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/V«9'5/*// 5=^,^7 ;,«£•
WJMTWOBTH-SMITH MATHEMATICAL SBBIES
PLANE TRIGONOMETRY
AND TABLES
BY
GEOEGE WENTWORTH
DAVID EUGENE SMITH
GINN AND COMPANY
COPYBIGHT, 1914, BY GEORGE WENTWORTH
AND DAVID EUGENE SMITH
ALL BIGHTS BBSBaVBD
225 JS
fPte gttew»iim_ i|rcgf
GINN AND COMPANY • PRO-
PRIETORS • BOSTON • U.S.A.
PEEFACE
In preparing a work to replace the Wentworth Trigonometry,
which has dominated the teaching of the subject in America for a
whole generation, some words of explanation are necessary as to the
desirability of the changes that have been made. Although the great
truths of mathematics are permanent, educational policy changes from
generation to generation, and the time has now arrived when some
rearrangement of matter is necessary to meet the legitimate demands
of the schools.
The principal changes from the general plan of the standard texts
in use in America relate to the sequence of material and to the
number and nature of the practical applications. With respect to
sequence the rule has been followed that the practical use of every
new feature should be clearly set forth before the abstract theory is
developed. For example, it will be noticed that the definite uses of
each of the natural functions are given as soon as possible, that the
need for logarithmic computation follows, that thereafter the secant
and cosecant assume a minor place, and that a wide range of prac-
tical applications of the right triangle awakens an early interest in
the subject. The study of the functions of larger angles, and of the
sum and difference of two angles, now becomes necessary to further
progress in trigonometry, after which the oblique triangle is con-
sidered, together with a large number of practical, nontechnical
applications.
The decimal division of the degree is explained and is used enough
to show its value^ but it is recognized that this topic has, as yet,
only a subordinate place. It seems probable that the decimal frac-
tion will in due time supplant the sexagesimal here as it has in other
fields of science, and hence the student should be familiar with its
advantages.
Such topics as the radian, graphs of the various functions, the
applications of trigonometry to higher algebra, and the theory of
trigonometric equations properly find place at the end of the course
in plane trigonometry. They are important, but their value is best
appreciated after a good course in the practical uses of the subject,
• • •
m
iv PREFACE
They may be considered briefly or at length as the circumstances
may warrant.
The authors have sought to give teachers and students all the
material needed for a thorough study of plane trigonometry, with
more problems than any one class will use, thus offering opportunity
for a new selection of examples from year to year, and allowing for
the omission of the more theoretical portions of Chapters IX-XII
if desired.
The tables have been arranged with great care, every practical
device having been adopted to save eye strain, all tabular material
being furnished that the student will need, and an opportunity being
afforded to use angles divided either sexagesimally or decimally, as
the occasion demands.
It is hoped that the care that has been taken to arrange all matter
in the order of difficulty and of actual need, to place the practical
before the theoretical, to eliminate all that is not necessary to a clear
understanding of the subject, and to present a page that is at the
same time pleasing to the eye and inviting to the mind will com-
mend itself to and will meet with the approval of the many friends
of the series of which this work is a part.
GEORGE WENTWORTH
DAVID EUGENE SMITH
CONTENTS
PLANE TEIGONOMETRY
CHAPTER PAGE
I. Trigonometric Functions of Acute Angles .... 1
XL Use of the Table of Natural Functions 27
III. Logarithms 39
IV. The Right Triangle 63
V. Trigonometric Functions of any Angle 77
VI. Functions of the Sum or the Difference of Two
Angles 97
VII. The Oblique Triangle 107
VIII. Miscellaneous Applications 133
IX. Plane Sailing 145
X. Graphs of Functions 151
XI. Trigonometric Identities and Equations 163
XII. Applications of Trigonometry to Algebra 173
The Most Important Formulas of Plane Trigonometry . . 185
PLANE TRIGONOMETRY
CHAPTER I
TRIOONOUKTSIC FUNCTIONS OF ACVTB ANGLES
•
1. The Nature of Arithmetic. In arithmetic we study computation,
tlie working with numbers. We may have a formula expressed
in algebraic symbols, such as a = bh, but the actual computation
involved in applying such a formula to a particular case is part
of arithmetic.
Arithmetic enters into all subsequent branches of mathematics. It plays
such an important part in trigonometry that it becomes necessary to introduce
another method of computation, the method which makes use of logarithms.
2. The Nature of Algebra. In algebra we generalize arithmetic.
Thus, instead of saying that the area of a rectangle with base 4 in.
and height 2 in. is 4 x 2 sq. in., we express a general law by saying
that a = bh. Algebra, therefore, is a generalized arithmetic, and the
equation is the chief object of attention.
Algebra also enters into all subsequent branches of mathematics, and its
relation to trigonometry will be found to be very close.
3. The Nature of Geometry. In geometry we study the forms and
relations of figures, proving many properties and effecting numerous
constructions concerning them.
Geometry, like algebra and arithmetic, enters into the work in trigonometry.
Indeed, trigonometry may almost be said to unite arithmetic, algebra, and
geometry in one subject.
4. The Nature of Trigonometry. We are now about to begin another
branch of mathematics, one not chiefly relating to numbers although
it uses numbers, not primarily devoted to equations although using
equations, and not concerned principally with the study of geometric
forms although freely drawing upon the facts of geometry.
Trigonometry is concerned chiefly with the relation of certain
lines in a triangle (trigon, "a triangle," + metrein, "to measure") and
forms the basis of the mensuration used in surveying, engineering,
mechanics, geodesy, and astronomy.
1
2 PLANE TRIGONOMETRY
S. How Angles are HeMured. For ordinary purposes angles can be
measured with a protractor to a degree of accuracy of about 30'.
For work out of doors it is customary to use a transit, an instra-
ment by means of which angles can be measured to minutes. By
turning the top of the transit to the
right or left, horizontal angles can be
measured on the horizontal plate. By
turning the telescope up or down, ver-
tical angles can be measured on the
vertical circle seen in the illustration.
For astronomical purposes, where great
care is necessary in measuring angles,
large circles are used.
The degree of accataicy required in meas-
uring an angle depends upon the nature of
the problem. We tbail now aBBume that, we
can measure angles in degreee, minutes, and
seconds, or in degrees and decimal parts of a degree. Thus 15° W is the same
as 15.5°, and 15° SCSS" is the same as 15^°+ ^^ of 1° or 15.51°.
The ancient Greek astronomers had no good symbols tor fractions. The best
system they could devise for close approximations was the so-called sezageramal
one, in which there appear only the numerators of fractions whose denomi-
nators are poweis of 60. This system seems to have been first suggested by the
Babylonians, but to have been developed by the Greeks. It is much inferior to
the decimal system that was perfected about 1000, but the world still continues
to nae it for the measure of angles and time. The decimal division of the angle
is, however, gaining ground, and in due time will probably replace the more
cumbersome one with which we are familiar.
In this book we shall use both the ancient and modem systems, but with the
chief attention to the former, since this Is still the n
FUNCTIONS OF ACUTE ANGLES 3
6. Functions of an Angle. In the annexed figure, if the line AR
moves about the point A in the sense indicated by the arrow, from
the position ^X as an initial position, it generates the angle A.
If from the points ByB\ B", ..., on AR, we let fall the perpen-
diculars EC, B^C\ B"C*', . . ., on AXy we form a series of similar
triangles ACBy AC'B', AC"B", and so on. The corresponding sides
of these triangles are proportional. That is,
BC B'C B"C"
AB
BC
AC
AB
AC
AB*
B'C
AC'
AB'
AB"
B"C"
AC"
AB"
= . . .,
AC' AC"
and similarly for the ratios
AB
= . . • ,
AC AC
BC BC AB
each of which has a series of other ratios equal to it.
AB AR AW
For example, — — = = .
^ ' BC RC B''C"
That is, these ratios remain unchanged so long as the angle remains
unchanged, hut they change a^s the angle changes.
Each of the above ratios is therefore z. function of the angle A.
As already learned in algebra and geometry, a magnitude which depends
upon another magnitude for its value is called a function of the latter mag-
nitude. Thus a circle is a function of the radius, the area of a square is a
function of the side, the surface of a sphere is a function of the diameter, and
the volume of a pyramid is a function of the base and altitude.
We indicate a function of x by such symbols as f(x), F(x), f\x),
and <l>(x), and we read these **/ of x, /-major of x, /-prime of «, and
phi of X " respectively.
For example, if we are repeatedly using some long expression like
X* + 3 cc* — 2 a:^ + 7 .r — 4, we may speak of it briefly as f(x). If we
are using some function of angle A, we may designate this as/(^).
If we wish to speak of some other function of A, we may write it
f'(A), F(A), or ^(^).
In trigonometry we shall make much use of various functions of an angle, but
we shall give to them special names and symbols. On this account the ordinary
function symbols of algebra, mentioned above, will not be used frequently in
trigonometry, but they will be used often enough to make it necessary that the
student should understand their significance.
1
^y^
ni
4 PLANE TRIGONOMETRY
7. The Six Functions. Since with a given angle A we may take
any one of the triangles described in § 6, we shall consider the
triangle ACB, lettered as here shown. b
It has long been the custom to letter in this way the <^
hypotenuse, sides, and angles of the first triangle con- ry'
sidered in trigonometry, C being the right angle, and the '^ i' '
hypotenuse and sides bearing the small letters corre-
sponding to the opposite capitals. By the sides of the ^ _
triangle is meant the sides a and 6, c being called the -^ '^ 4* ^
hypotenuse. The sides a and 6 are also called the legs of the triaiigle^ par-
ticularly by early writers, since it was formerly the custom to represent the
triangle as standing on the hypotenuse.
The ratios ->->"7>->7> and - have the following names :
c c a a
ft , ''
- is called the sine of A, written sini4 :
e
- is called the cosine of A, written C0Si4;
( - is called the tangent of A, written tan^ ;
- is called the cotangent of A , written cot A ;
\ c.
\ 7 is called the secaivt of A, written sec -4 ;
o
ft
- is called the cosecant of A. written csc^
a ■
That is,
• . ^ opposite side ^ h adiacent side
sm^ = - = -i*-^^ — 7 > cos^ = - = -h' — >
c hypotenuse c hypotenuse
a opposite side ^ , h adiacent side
tani4=T = -ff . .. > cotA=- = — ^ — - — tt-j
h adjacent side a opposite side
c hypotenuse , c hypotenuse
sec ^ = 7 = --jT- — ■ ., > csc^ = - = ^^ .^ — TT- •
h adjacent side a opposite side
These definitions must be thoroughly learned, since they are the foundation
upon which the whole science is built. The student should practice upon them,
with the figure before him, until he can tell instantly what ratio is meant by
sec^, cot-4, sin^, and so on, in whatever order these functions are given.
There are also two other functions, rarely used at present. These are the
versed sine -4 = 1— cos^, and the coversed sine -4 = 1 — sin ^. These defini-
tions need not be learned at this time, since they will be given again when the
functions are met later in the work.
FUNCTIONS OF ACUTE ANGLES 5
Bzercise 1. The Six Functions
1. In the figure of § 7, sinB = -• Write the other five functions
c
of the angle B,
2. Show that in the right triangle A CB (§7) the following
relations exist:
sin A = cos B, cos A = sin JB, tan A = cot By
cot A = tan JB, sec A = esc By esc A = sec 5.
/SCote which of the follomng is the greater :
V 3. sin^ or tan^. ^^ 5. sec^ or tan^l.
4. cos^ or cot^. 6. csc^ or cot^.
FiTid the valves of the six functions of A, if a, J, c respect toe/ (^
have ifie following values :
— 7. 3, 4, 5, /^ 9. 8, 16, 17. 11. 3.9, 8, 8.9.
*< 8. 6, 12, 13. 10. 9, 40, 41. 12. 1.19, 1.20, 1.69.
" J^. What condition must be fulfilled by the lengths of the three
lines a, by c (§ 7) to make them the sides of a right triangle ? Show
that this condition is fulfilled in Exs. 7-12.
Find the values of the six functimis of A, if Oy h, c respectively
have the following values :
14. 2 w, 71* — 1, n* + 1. 16. 2 mil, m^ — w*, m* -h n^.
n^ — 1 n^ + 1 .«2 mn m^ + n^
15. n, — 7ri~* — 71 — • 17. y 7fi -^ n, •
J J in — n m — n
18. As in Ex. 13, show that the condition for a right triangle is
fulfilled in Exs. 14-17.
Qiven a^-\-b^ = <?y find the six functions of A when :
•19. a=:b, ^^^20. a = 2b. 21. a = |c.
Given a^-\-b^ = (?y find the six functions of B when :
*^22. a = 24, b = 143. 24. a = 0.264, c = 0.265.
23. b = 9.6, c = 19.3. 25. b = 2 y/pqy c=p-\-q.
Given a^ -{-b^ = (?y find the six functions of A and also the six
functions of B when :
26. a = y/p^ -h q\ b = V2p^. 27, a = y/p^ -\- p, c=p + l.
V 6
<..
PLANE TRIGONOMETRY
iw the right triangle ACBy as shmtm in § 7:
2^. Find the length of side a if sin^ = f, and c = 20.6. ^ ^ ^^^^
*^ 39. Find the length of side b if cos^ = 0.44, and c = 3.6.
"^30. Find the length of side a if tan^ = 3f, and b = 2-^.
. ' 31. Find the length of side b if eot^ = 4, and a = 1700.
^32. Find the length of the hypotenuse if sec .4 = 2, and ^ = 2000
33. Find the length of the^ypotenuse if esc ^ = 6.4, and a = 35.6.
Find the hypotenuse and oth^r side of a right triangle, given :
\
N
\
36. b = 4:, cscA = If.
\\
34. b = 6y tan^ = |.
o. ^ 35. a = 3.6, cos^ = 0.6. ^37. b = 2, sin^ = 0.6.
^"^ 38. The hypotenuse of a right triangle is 2.6 mi., sin A = 0.6, and
^ cos ^ = 0.8. Compute the sides of the triangle.
"^""^S. Construct with a protractor the angles 20®, 40®; and 70®;
determine their functions by measuring the necessary lines and
compare the values obtained in this way with the more nearly
correct values given in the following table :
20°
40°
70°
sin
cos
tan
cot
sec
CSC
0.342
0.643
0.940
0.940
0.766
0.842
0.864
0.839
2.747
2.747
1.192
0.364
1.064
1.806
2.924
2.924
1.666
1.064
Findj by means of the above table, the sides and hypotenuse of a
^ right triangle, given:
^"''^40. ^ = 20®, c = l.
41. A = 20®, c = 4.
42. A = 20®, c = 3.6.
43. A = 20®, c = 4.8.
44. ^ = 20®, c=7f
45. A = 40®, c =1. N. 60. A = 70®, c = 2.
46. A = 40®, c=3.fc,/\51. ^ = 70®,a=2.
47. A = 40®, c=7. ^ ^52. ^ = 70®, ft = 2.
48. A = 40®, c =10.7. '>53. A = 70®, a = 26.
49. A = 40®, c = 260. -54. A = 70®, b =160.
^55. By dividing the length of a vertical rod by the length of its
horizontal shadow, the tangent of the angle of elevation of the sun
at that time was found to be 0.82. How high is a tower, if the
length of its horizontal shadow at the same time is 174.3 yd. ?
''**'56. A pin is stuck upright on a table top and extends upward
1 in. above the surface. When its shadow is J in. long, what is the
tangent of the angle of elevation of the sun ? How high is a tele-
graph pole whose horizontal shadow at that instant is 21 ft. ?
FUNCTIONS OF ACUTE ANGLES 7
8. Functions of Complementary Angles. In the annexed figure we
see that B is the complement of A ; that is, B = 90° —A, Hence,
sin ^ = - = cos 5 = cos (90° — A),
cos ^ = - = sin B = sin (90° — A),
tan^ = I = cot 5 = cot (90° - A),
cot ^ = - = tan5 = tan (90° — ^),
\ sec ^ = - = CSC B = CSC (90° —A)j
\ CSC y4 = - = sec 5 = sec (90° —-A),
a ^
That is, each function of an a^ute angle is equal to the eo-named
function of the complementary angle,
Co-sine means complement'* 8 sine, and similarly for the other co-functions.
It is therefore seen that sin 76® = cos (90® — 76®) = cos 16®, sec 82® SC =
CSC (00® - 82® SO') = CSC 7® 30', and so on.
Therefore, any function of an angle between 46^ and 90^ may he
found by talcing the co-wimed function of the complemerUary angle,
which is between ff* and 45^,
Hence, we need never have a direct table of functions beyond 46®. We shall
presently see (§ 12) that this is of great advantage.
Exercise 2. Functions of Complementary Angles
EocpresB as functions of the complemeTvtary angle :
1. sin 30°. 5. sin 50°. 9. sin 60°. 13. sin 75° 30'.
2. cos 20°. 6. tan 60°. 10. cos 60°. 14. tan 82° 45'.
3. tan 40°. 7. sec 75°. 11. tan 45°. 15. sec 68° 15'.
4. sec 25°. 8. esc 85°. 12. sec 45°. ie. cos 88° 10'.
Express a>s functions of an angle less than 45^ :
17. sin 65°. 20. cos 52°. 23. sin 89°. 26. sin 77^°.
18. tan 80°. 21. cot 61°. 24. cos 86°. 27. cos82j°.
19. sec 77°. 22. esc 78°. 26. sec 88°. 28. tan 88.6°.
Find Ay given the following relations:
29. 90° - .4 = .4. 31. 90° - .4 = 2 A,
30- coSi4 = dim A, 32. cos^l = sin2ii.
Jf/V'^v^ J-ryA^ /Jr- ^a^T^u^ <?^^>^!C^ ^ 0^^ (
8
PLAJNE TKIGOiffOMETRY
9. Functions of 46**. The functions of certain angles, among them
45°, are easily found. In the isosceles right triangle A CB we have
/I = B = 46°, and a^b. Furthermore, since a^-\-b!^ = c^, we have
2 a* = c", a V2 = c, and a = ^c V2. Hence,
/"sin46° = cos45° = i-^ = H V2;
{
^"^ tan 45° = cot 46° = ^ = 1;
aV2
V
sec 45° = esc 45° =
= V2.
a
We have therefore found all six functions of 45". For purposes of computa-
tion these are commonly expressed as decimal fractions. Since V2 = 1.4142 +,
we have the following values :
sin 46° = 0. 7071, cos 46° = 0. 7071,
tan 46° = 1, cot 46° = 1,
sec 46° = 1.4142, esc 46° = 1.4142.
10. Functions of 30° and 60°. In the equilateral triangle AA^B
here shown, BC i s the p erp endicular bisector of the base. Also,
i = Jc, and a = Vc^-ft« = Vc2- Jc^ = JcV3. Hence,
y^ sin 30° = cos 60*
cos 30° = sin 60° = - =
tan 30° = cot 60° = - =
\^ cot 30°
h
c
a
c
h^
a
1
iV3;
V3 3^^"'
sec 30°
CSC 30°
tan 60°
CSC 60°
sec 60°
= f =V3;
a
l
c
a
c
I
b C
icV3 3
2.
?v5,
The sine and cosine of 80^, 46°, and 60° are easily remembered, thus :
8in30°=i\/l, sin46°=iV2, sin60°=J\/3;
cos30°=jV3, cos46°=jV2, co860°=jVi.
The functions of other angles are not so easily computed. The computation
requires a study of series and is explained in more advanced works on mathe-
matics. For the present we assume that the functions of all angles have beep
computed and are available, as is really the case.
FUNCTIONS OF ACUTE ANGLES 9
Exercise 3. Functions of 30^, 45^, and 60**
GHven v^ = 1,7320^ express as decimal fractions the following (^
1. sin 30*. 4. cot 30*. 7. sin 60*. 10. cot 60*.
2. cos 30*. 5. sec 30*. 8. cos 60*. 11. sec 60*.
3. tan 30*. 6. esc 30*. 9. tan 60*. 12. esc
Write the ratios of the following, simplifying the results :
13. sin 45* to sin 30*. ^19. sin 30* to sin 60*.
14. cos 45* to cos 30*. ^20. cos 30* to cos 60*.
16. tan 46* to tan 30*. \jl. tan 30* to tan 60*.
16. cot 45* to cot 30*. 22. cot 30* to cot 60*.
^17. sec 45* to sec 30*. 23. sec 30* to sec 60*.
^18. CSC 45* to CSC 30*. 24. esc 30* to esc 60*.
Express as functions of angles less than 45^:
^25. sin 62* IV 40". 29. sin 75.8*.
\
\
26. tan 75* 28' 35". ^30. cos 82.75*.
27. sec 87* 32' 51". ^31. tan 68.82*.
28. cos 88* 0' 27". 32. sec 85.95*.
Find Aj given the following relations:
33. 90* -A = 45* - ifA, 38. cos^ = sin(45* - ^.4)
34. 90*-J^=A. ^^9. cotJ.4 = tan^.
35. 45* '\-A = 90* -A. 40. tan(45* +.4)= eot^.
3G. 90* — 4^ = ^. ^^41. cos 4^= sin ^.
^37. 90*— ^ = n.4. 42. cot.4 = tan n.4.
^43. By what must sin 45* be multiplied to equal tan 30* ?
44. By what must sec 45* be multiplied to equal esc 30* ?
•*^5. By what must cos 45° be multiplied to equal tan 60* ?
46. By what must esc 60* be divided to equal tan 45* ?
47. By what must esc 30° be divided to equal tan 30* ?
48. What is the ratio of sin 45* see 45* to cos 60* ?
49. What is the ratio of cos 45* esc 45* to cos 30* esc 30* ?
50. What is the ratio of sin 45* sin 30* to cos 45* cos 30* ?
51. What is the ratio of tan 30* cot 30* to tan 60* cot 60* ?
52. From the statement tan 30* = ^ VS find cot 60*.
10 PLANE TKIGONOMETKV
11. Values of the Functions. The values of the functions have
been computed and tables constructed giving these values. One
of these tables is shown on page 11 and will suffice for the work
required on the next few pages.
This table gives the values of the functions to four decimal places for every
degree from 0° to 90°. All such values are only approximate, the values of the
functions being, in general, incommensurable with unity and not being ex-
pressible by means of common fractions or by means of decimal fractions with
a finite number of decimal places.
12. Arrangement of the Table. As explained in § 8, cos 45^ = sin 45^,
cos 46® = sin 44®, cos 47® = sin 43®, and so on. Hence the column
of sines from 0® to 45® is the same as the column of cosines from
45® to 90®. Therefore
In finding the functions of angles from 0^ to 45^ read from the top
cUfwn ; in finding the functions of angles from 4S^ to 90^ read ftom
the bottom^ up.
Exercise 4. Use of the Table
From the table on page 11 find the vataxf^^^JJie foUotving :
1. sin 5®. 9. cos 6®. 17. cot 5®. 25. secO®.
2. sin 14®. 10. sin 84®. 18. tan 86®. 26. esc 90®.
3. sin 21®. 11. cosl4®. 19. cot 11®. 27. secl5®.
4. sin 30®. 12. sin 76®. 20. tan 79®. 28. esc 76®.
5. cos 86®. 13. cos 24®. 21. tan 21®. 29. esc 12®.
6. cos 76®. 14. sin 66®. 22. cot 69®. 30. sec 78®.
7. cos 69®. 16. cos 35®. 23. tan 45®. 31. esc 44®.
8. cos 60®. 16. sin 65®. 24. cot 45®. 32. sec 46®
^ 33. Find the difference between 2 sin 9® and sin (2 x 9®).
\ 34. Find the difference between 3 tan 5® and tan (3 x 5®).
. 35. Which is the larger, 2 sec 10® or sec (2 x 10®) ?
.J 3G. Which is the la4:ger, 2 esc 10® or esc (2 x 10®)?
"" 37. Which is the largeT;%iK)s 15® or cos (2 x 15®)?
. 38. Compare 3 sin 20® with sin (3 x 20®); with sin (2 x 20®).
39. Compare 3 tan 10® with tan (3 x 10"*); with tan (2 x 10®).
^' 40. Compare 3 cos 10® with cos (3 x 10®); with cos (2 x 10®).
"^41. Is sin (10® -h 20®) equal to sin 10® -f- sin 20® ?
42. When the angle is increased from 0® to 90® which of the six
functions are increased and which are decreased ?
FUNCTIONS OF ACUTE ANGLES
11
Table of Trigokometbig Functions for every Degree
FROM 0** TO 90*
Angle
Bin
COB
tan
cot
BCC
CSC
0°
.0000
1.0000
.0000
GO
1.0000
GO
oo°
1°
.0175
.9998
.0175
57.2900
1.0002
57.2987
89°
2°
.0349
.9994
.0349
28.6363
1.0006
28.6537
88°
3°
.0523
.9986
.0524
19.0811
1.0014
19.1073
87°
4°
.0698
.9976
.0699
14.3007
1.0024
143356
86°
5**
.0872
.9962
.0875
11.4301
1.0038
11.4737
85°
6°
.1045
.9945
.1051
9.5144
1.0055
9.5668
84°
7°
.1219
.9925
.1228
8.1443
1.0075
8.2055
83°
8°
.1392
.9903
.1405
7.1154
1.0098
7.1853
82°
9°
.1564
.9877
.1584
6.3138
1.0125
6.3925
81°
io»
.1736
.9848
.1763
5-6713
1.0154
5.7588
80°
11°
.1908
.9816
.1944
5.1446
1.0187
5.2408
79°
12°
.2079
.9781
.2126
4.7046
1.0223
4.8097
78°
• 13°
.2250
.9744
.2309
4.3315
1.0263
4.4454
77°
14°
.2419
.9703
.2493
4.0108
1.0306
4.1336
76°
15°
.2588
.9659
.2679
3.7321
1.0353
3.8637
75°
16°
.2756
.9613
.2867
3.4874
1.0403
3.6280
74°
17°
.2924
.9563
.3057
3.2709
1.0457
3.4203
73°
18°
.3090
.9511
.3249
3.0777
1.0515
3.2361
72°
19°
.3256
.9455
.3443
2.9042
1.0576
3.0716
71°
20»
.3420
.9397
.3640
2.7475
1.0642
2.9238
70°
21°
.3584
.9336
.3839
2.6051
1.0711
2.7904
69°
22°
.3746
.9272
.4040
2.4751
1.0785
2.6695
68°
23°
.3907
.9205
.4245
2.3559
1.0864
2.5593
67°
24°
.4067
.9135
.4452
2.2460
1.0946
2.4586
66°
25°
.4226
.9063
.4663
2.1445
1.1034
2.3662
65°
26°
.4384
.8988
.4877
2.0503
1.1126
2.2812
64°
27°
.4540
.8910
.5095
1.9626
1.1223
2.2027
63°
28°
.4695
.8829
.5317
1.8807
1.1326
2.1301
62°
29°
.4848
.8746
.5543
1.8040
1.1434
2.0627
61°
30**
.5000
.8660
.5774
1.7321
1.1547
2.0000
60°
31°
.5150
.8572
.6009
1.6643
1.1666
1.9416
59°
32°
.5299
.8480
.6249
1.6003
1.1792
1.8871
58°
33°
.5446
.8387
.6494
1.5399
1.1924
1.8361
57°
34°
.5592
.8290
.6745
1.4826
1.2062
1.7883
56°
•
36°
.5736
.8192
.7002
1.4281
1.2208
1.7434
55°
36°
.5878
.8090
.7265
1.3764
1.2361
1.7013
54°
37°
.6018
.7986
.7536
1.3270
1.2521
1.6616
53°
38°
.6157
.7880
.7813
1.2799
1.2690
1.6243
52°
39°
.6293
.7771
.8098
1.2349
1.2868
1.5890
51°
40°
.6428
.7660
.8391
1.1918
1.3054
1.3250
1.5557
50°
41°
.6561
.7547
.8693
1.1504
1.5243
49°
42°
.6691
.7431
.9004
1.1106
1.3456
1.4945
48°
43°
.6820
.7314
.9325
1.0724
1.3673
1.4663
47°
44°
.6947
.7193
.9657
1.0355
1.3902
1.4396
46°
45°
.7071
.7071
1.0000
1.0000
1.4142
1.4142
45°
cos
sin
cot
tan
CSC
sec
Angle
f.
i
PLAKE TRIGONOMETRY
13. Reciprocal Functions. Considering the definitions of the six
functions, we see that, since
sin^ = ->
c
C0Sii= -)
c
tan^=*7>
. c
CSC i4 = - >
a
^ec^ = ^,
cot ^ = - >
a
The sine is the reciprocal of the cosecant, the cosine is the reciprocal
of the secant, and the tangent is the reciprocal of the cotangent.
