9 — i -T 7—; — ... (« > i)
22 W2 244 tt4 2466 tt6
III /
107. tanh"1 u = u + - w -\ — «6 H — w + . . . (« < i)
3 5 7
i oo . seen w *~ \^V/JI.A — AVCL
M « 22 244 2466
II II I^I 1351 / '\
no. csch-'* = sinh"1- ----_+--_- --- —, + . . .
= logf +I^_I3±4 + I3|^_ B-< (tt2<1)
0 U 22 244 2466
FORMULAS
E. DERIVATIVES
deu
in. -7- = eu.
du
. loge U I
112. d j = - •
(III U
dav dv .
113. -r— = a", -r- .log, a.
du du
114. -3 — = uu (i + loge «).
J sinh M
iis. : = coshtt.
du
d cosh M . ,
116. •: = smhtt.
du
d tanh u
117. —-j = sech2«.
d coth u
118. ; = — csch2 M.
du
d sech u
no. ; =— sechw. tanhM.
du
d csch u
1 20. — ; = — csch u. coth u.
du
121
d sinh"1 u
du ^#2 _|_ j
J cosh"1 M i
122. — = —
123.
du
d tanh"1 u
d coth"1 u
124. -j =
du
d sech"1 u
125. ; = —
du u
126.
du u ^ + z
dgd u
127. —3 — = sechw.
du
d gd-1 «
128. —^3 = secw.
du
[232]
FORMULAS
F. INTEGRALS. (INTEGRATION CONSTANTS ARE OMITTED.)
1 29. I sinh u du = cosh u.
130. I cosh udu = sinh u.
131. I tanh udu = log cosh «.
132. I coth udu = log sinh «.
X33- / sech udu = 2 tan"1 eu = gd u.
134. I csch udu = log tanh - -
135. / sinh" u du = - sinh "-' M. cosh u — — / sinh n~2 u du,
J w n J
I w _|_ 2 /*
= — ; — sinh "4l M cosh w I sinh n+2 M rfw.
» + i w + i J
136. I cosh" udu = - sinh w. cosh n-1 M + / cosh n~? w dw,
J w w J
sinh u coshn+1 u -\ — ^ / coshn+2 « du.
n + i
137. I u sinh udu = u cosh M — sinh M.
138. I u cosh udu = u sinh w — cosh «.
139. / w2 sinh udu = (u2 + 2) cosh u — 2 u sinh «.
140. I M" sinh udu = un cosh « — ww""1 sinh u + n(n — i) I «n~s si
141. I sinh2 w JM = \ (sinh M cosh u — u).
142. I sinh «. cosh u du = \ cosh (2 «).
^
143. / cosh2 u du = % (sinh u cosh « + M).
[233]
FORMULAS
144. I tanh2 u du = u — tanh w.
145. I coth2 u du = u — coth u.
146. I sech2«rf« = tanhw.
147. I sech3 M du = i sech w tanh u + \ gd u.
148. I csch2 udu=— coth «.
149. / sinh-1 u du = u sinh-1 « — (i + w2)*.
150. I cosh-1 u du = u cosh"1 u — (w2 — i)*.
151. / tanh-1 udu = u tanh-1 w + i log (i — w2).
152. T« sinh-1 u du = i (2 w2 + i) sinh-1 u - u (i + «2)* .
153- TM cosh-1 u du = l\ (2 u2 - i) cosh-1 u - u (u? - i)» .
154. i (cosh a + cosh w)"1 du = 2 csch o. tanh"1 (tanh ^ w. tanh £ a),
= csch a log cosh £ (w + a) - log cosh \(u - a) .
155. I (cos a + cosh u)~l du - 2 esc a. tan-1( tanh $ w. tan \ a).
156. / (i + cos a. cosh w)"1 du = 2 esc a. tanh-1 (tanh i w. tan £ a).
157. I sinh « cos «•</« = i (cosh M. cos u + sinh «. sin «).
158. / cosh «. cos udu= \ (sinh w. cos u + cosh w. sin «).