That is,
/ Sin A = 7 ) cos^ = 7 9 tajiA = — — — >
\ CSC A sec A cot A
^ 1 ,'11
t» CSCA=—. -) seeder ■ ■ , > COt Jf = -•
■■■ J sin A cos A *y^ t&nA
Hence sin -4 esc -4 = 1, cos -4 sec A = 1, and t9JiAcotA = l. For example,
from the table on page 11 we find sin 27° esc 27° thus :
sin 27° = 0.4540. ^
csc27° = 2.2027.
Therefore gin 27° esc 27° = 0.4540 x 2.2027
= 1.00002580, or approximately 1.
We have shown that sin -4 esc -4 =1 exactly, but the numbers given in the
table are, as jbefH/re stated, correct only to four decimal places.
Exercise 5. Use of the Table
Umig^the values given in the table on page 11, show as above that
the follcf&ing are reciprocals :
1. sin 30*^, CSC 30*. 4. sin 10*, csclO^ 7. sin75^csc75^
2. sin25^csc25^ 5. tanlO^ cotlO^ 8. COSTS**, sec 76^
3. cos 36^ sec 35*. 6. cos 10*, sec 10*. 9. tan 75*, cot 75*.
1;0; From the table show that the ratio of sin 20* esc 20* to tan 50*
cot 50* is 1.
11. Similarly, show that cos 40* sec 40* : tan 70* cot 70* = 1.
In the right triangle ACB, as shown in §7 :
'^ 19. Find the length of side a if ^ = 30*, and c = 75.2.
13. Find the length of side a if ^ = 45*, and c = 1.414.
^^ 14. Find the length of side ^ if A = 30*, and c = 115.47.
15. Find the length of side a\i A = 60*, and b = 34.64.
16. Find the length of side ft if ^ = 60*, and c = 25.72.
17.' Find the length of side a if ^ = 30*, and c = 45.28.
I
t
\
yCjTv^ y^ A^ '"(^^^^
FUNCTIONS OF ACUTE ANGLES 13
14. Other Relations of Functions. Since, from the figure iu § 7,
a^ + ^^ = (r*, we have ^a ^2
or (T) sin'^ + cos'^ = 1.
It is customary to write sin^^ for (sin-4)2, and similarly for the other
functions.
This formula is one of the most important in trigonometry and
should be memorized. From it we see that
sm
in ^ = Vl — cos^^,( ^/cos A = Vl — sin^^.
Furthermore, since tan^ = -> sin^ = -> and cos^ = -> it follows
*^^^ - , sinil ""
tanil =
©
1 -f- ^ = -^. Hence we see that
\fy cos -4 y' >.
/^ This is also an important fonnula to be memorized. From it we see that \ /Z-<^<
/ tan A cos-4 = sin A, and, in general, that we can find any one of the functions, / ^
I sine, cosine, or tangent, given the other two. ^
Furthermore, from the same equation a* + ^ = c^ we see that
f^, 1 + tan'il = secM.
In a similar manner we may prove that 1+ i = -^J whence we
have the formula ^ 1 + cot'il = csc'il. '
These two formulas should be memorized. '
From these formulas the following relations can easily be deduced :
sin X = cos X tanx = cos ar/cot x = tanx/sec x,
cos X = cot X sin x = cot a;/csc x = sin x/tana;.
^Je^TLx = sin x sec x = sin ar/cos x = sec aj/csc x.
cot X =i-^c X cos X = CSC x/secx = cos a;/sin x,
sec X =^|Pfin X esc x = tan aj/sin x = esc a;/cot a;.
CSC X = s^^j^cot a; = sec x/tanx = cot aj/cos x.
It is often convenienmo recall these relations, and this can be done by the
aid of a simple mnemon^ : „0Sfti
^ sinx secx
cosx cscx
"cotx
the above diagram, any function is equal to the prodvct of the two adjacerd
inctions, or to the quotientittf either adjacent function divided by the one beyond it.
<
\
^.-
PLANE TRIGONOMETRY
16. Practical Use of the Sine. Since by definition we nave
- = sin^,
c
we see that a = c sin^.
We might also derive the equation
c =
a
sin^
But since = esc A (§ IS), it is easier at present to use
sin-d.
c = a csc^,
and this will be .considered when we come to study the cosecant.
1. Given c = 38 and A = 40**, find a.
As above,
From the table,
and
a = csin^.
sin 40*'= 0.6428
c= 38
61424
19 284
c sin ^ = 24.4264
But since the table on page 11 gives only the first four figures of sin 40°, we
can expect orUy the first four fi^/ures of the result to be correct. We therefore
say that a = 24.43 — . If the third decimal place were less than 5, the value
of a would be written 24.42 +.
Some check should always be applied to the result. In this case we may
proceed as follows : 24.4264 h- 38 = 0.6428, which is sin 40°.
2. Given c = 10 and a = 6.293, find^.
Since
6.293
^ • A
c
we have
10
0.6293 = sin^.
Looking in the table we see that
0.6293 = sin 39** ;
whence A — 39**.
3. Given a = 4.68J and A = 22**, find c.
As stated above, c may be found from the formula a = c sin ^ by
using a and sin^, although we shall later use the cosecant for this
purpose. Substituting the given values, we have
4.68i = ^ sin 22^
or 4.6826 = 0.3746 c.
Dividing by 0.3746, 12.6 = c.
What check should be applied here and in Ex. 2 ?
4
FUNCTIONS OF ACUTE ANGLES
16
Bzercise 6. Use of the Sine
Find a to four figures^ given the following :
1. c = 10, ^ = 10*.
2. c = 15, ^ = 16".
QFind Aj given the following :
\. c = 10, a = 2.079.
^6. c = 20, a = 6.840.
3. = 58, A = 45**.
4. = 75, ^ = 50^
A ^7. c = 2, a=l.
^8. c = 50, a = 34.1.
/^.
ir /-VJ
•■r
r*
2586.
9. A 50-foot ladder resting against the
side of a house reaches a point 25 ft. from
the ground. What angle does it make with
the ground?
In all such cases the ground should be considered level and the side of the
building should be considered vertical unless the contrary is expressly stated.
^^ 10. From the top of a rock a cord is
stretched to a point on the ground, making
an angle of 40® with the horizontal plane.
The cord is 84 ft. long. Assuming the cord
to be straight, how high is the rock?
\ 11. Find the side of a regular decagon in-
scribed in a circle of radius 7 ft.
What is the central angle? What is half of this
angle ? Find BC and double it. By this plan we can
find the perimeter of any inscribed regular polygon,
given the radius of the circle. In this way we could
approximate the value of ir. For example, we see that the semiperimeter of a
polygon of 90 sides in a unit circle is 90 x sin 2°, or 90 x 0.0349, or 8.141.
12. The edge of the Great Pyramid is
609 ft. and makes an angle of 52® with the
horizontal plane. What is the height of the
pyramid ?
"'^^ 13. Wishing to measure BC, the length of a
pond, a surveyor ran a line CA at right angles
to BC, He measured AB and Z.A, finding
that ^5=928 ft., and ^ = 29^ Find the
length of BC.
In practical surveying we would probably use an oblique triangle, although
the work as given here is correct. The oblique triangle is considered later.
16
PLANE TRIGONOMETRY
16. Practical Use of the Cosine. Since by definition we have
we see that
- = cos^,
c
h = cco^A,
1. Given c = 28 and ^ = 46*, find b.
From the table, cos 46° = 0.6947
and c = 28
6 6676
13 894
19.4616
Hence, to four figures, h = 19.46.
2. Given c = 2 and «» = 1.9022, find A.
- = cos A,
c
Since
we have 1.9022 -v- 2 = 0.9611 = cos A.
From the table, 0.9611 = cos 18°.
Hence A = 18°.
What check should be applied here and in Ex. 1 ?
Exercise 7. Use of the Cosine
Fiifhd b to four figures,, given the following :
1. c = 11, A = 10°.
[^. c = 14, A = 16°.
3. c = 28, ^ = 24°.
4. c = 4:l,A= 39°.
5. c = 75,A = 42°.
Find A, given the follovring :
11. c = 10, «> = 9.848.
^^12. c = 20, «> = 19.126.
13. c = 40, «> = 36.952.
14. c = 17.6, «» = 8.8.
15. c = 600, b = 227.
21. A flagstaff breaks off 22 ft. from the top and, the parts still
holding together, the top of the staff reaches the earth 11 ft. from
the foot. What angle does it make with the ground ?
6.
c
= 2.8,
A =
= 48°.
L^
c
= 9.7,
A =
= 62°.
8.
c
= 11.2,
A =
= 68°.
9.
c
= 12.6,
A =
= 67°.
10.
c
= 28.26
,A =
= 76°.
I<f6.
c
= 600,
b =
: 206.2.
17.
c
= 200,
b =
117.66.
18.
c
= 187,
b =
93i.
19.
c
= 300,
b =
102|.
20.
c
= 1000,
b =
104^.
FUNCTIONS OF ACUTE ANGLES 17
f5. Wishing to measure the length of a pond,
a class constructed a right triangle as shown in
the figure. If AB = 640 ft. and A = 60*, required
the distance A C.
23. In the same figure what is the length of
AC when AB = 600 ft. and ^ = 40* ?
24. In the same figure, if ^C= 731.4 ft. and AB= 1000 ft., what
is the size of angle A ?
25. A regular hexagon is inscribed in a circle of
radius 9 in. How far is it from the center to a side ?
Having found this distance, the apothem^ and knowing
that a side of the regular hexagon equals the radius, we
can find the area, as required in Ex. 26.
26. What is the area of a regular hexagon inscribed in a circle of
radius 8 in. ?
l^^Jffhr A ship sails northeast 8 mi. It is then how many miles to the
east of the starting point ?
Northeast is 45° east of north. In all such cases in plane trigonometry the
figure is supposed to be a plane. For long distances it would be necessary to
considei^ spherical triangle.
U^iB. Some 16-foot roof timbers make an angle of
30® with the horizontal in an A-shaped roof, as
shown in the figure. Find AA\ the span of the roof.
29. An equilateral triangle is inscribed in a circle of radius 12 in.
How far is it from the center to a side ?
\^yJ9(C K crane AB, 30 ft. long, makes an angle
of X degrees with the horizontal line AC. Find
the distance A C when x = 20 ; when x = 45 ; when
aj = 66 ; when aj = ; when x = 90.
31. In Ex. 30 what angle does the crane make with the horizontal
when ^C = 15 ft.? when ^C = 30 ft.? ^I ^ ^. , D
32. The square ^i^, of which the side is 200 ft.,
is inscribed in the square CM. ^C is 181.26 ft.
Required the angles that the sides of the small
square make with the large one.
33. In Ex. 32 find the required angles when
AB = 15 in. and 5C = 7^ in.; when ^JS = 20 in. and BC = 10.3 in.
34. The edge of the Great Pyramid is 609 ft., and it makes an angle
of 62** with the horizontal plane. What is the diagonal of the base ?
18
PLANE TRIGONOMETRY
17. Practical Use of the Tangent. Since by definition we have
a
— = tanil.
we see that a=zh tan^l.
Given ft = 12 and A = 35®, find a.
From the table, tan 36* = 0.7002
ft = 12
14004
7 002
8.4024
Hence, to four fignres, a = 8.402.
The figures 1, 2, • • • , 9 are often spoken of ks MgnjficarA figures. In 8.402 the
zero is, however, looked upon as a significant figure, hut not in a case like
12,560. The first four significant figures in 0.6706067 are 6706.
18. Angles of Elevation and Depression. The angle of elevation^ or
the angle of depression, of an object is the angle which a line from
the eye to the object makes
with a horizontal lihe in the
same vertical plane.
Thus, if the observer is at O, z
is the angle of elevation of B, and
y is the angle of depression of C,
In measuring angles with a
transit the height of the instru-
ment must always be taken into account. In stating problems, however, it is not
convenient to consider this every time, and hence the angle is supposed to be
taken from the level on which the instrument stands, unless otherwise stated.
1. From a point 6 ft. above the ground and 160 ft. from the foot
of a tree the angle of elevation of the top is observed to be 20®.
How high is the tree?
We have a = b tan A
= 160 tan 20°
= 160 X 0.3640
= 64.6.
Hence the height of the tree is 64.6 ft. + 6 ft., or 69.6 ft.
2. From a point ^ on a cliff 60 ft. high, including the instrument,
the angle of depression of a boat 5 on a lake is observed to be 25°.
How far is the boat from C, the foot of the cliff ? ^^i ^^
We have ZBA (7=66°. Hence B(7= 60 tan 66°. From the
table, tan 66° = 2.1446. Hence BC = 60 x 2.1446 = 128.67.
FUNCTIONS OF ACUTE ANGLES 19
Bzercise 8. Use of the Tangent
Find a to four significant figures, given the follomng :
'1. ft = 37, .4 = 18^ 6. ft = 4.8, A = 51**.
2. ft = 26, ^ = 23^ 7. ft = 9.6, A = 57^
.3. ft = 48, ^ = 31^ * 8. ft = 23.4, A = 62^
^4. ft = 62, .i = 36^ 9. ft = 28.7, A = 75^
6. ft = 98, ^ = 45^ 10. ft = 39.7, ^ = 86^
JJViu? -4, ^ven the follomng :
11. a = 6, ft = 6. * 14. a =13.772, ft = 40.
12. a = 0.281, ft = 2. ^16. a = 2.424, ft = 6.
13. a = 4.752, ft = 30. 16. a = 20.503, ft = 10.
^17. A man standing 120 ft. from the foot of a clmrch spire finds
that the angle of elevation of the top is 50®. If his eye is 5 ft. 8 in.
from the ground, what is the height of the spire ?
18. When a flagstaff 55.43 ft. high casts a shadow 100 ft. long
on a horizontal plane, what is the angle of elevation of the sun ?
19. A ship S is observed at the same instant
from two lighthouses, L and L\ 3 mi. apart.
Z.VLS is found to be 40® and ZZZ'5 is found to
be 90**. What is the distance of the ship from V ?
What is its distance from L ?
20. From the top of a rock which rises vertically, including the
instrument, 134 ft. above a river bank the angle of depression of
the opposite bank is found to be 40®. How wide is the river?
21. An A-shaped roof has a span ^^4 'of 24 ft. The
ridgepole R is 12ft. above the horizontal line AA\
What angle does AR make with AA^'i with i2J' ?
with the perpendicular from i? on ^^' ?
"^22. The foot of a ladder is 17 ft. 6 in. from a wall, and the ladder
makes an angle of 42® with the horizontal when it leans against
the wall. How far up the wall does it reach ?
23. A post subtends an angle of 7® from a point on the ground
50 ft. away. What is the height of the post ?
24. The diameter of a one-cent piece is f in. If the coin is held
so that it subtends an angle of 40® at the eye, what is its distance
from the eye ?
20
PLANE TRIGONOMETRY
19. Practical Use of the Cotangent. Since by definition we hare
we see that
- = QOtA.
a
b = acot^.
For example, given a =* 71 and A = 28%
find b.
From the table, cot 28' = 1.8807
and a = 71
18807
131 649
133.6297
Hence, to four significant figures, b = 133.5.
What check should be applied in this case ?
X
Exercise 9. Use of the Cotangent
Find b to four significant figures^ given the following :
1. a = 29, A= 48^ 5. a = 425, A = 38^
2. a = 38, A= 12\ 6. a = 19j, A = 36^
3. a = 66, A= 19^, 7. a = 24.8, A = 43^
4. a = 72, A= 40^ 8. a = 266.8, A = 76*.
Find A^ given the following :
9. a =72, 6 = 72. 10. a = 60, ft =128.67.
11. How far from a tree 60 ft. high must a person lie in order to
see the top at an angle of elevation of 60** ?
12. From the top of a tower 300 ft. high, in-
cluding the instrument, a point on the ground
is observed to have an angle of depression of
36®. How far is the point from the tower ?
13. From the extremity of the shadow cast by a church spire
160 ft. high the angle of elevation of the top is 63®. What is the
length of the shadow ?
14. A tree known to be 60 ft. high, stand-
ing on the bank of a stream, is observed
from the opposite bank to have an apgle of
elevation of 20®. The angle is measured
on a line 6 ft. above the foot of the tree. How wide is the stream ?
300
FUNCTIONS OF ACUTE ANGLES 21
20. Practical Use of the Secant. Since by definition we liave
7 = sec-4,
we see that c = 5 sec -4.
For example, given ft = 15 and A = 30®, find c.
From the table, sec 30**= 1.1547
and h = 15
5 7735
11547
17.3206
Hence, to four significant figures, c = 17.32.
Exercise 10. Use of the Secant
Find c to four significant figures^ given the following :
1. ft = 36, ^ = 27**. 4. ft = 22J, A = 48^
2. ft = 48, ^ = 39^ 5. ft = 33.4, A = 53^
3. ft = 74, ^ = 43^ 6. ft = 148.8, A = 64^
Find Ay given the following :
7. ft = 10, c = 13J. 8. ft = 17.8, c = 35.6.
9. A ladder rests against the side of a build-
ing, and makes an angle of 28° with the ground.
The foot of the ladder is 20 ft. from the building.
How long is the ladder ? ^^"^ ^
10. From a point 50 ft. from a house a wire is stretched to a
window so as to make an angle of 30° with the horizontal. Find
the length of the wire, assuming it to be straight. ^fi
11. In measuring the distance AB si surveyor
ran the line AC, making an angle of 50° with AB,
and the line BC perpendicular to AC, He meas-
ured A C and found that it was 880 ft. Required
the distance AB. a^ — ■ — r
12. From the extremity of the shadow cast by a tree the angle of
elevation of the top is 47°. The shadow is 62 ft. 6 in. long. How
far is it from the top of the tree to the extremity of the shadow ?
13. The span of this roof is 40 ft., and the roof
timbers AB make an angle of 40° with the hori-
zontal. Find the length of AB,
PLAKE TRIGONOMETBY
31. Practical Uie of the Cosecant. Since by definition we have
- = osc-4,
a
we see that c = a esc -4.
For example, given a = 22 and A = 35**,
find c.
From the table, esc 35** = 1.7434
and a = 22
3 4868
34 868
38.3548
Hence, to four significant figures, c = 38.35.
Check. Since - = sin^, 22 -- 88.85 = 0.5786 = sin 85<».
c
Exercise 11. Use of the Cosecant
Find c to four aignificant figures^ given the following :
X
1. a = 24, ^ = 29^
2. a=:36, ^=4r.
3. a = 56, ^ = 44^
Find Aj given the following :
7. a s= 10, c = 11.126.
8. a =: 13, c = 27.6913.
4. a = 56^, A = 61^
5. a = 75.8, A = 69**.
6. a = 146.9, ^ = 74^
9. a = 5^, c = 6.0687.
10. a = 75, c = 106.065.
11. Seen from a point on the ground the angle of elevation of an
aeroplane is 64**. If the aeroplane is 1000 ft. above the ground, how
far is it in a straight line from the observer ?
12. A ship sailing 47** east of north changes its latitude 28 mi. in
3 hr. What is its rate of sailing per hour ?
13. A ship sailing 63** east of south changes its latitude 45 mi. in
5 hr. What is its rate of sailing per hour ?
14. From the top of a lighthouse 100 ft., including the instrument,
above the level of the sea a boat is observed under an angle of depres-
sion of 22**. How far is the boat from the point of observation ?
15. Seen from a point on the ground the angle of elevation of the
top of a telegraph pole 27 ft. high is 28**. How far is it from the
point of observation to the top of the pole ?
16. What is the length of the hypotenuse of a right triangle of
which one side is llf in. and the opposite angle 43** ?
FUNCTIONS OP ACUTE ANGLES
23
22. Functions as Lines. The functions of an angle, being ratios, are
numhers ; but we may represent them by lines if we first choose a unit
of length, and then construct right tri-
angles, such that the denominators of
the ratios shall be equal to this unit.
Thus in the annexed figure the
radius is taken as 1, the circle then
being spoken of as a unit circle. Then
. Drawing the four perpendiculars as
shown, we have:
sma; = — =JfP;
tana5 = --— r ^ AT\
OA
OT
secaj = -7-7 = or;
OA
cosx = •— ■ = 0M\
BS
cota; = — = 55;
OS ^„
csca; = -7-^ = 05.
OH
In each case we have arranged the fraction so that the denominator is 1.
TLJfP A T»
For example, instead of taking — — for tan x we have taken the equal ratio —— ,
because OA = 1. ^^ ^^
OP 08
Similarly, instead of taking for cscx we have taken the equal ratio — — ,
because OB = 1. ^-^ ^^
This explains the use of the names tangent and secant, A T being
a tangent to the circle, and OT being a secant.
Formerly the functions were considered as lines instead of ratios and received
their names at that time. The word sine is from the Latin sintLS, a translation
of an Arabic term for this function.
We see from the figure that the sine of the complement oi x
is NPf which equals OM; also that the tangent of the complement
of X is BSy and that the secant of the complement of x is OS.
Exercise 12. Functions as Lines
1. Represent by lines the functions of 45®.
2. Represent by lines the functions of an acute angle greater
than 45®.
UsiTig the above figure, determine which is the greater :
3. sin a; or tan x. 5. sec a; or tana?. 7. cos a; or cotaj.
4. sin x or sec x. 6. esc x or cot x. 8. cos x or esc a^
24 PLANE TKIGONOMETBY
Construct the angle Xj given the following :
9. tan a: = 3. ^ 11. cosa: = ^. ^ 13. jis.a; = 2 cos 05.
10. CSC X = 2, 12. sin x = cos x, ^ 14. 4 sin a; = tan x.
16. Show that the sine of an angle is equal to one half the chord
of twice the angle in a unit circle.
16. Find x if sin x is equal to one half the side of a regular deca-
gon inscribed in a unit circle.
Given x andf ytX-\-y being less than 90°, construct a line equal to
•17. sin (a; -h y)— sin a?. 20. cos x — cos (a; -|- y),
^^vJ.8. tan (a; -\- y) — tan x, 21. cot x — cot (a: -|- y).
19. sec (x + y)— sec x. 22. esc x — esc (x + y),
23. tan(a; + y)— sin (x + y)-\- tan x — sin x.
Given an angle Xj construct an angle y such that :
^24. sin y == 2 sin a:. ^^^^ 28. tany = 3tana;.
26. cos y = i cos X, 29. sec y = esc x.
26. sin y b= cos x. 30. sin y = ^ tan x,
27. tany = cota;. 31. siny = ftana:.
32. Show by construction that 2 sin ^ > sin 2^, when A < 45*.
33. Show by construction that cos A< 2 cos 2^, when A< 30*.
34. Given two angles A and B, A-\-B being less than 90**; show
that sin(^ + 5) < sin A + sin B,
35. Given sin a; in a unit circle; find the length of a line in a
circle of radius r corresponding in position to sin a;.
36. In a right triangle, given the hypotenuse c, and sin^=m;
find the two sides.
37. In a right triangle, given the side 5, and tan A=m; find the
other side and the hypotenuse.
Construct, or show that it is impossible to construct, the angle x,
given the following :
38. sin a; = J. ^^1. cos a; = 0. 44. tana; = J.
39. sin a; =1. 42. cos aj = J. 46. cot a: = ^.
40. sin a; = |. 43. cos x = ^, 46. sec aT^ ^.
47. Using a protractor, draw the figure to show that sin 60* •=;
cos (i of 60*), and sin 30* = cos (2 x 30*).
%'Lt'tr- U-^^ £■ A^-?^
FUNCTIONS OF ACUTE ANGLES
25
e see that
23. Cluuiges In the Fanctloiia. If we suppose AAOP, or x, to in-
crease gradually to 90°, the sine MP increases to AfP', M"P", and so
on to OB.
That is, the riue increases from 6' for the
angle 0", to 1 tor the angle 00°. Hence and
1 are called the limiting vabtes of the sine.
Similarly, AT and OT gradually in-
crease in length, while OM, SS, and OS
gradually decrease. That ia,
As an acute angle increases to 90°, its
sine, tangent, and secant also increase, while
its eosine, cotangent, and cosecant decrease.
It we suppose x to decrease to 0°, OF coin-
cide* with OA and is parallel to BS. Therefore
MP and AT vanish, OM becomes equal to OA, wliile BS and OS are each
infinitely long and are represented in value by the symbol uoJ Similarly, i
may consider the changes as x increases from 0° to 90°.
Hence, as the angle x increases from 0° to 90°, i
sin 3! increases from'O to 1,
COS X decreases from 1 to 0,
tan X increases from to oo,
cot X decreases from oo to 0,
sec X increases from 1 to oo,
CSC x decreases from oo to 1.
W« also s«e that
sines and cosines are never greater than 1 ;
secants and cosecants arc never less than 1 ;
tangents and cotangents may have any values from to («.
In particular, for the angle 0°, we have the following values :
sin 0° = 0, tan O" = 0, . sec 0° =1,
003 0°=!, cot 0° = 00, CSC 0° = 00.
For the angle 90° we have the following values :
sin 90° = 1, tan 90° = oo, sec 90° = oo,
cos 90° = 0, cot 90° = 0, CSC 90° = 1.
By reference to the figure and the table it is apparent that the functions of
45° are never equal to half of the corresponding functions of Wf> Thus,,' ''
tin ^' = 0.7011, tan46° = l, sec46° = 1.4142.
Om4^=0.7071. cot46'< = l, 'csc4fi° = 1.4142.
26 PLAKE TRIGONOMETRY
Exercise 13. Functions as Lines
1. Draw a figure to show that sin 90* = 1.
2. What is the value of cos 90' ? Draw a figure to show this.
3. What is the value of sec 0' ? Draw a figure to show this.
4. What is the value of tan 90' ? Draw a figure to show this.
6. What is the value of cot 90* ? Draw a figure to show this.
6. As the angle increases, which increases the more rapidly, the
sine or the tangent ? Show this by reference to the figure.
7. If you double an angle, does this double the sine ? Show this
by ?iBf erence to the figure.
8. If you bisect an angle, does this bisect the tangent ? Prove it.
9. State the angle for which these relations are true :
sin X = cos X, tan x = cot x, sec x = esc x.
Show this by reference to the figure.
10. If you know that sin 40' 15'= 0.6461, and cos 40' 15'= 0.7632,
and that the difference between each of these and the sine and cosine
of 40' 15' 30" is 0.0001, what is sin 40' 15' 30" ? cos 40' 15' 30" ?
11. If^ou know that tan 20' 12' is 0.3679, and that the difference
between this and tan 20' 12' 15" is 0.0001, what is tan 20' 12' 15" ?
12. If you know that cot 20' 12' is 2.7179, and that the difference
between this and cot 20' 12' 15" is 0.0006, what is cot 20' 12' 15" ?
13. If you know that tan 66.5' is 2.2998, and that the difference
between this and tan 66.6' is 0.0111, what is tan 66.6' ?
14. If you know that cos 57.4' is 0.5388, and that the difference
between this and cos 57.5' is 0.0015, what is cos 57.5' ?
Draw the angle xfor which the functions have thefollomng values
and state (jpage IT) to the nearest degree the value of the angle :
15. sin X = 0.1. 21. tan x = 0.1. 27. sec a; = 1.2.
16. sin a: = 0.4. 22. tana: = 0.23. 28. seca; = 1.3.
17. sin X = 0.7. 23. tana; = 0.4. 29. sec a; = 1.7.
18. cos X = 0.9. 24. cot X = 4.0. 30. esc x = 2.0.
19. cosaj = 0.8. 25. cot a; = 2.9. 31. csca; = 3.6.
20. cos X = 0.7. 26. cot x = 0.9. 32. esc a; = 1.66.
33. Find the value of sin x in the equation sin x : 1-1.5 = Ol
smx
Which root is admissible ? Why is the other root impossible ?
<*>■■
CHAPTER II
USB OF TH£ TABLE OF NATURAL FUNCTIONS
24. Sexagesimal and Decimal Fractions. The ancients, not having
developed the idea of the decimal fraction and not having any con-
venient notation for even the common fraction, used a system based
upon sixtieths. Thus they had units, sixtieths, thirty-six hun-
dredths, and so on, and they used this system in all kinds of theo-
retical work requiring extensive fractions.
For example, instead of 1^^ they would use 1 28', meaning l|-| ; and instead
of 1.61 they would use 1 SC 86'^ meaning l|^ + gfl^. The symbols for de-
grees, minutes, and seconds are modern.
We to-day apply these sexagesimaZ (scale of sixty) fractions only
to the measure of time, angles, and arcs. Thus
3 hr. 10 min. 15 sec. means (3 + -1^ + ^\Iq) hr.,
and 3* 10' 15" means (3 + 1§ + ^^y.
In medieval times the sexagesimal system was carried farther than this. For
example, 8 10' 20'' 80"' 45*^ was used for 8 + ^ + ^ + — + -^. Some
*^ ' 60 602 608 60*
writers used sexagesimal fractions in which the denominators extended to 60^'
Since about the year 1600 we have had decimal fractions with
which to work, and these have gradually replaced sexagesimal frac-
tions in most cases. At present there is a strong tendency towards
using decimal instead of sexagesimal fractions in angle measure. On
this account it is necessary to be familiar with tables which give
the functions of angles not only to degrees and minutes, but also to
degrees and hundredths, with provision for finding the functions also
to seconds and to thousandths of a degree. Hence the tables which
will be considered and the problems which will be proposed will in-
volve both sexagesimal and decimal fractions, but with particular
attention to the former because they are the ones still commonly used.
The rise of the metric system in the nineteenth century gave an
impetus to the movement to abandon the sexagesimal system. At the
time the metric system was established in France, trigonometric tables
were prepared on the decimal plan. It is only within recent years,
however, that tables of this kind have begun to come into use.
27
28 PLANE TRIGONOMETRY
25. Sexftgeainul Table. The following is a, portion of a page from
tlie Wentworth-Smith Trigonometric Tables ;
41- 42'
'
ain coa tao cot
'
6561 7547 8693 1.1504
fiO
6563 75+5 S698 1,1497
^y
6S6S 7543 8703 1.1490
w
6567 7541 8708 1.1483
17
6569 7539 S713 1,1477
6572 7538 8718 1.1470
bb
'
Bin COB tan cot
'
6691
7411
wn4
1.1106
m'
1
frtS4(
74-»l
1IHI9
l.llOO
St
'/.
6696
H/.'i
■Xll.S
1.1093
58
A
6fi9H
74?/.
m?.n
1.1087
S7
4
6700
V4'/4
i\m
1.1080
S6
G
6702
7422
yo30
1.1074
»
' CM
•in
eot
-I'l
48" 47"
The functions of 41" and any number of minutes are foimd by
reading down, under the abbreviations sin, cot, tan, cot.
For example. Bin 41° = 0.6561, sin 42° = O.S601,
0D841°2' = 0.7643, cos42° =0.7431,
t»n 41" 4' = 0.8713, tan 42° 3' = 0.9020,
cot41°6' = l.H70, cot 42" 5' = 1.1074.
Decimal points are usually omitted fn the tables when it is obvious where
thejr should be placed.
The secant and cosecant are seldom given in tables, being reciprocals of
tlie codne and Mne. We Bball presentlj see that we rarel; need them.
Since sin 41" 2' is the same as cos 48° 58' (§ 8), we may use the
same table for 48' and any number of minutes by reading up, above
the abbreviations cos, sin, cot, tan.
For example, cos48''56' = 0.B572,
sin 48° 66' = 0.7539,
cot48°58' = 0.8703,
tan 48° 6^= 1.1407,
coa47° 56' = 0.6702,
sin 47° 56' = 0.7424,
cot47° 57' = 0.8020,
tan 47° 59'= 1.1100.
Trigonometric tables are generally arranged with the degrees from
0° to 44° at the top, the minutes being at the left; and with the
degrees from 45° to 89° at the bottom, the minutes being at the right.
Therefore, in looking for functions of an angle from 0° to 44° 59',
look at the top of the page for the degrees and in the left column
for the minutes, reading the number below the proper abbreviation.
For functions of an angle from 45° to 90° (89" 60"), look at the bot-
tom of the page for the degrees and in the right-band column for
the minutes, reading toe number above the proper abbreviati<m.
NATURAL FUNCTIONS 29
Exercise 14. Use of the Sexagesimal Table
From the table on page 28 find the values of the following :
1. cos4r. 6. sin 48° 59'. 11. sin 42' 4'.
2. tan42^ 7. sin 47' 58'. 12. cos 47' 56'.
3. cos 41' 1'. . 8. cos 48' 59'. 13. tan 41' 3'.
4. tan 42' 2'. 9. cos 47' 59'. 14. cot 48' 57'.
5. cos 41' 5'. 10. cos 48' 57'. 15. tan 48' 57'.
In the right triangle A CB, in which C = 90° :
"^16. Given c = 27 and A = 41' 3', find a.
17. Given c = 48 and ^ = 42' 4', find a.
18. Given c = 61 and A = 41' 2', find b,
"\l9. Given c = 72 and ^ = 42' 3', find b.
20. Given ft = 24 and ^ = 41' 3', find a.
21. Given ft = 28 and ^ = 42' 4', find a.
'22. Given a = 42 and ^ = 41' 1', find ft.
23. Given a = 60 and ^ = 42' 4', find ft.
24. Given c = 86 and ^ = 48' 56', find a.
25. Given c = 92 and ^ = 48' 57', find a.
26. Given ft = 45 and ^ = 47' 55', find a.
27. Given ft = 85 and ^ = 47' 59', find a.
^28. Given a = 86 and A = 48' 56', find ft.
29. Given a = 98 and ^ = 47' 58', find ft.
30. Given ft = 67 and c = 100, find A.
31. A hoisting crane has an arm 30 ft. long. When the arm makes
an angle of 41' 3' with x, what is the length oi y?
what is the length of a: ?
32. In Ex. 31 suppose the arm is raised until
it makes an angle of 41' 5' with x, what are then
the lengths of y and x? - »
33. From a point 128 ft. from a building the angle of elevation
of the top is observed, by aid of an instrument 5 ft. above the ground,
to be 42' 4'. What is the height of the building ?
34. From the top of a building 62 ft. 6 in. high, including the
instrument, the angle of depression of the foot of an electric-light pole
is observed to be 41' 3'. How far is the pole from the building ?
\:
30
PLAKB TRIGONOMETRY
26. Decimal Table. It would be possible to have a decimal table
of natural functions arranged as follows :
o
sin cos tan cot
o
0.0
0000 1.0000 0000 00
90.0
0.1
0017 1.0000 0017 573.0
89.9
0.2
0035 1.0000 0035 286.5
89.8
0.3
0052 1.0000 0052 191.0
89.7
0.4
0070 1.0000 0070 143.2
89.6
0.5
0087 1.0000 0087 114.6
89.5
o
sin cos tan cot
o
4.0
0698 9976 0699 14.30
86.0
4.1
0715 9974 0717 13.95
85.9
4.2
0732 9973 0734 13.62
85.8
4.3
0750 9972 0752 13.30
85.7
4.4
0767 9971 0769 13.00
85.6
44(
0785 9%9 0787 12.71
S6Ji
o
cos sin cot tan
o
• • •
• •• ••• ••• •••
...
o
cos sin cot tan
o
Since, however, the decimal divisions of the angle have not yet become com-
mon, it is not necessary to have a special table of this kind. It is quite con-
venient to use the ordinary sexagesimal table for this purpose by simply
referring to the Table of Conversion of sexagesimals to decimals and vice versa.
This table is given with the other Wentworth-Smith tables prepared for use
with this book. Thus if we wish to find sin 27.76°, we see by the Table of
Conversion that 0. 75° = 45', so we simply look for sin 27° 45'.
For example, using either the above table or, after conversion to sexagesimals,
the common table, we see that :
sin 0.4° = 0.0070,
cos 4.1° = 0.9974,
tan0.6° = 0.0087,
cot 4.3°= 18.30,
sin 85.5° = 0.9969,
cos 86.5° = 0.0785,
tan 85.8°= 18.62,
cot 85.9° = 0.0717.
Exercise 15. Use of the Decimal Table
From the above table find the values of the following :
1~ sin 0.5®.
2. tan 0.4°.
3. sin 4°.
4. cos 4.2°.
6. tan 4.5°.
6. sin 4.1°.
7. cos 4.3°.
8. tan 4.4°.
9. cot 4.5°.
10. cot 4.2°.
11. sin 85.7°.
12. sin 85.9°.
13. cos 85.6°.
14. tan 85.9°.
15. cot 85.6°.
16. sin 89.5°.
17. cos 85.9°.
18. tan 89.6°.
19. cot 89.7°.
20. cot 85.8°.
21. The hypotenuse of a right triangle is 12.7 in., and one acute
angle is 85.5°. Find the two perpendicular sides.
22. From a point on the top of a house the angle of depression of
the foot of a tree is observed to be 4.4°. The house, including the
instrument, is 30 ft. high. How far is the tree from the house ?
23. A rectangle has a base 9.5 in. long, and the diagonal makes an
angle of 4.5° with the base. Find the height of the rectangle and the
length of the diagonal.
NATURAL FUNCTIONS 81
37. Interpolation. So long as we wish to find the functiofns of an
acute angle expressed in degrees and minutes^ or in degrees and
tenths, the tables already explained are sufficient. But when the
angle is expressed in degrees, minutes, and seconds, or in degrees
and hundredths, we see that the tables do not give the values of the
functions directly. It is then necessary to resort to a process called
interpolation.
Briefly expressed, in the process of interpolation we assume that
sin 42J® is found by adding to sin 42** half the difference between
sin 42** and sin 43^
In general it is evident that this is not true. For example, in
the annexed figure the line values of the functions of 30° and OOP
are shown. It is clear that sin 30° is more than half sin 60°, that
tan 30° is less than half tan 60°, and that sec 30° is more than half
sec 60°. This is also seen from the table on page 11, where
sin 80° = 0.5000, tan 30° = 0.5774, sec 30° = 1.1647,
sin 60° = 0.8660, tan 60° = 1.7321, sec 60° = 2.0000.
For angles in which the changes are very small^ interpolation gives
results which are correct to the number of decimal places given in
the table.
For example, from the table on page 11 we have
sin 42° = 0.6691
sin 41° = 0.6561
Difference for 1°, or 60", = 0.0130
Difference for V = ^ of 0.0130 = 0.0002.
Adding this to sin 41°, we have
sin 41° V = 0.6563,
a result given in the table on page 28.
But if we wish to find tan 89.6° from tan 89.5° and tan 89.7°, we cannot
use this method because here t?ie changes are very great^ as is always the case
with the tangents and secants of angles near 90°, and with the cotangents and
cosecants of angles near 0°. Thus, from the table on page 80,
tan 89.7° =191.0
tan89.5°= 114.6
Difference for 0.2° = 76.4
Difference for 0.1° = 38.2
Adding this to tan 89.5°, tan 89.6° = 152.8,
whereas the table shows the result to be 143.2.
When cases arise in which interpolation cannot safely be used, we resort to
the use of special tables that give the required values. These tables are
explained later. Interpolation may safely be used in all examples given in
the early part of the work.
32 PLANE TRIGONOMETRY
28. Interpolation applied. The following examples will illustrate
the cases which arise in practical problems. The student should
refer to the Wentworth-Smith Trigonometric Tables for the func-
tions used in the problems.
1. Find sin 22° 10' 20".
From the tables, sin 22° 11' = 0.3776
sin 22° 10' = 0.3778
Difference for 1', or 60", the tabular difference = 0.0003
Difference for 20'' is J J of 0.0003, or 0.0001
Adding this to sin 22° 10', we have
sin 22° 10' 20" = 0.3774
2. Find cos 64° IV 30".
From the tables, cos 64° 17' = 0.4339
cos 64° 18' = 0.4337
Tabular difference = 0.0002
Difference for 30" is gg of 0.0002, or 0.0001
Since the cosine decreases as the angle increases we must subtract 0.0001
from cos 64° 17', which gives us
cos64° 17' 30" = 0.4338
3. Find tan 37.54°.
By the Table of Conversion, 0.64° = 32' 24".
From the tables, tan 37° 33' = 0.7687
tan 37° 82' = 0.7683
Tabular difference = 0.0004
Difference for 24" is f J, or 0.4, of 0.0004 = 0.0002
Adding this to tan 37° 32', we have
tan 37.54° = tan 37° 32' 24" = 0.7685
4. Given sin x = 0.6456, find x.
Looking in the tables for the sine that is a little less than 0.6456, and for the
next larger sine, we have
0.6467= sin 40° 13'
0.6455 = sin 40° 12'
0.0002 = tabular difference
Therefore x lies between 40° 12' and 40° 13'.
Furthermore, 0.6456 = sinjc
0.6465 = sin 40° 12'
0.0001 = difference
But 0.0001 is \ of 0.0002, the tabular difference, so that x is halfway from
40° 12' to 40° 13'. Therefore we add ^ of 60", or 30", to 40° 12'.
Hence x = 40° 12' 30".
We interpolate in a similar manner when we use a decimal table.
NATURAL FUNCTIONS
33
Szercise 16. Use of the Table
Find the value% of the following :
V
1. sin 27** 10
2. sin 42* 15
3. sin 56* 29
4. sin 65* 29
5. cos 36* 14
6. cos 43* 12
7. cos 64* 18
8. tan 28* 32
9. tan32*41
10. tan42* 38
30".
30".
40".
40".
30".
20".
45".
20".
30".
30".
11. tan 52* 10' 46".
12. tan 68* 12' 45".
13. tan 72* 15' 50".
>^--n4. tan 85* 17' 45".
15. tan 86* 15' 50".
16. cot 5* 27' 30".
17. cot 6* 32' 45".
18. cot 7* 52' 50".
—19. cot 8* 40' 10".
— 20. cot 9* 20' 10".
21. Given sin a; = 0.6391, find x. Then find cos x.
'22. Given sin a: = 0.7691, find x. Then find cosa;.
— -183. Given cos x = 0.3174, find a-. Then find sin a:.
fy ^ 24. Given tan x = 2.8649, find x. Then find cot x.
— -26. Given tan a: = 5.3977, find a;. Then find cot a:.
First converting to sexagesimals^ find the follotving :
26. sin 25.5*. 31. cos 78.52*. 36. cos 11.25*.
27. sin 25.55*. ^^32. tan 78.59*. 37. cot 12.32*.
..^^28. sin 32.75*5jrtj^* 33. cos 81.43*. 38. cot 13.54*.
29. sin 41.65*. 34. tan 82.72*. 39. cot 15.48*.
30. sin 64.75*. 36. tan 84.68*. ^ — —40. cot 16.62*.
Find the value of o^n each of the following equations :
41. sina: = 0.5225.^Nl5. cos a; = 0.7853. 49. tan a; = 2.6395.
42. sin X = 0.5771. 46. cos x = 0.7716. 50. tan a: = 4.7625.
-43. sin a: = 0.6601. 47. cosaj = 0.9524. 51. tana: = 4.7608.
44. sin X = 0.7023. 48. cos x = 0.7115. 52. cot x = 3.7983.
53. If sinaj = 0.6431, what is the value of cosa; ?
■^ 54. If cos X = 0.7652, what is the value of sin a; ?
^^55. If tan X = 0.6827, what is the value of sin x ?
56. If tanaj = 0.6537, what is the value of a; ? of cota; ?
57. If cota =r 1.6550, what is the value of a; ? of tana? ? Verify
the second result by the relation tan x = 1/cot x.
84 PLAKE TRIGONOMETRY
29. Application to the Right Triangle. In §§ 15-21 we learned
how to use the several functions in finding various parts of a right
triangle from other given parts, the angles being in exact degrees.
In §§ 26-28 we learned how to use the tables when the angles were not
necessarily in exact degrees. We shall now review both of these phases
of the work in connection with the solution of the right triangle.
In order to solve a right triangle, that is, to find both of the acute
angles, the hypotenuse, and both of the sides, two independent parts
besides the right angle must be given.
In speaking of the sides of a right triangle it should be repeated that we shall
refer only to sides a and 6, the sides which include the right angle, using the
word hypotenuse to refer to c. It will be found that there is no confusion in
thus referring to only two of the three sides by the special name sides.
By ind^endent parts is meant parts that do not depend one upon another.
For example, the two acute angles are not independent parts, for each is equal
to 90° minus the other.
The two given parts may be :
1. An acute angle and the hypotenuse.
That is, given A and c, or B and c. If ^ and c are
given, we have to find a and &. The angle B is known
from the relation B = 90° — A, If B is given, we can
find A from the equation A = 90° — B,
2. An acute angle and the opposite side.
That is, given A and a, or B and 6. If A and a are given, we have to find
Bj 6, and c, and similarly for the other case.
3. An acute angle and the adjacent side.
That is, given A and 6, or B and a. If A and b are given, we have to find B,
a, and c, and similarly for the other case.
4. The hypotenuse and a side.
That is, given c and a, or c and h. If c and a are given, we have to find A, B,
and 6, and similarly for the other case.
6. The two sides.
That is, given a and 6, to find A, B, and c. Using side to include hypotenuse,
we might combine the fourth and fifth of these cases in one.
In each of these cases we shall consider right triangles which
have their acute angles expressed in degrees and minutes, in de-
grees, minutes, and seconds, or in degrees and decimal parts of a
degree In this chapter the angles are given and required only to
the nearest minute.
NATUBAL FUNCTIONS
85
30. OiTen an Acute Angle and the Hypotenuse. For example, given
4 =43° 17', == 26, find B, a, and h.
1. 5=90'-X = 46*43'.
2. - = sm^ ; .•. a = c sin-4.
c
3. - = cos-4; .*. 6 = ccoSil.
e
e
a =
0.6856
26
41136
13 712
17.8256
17.83
C08^
ft =
0.7280
26
4 3680
14 560
18.9280
18.93
As usual, when a four-place table is employed, the result is given to foui
figures only. The check is left for the student.
31. Given an Acute Angle and the Opposite Side. For example; given
A =13** 58', a =15.2, find B, b, and e,
1. 5=90** -^=76* 2'.
2. -=scot-4: ,\b = aGotA,
a
8.2
e
= sin^; .*. c = -7
a
QinA
a =15.2, cot ^ = 4.0207
4.0207
15.2
80414
20 1035
40 207
b = 61.11464
= 61.11
as
1
^ b
C
a = 15.2, sin A =
0.2414
62.97
=
2414)152000.00
14484
7160
4828
23320
21726
In divldihg 15.2 by 0.2414, we adopt the modem plan of first multiplying
each by 10,000. Only part of the actual division is shown.
Instead of dividing a by sin -4. to find c, we might multiply a by esc ^, as on
page 22, except that tables do not generally give the cosecants. It will be seen
in Chapter III that, by the aid of logarithms, we can divide by sin A as readily
as multiply by cscul, and this is why the tables omit the cosecant.
36
PLANE TRIGONOMETEY
32. Given an Acute Anj^le and the Adjacent Side. For example, ^ven
A == 27** 12', b = 31, find B, a, and c.
1. 5= 90** -^=62*^48'.
2. - = tan^; ,\ a = btSinA,
b
3. - = cos^; .'.<? = 7*
c cosA
tan^
b
a =
0.6139
31
6139
15 417
15.9309
15.93
& = 31, cos ^=0.8894
34.85 = e
8894)310000.00
26682
43180
35576
We might multiply b by sec -4. instead of dividing by cos -4. The reason for
not doing so is the same as that given in § 31 for not multiplying by esc -4.
33. Given the Hypotenuse and a Side.
c = 63, find Ay -B, and b.
1. sm^= -•
c
2. ^5=90**-^.
For example, given a
B
47,
3. b = y/?^
a
= V(c + a) (c — a).
In the case of Vc^ — a^ we can, of course, sciiiare c, square a, take the dif-
ference of these squares, and then extract the square root. It is, however, easier
to proceed by factoring d^ — a^ as shown. This will be even more apparent when
we come, in Chapter III, to the short methods of computing by logarithms.
a = 47, c = 63
0.7460
63)47.0000
441
2 90
2 62
380
378
sin^ = 0.7460
.-.^ = 48** 15'
.-. 5=4r46'
e-\- a
= 110
e — a
= 16
660
110
'-a«
= 1760
.-.*«
= 1760
.'.b
= V1760
= 41.95
NATURAL FUNCTIONS
37
34. Given the Two Sides. For example, given a = 40, 5 = 27, find
A^ B, and c.
1. tan^= 7*
2. ^=90** -.4.
3. c = Va^ 4- 61 ^ 5^27~Cr
Of course c can be found in other ways. For example, after finding tan J. we
can find A^ and hence can find sin^. Then, because sin -4 = a/c^ we have
c = a/sin ^. When the numbers are small, however, it is easy to find c from
the relation given above.
a^ = 1600
^2= 729
a = 40, 6 = 27
1^ = 1.4815
tan ^=1.4815
.•.^ = 55'59'
.•.^=34M'
c" = 2329
.•.c=V2329
= 48.26
35. Checks. As already stated, always apply some check to the
results. For example, in § 34, we see at once that a^ = 1600 and b^
is less than 30^, or 900, so that c^ is less than 2500, and c is less
than 50. Hence the result as given, 48.26, is probably correct.
We can also find B independently.
For since
tan5 = -,
we see that
tan5=fj = 0.6750,
and therefore that
B = 34° 1'.
Exercise 17. The Right Triangle
Solve the right triangle A CB^ in which C = 90^^ given :
1. a = 3, & = 4.
2. a = 7, c = 13.
3. a = 5.3, A = 12° 17'.
4. a =10.4, 5=43^18'.
6. c = 26, ^ = 37** 42'.
6. c =140, 5= 24° 12',
7. 5=19, c = 23.
8. 5 = 98, c = 135.2.
9. 5 = 42.4, ^ = 32° 14'.
10. h =200, B= 46° 11.
11. a = 95, & = 37.
12. a = 6, c = 103.
13. a = 3.12, 5= 5° 8'.
14. a = 17, c = 18.
15. c = 57, ^ = 38° 29'.
16. a 4- c = 18, 5 = 12.
17. a + c = 90, 5 = 30.
18. a + c = 46, 5 = 30.
88 PLANE TEIGONOMETRY
Solve the right triangle AC By in which C =s SO**, given :
19. a = 2.5, A = SS** 10' 30". 26. a = 48, ^ = 26.6*.
20. a = 6.7, A = 42* 12' 30". 27. c = 26, ^ = 24.6*.
21. a = 6.4, 5= 29* 18' 30". 28. c = 40, ^ = 32.66*.
22. a = 7.9, JB= 36* 20' 30". 29. c = 80, ^ = 66.61*.
23. c = 6.8, A = 29* 42' 30". 30. c = 76, ^ = 63.46*.
24. c = 360, A = 34* 20' 30". 31. a = 46, 5= 60.69*.
26. ^i = 260, ^ = 41* 10' 40". 32. ^ = 90,^=68.26*.
33. Each equal side of an isosceles triangle is 16 in., and one of
the equal angles is 24* 10'. What is the length of the base ?
34^ Each equal side of an isosceles triangle is 26 in., and the ver-
tical angle is 36* 40'. What is the altitude of the triangle ?