159. I sinh w. sin M rfw = 5 (cosh u. sin « — sinh u. cos w).
160. I cosh w. sin M du = i (sinh w. sin u — cosh «. cos «).
[234]
FORMULAS
161. I sinh (mu) sinh (nu) du
— -2 m sinh (nu) cosh (mu) — n cosh (nu) sinh (w«) .
m - n L J
162. I cosh (WM) sinh (nu) du
m sinh (nu) sinh (w«) — w cosh (nu) cosh (w«)
w n L J
163. I cosh (WM) cosh («M) rfw
= — 5 m sinh (mu) cosh (WM) — n sinh (««) cosh (mu) .
m — n \_
164. / sinh u tanh u du = sinh u — gd u.
/u
cosh u coth udu = cosh M + log tanh — •
1 66. I sec M du — gd"1 «.
167. I sec3 <j>d<t> = I (i + tan2 </>)* J tan <£ = £ sec <£ tan 0 + | gd-1 0,
= | tan $ (i + tan2 <£)* + \ sinh-1 (tan 0). Here <f> = gd u.
r du .u r du
168. I , „ , ,.. = sinh-1-- f -p-j - 5T7— = sin"1
J (u2 + a2)* a J (a2 — «
n
')* """ a'
r du ,u r — du .u
l6o. I — 7-S - ^T = COSh"1 - • I —r-~ -- 5rr- = COS 1— •
J (u2 — a2)* a J (a2 — «2)* a
/<f w i . u r du * i A . «
7-5 -- 5x — = - tanh"1 - • , , , — = - tan-1 - •
(a2 - w2)u<o a « J a2 + w2 a a
/— J M i ,« C — du i .«
r~5— v - = - coth-1 — / - r^—i — = ~ cot -*
(w2-a2)u>a a a J a2 + «2 a a
/— d« i . « f du i u
— — - — - = - sech-1 - • I — 7-5 - ^T- = — sec * - •
M (a2 — «2)* a a J u (u2 — a2)* a a
/— d« i , u f — du i u
u(a* + u2)* =aCSCh a" J « («2 - a2) = a C! a
/du i , aw + b
. , , — TT = = sinh-1 7 -- rjrr- , a positive, ac > b2;
(au2 + 2bu + c)* v« C« - P)1
i , au + & ... ^ ,,
= -= cosh"1 ^75— — y , a positive, ac < b2;
= — — cos"1 , ^U X1> a negative.
(b2 — ac)»
[235]
FORMULAS
/
du
(au2
tan
au + b
(ac - 62)* (ac -
— i •" v-i au + b
(b* - a,
tanlr
(b2 - ac)*
ac > b*;
ac < b2,
au + b < (b2 — ac)*.
ac < 62,
au + 6 > (62 - ac)*.
176.
J (a-
u) (u - 6)* (a - 6)*
— b
a — b
or
or
— 2 \u — b
tan-1
(b - a)*
2
(a - 6)i
coth-1
I77<
f
J (a-
du
b~=~a'
u — b
a — b
tanh"1
u) (b - w)* (b - a)*
2 \b — u
or
or
TT - rr
(b — a)*
-. - TTJ
(a — b)*
coth-1
tan
a
- -
a — b
(The real form is to be taken.)
b — u
b — a
(The real form is to be taken.)
178.
179.
1 80.
181.
182.
183-
184.
185.
= i w («2 — a2)* — | a2 cosh"1 —
a
«2)* JM = ^ u (a2 - «2)i + ^ a2 sin"1 -
o>
du=
a2)*
/<«'-
JV-
JV +
//flu
** u~~a''
/gOU
utf*u du = -j (aw — i).
a
/Um MU yy> f*
um-(Pu du = I um~l e?u du.
a aj
/e?u du _ i e°" /^e°u a7!* 1
um m — i [ « ""^ «/ w*""1 J
//T&«
a^rfw =
6 log a
[236]
FORMULAS
C _auun _ nau un~l n (n — i) au un ~2
J log a (log a)2 (log a)3
n (n — i) (n — 2) • • 2. i a"
(log a)n+1
l8? ra"^M = a" r E iqgg _ (log a)2
J un n — i [ « "-1 (w — 2) w""2 (w — 2) (w - 3) «»-;
(log a) "-1 ra"du 1
(w- 2) (w- 3) .... 2.iJ ~u~ \'
Cau du _ . (u log a)2 (« log a)3
J U 2.2 I 3.3 !
du eu
du
- log ( V a + bemu + V"a) .