35. Each equal side of an isosceles triangle is 26 in., and one of
the equal angles is 32* 20' 30". What is the length of the base ?
36. Each equal side of an isosceles triangle is 60 in., and the ver-
tical angle is 60* 30' 30". What is the altitude of the triangle ?
37. Eind the altitude of an equilateral triangle of which the side
is 60 in. Show three methods of finding the altitude.
38. What is the side of an equilateral triangle of
which the altitude is 62 in. ?
39. In planning a truss for a bridge it is necessary
ta have the upright BC = 12 ft., and the horizontal
XC = 8 ft., as shown in the figure. What angle does .^ — - ,^ -
AB make with ^ C ? with BC ?
40. In Ex. 39 what are the angles if AB = 12 ft. and ^C= 9ft. ?
41. In the figure of Ex. 39, what is the length of BC if ^JB= 16 ft.
and « = 62* 10'?
^42. Two angles of a triangle are 42* 17' and 47* 43' respectively,
and the included side is 26 in. Eind the other two sides.
43. A tangent AB, drawn from a point ^ to a circle, makes an angle
of 61* 10' with a line from A through the center. If AB = 10 ft., what
is the length of the radius ?
44. How far from the center of a circle of radius 12 in. will a
tangent meet a diameter with which it makes an angle of 10* 20'?
46. Two circles of radii 10 in. and 14 in. are externally tangent.
What angle does their line of centers make with their common
exterior tangent?
CHAPTER III
LOGARITHMS
36. Importance of Logaritluns. It has already been seen that the
trigonometric functions are, in general, incommensurable with unity.
Hence they contain decimal fractions of an infinite number of places.
Even if we express these fractions only to four or five decimal places,
the labor of multiplying and dividing by them is considerable. For
this reason numerous devices have appeared for simplifying this
work. Among these devices are various calculating machines, but
none of these can easily be carried about and they are too expensive
for general use. There is also the slide rule, an inexpensive instru-
ment for approximate multiplication and division, but for trigono-
metric work this is not of particular value because the tables must be
at hand even when the slide nde is used. The most practical device
for the purpose was invented early in the seventeenth century and
the credit is chiefly due to John Napier, a Scotchman, whose tables
appeared in 1614. These tables, afterwards much improved by
Henry Briggs, a contemporary of Napier, are known as tables of
logarithms, and by their use the operation of multiplication is re-
duced to that of addition; that of division is reduced to subtraction ;
raising to any power is reduced to one multiplication; and the
extracting of any root is reduced to a single division.
For the ordinary purposes of trigonometry the tables of functions
used in Chapter II are fairly satisfactory, the time required for
most of the operations not being unreasonable. But when a problem
is met which requires a large amount of computation, the tables of
natural functions, as they are called, to distinguish them from the
tables of logarithmic functions, are not convenient.
For example, we shall see that the product of 2.417, 8.426, 517.4, and 91.63
can be found from a table by adding four numbers which the table gives.
In the case of -^ — x — '— x we shall see that the result can be found
62.9 5.28 9283
from a table by adding six numbers.
Taking a more difficult case, like that of i j— x ^'^ , we shall see that it
\711 0.379
is necessary merely to take one third of the sum of four numbers, after which
the table gives 1*8 the result.
39
40 PLAKE TEIGONOMETRY
37. Logarithm. The exponent of the power to which a given num-
ber, called the basey must be raised in order to be equal to another
given number is called the logarithm of this second given number.
For example, since 10^ = 100,
we have, to the base 10, 2 = the logarithm of 100.
In the same way, since 10^ = 1000,
we have, to the base 10, 3 = the logarithm of 1000.
Similarly, 4 = the logarithm of 10,000,
5 = the logarithm of 100,000,
and so on, whatever powers of 10 we take.
In general, if 6* = JV,
then, to the base 6, x = the logarithm of N,
38. Symbolism. For " logarithm of N " it is customary to write
^^ log iV." If we wish to specify log N to the base by we write logj^iV,
reading this " logarithm of iV to the base 5."
That is, as above, log 100 = 2, log 10,000 = 4,
log 1000 = 3, log 100,000 = 5,
and so on for the other powers of 10.
39. Base. Any positive number except unity may be taken as the
base for a system of logarithms, but 10 is usually taken for purposes
of practical calculation.
Thus, since 2' = 8, logj 8 = 3 ;
since 3* = 81, log8 81 =4;
and since 5* = 626, . logg 626 = 4.
It is more convenient to take 10 as the base, however. For since
102 = 100 and lO^ = 1000,
we can tell at once that the logarithm of any number between 100 and 1000
must lie between 2 and 3, and therefore must be 2 + some fraction. That is,
by using 10 as the base we kno^ immediately the integral part of the logarithm.
When we write log 27, we mean log^Q27 ; that is, the base 10 is to be under-
stood unless some other base is specified.
Since log 10 = 1, because 10^ = 10,
and log 1 = 0, because 10° =1,
and log To = "■ 1> because 10" ^ = y^,
we see that the logarithm of the base is always ly the logarithm, of 1
is always zerOy and the logarithm of a proper fraction is negative.
That this is true for any base is apparent from the fact that
61 = 6, whence logt 6=1;
6° = 1, whence logjl =0;
6- " = — » whence loa— = — n.
6» ^6»
LOGARITHMS 41
Exercise 18. Logarithms
1. Since 2* = 32, what is log^ 32 ?
2. Since 4^ = 16, what is log^ 16 ?
3. Since 10* = 10,000, what is log 10,000 ?
Write the following logarithms:
4. log3l6. 8. logg243. 12. logg36. 16. log 100.
6. log264. 9. log3 729. 13. log^343. 17. log 1000.
6. log3l28. 10. log^266. 14. logg512. 18. log 100,000.
7. log2266. 11. loggl25. 15. logg6561. 19. log 1,000,000.
20. Since 10-^ = to,ot 0.1, what is log 0.1 ?
21. What is log y^, or log 0.01 ? log 0.001 ? log 0.0001 ?
22. Between what consecutive integers is log 52? log 726?
log 2400? log 24,000? log 176,000? log 175,000,000 ?
2^. Between what consecutive negative integers is log 0.08 ?
log 0.008? log 0.0008? log 0.1238? log 0.0123? log 0.002768?
24. To the base 2, write the logarithms of 2, 4, 8, 64, 512, 1024,
ii 1 _i_ 1 1
> TT > T2> 64> 128> 256*
25. To the base 3, write the logarithms of 3, 81, 729, 2187, 6561,
9f Of 27 > TT> "STa » 7 29 ' 2 187*
26. To the base 10, write the logarithms of 1, 0.0001, 0.00001,
10,000,000, 100,000,000.
Write the consecutive integers between which the logarithms of
the following numbers lie :
27. 75. 31. 642. 35. 7346. 39. 243,481.
28. 76.9. 32. 642.75. 36. 7346.9. 40. 5,276,192.
29. 76.06. 33. 642.005. 37. 7346.09. 41. 7,286,348.5
30. 82.96. 34. 793.175. 38. 9182.735. 42. 19,423,076.
Show that the following statements are true:
43. log,4 + log28 + log2l6 + log264 + log22 + log232 = 21.
44. log33 + loggO + logg81 + log3 729 + log327 + log3243 = 21.
45. log^ll + log,, 121 + log,, 1331 + log,, 14,641 = 10.
46. log 1 + log 10 + log 1000 + log 0.1 -f log 0.001 = 0.
47. log 1 -f log 100 + log 10,000 + log 0.01 + log 0.0001 = 0.
48. log 10,000 - log 1000 + log 100,000 - log 100 = 4.
42 PLANE TRIGONOMETRY
40. Logarithm of a Product. The logarithm of the prodttct of two
numbers is equal to the sum of the logarithms of the numbers.
Let A and B be the numbers, and x and y their logarithms. Then,
taking 10 as the base and remembering that x = logA^ and y = logB,
we have ^ = Iqx^
and ^ = 10^.
Therefore AB =^10^+^,
and therefore log ^5= x + 3/
= log^ + logJB.
The proof is the same if any other base is taken. For example,
if X = logb A, we have A = b';
and if y = logb B, we have B = l)V,
Therefore AB = b^ + y,
and \ogbAB==x-\-y
= logb A + logb B,
The proposition is also true for the product of more than two numbers, the
proof being evidently the same. Thus,
log^^C = log^ + log5 + log 0,
and so on for any number of factors.
41. Logarithm of a Quotient. The logarithm of the quotient of two
numbers is equal to the logarithm of the dividend minus the logarithm
of the divisor.
For if A = 10=",
and B=^W,
then 4 = 1^"^
B
and therefore log — = a; — y
B
= log^— logB.
This proposition is true if any base h is taken. For, as in § 40,
A ^
B '
and therefore log& — = x — y
B
= logb -4. — logb B.
It is therefore seen from §§40 and 41 that if we know the logarithms of all
numbers we can find the logarithm of a product by addition and the logarithm
of a quotient by subtraction. If we can then find the numbers of which these
results are the logarithms, we shall have solved our problems in multiplication
and division by merely adding and subtracting.
LOGARITHMS 48
42. Logarithm of a Power. The logarithm of a power of a number
is equal to the logarithm, of the number m,ultiplied by the exponent
For if A=: 10^, ^
raising to the pth. power, A"* = lO^"".
Hence log A'* =px
= plogA,
This is easily seen by taking special numbers. Thus if we take the base 2,
we have the following relations :
Since 2* = 82, then log2 82 = 6;
and since (26)2 _ 322 ^ 1024, then logg 1024 = 2-6
= 2 log2 82.
That is, logg 322 _ 2 logg 32.
43. Logarithm of a Root. The logarithm of a root of a number is
equal to the logarithm of the number divided by the index of the root.
For if ^ = 10^,
1 X
taking the Hh root, A'^ = \0'\
1 X
Henoe log^** = -
r
log A
The propositions of §§ 42 and 48 are true whatever base is taken, aa may
easily be seen by using the base h.
From §§ 42 and 48 we see that the raising of a number to any power, integral
or fractional, reduces to the operation of multiplying the logarithm by the ex-
ponent (integral or fractional) and then finding the number of which the result
is the logarithm.
Therefore the operations of multiplying, dividj[ng, raising to powers, and
extracting roots will be greatly simplified if we can fiiid the logarithms of num-
bers, and this will next be considered.
44. Characteristic and Mantissa. Usually a logarithm consists of
au integer plus a decimal fraction.
The integral part of a logarithm is called the characteristic.
The decimal part of a logarithm is called the mantissa.
Thus, if log 2358 = 8.37162, the characteristic is 3 and the mantissa 0.37162.
This means that lOS-snea = 2368, or that the 100,000th root of the 337,162d
power of 10 is 2363, approximately.
It must always be recognized that the mantissa is only an approximation,
correct to as many decimal places as are given in the table, but not exact.
C-omputations made with logarithms give results which, in general, are correct
only to a certain number of figures, but results which are suflftciently near the
correct result to answer the purposes of the problem.
44 PLANE TRIGONOMETRY
45. Finding the Characteristic. Since we know that
10« = 1000 and 10* = 10,000,
therefore 3 = log 1000 and 4 = log 10,000.
Hence the logarithm of a number between 1000 and 10,000 lies
between 3 and 4, and so is 3 plus a fraction. Thus the characteristic
of the logarithm of a number tetween 1000 and 10,000 is 3.
Likewise, since
10-3 = 0.001 and 10-2 = 0.01,
therefore — 3 = log 0.001 and — 2 = log 0.01.
Hence the logarithm of a number between 0.001 and 0.01 lies
between — 3 and — 2, and so is — 3 plus a fraction. Thus the char-
acteristic of the logarithm of a number between 0.01 and 0.001 is — 3.
Of course, instead of saying that log 1476 is 8 + a fraction, we might say that
it is 4 — a fraction; and instead of saying that log 0.007 is — 3 + a fraction,
we might say that it is — 2 — a fraction. For convenience, however, t?ie man-
tissa of a logarithm is always taken as positive, but the characteristic may be
either positive or negative.
46. Laws of the Characteristic. From the reasoning set forth in
§ 45 we deduce the following laws :
1. The characteristic of the logarithm of a number greater than 1
is positive and is one less than the number of integral places in the
number.
For example, log 75 = 1 + some mantissa,
log 472.8 = 2 + some mantissa,
and log 14,800.75 = 4 + some mantissa.
2. The characteristic of the logarithm, of a number between and 1
is negative and is one greater than the number of zeros between the
decimal point and the first significant figure in the number.
For example, log 0.02 = — 2 + some mantissa,
and log 0.00076 = — 4 + some mantissa.
The logarithm of a negative number is an imaginary number, and hence such
logarithms are not used in computation.
47. Negative Characteristic. If log 0.02 = — 2 + 0.30103, we cannot
write it — 2.30103, because this would mean that both mantissa and
characteristic are negative. Hence the form 2.30103 has been chosen,
which means that only the characteristic 2 is negative.
That is, 2.30103 =-2 + 0.30103, and 6.48561 =— 5 + 0.48661. We may also
write 2.30103 as 0.30103 — 2, or 8.30103 — 10, or in any similar manner which
will show that the characteristic is negative.
LOGAEITHMS 45
48. Mantissa independent of Decimal Point. It may be shown that
108.87107^ ^50 . whence log 2350 = 3.37107.
Dividing 2360 by 10, we have
lQ8.87io7-i^ 102.87107 ^ 236 } whencc log 236 = 2.37107.
Dividing 2360 by 10*, or 10,000, we have
lQ8.87io7-4^ lQi.min ^ 0.236 ; whence log 0.236 = 1.37107.
That is, the mantissas are the same for log 2360, log 236, log 0.236,
and so on, wherever the decimal points are placed.
77ie mantissa of the logarithm of a number is unchanged by any
change in the position of the decimal point of the numier.
This is a fact of great importance, for if the table gives us the mantissa of
log 285, we know that we may use the same mantissa for log 0.00236, log 2.36,
log 23,600, log 235,000,000, and so on.
£zercise 19. Logarithms
Write the characteristics of the logarithms of the following :
1. 76. 6. 2678.
2. 76.4. 7. 267.8.
3. 764. 8. 26.78.
4. 7.64. 9. 2.678.
5. 7640. 10. 26,780.
Griven 3.58681 as the logarithm of 3862^ find the following :
21. log 38.62. 24. log 38,620. 27. log 0.3862.
22. log 3.862. 25. log 386,200. 28. log 0.03862.
23. log 386.2. 26. log 38,620,000. 29. log 0.0003862.
Given I.677M as the logarithm of 0.4736 j find the following .
30. log 4766. 32. log 47,660. 34. log 0.04766.
31. log 4.766. 33. log 47,560,000. 35. log 0.00004766
Gfiven 3.40603 as the logarithm of 2S47, find the following :
36. log 2.647. 38. log 0.2647. 40. log 26,470.
37. log 26.47. 39. log 0.002647. 41. log 26,470,000.
Oiven 1.39794 as the logarithm of 25^ find the following :
42. log 2f 44. log 0.26. 46„ log 26,000. .
43. logj. 45. log 0.026. 47. log 26,000,00a
11.
0.8.
16.
0.0007.
12.
0.08.
17.
0.0077.
13.
0.88.
18.
0.00007.
14.
0.886.
19.
0.10007.
16.
0.005.
20.
0.07007.
46
PLAKB TRIGONOMETEY
49. Using the Table. The following is a portion of a page taken
from the Wentworth-Smith Logarithmic and Trigonometric Tables :
250 — 300
N
12 8 4
5 6 7 8 9
250
251
252
253
254
255
39 794 39811 39829 39 846 39863
39%7 39985 40002 40019 40037
40140 40157 40175 40192 40 209
40312 40329 40346 40364 40381
40 483 40 500 40 518 40 535 40 552
40 654 40 671 40 688 40 705 40 722
39881 39898 39915 39933 39950
40054 40071 40088 40106 40123
40 226 40 243 40 261 40 278 40 295
40398 40415 40432 40449 40466
40 569 40 586 40603 40620 40637
40 739 40 756 40 773 40 790 40807
Only the mantissas are given ; the characteristics are always to be
determined by the laws stated in § 46. Altoays write the characteristio
at once, before writing the mantissa.
For example, looking to the right of 251 and under 0, and writing the proper
characteristics, we have
log 251 =2.89967, log 25.1 =1.39967,
log 2510 = 3.39967, log 0.0251 = 2.39967.
The first three significant figures of each number are given under
N, and the fourth figure under the columns headed 0, 1, 2, . . . , 9.
For example, log252.1 = 2.40157, logO.2647 = 1.40603,
log 25.26 = 1.40226, log 2649 = 3.40637.
Furthermore, log 261.1 = 2.39985 — , the minus sign being placed beneath
the final 6 in the table to show that if only a four-place mantissa is being used
it should be written 3998 instead of 3999.
The logarithms of numbers of more than four figures are found by
interpolation, as explained in § 27.
For example, to find log 25,314 we have
. log 26,320= 4.40346
log 25,310 = 4.40329
Tabular difference = 0.00017
A
0.000068
Difference to be added = 0.00007 ^
Adding this to 4.40329, log 25314 = 4.40336
In general, the tabular difference can be foimd so easily by inspection that
it is unnecessary to multiply, as shown in this example. If any multiplication is
necessary, it is an ea«y matter to turn to pages 46 and 47 of the tables, where
will be found a table of proportional parts. On page 46, after the number 17 ia
the column of differences (D), and under 4 (for 0.4), is found 6.8. In the same
way we can find any decimal part of a difference.
LOGARITHMS 47
Exercise 20. Using the Table
U%ing the tahle^ find the logarithms of the following :
1.2. 9.3485. 17.0.7. X26. 12,340. ^
2. 20. 10. 4462. 18. 0.75. 26. 12,345.
3. 200. 11. 5581. 19. 0.756. 27. 12,347.
4. 0.002. 12. 7007. 20. 0.7567. 28. 123.47.
6. 2100. 13. 5285. 21. 0.0255. 29. 234.62.
6. 2150. 14. 68.48. 22. 0.0036. 30. 41.327.
7 2156. 15. 7.926. 23. 0.0009. 31. 56.283.
8. 2.156. 16. 834.8. 24. 0.0178. ^32. 0.41282.
33. In a certain computation it is necessary to find the sum of the
logarithms of 45.6, 72.8, and 98.4. What is this sum ?
34. In a certain computation it is necessary to subtract the loga-
rithm of 3.84 from the sum of the logarithms of 52.8 and 26.5.
What is the resulting logarithm ?
Perform the following operations :
35. log 275 -f log 321 -f log 4.26 + log 3.87 4- log 46.4.
36. log 2643 + log 3462 + log 4926 -f- log 5376 -|- log 2194.
37. log 51.82 + log 7.263 -f log 5.826 + log 218.7 + log 3275.
38. log 8263 -f log 2179 + log 3972 - log 2163 - log 178.
39. log 37.42 + log 61.73 + log 5.823 - log 1.46 - log 27.83.
40. log 3.427 + log 38.46 + log 723.8 - log 2.73 - log 21.68.
41. In a certain operation it is necessary to find three times
log 41.75. What is the resulting logarithm ?
42. In a certain operation it is necessary to find one fifth of
log 254.8. What is the resulting logarithm?
Perform the following operations:
43. 2 X log 3. 50. ilog2. 57. 0.3 log 431.
44. 3 X log 2. 51. i log 2000. 58. 0.7 log 43.19.
45. 3 X log 25.6. 52. i log 3460. 59. 0.9 log 4.007.
46. 5 X log 3.76. 53. ilog24.7G. 60. 1.4 log 5.108.
47. 4 X log 21.42. 54. i log 368.7. 61. 2.3 log 7.411.
48. 5 X log 346.8. 55. flog 41.73. 62. | log 16.05.
49. 12 X log 42.86. 56. flog 763.8. 63. | log 23.43.
48 PLANE TRIGONOMBTEY
50. Antilogarithm. The number corresponding to a giyen logarithm
is called an antilogarithm.
For ** antilogarithm of N " it is customary to write ^ antilog N.^*
Thus if log 25.31 = 1.40320, antilogl. 40329 = 25.81. Similarly, we see that
antilog 5.40329 = 253,100, and antilog 2.40329 = 0.02531.
51. Finding the Antilogarithm. An antiloga^rithm is found from
the tables by looking for the number corresponding to the given
mantissa and placing the decimal point according to the character-
istic. For example, consider the following portion of a table :
550 — 600
N
12 8 4
5 6 7 8 9
650
551
74 036 74 044 74 052 74 060 74 068
74115 74123 74131 7413974147
74 076 74 084 74 092 74 099 74 107
74 155 74 162 74 170 74 178 74 186
If the mantissa is given in the table, we find the sequence of the
digits of the antilogarithm in the column under N. If the mantissa
is not given in the table, we interpolate.
1. Find the antilogarithm of 5.74139.
We find 74139 in the table, opposite 551 and under 3. Hence the digits of the
number are 5513. Since the characteristic is 5, there are six integral places,
and hence the antilogarithm is 551,300. That is,
log 551,300 = 6.74139,
or antilog 5.74139 = 551,300.
2. Find the antilogarithm of 2.74166.
We find 74170 in the table, opposite 551 and under 7.
Iog0.05517 = 2.74170
log0.05516 = 2.74162
Tabular difference = 0.00008
Subtracting, we see that, neglecting the decimal point, the tabular difference
is 8, and the difference between log x and log 0.05516 is 4. Hence x is | of the
way from 0.05616 to 0.06617. Hence x = 0.065165.
3. Find the antilogarithm of 7.74053.
We find 74060 in the table, opposite 650 and under 3.
Iog55,030,000 = 7.74060
log 55,020,000 = 7.74062
Tabular difference = 0.00008
Reasoning as before, ou is J of the way from 56,020,000 to 56,030,000.
Hence, to five significant figures, x = 56,021,000.
In general, the interpolation gives only one additional figure correct ; that i£,
with a table like the one above, the sixth figure will not be correct if found by
interpolation.
LOGARITHMS
49
Exercise 21. Antilogarithms
7 the antilogarithms of the follomng :
17. 0.23305.
18. 1.43144.
4.0. 1.58041.
21. 3.t'3490.
22. 4.63492.
23. 0.63994.
24. 2.69085.
Cr^
0L.
. 0.4771
2. 3.4771L
3. 3.47712.
4. 2.48359.
6. 4.56844.
6. 1.66276.
7. 2.66978.
8. 5.74819.
0. 3.74076.
\ 2.7f:3^5.
1 i'S4y7.
12. 1.81954.
13. 0.82575.
14. 0.88081.
15. 9.89237.
16. 7.90282.
26. 8.77425.
26. 4.82966.
27. 3.83547.
28. 2.83604.
29. 4.88960.
30. 2.89523
31. 3.89858.
32. 0.93223.
33. If the logaxithm of the product of two numbers is 2.94210,
what is the product of the numbers ?
34. If the logarithm of the quotient of two numbers is 0.30103,
what is the quotient of the numbers ?
35. If we wish to multiply 2857 by 2875, what logarithms do we
need ? What are these logarithms ?
36. If we know that the logarithm of a result which we are seek-
ing is 3.47056, what is that result ?
37. If we know that log VO.000043641 is 3.81995, what is the
value of VO.000043641 ?
38. If we know that log ^0.076553 is 1.81400, what is the value
of <^0.076553 ?
39. The logarithm of V8322 is 1.96012. Find V8322 to three
decimal places.
40. The logarithm of the cube of 376 is 7.72557. Find the cube
of 376 to five significant figures.
41. If we know that log 0.003278^ is 5.03122, what is the value
of 0.003278^ ?
42. Find twice log 731, and find the antilogarithm of the result.
43. Find the antilogarithm of the sum of log 27.8 + log 34.6 +
log 367.8.
Find the antiiogarithms of the following :
44. log 7 + log 2 - log 1.934. 47. 5 log 27.83.
45. log 63 + log 5.8 - log 3.415. 48. 2.8 log 5.683.
46. log 728 + log 96.8 - log 2.768. 49. f (log 2 -f log 4.2).
60 PLANE TRIGONOMETRY
J 2. Multiplication by Logarithms. It has been shown (§40) that
■ogarithm of a product is equal to the sum of the logarithms of
the numbers. This is of practical value in multiplication.
Find the product of 6.16 x 27.05.
From the tables, log 6.15 =0.78888
log 27.06 = 1.43217
logx =2.22105
Interpolating to find the value of x, we have
log 166.4 = 2.22116 logx = 2.22106
log 166.3 = 2.22089 log 166.3 = 2.22089
26 16
Annexing to 166.3 the fraction ^ , we have
X = 166.3^1
= 166.36,
the interpolation not being exact beyond one figure.
If we perform the actual multiplication, we have 6.16 x 27.05 = 166.3676, or
166.36 to two decimal places.
Exercise 22. Multiplication by Logarithms
Umig logarithms, find the following products :
1. 2 X 5. 11. 2 X 60. 21. 36.8 x 28.9.
2. 4 X 6. 12. 40 X 60. 22. 62.7 x 41.6.
3. 3x6. 13. 3 x 500. 23. 2.76 x 4.84.
4. 6x7. 14. 50 X 70. 24. 5.25 x 3.86.
5. 2 X 4. 15. 2 x 4000. 25. 14.26 x 42.35.
6. 3 X 7. 16. 30 X 700. 26. 43.28 x 29.64.
7. 2 X 6. 17. 200 X 60. 27. 629.6 x 348.7.
8. 3 X 6. 18. 30 X 600. 28. 240.8 x 46.09.
9. 7x8. 19. 7 X 80,000. 29. 34.81 x 46.26.
10. 2x9. 20. 200 X 900. 30. 5028 x 3.472.
31. Taking the circumference of a circle to be 3.14 times the
diameter, find the circumference of a steel shaft of diameter 5.8 in.
32. Taking the ratio of the circumference to the diameter as given
in Ex. 31, find the circumference of a water tank of diameter 36 ft.
Using logarithms^ find the following products :
33. 2x3x5x7. 36. 43.8 x 26.9 x 32.8.
34. 3x6x7x9. 37. 627.6 x 283.4 x 4.196.
35. 6 X 7 X 11 X 13. 38. 7.283 x 6.987 x 5.437.
LOGARITHMS 51
53. Negative Characteristic. Since the mantissa is always positive
(§ 45), care has to be taken in adding or subtracting logarithms it
which a negative characteristic may occur. In all such cases it is
better to separate the characteristics from the mantissas, as shown
in the following illustrations :
1. Add the logarithms 2.81764 and 1.41283.
Separating the negative characteristic from its mantissa, we have
2.81764 = 0.81764 - 2
1.41283 = 1.41288
Adding, we have 2.23047 — 2
= 0.23047
2. Add the logarithms 4.21255 and 2.96245.
Separating both negative characteristics from the mantissas, we have
4.21266 = 0.21266 - 4
2.96246 = 0.96246 - 2
Adding, we have 1.17600 — 6
= 6.17600
Exercise 23. Negative Characteristics
Add the following logarithms :
1. 2.41283 + 5.27681. 6. 2.63841 + 1.36158.
2. 2.41283 + 5.27681. 7. 2.41238 + 3.62701.
3. 2.41283 + 5.27681. 8. 5.58623 + 6.41387.
4. 0.38264 -h 4.71233. 9. 6.41382 + 7.58617.
6. 0.57121 -+- 1.42879. 10. 4.22334 + 3.77666.
Using logarithms^ find the following products :
11. 256 X 4875. 18. 0.725 x 0.3465.
12. 2.56 X 48.75. 19. 0.256 x 0.0875.
13. 0.256 X 0.4875. 20. 0.037 x 0.00425.
14. 0.0256 X 0.004875. 21. 47.26 x 0.02755.
15. 0.1275 X 0.03428. 22. 296.8 x 0.1283
16. 0.2763 X 0.4134. 23. 45,650 x 0.0725.
17. 0.00025 X 0.00125. 24. 127,400 x 0.00355.
25. Given sin 25.75^ = 0.4344, find 52.8 sin 25.75^
26. Given cos 37.25^ = 0.7960, find 42.85 cos 37.25*.
27. Given tan 30* 50' 30'' = 0.5971, find 27.65 tan 30* 50' 30".
52 PLANE TRIGONOMETRY
54. Division by Logarithms. It has been shown (§ 41) that the
logarithm of a quotient is equal to the logarithm of the dividend
minus the logarithm of the divisor.
Care must be taken that the mantissa in subtraction does not
become negative (§ 45).
1. Using logarithms, divide 17.28 by 1.44.
•From the tables, logl7.28 = 1.23764
logl.44 = 0.16886
1.07918
= logl2
Hence 17.28 -5- 1.44 = 12.
2. Using logarithms, divide 2603.5 by 0.015998.
log 2003.5 =3.41666
log0.016998 = 2.20407
Arranging these in a form more convenient for subtracting, we have
log 2603.5 =3.41656
log 0.015998 = 0.20407 - 2
3.21149 + 2
= 5.21149 = log 162,740
Hence 2603.6 -f- 0.016998 = 162,740.
3. Using logarithms, divide 0.016502 by 127.41.
log 0.016502 = 2.21753 = 8.21753 - 10
log 127.41 = 2.10520 = 2.10520
6.11233-10
= 4.11233 = log 0.00012952
Hence 0.016502 -4- 127.41 = 0.00012952.
Here we increased 2.21753 by 10 and decreased the sum by 10. We might
take any other number that would make the highest order of the minuend
larger than the corresponding order of the subtrahend, but it is a convenient
custom to take 10 or the smallest multiple of 10 that will serve the purpose.
4. Using logarithms, divide 0.000148 by 0.022922.
log 0.000148 = 4.17026 = 16.17026 - 20
log 0.022922 = 2.36025 = 8.36025 - 10
7.81001 - 10
= 3.81001 = log 0.0064567
Hence 0.000148 -^ 0.022922 = 0.0064567.
5. Using logarithms, divide 0.2548 by 0.05513.
log 0.2648 = 1.40620 = 9.40620 - 10
log 0.06513 = 2.74139 = 8.74139 - 10
0.66481
= log 4.6218
Hence 0.2648 -f- 0.05513 = 4.6218.
LOGARITHMS 53
Exercise 24. Division by Logarithms
Add the following logarithrm:
1. 2.14755 -+- 3.82764. 6. 4.18755 -+- 2.81245.
2. 4.07256 -f 1.58822. 6. 6.28742 + 3.41258.
3. 0.21783 + 1.46835. 7. 4.21722 + 4.78278.
4. 0.41722 4- 3.28682. 8. 6.28720 -+- 3.71280.
9. Find the sum of 2.41280, 4.17623, 5.26453, 0.21020, 7.36423,
2.63577, 6.41323, and 3.28740.
From the first of these logarithms subtract the second :
10. 0.21250, 2.21250. 14. 4.17325, 2.17325.
11. 0.17286,3.27286. 15. 6.82340,3.71120.
12. 2.34222, 6.44222. 16. 3.14286, 1.14000.
13. 3.14726, 1.25625. 17. 3.27283, 5.56111.
Using logarithms^ divide as follows:
18. 10 H- 2. 26. 25,284-4-301. 34 59.29^0.77.
19. 16 -4- 3. 27. 51,742 -5- 631. 35. 2.451 -t- 190.
20. 15 H- 6. 28. 47,348 -5- 623. 36. 851.4 ^ 0.66.
21. 12 ^ 3. 29. 19,224 ^ 540. 37. 0.98902 -f- 99.
22. 12^4. 30. 37,960-^520. 38. 0.41831^5.9.
23. 60-5-12. 31. 84,640^920. 39. 0.08772^4.3.
24. 76 -J- 2b. 32. 65,100 ^ 620. 40. 0.02275 -s- 0.35.
25. 125 -f- 26, 33. 45,990 ^ 730. 41. 0.02736 ^ 0.057
Using logarithms^ divide to four significant figures :
42. 16^7. 45. 26.4^13.8. 48. 17.626-5-3.4.
43. 7 ^ 15. 46. 4.21 ^ 3.75. 49. 43.826 ^ 0.72.
44. 0.7 H^ 150. 47. 63.25^4.92. 50. 6.483^8.4.
Taking log 3.1416 as 0.49716 and interpolating for six figures
on the same principle as for five^ find the diameters of circles with
circumferences as follows :
51. 62.832. 53. 2199.12. 55. 28,274.2. 57. 376,992
52. 157.08. 54. 2513.28. 56. 34,557.6. 58. 0.031416.
59. By using logarithms find the product of 41.74 x 20.87, and
the quotient of 41.74 -5- 20.87.
64 PLANE TRIGONOMETRY
55. Cologarithm. The logarithm of the reciprocal of a number is
called the cologarithm of the number.
For " cologarithm of N " it is customary to write " colog A^."
By definition colog x = log - = logl — logx (§41). But log 1=0.
X
Hence we have colog x = — log x.
To avoid a negative mantissa (§ 45) it is customary to consider that
colog a; = 10 — log a; — 1 0,
since 10 — logo? — 10 is the same as — logat.
For example, colog 2 = — log 2 = 10 — log 2—10
= 10 - 0.30103 - 10
= 0.69897 - 10 = 1.69897.
56. Use of the Cologarithm. Since to divide by a number is the same
as to multiply by its reciprocal, instead of suhtractlng the logarithm
of a divisor we may add its cologarithm.
The cologarithm of a number is easily written by looking at the logarithm
in the table. Thus, since log 20 = 1.30103, we find colog 20 by subtracting this
from 10.00000 — 10. To do this we begin at the left and subtract the number
represented by each figure from 9, except the right-hand significant figure,
which we subtract from 10. In full form we have
10.00000 - 10 = 9. 9 9 9 9 10 - 10
log20= 1.30103 = 1. 3 1 3
colog 20= 8. 6 9 8 9
Similarly, we may find colog 0.03952 thus :
10.00000 - 10 = 9. 9 9 9 9
log 0.03962= 2.59682 = 8. 6 9 6 8
colog 0.03952= 1. 4 3 1
Practically, of course, we would find log 0.03952 and subtract mentally.
Exercise 25. Cologarithms
Write the cologarithms (f the following numbers:
7
- 10 = 2.69897
10
-10
2
-10
8
= 1.40318
1.
25.
5. 3751.
9.
0.5.
13.
3.007.
2.
130.
6. 427.3.
10.
0.72.
14.
62.09.
3.
27.4.
7. 51.61.
11.
0.083.
15.
0.0006.
4.
5.83.
8. 7.213.
12.
0.00726.
16.
O.OOOOT
17. What number has for its cologarithm ?
18. What number has 1 for its cologarithm ?
19. What number has oo for its cologarithm?
20. Find the number whose cologarithm equals its logarithm.
J?^^
LOGARITHMS' 56
57. Advantages of the Cologarithm. If, as is not infrequently the
case in the computations of trigonometry and physics, we have the
product of two or more numbers to be divided by the product of
two or more different numbers, the cologarithm is of great advantage.
Using logarithms and cologarithms, simplify the expression
17'.28 X 6.25 X 16.9
1.44 X 0.25 X 1.3
This is so chosen that we can easily verify the answer by cancellation.
By logarithms we have,
log 17.28= 1.23764
log 6.25 =0.79688
log 16.9 = 1.22789
cologl.44 =9.84164-10
cologOlie =0.60206
colog 1.3 = 9.88606 - 10
3.69107 = log 3900.1
In a long computation the fifth figure may be in error.
Exercise 26. Use of Cologarithms
Udng cologarithms J Jivd the value of the following to five figures :
3x2 172.8 X 1.44 435 x 0.2751
4 xl.5' 0.288 X 0.864* 2.83 x 1.045'
8x9 57.5 X 0.64 50.05 x 2.742
3x4* • 1.25 X 820 * * 381.4 x 2.461* %
6x12 1.28 X 13.4 1 ^ 50730 x 2.875
3 X 8 * 1.49 X 6.4 ' i/ 34.48 x 1.462
^ 4 X 24 _ 5.48 X 0.198 J v 3.427 x 0.7832
4. TT tt: • 13. ^ ^^ ^^ . • r 253.
A
12 X 16 • 3.96 X 27.4 « * 3.1416 x 0.0081
12 X 15 _ 1.176 X 10.22 ^„ 27.98 x 32.05
5. ■:: 7r:r' 14. -tt-t: ^ ^^ ■ 23.
9 X 20 • 14.6 X 3.92 * 0.48 x 0.00062
12x28 3 X 11 X 17 2.1 X 0.3 x 0.11
• '8 X 21 * • 7 X 13 ' "^ ' 17 X 0.05
^ 3x22 ,^ 16x23 „^ 1.1x3.003
7. tt; :^- 16. r = -r- 25.
18 X 33 3 X 7 X 41 0.2 x 0.07112
^11x13 ^^ 23 X 39 X 47 ^^ 0.0347x0.117
17x19 17x33x53 3x11x170
„ 15 X 17 , „ 0.2 X 0.3 „„ 528.4 x 3 .001
11x13 • 0.11xl7i 7.03x0.7281
66 PLAlfE TRIGONOMETRY
58. Raising to a Power. It has been shown (§ 42) that the logarithm
of a power of a number is equal to the logarithm of the number
multiplied by the exponent.
1. Find by logarithms the value of II'.
From the tables, log 11 = 1.04139
Multiplying by 3, 3
log 11» = 3.12417
= log 1331.0
That is, 11» = 1331.0, to five figures. Of course we see that 11» = 1331 exactly,
log 1331 being 3.12418. The last figure in log 11^ as found in the above multi-
plication is therefore not exact, as is frequently the case in such computations.
As usual, care must be taken when a negative characteristic
appears.
2. Find by logarithms the value of 0.2413*.
From the tables, log 0.2413 = 0.38266 - 1
Multiplying by 6, ^
log 0.24136= 1.91280-6
= 4.91280
= log 0.00081808
Hence 0.2413* = 0.00081808, to five significant figures.
As on page 18, we use the expression ''significant figures" to indicate the
figures after the zeros at the left, even though some of these figures are zero.
Exercise 27. Raising to Powers
By logarithms^ find the value of each of the following to five
significant figures:
17. 26\ 26. l.ll 33. 12.65^
18. 2b\ 26. 2.r. 34. 34.75».
19. 1251 27. 0.1^. 36. 1.275».
20. 625». 28. 0.2^. 36. 0.1254".
21. 1750*. 29. 0.71 37. 0.4725^
22. 27752. 30. 0.07«. 38. 0.01234^
23. 3146*. 31. 0.37*. 39. 0.00275^.
24. 4135*. 32. 5.37*. 40. 0.000355^.
41. If log TT = 0.49715, what is the value of tt* ? of tt* ?
42. Using log TT as in Ex. 41, what is the value of ttt when r = 7 ?
of irr^ when r = 7? of J m* when r = 9 ?
1.
2'.
9.
l*'.
2.
2».
10.
7».
3.
2».
11.
9'.
4.
2»».
12.
8".
6.
3^
13.
11'.
6.
3«.
14.
15«.
7.
4"'.
16.
1.5».
8.
6».
16.
17*.
^
LOGARITHMS 57
59. Fractional Exponent. It has been shown (§ 43) that the log-
arithm of a root of a number is equal to the logarithm of the number
divided by the index of the root. This law may, however, be com-
bined with that of § 58, since a* means Va, and a' means Vo^.
The law of § 58 therefore applies to roots or to powers of roots; the
exponent simply being considered fractional.
1. Find by logarithms the value of Vi, or 4*.
From the tables, log 4 = 0.60206
Dividing by 2, 2 )0.60206
log Vi, or log 4J, = 0.30108
= log2
Hence Vi, or 4*, is 2.
2. Find by logarithms the value of 8*.
From the tables, log 8 = 0.90309
Multiplying by §, log sJ = 0.60206
= log4
Therefore 8* = 4.
3. Find by logarithms the value of 0.127*.
From the tables, log 0.127 = 0.10380 — 1.
Since we cannot divide — 1 by 6 and get an integral quotient for the new
characteristic, we add 4 and subtract 4 and then have
log 0.127 = 4.10380 -6
Dividing by 5, log 0. 127* = 0.82076 - 1
= log 0.66186
Hence 0.127*, or \^0.127, is 0.66186.
We might have written log 0.127 = 9.10380 - 10, 14.10380 - 16, and so on.
Exercise 28. Extracting Roots
By logarithmSy find the value of each of the follovnng :
9. Vll. 13. 0.3*. 17. 127.8*.
10. -V^, 14. 0.05*. 18. 2.475*.
11. ^^. 15. 0.0175*. 19. 5.135*.
12. ^5^100. 16. 0.0325*. 20. 0.00125*.
21. If log IT = 0.49715, what is the value of Vtt ? of -^ ?
22. Using the value of log ir given in Ex. 21, what is the value of
vi ? of TT* ? of 7r* ? of ir-i ? of tt"* ? of tt-^"* ?
1.
V2.
5. 2*.
2.
</5.
6. si.
3.
^.
7. si.
4.
5^25.
8. 7*.
68 PLANE TRIGONOMETRY
60. Exponential Equation. An equation in which the unknown
quantity appears in an exponent is called an exponential equation.
Exponential equations may often be solved by the aid of loga-
rithms.
1. Given 5* = 625, find by logarithms the value of x.
Taking the logarithms of both sides, we have (§ 42)
2 log 5 = log 625
log 626
Whence x
log 5
3 7Qf;ft»
... , e^ ^«e 0.69897
Check, 6* = 626.
Ill all such cases bear in mind that one logarithm must actually be divided
by the other. If we wished to perform this division by means of logarithms,
we should have to take the logarithm of 2.79688 and the logarithm of 0.69897,
subtract the second logarithm from the first, and then find the antilogarithm.
We may apply this principle to certain simultaneous equations.
2. Solve this pair of simultaneous equations
2* . 3" = 72 (1)
4* . 2ly = 46,656 (2)
Taking the logarithms of both sides, we have {§§ 40, 42)
X log 2 + 2/ log 3 = log 72, (3)
and X log 4 + y log 27 = log 46,666. (4)
Then, since log 4 = log 2^ = 2 log 2,
and log 27 = log 3» = 8 log 8,
we have 2 x log 2 + 3 y log 3 = log 46,666. (5)
Eliminating x by multiplying equation (3) by 2 and subtracting from equa-
tion (6), we have
_ log 46666 - 2 log 72
^~ logs
4.66890-2 X 1.86733
0.47712
_ 0.96424 _
~ 0.47712 ~
We may substitute this value of y in (1), divide by 3^, and then find x by
taking the logarithms of both sides. It will be found that x = 3.
We may check by substituting in (2).
In the same way, equations involving three or more unknown
quantities may be solved. Although the exponential equation is
valuable in algebra, as in the solution of Exs. 22, 23, 25, and 26 of
Exercise 29. we rarely have need of it in trigonometry.
LOGARITHMS 59
Exercise 29. Exponential Equations
By logarithms^ solve the following exponential equations :
1. 2' = 8. 6. 2^ = 19. 11. 2-' = \,
2.3^=81. 7.3^=75. 12.2-^ = 0.1.
3. 6^ = 625. 8. 5^ = 1000. 13. 0.3"* = 0.9.
4. 4* = 256. 9. 4^ = 2560. 14. 2' + ^ = 3^"^
5. 11^ = 1331. 10. IP =1500. 15. 9* + * = 53,143.
Solve the following simultaneous equations :
16. a*+«' = a* 18. 3^ • 4*' = 12 20. 2* . 5*' = 200
a'-y = a^ 5^. 7*^ = 35 3^ . 3*^ = 243
17. m2^+«' = m" 19. 2^ • 3^ = 36 21. 2^ • 8*' = 256
n8^-i' = ri" 4^.5«' = 400 8* . 32«' = 65,536
Solve the following equations by logarithms :
22. a=^p{l'{-ry. 25. a=^(l-f-r^)^
23. l = ar^-''^. 26. s(r —1) = ar' — a.
24. 2^+2x ^ 8 27. 3^- ^+i = 27.
Perform the following operations ly logarithms :
2A7 X 84.96 / 5.75 x 3.428 \^
34.8 X 96.55' V59.62 x 48.08/ *
4 f 42.4 X 0.075 6|/ 0.07 x 0.00964 Y
^^* \3.64 X 0.009' \ V 3.426 x 0.875 / '
32. To what power must 7 be raised to equal 117,649 ?
33. To what power must a be raised to equal b ?
34. To what power must 5 be raised to equal n ?
35. Find the value of x when >/9 = 3 ; when ">/2 = 1.1 ; when
</2 = 1.414 ; when "v^ = 1.73.
36. Find the value of x when "v^ = 3 ; when "Va = b ; when
-v^ = a ; when ^^"1331 = 11 ; when ^20736 = 12.
37. Solve the equations
■\/y = a
x+l/- _
I
8
38. What value of x satisfies the equation 0^+^*+* = Va?
^L>u*^>^
/
60 / PLANE TKIGONOMETRY
61. Logarithms of the Functions. Since computations involving
trigonometric functions are often laborious, they are generally .per-
formed by the aid of logarithms. For this reason tables have been
prepared giving the logarithms of the sine, cosine, tangent, and
cotangent of the various angles from 0® to 90® at intervals of 1'.
The functions of angles greater than 90® are defined and discussed
later in this work when the need for them arises.
Logarithms of the secant and cosecant are usually not given for the reason
that the secant is the reciprocal of the cosine, and the cosecant is the reciprocal
of the sine. Instead of multiplying by secx, for example, we may divide by
cosx ; and when we are using logarithms one operation is as simple as the other,
since multiplication requires the addition of a logarithm and division requires
the addition of a cologarithm.
In order to avoid negative characteristics the characteristic of
every logarithm of a trigonometric function is printed 10 too large,
and hence 10 must be subtracted from it.
Practically this gives rise to no confusion, for we can always tell by a result
if a logarithm is 10 too large, since it would give an antilogarithm with 10
integral places more than it should have. For example, if we are measuring
the length of a lake in miles, and find 10.30103 as the logarithm of the result,
we see that the characteristic must be much too large, since this would make
the lake 20,000,000,000 mi. long.
It would be possible to print 2.97496 for log sin 6° 25', instead of 8.97496,
which is 10 too large. It would be more troublesome, however, for the eye to
detect the negative sign than it would be to think of the characteristic as
10 too large.
On pages 56-77 of the tables the characteristic remains the same throughout
each column, and is therefore printed only at the top and bottom, except in
the case of pages 58 and 77. Here the characteristic changes one unit at the
places marked with the bars. By a little practice, such as is afforded on pages
61 and 62 of the text, the use of the tables will become clear.
On account of the rapid change of the sine and tangent for very
small angles log sin aj is given for every second from 0" to 3' on
page 49 of the tables, and log tan x has identically the same values
to five decimal places. The same table, read upwards, gives the
log cos X for every second from 89® 57' to 90®. Also log sin x,
log tan x, and log cos x are given, on pages 50-55 of the tables, for
every 10" from 0" to 2®. Reading from the foot of the page, the
cofunctions of the complementary angles are given.
On pages 56-77 of the tables, log sin a, log cos a;, log tan Xy and
log cot 05 are given for every minute from 1® to 89®. Interpolation
in the usual manner (page 31) gives the logarithmic functions for
every second from 1® to 89®.
LOGARITHMS 61
62. Use of the Tables. The tables are used in much the same wav
as the tables of natural functions.
For example, logsiD 5^25^ =8.97496 — 10 Page 68
log tan 40P 65' = 9.93789 - 10 Page 76
log cos 62° 20^ = 9. 78609 - 10 Page 74
log cot 88'* 69^ = 8.24910 - 10 Page 66
logsin 0« 28' 40^' = 7.92110 - 10 Page 61
log sin 0° r 62'' = 6.73479 -10 Page 49
Furthermore, if log cot x = 9,66910 — 10, then x = 70° 6'. * Page 66
Interpolation is performed in the visual manner, whether the angles
are expressed in the sexagesimal system or decimally.
1. Find log sin 19* 50' 30".
From the tables, log sin 19° 60' = 9.63066 — 10, and the tabular difference
is 36. We must therefore add |^ of 36 to the mantissa, in the proper place.
We therefore add 0.00018, and have logsin 19° 60' 30" = 9.63074 — 10.
2. Find log tan 39.75^
From the tables, log tan 39.7° = 9.91919 — 10, and the tabular difference is
164. We therefore add 0.6 of 164 to the mantissa, in the proper place. Adding
0.00077, we have log tan 39. 76° = 9.91996 - 10.
Special directions in the case of very small angles are given on
page 49 of the tables. It should be understood, however, that we
rarely use angles involving seconds except in astronomy.
»
If the function is decreasing, care must be taken to subtract instead
of add in making an interpolation.
3. Find log cos 43* 45' 15".
From the tables, log cos 43° 46' = 9.86876 — 10, and the tabular difference is
12. Taking 1| of 12, or J of 12, we have 0.00003 to be avbtracted.
Therefore log cos 43° 46' 16" = 9.86873 - 10.
4. Given log cot x = 0.19268, find x.
From the tables, log cot 32° 41' = 10.19276 - 10 = 0.19276.
The tabular difference is 28, and the difference between the logarithm 0.19276
and the given logarithm is 7, in each case hundred-thousandths. Hence there is
an angular difference of -^ of 1', or J of 1', or 16". Since the angle increases as
the cotangent decreases, and 0.19268 is less than 10.19276 — 10, we have to
add 16" to 82° 41', whence x = 32° 41' 16".
6. Given log tan x = 0.26629, find x.
From the tables, log tan 61° 33' = 10.26614 - 10 = 0.26614.
The tabular difference is 30, and the difference between the logarithm
0.26614 and the given logarithm is 16, in each, case hundred-thousandths.
Henc6 there is an angular difference of |4 o^ l'» or 30". Since/(x) is increasing in
this caee, and z is also mcreasing, we add 30" to 61® 83'. Hence x = 61° 33' 30"
62
PLANE TRIGONOMETRY
Exercise 30. Use of the Tables
Find the value of each
1. logsin27^ 16.
2. logsin69^ 17.
3. logcos36^ 18.
4. logcos48^ 19.
5. log tan 75^ 20.
6. log tan 12^ 21.
7. logcotl5^ 22.
8. log cot 78°. 23.
9. log sin 9"* 15'. 24.
10. log cos 8° 27'. 25.
11. log tan 7° 56'. 26.
12. log cot 82^'. 27.
13. log sin 4.5°. 28.
14. log cos 7.25°. 29.
15. log tan 9.75°. 30.
of the following :
log cos 42° 45".
log tan 26° 15".
log cot 38° 30".
log sin 21° 10' 4".
log sin 68° 49' 56".
log cos 15° 17' 3".
log cos 74° 42' 57".
log tan 17° 2' 10".
logtan26°3'4".
log cot 48° 4' 5".
log cot 4° 10' 7".
log sin 34° 30".
log sin 27.45°.
log tan 56.35°.
log cos 48.26°.
31. log sin 0° 1' 7".
32. log sin 1° 2' 5".
33. Iogtan0°2'8".
34. log tan 2° 7' 7".
36. log cos 89° 50' 10"
36. log cos 89° 10' 45"
37. log cot 89° 15' 12"
38. log cot 89° 25' 15"
39. logsinl°l'l".
40. log cos 88° 58' 59".
41. logtan2°27'25".
42. log cot 87° 32' 45".
43. log sin 12° 12' 12".
44. log cos 77° 47' 48".
45. log tan 68° 6' 43".
Find the value
is 10 too large :
46. log sin a; =
47. log sin X =
48. log sin X =
49. log sin X =
50. log cos X =
51. log cos OJ =
52. log cos a; =
63. log cos aj =
64. log tana; =
55. log tana; =
66. log tana; =
67. log cot a; =
68. log cot X =
ofxj given the following logarithms, each of which
9.11570.
9.72843.
9.93053.
9.99866.
9.99866.
9.93053.
9.71705.
9.80320.
9.90889.
10.30587.
10.64011.
9.28865.
9.56107.
59. log sin a;
60. log sin a;
61. log sin a;
62. log sin a;
63. log cos a;
64. log cos a;
65. log cos a;
66. log tana;
67. log tana;
68. log tan a;
69. log cot a;
70. log cot a;
71. log cot a;
9.53871.
9.72868.
9.88150.
9.89530.
9.90151.
9.80070.
9.99483.
9.18854.
10.18750.
10.06725.
10.10134.
11.44442.
7.49849*
CHAPTER IV
THE RIGHT TRIANGLE
63. Given an Acute Angle and the Hypotenuse. In § 30 the solution
4
of the right triangle was considered when an acute angle and the
hypotenuse are given. We may now consider this case and the follow-
ing cases with the aid of logarithms. For example,
given A = 34° 28', c = 18.75, find B, a, and k
1. 5 = 90^-^ = 55^32'.
^ ^ • ^ 'A
2. - = sin^: .'.a = csmA.
c
3. - = cos^ ; .'.i = ccos^.
c
log a = log c -f- log sin A
log c = 1.27300
log sin A = 9.75276
log a = 1.02576
.-. a = 10.611
10
log^ = log c 4- log cos^
log c = 1.27300
log cos A = 9.91617 - 10
log^> = 1.18917
.•.^» = 15.459
= 15.46
= 10.61
Check. 10.612 + 16.462 = 861.68, and 18.762 := 351.56.