/ueu du e"
(i + «)2 ~TT« "
/e°ulog«
»
195. I logudu = u log u — u.
196. Jumlogudu = um+l [^^ ~ (w+l)2J-
197. I (log «)" du = u (log w)n — n I (log w)"-1 rfw.
r /i w>n+1 (l°g M)" n C m n ^n-
198. / «m (log u)ndu = - I wm (log «)"
«y m + i w -f- i«7
20°-
(log «)n </M _ (log u)n+l ^
u n + i
dw , (logw)2 . (log«)
[237]
FORMULAS
r du u
2OI
2O2
/du u i r du
(log «)n ~ (w - i) (log tt)"-1 « - ij (logtt)"-1'
/MTO o# um+l m + i /* um du
., i *^~ — -4— ' I *
(log «)n (» — i) (log tt)""1 n — i J (log «)n~
/«m a"« /^e~tf
— = I — Jy, where y = — (m -f i) log w.
log u J y
/du
— := log (log u).
u log u
/du i
_ — .
u (log u)n (n — i) (log w)n~
206. I (a + bu)m log udu =
— r (a + to)*+1log u — I —
o (w + i) L J w J
207. I «m log (a + to) O"M =
- — u"*1 log (fl + to) - 6 / —
w+iL J a + bu \
208. I — - =
J u
bu
/logM^M _ I logW /* </
(a + to)* " 6 (f» - i) L ~ (a + to)"1-1 + J w(a +
/log udu i ' , / , T N i Tlog (a + to)
^-^==_
/, (a + to)2 .
(a + bu) log u du = - - log «
20
f(a + bu)2 . a2 log u
(a + bu) log u du = - — ; — - log u ^ au —
fk
212. I 7-
J (a
log M C?M
+ to)* ~
(log u — 2) V(« + to) + V a log ( Va + to + V a)
* L
- V a log ( V a, 4- to — V a) , if a> o,
= -T- (log M — 2) V (& + bu) + 2 V — a tan"1 -vl - — , if a < o.
[238]
FORMULAS
213. Ce~<**du = -^ =— r(i).
Jo 2 a 20,
214. ru»e-*«du=T(n+J = ^
Jo an+l an
«s. r»-r~ ^=i
Jo
-
+1
.
2 n a
2i6. I e "*~ ^r du = - - \ IT • a > o.
. I e~"*~ & du = - A| IT
Jo 2 >
/*" i [TT
217. ' e""" V M d« = -- A - '
Jo 2n V n
/*e-»u 17
==~du = \ — •
i Vw > w
f00 ^M 7T
219. . , , - r = - •
J o sinh (w«) 2 w
f00 M^M 7T2
. . , ( - r- = - 5-
J o smh (nu) 4 wj
220
221
f*it /»IT
sinh (mu) . sinh (««) du = I cosh (WM) . cosh (»«) <
*/ o Jo
= o, if m is different from n.
/it (*i* i^
cosh2 (mu) du =— I sinh2 (w«) </w = — •
i Jo 2
£it
sinh (WM) JM = o.
*
rir
224. I cosh (mu) du = o.
J o
/tf
sinh (w«) cosh (nu) du = o.
-it
pit
226. I sinh (mu) cosh (ww) </M = o.
J o
T1 logM , 7T2
227. - du = *
Jo i - « 6
/"! log U IT2
228. I . — - — du =
Jo I + U 12
Cl log u ,
229. '— du = —
Jo i-u2
*.
8
[239]
FORMULAS
r1 . / I + U \ du 7T*
230. log { ) • — = —
Jo 3 \i - u) u 4
du
-u«)du
233. / (logw)n<fo= (-
J 0
r/. i\»
Jo (log-j
237.
[240]
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