This solution may be compared with the one on page 36. In this case there
is a gain in using logarithms, since we avoid two multiplications by 18.76.
The result is given to four figures (two decimal places) only, the length of c
having been given to four figures (two decimal places) only, and this probably
being all that is desired. In general, the result cannot be more nearly accurate
than data derived from TneasuremerU.
Consider also the case in which A = 72° 27' 42", c = 147.35, to
find By a, and b as above.
log a = log e -f- log sin A
log e = 2.16835
log sin^ = 9.97933 - 10
log a = 2.14768
.-. a = 140.50
log b = log c -f- log cos A
log c = 2.16835
log cos A = 9.47906 - 10
log b = 1.64741
.\ b = 44.403
Check. What convenient check can be applied in this case ?
63
64
PLANE TRIGONOMETRY
64. Given an Acute Angle and the Opposite Side. For example, given
A = 62* 10', a = 78, find B, h, and c.
1. 5 = 90° - A = 27* 60'.
2. - = coti4 ; ,\h = a QOtA.
a
a
a
3. - = sin^;
c
.'.a = csiuil, andc = . .
log h = log a -f- log coti4
loga = 1.89209
log cot A = 9.72262 - 10
log h = 1.61471
.-. 6 = 41.182
= 41.18
log c = log a + colog sin A
log a = 1.89209
colog 8inA= 0.05340
log c = 1.94649
.-. c = 88.204
= 88^(J
Check. 88.202 _ 41. 132 = 6083 +, and 782 _. 6084.
This solution should be compared with the one given in § 31, page 36. It will
be seen that this is much shorter, especially as to that part in which c is found.
The difference is still more marked if we remember that only part of the long
division is given in § 31.
65. Given an Acute Angle and the Adjacent Side. For example,
given A = 60* 2\ h = 88, find B, a, and c,
1. 5=90*-i4 = 39*68'.
d
2. 7=tan^; .•.a = otan^,
3. - = coSi4;
c
J.
,\h = ccoSi4, and c =
cos^
log a
log 6
log tan i4
log a
.'. a =
log h 4- log tan A
1.94448
10.07670 - 10
2.02118
105.00
logc
log^»
colog COS i4
logc
b— 88 O
log h 4- colog COS -4
1.94448
0.19223
.-. c =
2,13671
137.00
Check. 1372 - 1062 - 7744^ and 882 = 7744.
This solution should be compared with the one given in § 32, page 36. Here
again it will be seen that a noticeable gain is made by using logarithms, partic-
ularly in finding the value of c
THE RIGHT TRIANGLE
66
66. Given the Hypotenuse and a Side. For example, given a = 47.55,
e = 58.4, find ^4, By and h,
1. s\uA = —
c
2. 5=90**--il.
3. - = coti4; ,\h ^ aootA,
a
We could, of course, find 6 from the equation 6 = V(c + ^(c — d), as in
§ 38, page 36. By taking 6 = a cot^, however, we save the trouble of first find-
ing c + a and c — a.
log sin .4 = le^^ -I- colog c
* loga = 1.67715
-^logc = 8.23359 - 10
log sinA = 9.91074 -10
.-.A =54^ 31'
.-. 5= 35^ 29'
log h = log a + log coti4
log a = 1.67715
log cot ^ = 9.85300-10
log h = 1.53015
.-. h = 33.896
= 33.90
Check, 58.42 _ 33.92 = 2261 +, and 47.552 = 2261 + .
This solution should be compared with the one given in § 33, page 36.
67. Given tke Two Sides. For example, given a = 40, 5 = 27, find
A, B, and c.
1. tani4= 7*
2. B= 90° -A.
a
a
3. - = sin i4 ;
c
.'.a = c sin^, and c = . ^
sin A
log tan A = log a + colog b
loga= 1.60206
colog b = 8.56864 - 10
logtan^= 10.17070 -10
.•.^=55° 59'
.•.5=34°1'
6-27
log c = log a -f- colog sin^
log a = 1.60206
colog sinA= 0.08151
logc = 1.68357
.-. c = 48.258
= 48.26
Check, 272 + 402 = 2329, and 48.262 = 2329 + .
This solution should be compared with the solution of the same problem given
in § 34, page 37. There is not much gained in this particular example because
the numbers are so small that the operations are easily performed.
66 PLANE TRIGONOMETRY
•
68. Area of a Right Triangle. The area of a triangle is equal to one
half the product of the base by the altitude ; therefore, if a and h
denote the two sides of a right triangle and S the area, then S=:^ab.
Hence the area may be found when a and b are known.
Consider first the case in which an acute angle and the hypotenuse
are given. For example, let ^ = 34® 28' and c = 18.75. Then, finding
log a and log ^ as in § 63, we have
log S = colog 2 -h log a -h log h
colog 2 = 9.69897 - 10
log a = 1.02576
log h = 1.18917
log S = 1.91390
.-. 5=82.016
= 82.02
Next consider the case in which the hypotenuse and a side are
given. For example, let c = 58.4 and a = 47.55. Then, finding log h
as in § 66, we have
log S = colog 2 -h log a 4- log b
colog 2 = 9.69897 - 10
loga = 1.67715
log b = 1.53015
log S = 2.90627
.\ S = 805.88
= 805.9
Finally, consider the case in which an acute angle and the opposite
side are given. For example, let A = 62° 10' and a = 78. Then,
finding log 6 as in § 64, we have
log S = colog 2 -h log a 4- log b
colog 2 = 9.69897 - 10
log a = 1.89209
logb = 1.61471
log S = 3.20577
.-.5 = 1606.1
= 1606
We can easily verify this result, since, from §64, a = 78 and 6 = 41.18;
whence J a6 = 1606, to four significant figures.
The case of an acute angle and the opposite side is treated in § 64 ; that of
an acute angle and the adjacent side in § 65 ; and that of the two sides in § 67
A
^x
THE RIGHT TRIANGLE 67
Exercise 31. The Right Triangle
Umig logarithms, solve the following right triangles, finding the
sides and ^^pmistto four figures, and the angles to minutes:
^ 1. a = 6, c = 12. 16. ^ = 2, iB= 3° 38'.
"^ 2. 5 = 4, ^ = 60°. 17. a = 992, 5= 76° 19'.
a = 3, .4 = 30°. 18. a = 73, B= 68° 62'.
4. a = 4, ? = 4. 19. a = 2.189, 5= 45° 26'.
6. a = 2, c = 2.89. / /) 20. «> = 4, ^ = 37° 56'.
c = 627, ^=23° 30'. 4 ^ c = 8590, a = 4476.-^^3
7. c = 2280, ^ = 28° 5'. ' 22. c = 86.53, a = 71.78.
8. c = 72.15, A = 39° 34'. 23. c = 9.35, a = 8.49.
9. c=l, ^=36°. ^ c = 2194, ^ = 1312.7. ->.
10. c = 200, 5= 21° 47'. (K 25. c = 30.69, b = 18.25.
11. c = 93.4, B= 76° 26'. 26. a = 38.31, b = 19.62.
12. a = 637, ^ = 4° 35'. J^ ^27. a = 1.229,§ b = 14.95. 8
13. a = 48.53, ^ = 36° 44'. 28. a = 415.3, ' b = 62.08.
14. a = 0.008, A = 86°. C ^29. a = 13.69, ji * = 16.92. A
15. b = 50.94, B= 43° 48'. a ^30. c = 91.92, c = 2.19. S
Compute the unknown parts and also the area, having given :
^31. a = 5, b = 6, A JBN36. c=68, X = 69° 54'. ^
jry ^32. a = 0.615, c = 70. S (4^37. c = 27, 5= 44° 4'. -^
fr^\33. ^=%/2, c=V3. A 38. a = 47, iB= 48° 49'.
V^^^^ 34. a = 7, A= 18° 14'. j3] C,39. ^» = 9, iB= 34° 44'.
0^^^- — 35. 6 = 12, ^=29°8'. CC.C'40. c = 8.462, 5= 86° 4'.
'41. Find the value of S in terms of c and A,
.42. Find the value of S in terms of a and ^.
---43. Find the value of S in terms of b apd A^
44. Find the value of S in terms of a and c,
45. Given S = 5S and a = 10, solve the right triangle.
46. Given S = 18 and 6 = 6, solve the right triangle.
'47. Given 5 = 12 and A = 29°, solve the right triangle.
48. Given S = 98 and c = 22, solve the right triangle.
fj — - "^49. Find the two acute angles of a right triangle if the hypote-
nuse is equal to three times one of the sides.
"» V, . ■ ■ "T ■
tf '^ -* ■ *- ^ - .^ ..
68 PLANE TRIGONOMETRY
60. The latitude of Washington is 38** 55' 15" N. Taking the
radius of the earth as 4000 mi., what is the radiug of the circle
of latitude of Washington ? What is the circum-
ference of this circle ?
In ail such examples the earth will be considered as
a perfect sphere with the radios as above given, unless
the contrary is stated. For more accurate data consult
the Table of Constants.
51. What is the difference between the length of a degree of lati-
tude and the length of a degree of longitude at Washington ?
Use the data given in Ex. 50.
52. Erom the top of a mountain 1 mi. high, overlooking the sea,
an observer looks toward the horizon. What is the angle of depres-
sion of the line of sight ?
In the figure the height of the mountain is necessarily
exaggerated. The angle is so small that the result can be
found by five-place tables only between two limits which
differ by 3' 40". ^
(^. At a horizontal distance of 120 ft. from the
foot of a steejie, the angle of elevation of the top is found to be
60® 30'. Find the height of the steeple above the instrument.
nS Erom the top of a rock which rises vertically 326 ft. out of
the water, the angle of depression of a boat is found to be 24°.
Eind the distance of the boat from^the base of the rock.
/S?". How far from the eye is a monument on a level plain if the
height of the monument is 200 ft. and the angle of elevation of '
the top is 3° 30' ?
f»6/ A distance ^jB of 96 ft. is measured along the bank of a river
from a point A opposite a tree C on the other bank. The angle ABC
is 21° 14'. Find the breadth of the river.
^^. What is the angle of elevation of an inclined plane if it rises
1ft. in a horizontal distance of 40 ft.?
58. Eind the angle of elevation of the sun when a tower 120 ft.
high casts a horizontal shadow 70 ft. long.
59. How high is a tree which casts a horizontal shadow 80 ft. in
length when the angle of elevation of the sun is 60° ?
60. A rectangle 7.5 in. long has a diagonal 8.2 in. long. What
angle does the diagonal make with the base?
THE RIGHT TRIANGLE
69
61. A rectangle 8^ in. long has an area of .49 J sq. in. Find the
angle which the diagonal makes with the base.
62. The length AB oi b, rectangular field A BCD is 80 rd. and the
width ^D is 60 rd. The field is divided into two equal parts by a
straight fence PQ starting from a point P on AD which is 15 rd.
from A, What angle does PQ make with AD?
63. A ship is sailing due northeast at the rate of 10 mi. an hour.
Find the rate at which she is moving due north, and also due east.
64. If the foot of a ladder 22 ft. long is 11 ft. from
a house, how far up the side of the house does the lad-
der reach?
65. In front of a window 20 ft. from the ground there
is a flower bed 6 ft. wide and close to the house. How
long is a ladder which will just reach from the outside
edge of the bed to the window?
66. A ladder 40 ft. long can be so placed that it will reach a win-
dow 33 ft. above the ground on one side of the street, and by tipping
it back without moving its foot it will reach a window 21 ft. above
the ground on the other side. Find- the width of the street.
67. From the top of a hill the angles of depression of two suc-
cessive milestones, on a straight, level road leading to the hill,
are 5"* and 16"*. Find the height of the hill.
68. A stick 8 ft. long makes an angle of 45® with
the floor of a room, the other end resting against the
wall. How far is the foot of the stick from the wall ?
69. A building stands on a horizontal plain.
The angle of elevation at a certain point on the
plain is 30®, and at a point 100 ft. nearer the
building it is 45°. How high is the building ?
70. From a certain point on the ground the angles of elevation
of the top of the belfry of a church and of the top of the steeple
are found to be 40® and 51® respectively. From a point 300 ft. fur-
ther off, on a horizontal line, the angle of elevation of the top of
the steeple is found to be 33® 45'. Find the height of the top of the
steeple above the top of the belfry.
71. The angle of elevation of the top C of an inaccessible fort
observed from a point A is 12®. At a point B, 219 ft. from A and
on a'line AB perpendicular to AC, the angle ABC is 61® 45'. Find
the height of the fort.
70
PLANE TRIGONOMETRY
69. The Isosceles Triangle. Since an isosceles triangle is divided
uy the perpendicular from the vertex to the base into two congruent
right triangles, an isosceles triangle is determined by any two parts
which determine one of these right triangles.
In the examples which follow we shall represent the parts of the
isosceles triangle ABC, among which the altitude CD is included,
as follows :
a = one of the equal sides,
c = the base, ^
h = the altitude,
A = one of the equal angles,
C = the angle at the vertex.
Eor example, given a and c, find A, C,
and h,
= i-^ = — .
a 2 a
1. COS A
2. C-\-2A = 180°; .'. C = 180^ - 2 A = 2(90° - A).
3. h may be found by any one of the following equations :
h^^^(^=-.a'
whence
or
or
- = sin^, whence h = a sin A ;
a
-— = tan^, whence A = J ctan^.
When G and h are known, the area can be found by the formula
S= ^ch
That is, S = ^c ' a sin A = ^ ac sin A,
or S = ^c ' ^c tan^ = J c^ tan^,
or 5 = J c V(a -f i c) (a — ^ c).
Consider also the case in which a and h are given, to find A, C.
Cy and S.
1. sin .4 = -, and hence A is known.
a
2. C = 2(90° —A), as above, and hence C is known.
3. J c = a cos A, and hence c is known.
4. S = ^ ch, and hence S is known.
We can also find S from any of its other equivalents, such as those given
above, or a^ sin J C sin^, each of which is easily deduced.
THE RIGHT TRIANGLE 71
Exercise 32. The Isosceles Triangle
Solve the following isosceles triangles :
1. Given a and A, find C, c, and h.
2. Given a and C, find A, c, and h,
3. Given c and ^, find C, a, and h,
^4. Given c and C, find A, a, and A<.
>,^^ 6. Given h and ^, find C, a, and c.
>^6. Given h and C, find A, a, and c.
7. Given a and ^, find ^, C, and c.
8. Given c and ^, find ^ , C, and a.
'w^ 9. Given a = 14.3, c = 11, find A, C, and A<.
Y 10. Given a = 0.295, ^ = 68° 10', find c, h, and 5.
^ "^11. Given c = 2.352, C = 69° 49', find a, h, and 5.
12. Given h = 7.4847, ^ = 76° 14', find a, c, and S,
13. Given c = 147, S = 2572.5, find A, C, a, and h,
^,^14. Given A< = 16.8, S = 43.68, find A, C, a, and c.
15. Given a = 27.56, A = 75° 14', find c, A, and 5.
Given an isosceles triangle^ ABC :
16. Find the value of S in terms of a and C.
17. Find the value of S in terms of a and A,
18. Find the value of S in terms of h and C.
19. A barn is 40 ft. by 80 ft., the pitch of the roof is 45°; find
the length of the rafters and the area of the whole roof.
20. In a unit circle what is the length of the chord subtending
the angle 45° at the center ?
21. The radius of a circle is 30 in., and the length of a chord is
44 in. ; find the angle subtended at the center.
22. Find the radius of a circle if a chord whose length is 5 in.
subtends at the center an angle of 133°.
23. What is the angle at the center of a circle if the subtending
chord is equal to f of the radius ?
'• 24* Find the area of a circular sector if the radius of the circle is
12 in., and the angle of the sector is 30°.
25. Find the tangent of the angle of the slope of an A-roof of a
building which is 24 ft. 6 in. wide at the eaves, the ridgepole being
10 ft. 9 in. above the eaves.
72
PLANE TRIGONOMETRY
70. The Regular Polygon. We have already considered a fei¥ cases
involving the regular polygon. It is evident from geometry that if
the polygon shown below has n sides, the angle of the right triangle
which has its vertex at the center is equal to J of 360°/n, or 180®/n.
The triangle may evidently be solved if the raidius of the circum-
scribed circle (r), the radius of the inscribed circle (h), or the side of
the polygon (c) is given.
In the exercises we shall let
n = number of sides,
c = length of one side,
r = radius of circumscribed circle,
h = radius of inscribed circle,
p = the perimeter,
S = the area.
Then, by geometry,
Ezetcise 33. The Regular Polygon
Find the remaining parts of a regular polygon^ given :
1. w =10, c =1. 3. 7i = 20, r = 20. 5. n =11, S = 20.
2. 7i=18,r=l. 4. 7i=8, A = l. ^6.71=7, 5=7.
7. The side of a regular inscribed hexagon is 1 in.; find the side
of a regular inscribed dodecagon.
"^"""^ 8. Given n and c, and represent by b the side of the regular
inscribed polygon having 2 n sides, find h in terms of n and c,
9. Compute the difference between the areas of a regular octagon
and a regular nonagon if the perimeter of each is 16 in.
10. Compute the difference between the perimeters of a regular
pentagon and a regular hexagon if the area of each is 12 sq. in.
11. Find the perimeter of a regular dodecagon circumscribed about
a circle the circumference of which is 1 in.
12. What is the side of the regular inscribed polygon of 100 sides,
the radius of the circle being unity ? What is the perimeter ?
13. What is the perimeter of the regular inscribed polygon of
360 sides, the radius of the circle being unity?
14. The area of a regular polygon of twenty-five sides is 40 sq. in ;
find the area of the ring included between the oirQixiftforeiices of the
inscribed and circumscribed circles.
p
^-^L<^
?
/^ •<^^ ^V
THE RIGHT TRIANGLE
78
Exercise 34. Review Problems
1. Prove that the area of the parallelogram here shown is equal
to ah sin^. • 2).
2. Two sides of a parallelogram are 5 in. and
6 in. respectively, and their included angle is
82** 46'. What is the area ? ^ ^
3. Two sides of a parallelogram are 9 ft. ^
and 12 ft. respectively, and their included angle is 74.5°. What is
the area?
4. Each side of a rhombus is 7.36 in., and one angle is 42® 27'.
What is the area ?
6. The area of a rhombus is 250 sq. in., and one of the angles
is 37° 26'. What is the length of each side ?
6. A pole BD stands on the top of a mound EC,
From a point A the angles of elevation of the top and
foot of the pole are 60° and 30° respectively. Prove
that the height of the pole is twice the height of the
mound.
7. A ladder 38 ft. long is resting against a wall. The foot of the
ladder is 7 ft. 2 in. from the wall. What is the height of the top of
the ladder above the ground ?
8. Erom a boat 1325 ft. from the base of a vertical cliff the angle
of elevation of the top of the cliff is observed to be 14° 30'. Find
the height of the cliff. ,
9. On the top of a building 60 ft. high there is a flagstaff BD.
From a point A on the ground the angles of elevation of B and B
are 30° and 46° respectively. Find the length of
the flagstaff and the distance ^C of the observer
from the building, as shown in the annexed figure.
Since — = tan 30° and = tan 46**, x can evidently
X X
be eliminated.
10. A man whose eye is 6 ft. 8 in. above the ground stands midway
between two telegraph poles which are 200 ft. apart. The elevation
of the top of each pole is 48° 60'. What is the height of each ?
11. The captain of a ship observed a lighthouse directly to the
east. After sailing north 2 mi. he observed it to lie 55^ 30' east of
south. How far was the ship from the lighthouse at the time of each
observation ?
74 PLANE TRIGONOMETRY
12. A leveling instrument is placed at A on the slope MN, and the
line M'N' is sighted to two upright rods. By measurement MM* is
found to be 12.8 ft., NN' to be 3.4 ft., and JkfiV' to ^
be 48.3 ft. Required the angle of the slope of MN.
and the distance MN.
13. A wire stay is fastened to a telegraph pole 6.8 ft. from the
ground and is stretched tightly so as to reach the ground 5.2 ft. from
the foot of the 'pole. What angle does the wire stay make with
the ground?
14. The top of a conical tent is 8 ft. 7 in. above the ground, and
the diameter of the base is 9 ft. 8 in. Find the inclination of the
side of the tent to the horizontal. Check the result by drawing the
figure to scale and measuring the angle with a protractor.
16. In this piece of iron construction work -BC = llin. and
AB makes an angle of 30° with BC. What is the length of ^C?
16. In Ex. 15 it is also known that BE and CD r-^r-*-^
are each 9 in. long and make angles of 60° with BC j^^^^^Sn
produced. What is the length of ED ? V^jiy^if
17. From the conditions given in Ex. 16, find the
length of CF.
18. The base of a rectangle is 14| in. and the diag-
onal is 19j in. What angle does the diagonal make with the base ?
Check the result by drawing the figure to scale and measuring the
angle with a protractor.
19. In constructing the spire represented in the figure below it is
planned to have AB= 4:2 it. and PM=92 ft. What angle of slope
must the builders give to AP?
20. In Ex. 19 find the length of ^P and find the
angle P,
21. In the figure of Ex. 19 the brace CD is put in
38 ft. above AB. What is its length ?
?2. The spire of Ex. 19 rests on a tower. A man
standing on the ground at a distance of 400 ft. from the
base of the tower observes the angle of elevation of P to be 25° 38',
the instrument being 5 ft. above the groimd. What is the height of
P above the ground ?
23. When the angle of elevation of the sun is 38.4®, what is the
length of the shadow of a tower 175 ft. high ?
THE EIGHT TRIANGLE
75
F E
24. Two men, M and N, 3200 ft. apart, observe an aeroplane A
at the same instant, and at a time when the plane MNA is vertical.
The angle of elevation at M is 41® 27' and the v
angle at N is 61° 42'. Required AB, the height of
the aeroplane.
Show that h cot 41° 27' + h cot 61° 42' is known, whence
h can be found.
25. A kite string 475 ft. long makes an angle of elevation of
49° 40'. Assuming the string to be straight, what is the altitude
of the kite ?
26. A steel bridge has a truss ADEF in which it is given that
AD = 20 ft., ^F= 6 ft. 8 in., and FE =12 ft., as
shown in the figure. Required the angle of slope
which AF makes with AD,
A B C
27. Two tangents are drawn from a point P to a
circle and contain an angle of 37.4°. The radius of the cii'cle is 6 in.
Find the length of each tangent and the distance of P from the center.
28. From the top of a cliff 96 ft. high, the angles of depression
of two boats at sea are observed, by the aid of an instrument 6 ft.
above the ground, to be 46° and 30° respectively. The boats are in a
straight line with a point at the foot of the cliff directly beneath the
observer. What is the distance between the boats ?
29. A carpenter's square BCA is held against the vertical stick
ED resting on a sloping roof AD^ as in the figure. It is found
that ^C=24 in. and CD = 11.6 in. Find the
angle of slope of the roof with the horizontal.
30. In Ex. 29 find the length of AD.
31. A man 6 ft. tall stands 4 ft. 9 in. from a
street lamp. If the length of his shadow is 19 ft.,
how high is the light above the street ?
32. The shadow of a city building is observed
to be 100 ft. long, and at the same time the shadow
of a lamp-post 9 ft. high is observed to be 5.2 ft. long. Find the
angle of elevation of the sun and the height of the building.
33. A man 5 ft. 10 in. tall walks along a straight line that passes
at a distance of 2 ft. 9 in. from a street light. If the light is 9 ft.
6 in. above the ground, find the length of the man's shadow when
his distance from the point on his path that is nearest to the lamp
is 3 ft. 8 in.
Ill Mil I III I
76 PLAKE TRIGONOMETRY
34. A. man on a bridge 35 ft. above a stream, using an instrument
5 ft. high, sees a rowboat at an angle of depression of 27® 30'. If
the boat is approaching at the rate of 2| mi. an hour, in how many
seconds will it reach the bridge ?
35. A shaft 0, of diameter 4 in., makes 480 revolutions
per minute. If the point P starts on the horizontal line OA .
how far is it above OA after ^ of a second ?
36. Assuming the earth to be a sphere with radius 3967 mi., find
the radius of the circle of latitude which passes through a place in
latitude 47° 27' 10" N.
37. When a hoisting crane AB, 28 ft. long, makes an angle of
23° with the horizontal AC, what is the length of ^C? Suppose
that the angle CAB is doubled, what is then
the length of ^ C ?
38. In Ex. 37 find the length of BC in
each of the two cases. ^f^ o
39. Wishing to measure the distance AB, a man swings a 100-foot
tape line about B, describing an arc on the ground, and then does the
same about A, The arcs intersect at C, and the ^ j .
angle A CB is found to be 32° 10'. What is the "\. 3^
length oiAB? ^\^ ^
40. From the top of a mountain 15,260 ft. high,
overlooking the sea to the south, over how many minutes of latitude
can a person see if he looks southward ? Use the assumption stated
in Ex. 36.
41. The length of each blade of a pair of shears, from the screw
to the point, is 6J in. • When the points of the open shears are 3J in.
apart, what angle do the blades make with each other ?
42. In Ex. 41 how far apart are the points when the blades make
an angle of 28° 46' with each other ? ^
43. The wheel here represented has eight spokes, ^
each being 19 in. long. How far is it from A to B?
from Bto D?
44. The angle of elevation of a balloon from a
station directly south of it is 60°. From a second station lying
5280 ft. directly west of the first one the angle of elevation is 45°.
The instrument being 5 ft. above the level of the ground, what is
the height of the balloon ?
CHAPTER V
TSIGONOHETRIC FUNCTIONS OF ANT ANGLE
71. Need for Oblique Angles. We have thus far considered only
right triangles, or triangles which can readily be cut into right tri-
angles for purposes of solution. There are, however, oblique triangles
which cannot conveniently be solved by merely separating them into
right triangles. We are therefore led to consider the functions of
oblique angles, and to enlarge our idea of angles so as to include
angles greater than 180®, angles greater than 360®, and even negative
angles and the angle 0®.
72. Positive and Negative Angles. We have learned in algebra that
we may distinguish between two lines which extend in opposite direc-
tions by calling one positive and the other negative.
For example, in the annexed figure we consider OX
as positive and therefore OX^ as negative. We also con-
sider OY as positive and hence OF' as negative. In gen-
eral, horizontal lines extending to the right of a point ^
which we select as zero are considered positive, and those
to the left negative. Vertical lines extending upward from
zero are considered positive, and those extending down-
ward are considered negative.
With respect to angles, an angle is consideTed positive if the rotat-
ing line which describes it moves counterclockwise, that is, in the
direction opposite to that taken by the hands of a
clock. An angle is considered negative if the rotat-
ing line moves clockwise, that is, in the same
direction as that taken by the hands of a clock.
Arcs which subtend positive angles are considered
positive, and arcs which subtend negative angles
are considered negative. Thus Z.AOB and arc AB are considered
positive; Z.AOB^ and arc AB' are considered negative.
For example, we may think of a pendulum as swinging through a positive
angle when it swings to the right, and through a negative angle when it swings
to the left. We may also think of an angle of elevation as positive and an angle
of depression as negative, if it appears to be advantageous to do so in the solu-
tion of a problem.
77
78 PLAJTE TRIGONOMETRY
73. Coordinates of a Point. In trigonometry, as in work with
graphs in algebra, we locate a point in a plane by meana of its
distances from two perpendicular lines.
These lines are lettered XX' and YY', and their point of Intersection 0,
the origin.
oblique a
Tbe lines are called the atet and the point of
In some branches of mathematics it is more
but in trigonometry reotongiUar axes are
used as here shown.
The distance of any point P from .
the axis XX', or tbe avaxis, is called
tbe ordinate of tbe point. Its distance
from the axis YY', or the y-axi8, is
called the abscissa of the point.
In the figure, j/ is the ordinate of P, and
X is the abscissa of P. The point P is rep-
resented by the symbol (i, y). In the figure
the side of each small square ma; be taken
to represent one unit, in which case P = {4, S), because its abscissa is 4 and
its ordinate 8. Following a helpful European custom, tbe points are indicated
hj small circles, so as to show more clearly when a line is drawn through them.
The abscissa and ordinate of a point are together called the coordi-
nates of the point.
74. Signs of the Coordinates. From § 73 we see that ahscissas to the
right of the y-axis are positive ; abscissas to the left of the y-axis are
negative; alginates above the x-axis are positive ; ordinates below the
X-axis are negative.
A point on the line TY has zero for its abscissa, and hence the abscissa may
l>e considered as either positive or negative and may be Indicated by ± 0. Simi-
larly, a point on the line XX' has ± for its ordinate.
75. Tlie Four Qtutdrants. The axes divide the plane into four parts
known as quadrants.
Because angles are generally considered as generated by the rotating line
moving counterclockwise, the four quadrants are named in a counterclookwiso
order. Quadrant XOY is spoken of as the first quadrant, YOX' as the second
quadrant, X'OY' as the third quadrant, and YOX as the fourth quadrant.
76. Signs of tlK Coordlnatee In tbe Several QuadrantB. From § 74>
we have the following rule of signs :
In quadrant I the abscissa is positive, the ordinate positive ;
In quadrant II the abscissa is negative, the ordinate positive ;
In quadrant III the a6saiasa is negative, the ordinate negative ;
In quadrant IV the abscissa is positive, the ordinate negative.
FUNCTIONS OF AJSY ANGLE 79
77. Plotting a Point. Locating a poiot, having given its coordi-
nates, is called plotting the point.
For example, Id the first of these figures the point (— 2, 4) Is shown in
quadrant II, the point (— 3, — 2) in quadrant III, and the point (1, — 1) in
quadrant IV.
In the second figure the point (— 2, D) is shown on OX', between quad-
rants II and III, and the point (1, 0) on OX, between quadrants I and IV.
In the third figure the point (0, 1) is shown on OF, between quadranw I and II,
and the point (0, — 3) on 01", between quadrants III and IV.
In ever; case the ori^n maj be designated as the point (0,0).
J}) Distance from the Origin, The coordinates of P being x and
e may form a right triangle the hypotenuse of which is the
distance of P from 0.
Representing OP by r.
I I
Since this may be written r — ± Vi' + y^, we see that
r may be considered as either positive or negative. It is
the custom, however, to consider the rotating iine which
forms the angie as positive. If r is produced through 0,
the production is considered as negative.
1. What is the distance of the point (3, 4) from the origin ?
r = V3» + 4^ = V26 = 6.
2. What is the distance of the point (— 3, — 2) from the origin ?
r = V(- S)= + (- 2)" = V9 + 4 = Vl8 = 8.81.
3. What is the distance of the point (5, — 5) from the origin ?
r = V5" + (- 6)» = v^ = 7.07.
4. What is the distance of the point (— 2, 0) from the origin?
r = V(- 2)" + 0* = Vi = 2,
as is evident from the conditions of Hie problem.
80 PLANE TRIGONOMETRY
I
Exercise 35. Distances from the Origin
Umig squared paper y or meaBuring with a ruler ^ plot thefolloW'
ing points :
1. (2, 3). 8. (- 3, 2). 15. (3, - 4). 22. (0, 0).
2. (3, 6). . 9. (- 3, 4). 16. (4, - 3). 23. (0, 2^).
3. (4, 4). 10. (- 5, 1). 17. (6, - 1). 24. (0, - 3^).
4. (2^, 3). 11. (- 4, 6). 18. (0, 7). 25. (4|, 0).
5. (3i, 4i). 12. (- 2, - 2). 19. (3, 0). 26. (5^, 0).
6. (^, 4i). 13. (- 3, - 5). 20. (0, - 4). 27. (- 2^, 0).
7. (6i, 3i). 14. (- 6, - 3). 21. (- 2, 0). 28. (- 3^, 0).
T'ini the distance of each of the following points from the origin :
29. (6, 8). 32. (1^, 2). 35. {f, Vs). 38. (0, 7).
30. (9, 12). 33. (i, 1). 36. (- 3, 4). 39. (6, 0).
31.(6,12). 34. (2i, 3). 37.(0,0). 40. (-12,-9).
41. Eind the distance from (3, 2) to (- 2, 3).
42. Find the distance from (- 3, - 2) to (2, - 3).
43. Find the distance from (4, 1) to (— 4, — 1).
44. Find the distance from (0, 3) to (- 3, 0).
45. A point moves to the right 7 in., up 4 in., to the right 10 in.,
and up 18f in. How far is it then from the starting point ?
46. A point moves to the right 9 in., up 5 in., to the left 4 in., and
up 3 in. How far is it then from the starting point ?
47. Find the distance from (- J, ^ V3) to (i, - ^ Vs).
48. A triangle is formed by joining the points (1, 0), (— ^, ^ Vs),
and (— i, — i Vs). Find the perimeter of the triangle. Draw the
figure to scale.
49. Find the area of the triangle in Ex. 48.
50. A hexagon is formed by joining in order the points (1, 0),
(i, i V3), (- i, i V3), (- 1, 0), {-h-i V3), (i, - i V3), and
(1, 0). Is the figure a regular hexagon ? Prove it.
51. A polygon is formed by joining in order the points (1, 0),
(iV2, iV2), (0, 1), (-iV2,iV2), (-1, 0), (-iV2, -iV2),
(0, — 1), (i V2, — i Vz), and (1, 0). Draw the figure, state the kind
of polygon, and find its area.
FUNCTIONS OF ANY ANGLE
81
79. Angles of any Magnitude. In the following figures, if the rotat-
ing line OP revolves about from the position OX, in a counterclock-
wise direction, until it again coincides with OX^ it will generate all
angles in every quadrant from 0° to 360°.
The line OX is called the initiaX side of the angle, and the line OP the ter-
minal side of the angle.
An angle is said to be an angle of that quadrant in which its
terminal side lies.
(^
XA
[
)
vd
IT /
/ly-
^
/
fVni
ivy
Angles between 0° and 90° are angles of quadrant I.
Angles between 00° and 180° are angles of quadrant II.
Angles between 180° and 270° are angles of quadrant III.
Angles between 270° and 360° are angles of quadrant IV.
The rotating line may also pass through 360°, forming angles from
360° to 720°. It may then make another revolution, forming angles
greater than 720°, and so on in-
definitely.
For example, in using a screwdriver
we turn through angles of 860°, 720°,
1080°, and so on, depending upon the
number of revolutions. In the same way,
the minute hand of a clock turns through 8640° in a day, and the drive wheel
of an engine may turn through thousands of degrees in an hour.
We might, if necessary, speak of an angle of 400° as an angle of quadrant I,
because its terminal side is in that quadrant, but we have no occasion to do so
in practical cases.
As stated in § 72, if the line OP is rotated clockwise, it generates
negative angles.
In this way we may form angles of — 40° or — 140°, as here shown, and the
rotation may continue until we have angles of — 360°, — 720°, — 1080°, — 1440°,
and so on indefinitely.
We shall have but little need for the
negative angle in the practical work
of trigonometry, but we shall make ex-
tensive use of angles between 0° and
180°, and some use of those between
180° and 360°.
82 PLANE TRIGONOMETRY
80. Functions of Any Angle. Since we have now seen that we may
have angles of any magnitude, it is necessary to consider their func-
tions. Although we must define these functions anew, it will be
Been that the definitions hold for the acute angles which we have
already considered.
-pj
_
-
T
-
r
-
--
X
/
'^
V
'^
*o
'
" r^
-_
_^
-
-
d
_
_
j— j ¥
#tt#
:'^y.-.-ll--^iy.
z.
™
In whatever quadrant the angle is, we designate it by A. We take
& point P, or (x, y), on the rotating line, and let OP = r. Then the
angle XOP, read counterclockwise, is the angle A. We then define
the functions as follows ;
oos^= - =
_ ordinate
distance'
abscissa
csc.4= -~r
distance
sin^ y ordinate
distance
abscissa
tan^
It will be seen that these deflnitlons are practically the same as tboee already
learned for angles in quadrant I. Their application to the other quadranta
ia apparent. The general definitions ndght have been given at first, but this
plan ofiers difficulties for a beginner which make it undesirable.
By counting the squares on squared paper and thus getting the lengths of
certfdn lines, the approximate values of the functions of any given angle maj be
found, but the exercise has no practical significance. The values of tliefunctioua
are determined by series, these being explained in works on the calculus.
FUNCTIONS OF ANY ANGLE 83
81. Angles determined by Functions. Given any function of an
angle, it is possible to construct the angle or angles which satisfy
the value of the function.
1. Given 8in^= |, construct the angle A.
If we take a, line parallel to X'X and
line OP, 6 units long, about O until P
in the annexed figure
OP = 6, PQ = S,
and likewiM OP = 5, P'Q- = 8.
Then si
op'
1 quadrant I:
OP'
>in quadrantll:
In other wordf, we have constructed two angles, each of which has J for
its dne.
Furthermore, we could construct an infinite number of such angles, for wo
see that 860° + A terminates in OP and has the same sine that A has, and that
the same maybe said of 360= + .4', 720° + ^, 720° + J.', 1080° + 4, and so on.
In general, therefore, the angle n x 360° + A has the same functions as j1, n
being any integer. Hence if we Itnow the value of any particular function of
an angle, the angle cannot be uniquely determined ; that Is, there is more than
one angle which satisfies the condition. In general, as we see, an infinite number
of angles will satisfy the given condition, although this gives no trouble because
only two of these angles can be less than 360°.
2. Given tanv4 = J, construct the angle A.
If we take an abscissa 1 and an ordinate 3, as in quadrant I of the figure,
we locate the point (3, i). Then angle ZOP has for its tangent |. But it is
evident that we may also locate the point (— 3, — 4) in quadrant III, and thus
find an angle between 1B0° and 270p whose tangent is ij.
82. Functions found from Other Functions. Given any function of an
angle, it is possible not only to construct the angle but also to find
the other functions.
For in Ex. 1 above, after constructing angles A and A', we see that
6
That is, if sin^ = |, thencoB^ = ± ^, tan^ — ± J
and cot A =± J.
86 PLANE TKIGONOMETKY
84. Variations in tlie Functions. A study of the line values of
the functions shows how they change as the angle increases from .
0° to 360=.
1. The Sine. In the first quadrant the sine MP
is poaitive, and increases from to 1 ; in the
second it remains positive, and decreases from
1 to 0; in the third it is negative, and increases
in absolute value from to 1; in the fourth it
is negative, and decreases in absolute value from
1 to 0. The absolute value of the sine varies,
therefore, from to 1, and its total range of values is from +1 to —1.
In the third quadrant the sine dtereasea Irom to —1, but the a&solute vaJue
(the value without reference to its sign) increases from to I, and ebnllarly
for other cases on this page in which the al)80lat« value is mentioned.
2. The Cosine. In the first quadrant the cosine OM is positive,
and decreases from 1 to 0; in the second it becomes negative, and
increases in absolute value from to 1 ; in the third it is negative,
and decreases in absolute value from 1 to 0; in the fourth it is
positive, and increases from to 1. The absolute value of the
cosine varies, therefore, from to 1.
3. The Tangent. In the first quadrant the tangent AT is positive,
and increases from to co ; in the second it becomes negative, and
decreases in absolute value from oo to ; in the third it is positive,
and increases from to oo ; in the fourth it is negative, and decreases
in absolute value from os to 0.
4. TJie Cotangent. In the first quadrant the cotangent BS is posi-
tive, and decreases from oo to 0; in the second it is negative, and
increases in absolute value from to oo ; in the third and fourth quad-
rants it has the same sign, and undergoes the same changes as in the
first and second quadrants respectively. The tangent and cotangent
may therefore have any values whatever, positive or negative.
5. The Secant. In the first quadrant the secant Oris positive, and
increases from 1 to oo ; in the second it is negative, and decreases in
absolute value from oo to 1 ; in the third it is negative, and increases
in absolute value from 1 to oo ; in the fourth it is positive, and decreases
from CO to 1,
6. The Cosecant. In the first quadrant the cosecant OS is positive,
and decreases from so to 1 ; in the second it is positive, and increases
from 1 to 00 ; in the third it is negative, and decreases in absolute
value from oo to 1; in the fourth it is negative, and increases in
absolute value from 1 to co.
FUNCTIONS OF ANY ANGLE
87
It is evident, therefore, that the sine can never be greater than 1
nor less than — 1, and that it has these limiting values at 90** and
270® respectively. We may also say that its absolute value can never
be greater than 1, and that it has its limiting value at 0® and 180®,
and its limiting absolute value 1 at 90° and 270®.
If we have an equation in which the value of the sine is found to be greater
than 1 or less than — 1, we know either that the equation is wrong or that an
error has been made in the solution.
Of course the values of the functions of 360° are the same as those of 0®,
since the moving radius has returned to its original position and the initial and
terminal sides of the angle coincide.
In the same way, the absolute value of the cosine cannot be greater
than 1, and it has its limiting value at 90® and 270®, and its limit-
ing absolute value 1 at 0® and 180®. Similarly we can find the
limiting values of all the other functions.
For convenience we speak of oo as a limiting value, although the function
increases without limit, the meaning of the expression in this case being clear.
Summarizing these results, we have the following table :
Function
0*»
90°
IW
270*»
860°
Sine
TO
+ 1
±0
-1
TO
Cosine
+ 1
±0
-1
TO
+ 1
Tangent
TO
±00
TO
±00
TO
Cotangent
Too
±0
Too
±0
Too
Secant
+ 1
±00
-1
Too
+ 1
Cosecant
Too-
+ 1
±00
-1
Too
Sines and cosines vary in value from + 1 to — 1 ; tangents and cotangents,
from + Qo to — Qo ; secants and cosecants, from + oo to + 1, and from — 1 to — oo .
In the table given above the double sign ± or T is placed before and oo .
From the preceding investigation it appears that the functions always change
sign in passing through or through oo ; and the sign ± or ^ prefixed to or oo
simply shows the direction from which the value is reached. For example, at OP
the sine is passing from — (in quadrant IV) to + (in quadrant I). At 90° the
tangent is passing from + (in quadrant I) to — (in quadrant II).
85. Functions of Angles Greater than 360®. The functions of 360° + aj
are the same in sign and in absolute value as those of x. If n is a
positive integer,
The functions of (n x 360® + x) are the same as those of x.
For example, the functions of 2200°, or 6 x 360° + 40°, are the same in sign
and in absolute value as the functions ol 40°,
88 PLANE TRIGONOMETRY
Exercise 37. Variations in the Functions
Represent tTie follomng functions hy lines in a unit circle:
1. sinl35^ 7. sin210^ 13. sinSOO^ 19. sin270^
2. cosl20^ 8. cos225^ 14. cos315^ 20. co8l80^
3. tanl50^ 9. tan240^ 16. tan330^ 21. tanl80^
4. cotl35^ 10. cot210^ 16. cot 300*. 22. cot 270°.
6. sec 120**. 11. 8ec225^ 17. sec315^ 23. seclSO^
6. CSC 150**. 12. csc240^ 18. esc 330**. 24. esc 270**.
25. Prepare a table showing the signs of all the functions in
each of the four quadrants.
26. Prepare a table showing which functions always ha,ve the
minus sign in each of the four quadrants.
Represent the follomng functions by lines in a unit circle:
27. sin 390**. 30. cos 390**. 33. sin 460°. 36. tan 475°.
28. tan 405**. 31. cot 405°. 34. sin 570°. 37. sec 705°.
29. sec 420°. 32. esc 420°. 36. sin 720°. 38. esc 810°.
Show by lines in a unit circle th^t:
39. sin 150° = sin 30°. 46. tan 120° = - tan 60°.
40. cos 150° = - cos 30°. 46. cot 120° = - cot 60°.
41. sin 210° = - sin 30°. 47. tan 240° = tan 60°.
42. cos 210° = - cos 30°. 48. cot 240° = cot 60°.
43. sin 330° = - sin 30°. 49. tan 300° = - tan 60°.
44. cos 330°= cos 30°, 60. cot 300° = - cot 60°.
61. Write the signs of the functions of the following angles:
340°, 239°, 145°, 400°, 700°, 1200°, 3800°.
62. How many values less than 360° can the angle x have if
sin x = + f , and in what quadrants do the angles lie ? Draw a figure.
63. How many values less than 720° can the angle x have if
cos aj = + §, and in what quadrants do the angles lie ? Draw a figure.
64. If we take into account only angles less than 180°, how many
values can x have if sin x = ^? if cos x = i? if cos a; = — f ? if
tana; = §? if cotaj = -7?
66. Within what limits between 0° and 360° must the angle x lie
if 008 05 =— J ? if cotaj = 4 ? if secoj = 80 ? if cscaJ = — 3 ?
FUNCTIONS OP ANY ANGLE 8d
66. Why may cot 360** be considered as either + oo or — oo ?
67. Find the values of sin 450**, tan 540^ cos 630°, cot 720**, sin 810**,
CSC 900^ cos 1800**, sin 3600°. *^
68. What functions of an angle of a triangle may be negative ?
In what cases are they negative ?
69. In what quadrant does an angle lie if sine and cosine are both
negative ? if cosine and tangent are both negative ? ^
60^ Between 0** and 3600** how many angles are there whose sines
have the absolute value f ? Of these sines how many ai*e positive ?
Compute the value% of the foUovnng expressions:
61. a sin 0** + h cos 90** - c tan 180**.
62. a cos 90** -ft tan 180** + c cot 90**.
63. a sin 90** -ft cos 360** +{a-h) cos 180**.
64. (a* - b^ cos 360** - 4 oft sin 270** + sin 360**.
66. (a^ + h") cos 180** + {a^ + lP) sin 180** + (a^ + h^ tan 135^
66. (a^ + 2ab + &«)sin90** +(a« -2ab + b^GOS 180**- 4a5 tan225**.
67. (a — b-\-c-d) sin 270**-(a - ^^ -h c - e^ cos 180** + a tan 360^
State the sign of each of the six functions of the following angles :
^^68. 75^ 70. 155**. ' 72. 275**. 74. 355**.
69. 125^ 71. 185**. 73. 325**. 76. - 65^
Find the four smallest angles that satisfy the following conditions :
^76. sin^ = f 78. sin^=iV3. 80. tan^=iV3.
77. cos -4 = ^ Vs. 79. cos -4 = ^. 81. tan^ = V3.
Find two angles less than 360° that satisfy the following conditions:
-^^2. sin^ = — ^. 84. sin-4 = — ^V2. 86. tan^ = — 1.
83. C0S-4 = — ^. 86. C0Si4 = — ^V2. 87. cot-4 = — 1.
If Aj Bj and C are the angles of any triangle ABC^ prove that :
88. COS ^A = sin ^(5 + C). 90. cos ^B = sin ^(^ + C).
89. sin^e=cosi(^ +5). 91. sin^^=cosi(5 + C).
As angle A increases from 0° to 360°^ trace the changes in sign
and magnitude of the following :
92. sin^ cos^. 94. sin-4— cos-4. 96. tan-4 4-cot^.
93. sini4 + cos-4. 96. sin -4 -f- cos -4. 97. tan-4— cot-4.
#
\>
%)
tLANE TElGOKOMETRY
86. Reduction of Functions to the First Quadrant. In the annexed
figure BB' is perpendicular to the horizontal diameter A A', and the
diameters PR and QS are so drawn as to
make ZAOP= Z.SOA, It therefore fol-
lows from geometry that A MOP, MOS,
NOQ, and NOR are congruent.
Considering, therefore, only the absolute
values of the functions, we have
sin^OP = sin^OQ = sin^Oie = sin^05,
cos^OP = cos^OQ = COH AOR = cos AOS,
and so on for the other functions.
Hence, For every acute angle there is an angle in each of the higher
quadrants whose functions, in absolute value, are equal to those of
this acute angle.
If we let /.AOP = x and Z.POB=^y, noticing that Z.AOP =
ZQOA'=ZA'OR=:Z.SOA =x, and ZPOB = ZB0Q = Z.ROB' =
Z.B'OS = y, and prefixing the proper signs to the functions (§ 83),
we have :
Angle" Tit'-QpadrInt JI
sin X
cos (ISO® — oj) = — cos X
tan (180® — 05) = — tan x
cot (180° — 05) = — cot cc
I sin (90** + y^ cos y
COS (90 + y) = — sill y
tan(90®-f-y) = -coty
cot (90® -h y) = - tan y
Angle in Quadrant III
sin (180® + x)=— sin x sin (270® — y) = — cos i/
cos (180® -^ x) =— cos x cos (270® — y) = — sin y
tan(180® + a;)= tana; tan(270®-y)= cot y
cot (180® + x)= cot X cot (270® -y)= tan y
Angle in Quadrant IV
sin (360® - a;) = - sin a; sin (270® + y) = - cos y
cos (360® — x) = cos X
tan (360® — x)=— tan x
cot (360® — a;) = — cot a;
cos (270® + y)= sin y
tan(270® + 2/)=-coty
cot(270® + y)=-tany
For example, sin 127° = sin (180° — 63°) = sin 63° = cos 87°,
sin 210° = sin (180° + 30°) = - sin 30° = - cos 60°,
and sin 860° = sin (360° - 10°) = - sin 10° = - cos 80°.
>yl
^-^
FUNCTIONS OF ANY ANGLE 91
It appears from the results set forth on page 90 that the functions
of any angle, however great, can he reduced to the functions of an
angle in the first quadrant.
For example, suppose that we have a polygon with a reentrant angle of
247° 30', and we wish to find the tangent of this angle. We may proceed by
finding tan (180° + x) or by finding tan (270° — x). We then have
tan 247° SO' = tan (180° + 67° 300 = ^an 67° SO',
and tan 247° SO' = tan (270° - 22° SO') = cot 22° 30'.
That these two results are equal is apparent, for
tan 67° SO' = cot (90° - 67° SO^ = cot 22° SO'.
It also appears that, for angles less than 180^, a given value of a
sine or cosecant determines two supplementary angles, one acute, the
other obtuse ; a given value of any other function determines only one
angle, this angle being acute if the value is positive and obtuse if the
value is negative.
For example, if we know that sinx = J, we cannot tell whether x = 30° or
150°, since the sine of each of these angles is J. But if we know that tan x = 1,
we kiijow that x = 46°.
Similarly, if we know that cotx =— 1, we know that x = 135°, there being
no other angle less than 180° whose cotangent is — 1.
Since sec x is the reciprocal of cos x and esc x is the reciprocal of sin x, and
since by the aid of logarithms we can divide by cosx or sinx as easily as we
can multiply by secx or cscx, we shall hereafter pay but little attention to the
secant and cosecant. Since the invention of logarithms these functions have
been of little practical importance in the work of ordinary mensuration.
Exercise 38. Reduction to the First Quadrant
Express the following as functions of angles less than 90^ :
1. sinlTO^ 11. sin275^ 21. sin 148° 10'.
2. cos 160°. 12. sin 345°. 22. cos 192° 20'.
3. tan 148°. 13. tan 282°. 23. tan 265° 30'.
cot 156°. 14. tan 325°. ^^24. cot 287° 40'.
5. sin 180°. 16. cos 290°. 26. sin 187°' 10' 3".
6. tan 180°. "Xie. cos 350°. 26. cos 274° 5' 14''.
7. sin 200°. 17. cot 295°. 27. tan 322° 8' 15".
"8. cos 225°. 18. cot 347°. 28. cot 375° 10' 3".
9. tan 258°. 19. sin 360°. 29. sin 147.75°.
10. cot 262°. 20. cos 360°. 30. cos 232.25°.
92
PLANE TEIGONOMBTRY
87. Functions of Angles Differing by 90^. It was shown in the case
of acute angles that the function of any angle is equal to the co-func-
tion of its complement (§ 8). B
That is, tan 28® = cot (90* - 2^) = cot 62<»,
sinx =s co8(90^ — x), and so on.
It will now be shown for all angles
that if two angles differ by 90^ , the funo- '^\
tions of either are equal in absolute value
to the co-functions of the other. ^^
In the annexed figure the diameters PR
and QS are perpendicular to each other,
and from P, Q, -R, and S perpendiculars are drawn to AA\ Then
from the congruent triangles OMP^ QHO, OKR, and SNO we see that
OM=QH=OK=SN,
and MP=OH=^KR = ON.
Hence, considering the proper signs (§ 83),
sin^OQ = cos^OP, cos^OQ =— sin^l OP,
WisiAOR = Gos AOQ, cos A OR = — sin A OQ,
sinAOS = G08A0R, cos AOS =:^ sin AOR.
In all these equations, if x denotes the angle on the right-hand
side, the angle on the left-hand side is 90** -f- x.
Therefore, if a; is an angle in any one of the four quadrants,
sin (90** -f- x) = cos at, cos (90** -f x) = — sin a; ;
and hence tan (90* + x) = — cot x, cot (90** + x) = — tan x.
It is therefore seen that the algebraic sign of the function of the resulting
angle is the same as that found in the similar case in § 86.
88. Functions of a Negative Angle. If the angle x is generated
by the radius moving clockwise from the initial position OA to the
terminal position OS, it will be negative (§ 72), and its terminal
side will be identical with that for the
angle 360** — aj. Therefore the functions
of the angle —x are the same as those
of the angle 360**— x; or
sin (— aj) = — sin aj,
cos (— x) = cos X,
tan (— a;) = — tan x,
cot (—/x) = — cot x.
FUNCTIONS OF ANY ANGLE 98
Exercise 39. Reduction of Functions
JEiipreaa the following as functions of angles less than 4S^ :
Xl. sinlOO^ 6. co8 95^ 9. tan 91**. 18. cot 94* 1^
2. sinl20^ 6. co8 97^ ^^10. tan99^ 14. cot 97** 2'.
3. sinllO^ 7. coslir. 11. tanll9^ 16. cot 98* 3'.
4. sinlSO^ 8. co8l27^ 12. tanl29^ 16. cot 99** 9'.
Express thefollomng as functions of positive angles :
17. sin (-3**). 21. cos (-87**). 26. tan (-200**).
18. sin (-9**). 22. cos (-95**). 26. cot (-1.5**).
19. sin (- 86**). 23. tan (-100**). 27. cot (- 7.8**).
20. cos (-75**). ^^ 24. tan (-150**). 28. cot (- 9.1**).
Find the following hy aid of the tables :
29. sin 178** 30'. 37. logsinl27.5^
30. cos 236** 45'. 38. log cos 226.4**.
31. tan 322** 18'. 39. log tan 327.8**.
32. cot 423** 15'. 40. logcot343.3^
33. sin (- 7** 29' 30"). 41. log sin 236** 13' 5".
34. cos (- 29** 42' 19"). 42. log cos 327** 5' 11".
35. tan (-172** 16' 14"). 43. log tan (-125** 27').
36. cot (- 262** 17' 15"). 44. log cot (- 236** 15').
^45. Show that the angles 42°, 138**, - 318^ 402**, and - 222** all
have the same sine.
46. Find four angles between 0** and 720** which satisfy the equa-
tion sin 05 =— ^ V2.
47. Draw a circle with unit radius, and represent by lines the
sine, cosine, tangent, and cotangent of — 325**.
48. Show by drawing a figure that sin 195** = cos (— 105**), and
that cos 300** = sin (- 210**).
49. Show by drawing a figure that cos 320** = — cos (— 140**), and
that sin 320** = - sin 40**.
50. Show by drawing a figure that sin 765** = J V2, and that
tan 1395** = - 1.
51. In the triangle ABC show that cos^ = — cos(5 4-C), and
that cos B = — cos {A + C).
94 PLANE TRIGONOMETEY
89. Relations of the Functions. Certain relations between the func-
tions have already been proved to exist in the case of acute angles
(§§ 13, 14), and since the relations of the functions of any angle to
the functions of an acute angle have also been considered (Si 80, 86,
86, 88), it is evident that the laws are true for any angle. These
laws are so important that they will now be summarized, and others
of a similar kind will be added.
These laws should be memorized. They will be needed frequently in the
subsequent work. The proof of each should be given, as required in § 14.
The ± sign is placed before the square root sign, since we have now learned
the meaning of negative functions.
To find the sine we have :
1
CSCiC
sm X = sin aj = ± Vl — cos^a;
To find the cosine we have :
1
sec 05
cos X = cos X = ± Vl — sin^a;
To find the tangent we have :
^1 ^ sinx
tanaj = — r" tana =
cot X cos X
x^^,. _ ■ sin a; Vl-cos^a;
tan aj — db -7==== tana; = ±
VI— sm^'a; cos a;
tana; = ± Vsec^a; — 1 tana; = sin a; sec a;
To find the cotangent we have :
1 ^ cos a;
cot X == cot X = -: —
tan X sm x
cos a; Vl— si
cot a; = ± =. cot a; = ±
sm'a;
Vl — cos^a; ~ sma;
cot a; = ± Vcsc^a; — 1 cot x = cos x esc x
To find the secant we have :
sec X = sec a; = ± Vl + tan^a?
cos a;
To find the cosecant we have :
1
csca; = -: — CSC a; = + Vl+ cot^a?
sm a;
FUNCTIONS OF ANY ANGLE
95
Exercise 40. Relations of the Functions
1. Prove each of the formulas given in § 89.
Prove the following relations:
tana;
2. sin a; = ±
3. cosaj = ±
Vl -h tan^aj
cot a;
Vl -h cot^aj
6. Find sin x in terms of cot x,
7. Find cos x in terms of tan x.
Prove the following relations:
10. tan X cos x = sin x.
4. tan a = ±
6. cot a; = ±
Vcsc^aj — 1
1
Vsec'^aj — 1
8. Find sec x in terms of sin x,
9. Find CSC x in terms of cos x.
\
\
14. cot^a; = cos^a; + cos'^a? cottar.
11. cos^a; = cot^a; — cot^aj cos^aj. . 15. cot^a; sec^a; = 1 -+• cot^a;.
12. tan^a? = sin^»^+ sin^a; tan^a;. >/^ 16. csc^aj — cot^aj = 1.
13. cos^a; + 2 sin^a; = 1 + sin^a;. 17. sec^a: + csc^aj = sec^aj csc^a;.
18. Show that the sum of the tangent and cotangent of an angle
is equal to the product of the secant and cosecant of the angle.
Recalling the values given on page 8, find the value of x when :
*^ 19. 2 cos X = sec x. x^6. tan ar = 2 sin x,
^^20. 4sinaj = csca. / 26. sec a; = V2 tan a?.
"^^1. sin-'a; = 3 cos^aj. / 27. sin^a; — cos a; = J.
22. 2 sin^aj + cos^a; = J. \ 28. tan^a; — sec a; = 1.
23. 3 tan^a; — sec^a; =1. V 29. tan^a; + csc^a; = 3.
24. tan aj + cot a; = 2. ^30. sin aj + VS cos a; = 2.
\31. Given (sin x -\- cos a)^ — 1 = (sin x — cos aj)^ + 1, find x.
32. Given 2 sin x = cos a;, find sin x and cos x.
33. Given 4 sin x = tan x, find sin x and tan x,
\ 34. Given 5 sin x = tan x, find cos x and sec x,
36. Given 4 cot a; = tanar, find the other functions.
36. Given sin a = 4 cos x, find sin x and cos x.
37. If sin X : cos x = 9 : 40, find sin x and cos x,
sin a:
38. From the formula tan x = ±
under which tan x = sin x.
Vl - sin»
find the condition
X
98
PLANE TRIGONOMETRY
92. The Proofs continued. In the proofs given on page 97, x, y,
and X + y were assumed to be acute angles. If, however, x and y
are acute but aj+y is obtuse, as shown in
this figure, the proofs remain, word for
word, the same as before, the only differ-
ence being that the sign of OF will be nega-
tive, as DG is now greater than OE, This, L
however, does not affect the proof. The
above formulas, therefore, hold true for all acute angles x and y.
Furthermore, if these formulas hold true for any two acute angles
X and y, they hold true when one of the angles is increased by 90®.
Thus, if for x we write x' = 90** + x, then, by § 87,
sin(x' + y)= sin(90'' + x + y) = ooa(x + y)
= cos X cos y — sin x sin y.
But by § 87, cos x = sin (90** + a?) = sin a',
and sina; = — cos (90® + a?) = — cos a;'.
Henoe^ by substituting these values,
sin (x' -h y) =• sin a;' cos y + cos x' sin y.
That is, § 90 holds true if either angle is repeatedly increased by 90^. It is
therefore true for all angles.
Similarly, by § 87,
cos (a;' + y)= cos (90** + a; -f y) = — sin(a; + y)
= — sin a; cos y — ' cos aj sin y
= cos a;' cos y — sin x* sin y,
by substituting cos a?' for — sin x and sin x' for cos x as above.
That is, § 91 also holds true if either angle is repeatedly increased by 90°.
It is therefore true for all angles.
Ezettise 41. Sines and Cosines
GHven 9in30'' = co860''=:^j co8 30'' = 8in60'' = jy/3, and »in43'
co94S^ = J V^, find the values of the following :
1. sin 15®.
. cos 15®.
3. sin 75®.
4. cos 75®,
6. sin 90®.
6. cos 90®.
7. sin 105®.
8. COS 105®,
9. sin 120®.
--10. cos 120®.
1. sin 135®.
12. cos 135®.
13. sin 150®.
^14. cos 150®.
15. sin 165®.
16. cos 166®,
SUM OE DIFFEEENCE OF TWO ANGLES 99
siu jL
93. Formula for tan (x 4- y). Since tan-4 = > therefore
, ^ sin (cc + y) sin x cos y + cos x sin y
tan (x + y) = — r— p^ = ^-^— : r-^y
^ ^^ cos (x + y) cos x cos y — smx sm y
whatever the size of the angles x and y (§ 92).
Dividing each term of the numerator and denominator of the
last of these fractions by cos x cos y, we have
sin a; , sin^
, . cos a; cosy
tan(aj + y) = : r-^-*
^ ^^ sma; smy
cos X cos y
^ ^ . sinaj ^ - siny
But since = tan x, and ^ = tan y,
cosoj cosy ^
we have tan(x + y) = ; — •
^ ^ 1 — tanxtany
This important formula should be memorized.
cos jL
94. Formula for cot (x+y). Since cot A = -: — - > therefore
^ ^ SID. A
cosfoj + y) cos X cos y — sin x sin y
cot(ir + y) = . ) T X = -^ ^ -'
^ ^^ sm(a; + y) sm aj cos y + cos a; sm y
whatever the size of the angles x and y (§ 92).
Dividing each term of the numerator and denominator of the
last of these fractions by sin a; sin y, and then remembering that
cos a;
cos 11
= cot X and — — - = cot y, we have
sm X sm y
, . cotxcoty — 1
cot (or + y) = —7 — . V. •
^ -' coty + cotx
This important formula should be memorized.
Exercise 42. Tangents and Cotangents
Given tan SO"" = 00160"" = |v^, cotSO'' = tan 60'' = V5, tan 4S^
= cot 45^ = i, jSwd ^A« value% of tliefollomng:
1. tan 15°. ^5. tan 90°. 9. tan 120°. ^13. tan 160°
2. cot 15°. 6. cot 90°. 10. cot 120°. 14. cot 150°.
3. tan 75°. 7. tan 105°. 11. tan 135°. 16. tan 165°
4. cot 75°. 8. cot 105°. 12. cot 135°. 16. cot 165°
100
PLANE TEIGONOMETRY
95. Fonna]afortiii(ar~j(). In this figure there axe shown two acute
angles, AOB = x and COB = y, with Z.AOC equal to a? — y ; two per-
pendiculars are let fall from C, and two from D.
The perpendiculars from D are BE and DO, HQ
being drawn to ¥C produced.
Then, considering the radius as unity, we have
sin (aj - y) = CF = Z>J5: - C G.
But BE = sin x • OB = sin aj cos y,
and (tC = cosa: • CD= cosajsiny.
Hence, by substituting these values of BE and (rC,
sin (x — y) = sin X cos y — cos X sin y.
Tills is one of the most important formulas and should be memorized.
96. Formula for cos (^—y)* Using the above figure we see that
cos(aj — y)= OF— OE + BG.
But OE = cos X • OB — cos x cos y,
and BG = sinaj • CB = sin'ZB siny.
Hence it follows that
co8(x~ y)=: cos xcosy-i- sin X siny.
This important formula should be memorized. The proof in §§ 95 and 96
refers only to acute angles, but the formulas are entirely general if due regard
is paid to the algebraic signs. The general proof may follow the method of
§ 92, or it may be based upon it ; the latter plan is followed in § 97.
97. The Proofs continued. Since aj = (a; — y) -|- y, we see that
sin aj = sin {(aj — y)-\-y) — sin {x — y) cos y -f- cos (a; — y) sin y,
cos X = cos {(x — y)'b y} = cos(aj — y)cos y — sin (a; — y)sin y.
Multiplying the first equation by cos y, and the second by sin y,
sin X cos y = sin (x — y) cos^y -|- cos (x — y) sin y cos y,
cos ac siny =— sin(a; — y)sin^ + cos(a; — y)siny cosy.
Hence sin x cos y — cos x sin y = sin (x — y) (sin^ + cos^).
But by § 14 sin^ -f- cosV =1.
Therefore sin (x — y)= sin a? cos y — cos x sin y.
Similarly, cos (x — y)= cos x cos y + sin a: sin y.
Therefore the formulas of §§ 95 and 96 are universally true
SUM OR DIFFERENCE OF TWO ANGLES 101
siu jL
98. Formula for tan (x — if). Since tan-4 = -> we have
tan (a; - y) = J ^
^ ^^ co8(aj — y)
COSii
sin X cos y — cos x sin y
cos a; cos y + sin a; sin y
Dividing numerator and denominator by cos a; cos y, as in § 93, we
obtain sin a; siny
cosa; cosy
. , sinx siny
1+
tan(a; — y) =
1 +
cos a; cosy
tanor— tanif
That is, tan(af— |f)i-- . ^
' V ^f l+tanxtan|f
This important formula should be memorized.
99. Formula for cot (;r — y). Following the plan suggested in § 98,
we may show that , .
•^ ^, . cos (aj — y)
cot(a; - y) = -r-) ^
^ ^^ sin (aj — y)
_ cosa; cosy + sina; siny
"" sin X cos y — cos aj sin y
cosaj cosy . ^
-: — • -7—^ +1
sm X sin y
^ •
cosy cos a;
sin y sin a;
r«i^ ^ . .r X cot;ccoty+l
That is, <»t(x-y) = --- ^-^^
* ^ -^ coty— cotx
This important formula should be memorized.
100. Summary of the Addition Formulas. Theformulasof§§ 90-99
may be combined as follows :
sin (x ±iy)=^ sin x cos y ± cos x sin y,
oos(a ± y) = cos a; cos y ip sin aj siny,
, , . tana:±tany
^<^=fcy) = lTten»teny '
^ ^^ cotyicota;
When the signs d: and T occur in the same formula we should be careful to
take the — of :f with the + of ±, That is, the upper signs are to be tiiken
together, and the lower signs are to be taken together.
102 PLANE TRIGONOMETRY
Exercise 43. The Addition Formulas
Qiven sin x = jyeo8x = ^, 9in y^^jcosy^^^ find the value of:
. sin(aj -f- 2^)' 3. cos(a3-|-y). 6« taii(aj -f- y).
2. siii(aj — y). 4. C08(a5 — y). 6. tan(aj — y).
J?y letting x = P0° iw the formulas^ find the following :
. sin(90^-y). 8. cos(90*-y). 9. tan (90^ - 2^).
Similarly^ hy mbstituting in the formulas^ find the following :
0. sm(90**-f2/). 17. cos (aj — 90^). 24. sin(-y).
1. sin(180* - y). 18. cos (x - 180^). 26. sin (45° - 3/).
12. sin(180° + 2^)
13. sin(270°-y)
14. sin(270°-f2/)
16. sin(360*-2/)
16. sin(360° + 2^)
19. cos (x - 270°). 26. cos (45° - 3/).
20. tan(aj - 90°). 27. tan (45° - y),
21. tan (aj - 180°). 28. cot (30° + y).
22. cot (x - 90°). 29. cot (60° - y).
23. cot (x - 180°). 30. cot (90° - y).
^51. If tan X = 0.5 and tan y = 0.25, find tan (x + y) and tan (x — y)
32. If tan x = 1 and tan y = J Vs, find tan(aj -f- y) and tan(x — y),
33. If tan a; = ^ and tan 3/ = ^, find tan (aj + y) and tan (aj — y),
and find the number of degrees in a; + y.
*^34. If tan x = 2 and tan y = J, what is the nature of the angle
x + y? Consider the same question when tan a; = 3 and tan y = J,
and when tan aj = a and tan y = 1/a.
36. Prove that the sum of tan (x — 45°) and cot (x -f- 45°) is zero.
36. Prove that the sum of cot (x — 45°) and tan (x -f- 45°) is zero.
37. If sin X = 0.2 VS and sin y = 0.1 VlO, prove that x-\-y = 45°
May X -{-y have other values ? If so, state two of these values.
38. Prove that if an angle x is decreased by 45° the cotangent of
the resulting angle is equal to r •
39. Prove that if an angle x is increased by 45° the cotangent of
the resulting angle is equal to — - — —7 •
° ° ^ cotaj + 1
1
40. If tanaj = t and tan y = > prove that tan (x-\'y) = 1.
41. If a right angle is divided into any three angles a., y, «, prove
, , , , 1 — xan y tan z
that tan x = -7 ~ •
tan^ + tauji;
SUM OR DIFFERENCE OF TWO ANGLES 103
101. Functions of Twice an Angle. By substituting in the formulas
for the functions ofx + y we obtain the following important for-
mulas for the functions of twice an angle :
sin 2 ;r= 2 sin or cos X,
cos 2 :r = cos* or — sin* ;r,
^ ^ 2tan;r
tan2;c=— — — — I
l*tan*;r
cot*;c— 1
cot2;r=
2cotx
Letting 2x = y we have the following useful formulas :
sin 2/ = 2 sin i y cos ^y, cos y = cos^ \y — siu^ \ y^
2taniy ^ cot^iy — 1
*«^ny = 3 — 7 — 1-7— » coty= ^ V,
^ l--tai?y^ ^ 2cotiy
Exercise 44. Functions of Twice an Angle
As suggested aiove, deduce the formulas for the following :
1. sin 2 a;. 2. cos 2 a;. 3. tan 2 a;. 4. cot2a:L
Find sin 2 x^ given the following values of sin x and cos x :
. sina;=4^ V2, cosx=^V2. 6. sin a; = 4^, cos aj = 4^ VS.
Find cos 2 Xj given the following values of sin x and cos x :
7. sin a; = 4^ Vs, cos x=^i. ^*^*6. sin aj = f , cos aj = f .
Find tanSx^ given the following values of tanx:
9. tanaj = 0.3673. 10. tana; = 0.2701.
Find cot 2 x, given the follomng values of cot x and tan x :
11. cot a; = 0.3673. 12. tan x = 0.2701.
Find sin2xy given the follomng values of sin x :
13. sina; = -|%^. """"""^^ 14. sina; = j-|-
*16. As suggested in § 101, find sin 3 a; in terms of sin a;.
16. As suggested in § 101, find cos 3 a; in terms of cos a;.
104 PLANE TRIGONOMETRY
102. Functions of Half an Angle. If we substitute iz for a; in the
fonnulas cos* a; + sin** = 1 (§ 14) and cos* a; — sin* a; = cos 2a; (§101),
so as to find the functions of half an angle, we have
cos*i« + sin*i» =1,
and coB^iz — aiD?iz =icoaz.
Subtracting, 2sin*i« =1— cos»;
whence sin 4 x = =fc -^ I "
1'=*^
In the above proof, if we add instead of subtract we have
2 coa^iz = 1 + cos « ;
. 1 , | 1+ cos^
whence cosgz = ±-^
Since tanA-^ = — 7—1 and coti« = -i — 7— > we have, by dividing,
2 Nl +
^ * _i_ ,-— cosz
cosz
and cotsz = ifcx -"^^^ •
5* All — cosz
These four formulas are important and should be memorized.
From the formula for tan ^ z can be derived a formula which is
occasionally used in dealing with very small angles. In the triangle
ACB we have
^ , ^ ll— cos-4 e \c--b
Exercise 45. Functions of Half an Angle
Griven gin 30^ = j, find the values of the folhmng :
1. sinl5^ 2. cos 15**. 3. tanl5^ 4. cot 15**. 5. cot7i^
Qiven tan 45^ =i, find the values of the follomng :
6. sin22.5**. 7. cos22.5^ 8. tan22.5°. 9. cot22.5^ 10. cot Hi**.
^11. Given sin a; = 0.2, find sin 4^ a; and cos^a;. £jr^ c^r^ <5^k^^v,^
12. Given cos x = 0.7, find sin ^a;, cos ^a;, tan ^05, and cot ^a;.
SUM OE DIFFEEENCE OF TWO ANGLES 106
103, Sums and Differences of Functions. Since we have (§§ 92, 97)
sin(a; -|- y) = sin x cos y -f- cos x sin y,
and sin (« — y) = sin x cos y — cos x sin y,
we find, by addition and subtraction, that
sin (a; H- y) + sin (a; — y) = 2 sin a cos y,
and sin (aj H- 2^) — sin (aj — 2^) = 2 cos a: sin y.
Similarly, by using the formulas for cos (x ± y), we obtain
cos (X'\'y^-\- cos (a; — y) == 2 cos a: cos y,
and co8(aj + y)— cos(a; — y) = — 2sina:siny.
By letting aj + y = -4, and x — y = ByWe have aj = ^^(^1 + 5), and
y = ^(^ — JB), whence
sin ii + sin B = 2 sin \{A + B) cob\{A ^ B),
siuii — sinB= 2coS|(A + B)sinJ(ii — B),
cosii + cosB= 2 cos \(A + B) cos \(A — B),
and cos ii — cos B=: — 2 sin^ii + B) sin \{A — B).
By division we obtain
sin-4 -f sinB ^ ,, ^ r.v i. 1 / . t.x
sin^ - sin^ = tani(^ -hB)coti(^ -^);
and since coti(^ -B) = g^^^^—^,
, sinii+sinB tan|(ii + B)
w^e nave aii^B^B^HHMMB^K^BH ■"» ^^k^m^b^hb^m^^^ •
sinii— sinB tan|(ii — B)
This is one of the most important formulas in^the solution of oblique trianglea
Exercise 46. Formulas
Prove the following formvlaa :
V , . ^ 2tanaj ^^V^„ 4. , sinaj
M. sm2a; = 3— TT — T" 3. tan4^aj = :r-; •
1 + tan-'a; IH- cos a;
rt 1 — tan^a; , ^ , sin a:
2. cos2a; = T— — 7 — s-- 4. cot^aj =
1 + tan^a; * 1 — cos aj
Arj\ If Ay B^C are the angles of a triangle^ prove that :
^^5. sini4 -|-sinJ5 + sinC= 4 cos ^-4 cos JBcos^C.
6. cos^ -f- cos-^-f- cosC = l + 4sin^^ sin^J5sin^C
7. tan^ H- tan J? + tan C = tan^ tan5 tan C.
106 PLANE TRIGONOMETRY
8. Given tan J aj = 1, find cos x,
9. Given cot ^ x = VS, find sin x,
sin 33* -f sin 3*
10. Prove that tan 18° = ..^ . _.
cos 33 -f cos 3
11. Prove that sin ^aj ± cos ^a; = Vl ± sinaj.
. « -r* ^1 ^ ^^ ^ ± tstn V ^ ^
12. Prove that — : — ^ = + tanaj tany.
cot X ± cot y ^
13. Prove that tan (45° — aj) = t-— r •
^ ^ 1 + tanaj
14. In the triangle ABC prove that
cot ^ ^ H- cot )^B -|- cot ^ C = cot \A cot \B cot \ C,
Change to a form involving products instead of sums, and hence
more convenient for computation hy logarithms :
16. cot X + tan X, 20. 1 + tan x tan y,
16. cot X — tan x. 21.1 — tan x tan y.
17. cot x + tan y, 22. cot x cot y -|- 1.
18. cot a; — tany. 23. cot a; coty — 1.
^ ^ 1 — cos 2 a; ^ ^ tan x 4- tan v
19^ . 24. ^ — ■ ^»
1 + cos 2 a; ' cot x -|- cot y
26. Prove that tan x + tan y = ^ ^ •
^ cos X cos y
26. Prove that cot y — cot a; = —r-^ — r-^ •
sm X sm y
27. Given tan (a; + y) = 3, and tana; = 2, find tan y,
9/8. Prove that (sin x -f- cos a;)^ = 1 -|- sin 2 a;.
'29. Prove that (sin x — cos a;)^ = 1 — sin 2 a;.
30. Prove that tan a; -f- cot a; = 2 esc 2 a;.
31. Prove that cot x — tan x = 2 cos 2 x esc 2 a;.
32. Prove that 2 sin2(45° - a;) = 1 - sin 2 a;.
33. Prove that cos 45° + cos 75° = cos 15°.
34. Prove that 1 + tan x tan 2x = tan 2 x cot a; — 1.
Prove the following formulas :
36. (cos X -f- cos yy + (sin a; + sin 3/)* = 2 + 2 cos (x — y).
36. (sin X + cos y)* + (sin y -f- cos a;)^ = 2 + 2 sin (x + y),
37. sin (a; + y) + cos (x — y) = (sin a; + cos x) (sin 2/ -f- cos y).
38. sin (a; + y) cos y — cos (x + 2/) sin y = sin a;.
CHAPTER VII
TH£ OBLIQUE TRIANGLE
104. Geometric Properties of the Triangle. In solving an oblique
triangle certain geometric properties are involved in addition to
those already mentioned in the preceding chapters, and these should
be recalled to mind before undertaking further work with trigono-
metric functions. These properties are as follows :
The angles opposite the equal sides of an isosceles triangle
are equal.
If two angles of a triangle are equals the sides opposite the equal
angles are equal.
If two angles of a triangle are unequal, the greater side is
opposite the greater angle.
If two sides of a triangle are unequal, the greater angle is
opposite the greater side.
A triangle is determined, that is, it is completely fixed in form
and size, if the following parts are given :
1. Two sides and the included angle.
2. Two angles and the included side.
3. Two angles and the side opposite one of them.
4. Two sides and the angle opposite one of them.
5. Three sides.
The fourth case, however, will be recalled as the ambiguous case, since the
triangle is not in general completely determined. If we have given ZA and
sides a and b in this figure, either of the triangles ABC
and ARC will satisfy the conditions.
If a is equal to the perpendicular from C on AB, how-
ever, the points B and R will coincide, and hence the two
triangles become congruent and the triangle is completely
determined.
The five cases relating to the determining of a
triangle may be, summarized as follows; A triangle is determined
when three irldeperident parts are given.
This excludes the case of three asgles, because they are not independent*
That is, A = 180° — (j5 + C), and therefore A depends upon B and C.
107
T!"*'^^^
I^w^
PLANE TRIGONOMETEY
Law of Sines. In the triangle A BCy using either of the figures
as nere shown, we have the following relations.
In either figure,
In the first figure,
and in the second figure,
7 = SiUil.
- = sinB.
a
- = sin(180* - B)
sssin^.
Therefore, whether h lies within or without the triangle, we
obtain, by division, the following relation:
a sinii
^""siiiB*
In the same way, by drawing perpendiculars from the vertices
A and B to the opposite sides, we may obtain the following relations :
h sin^
c sinC
a sin^
c sinC
and
This relation between the sides and the sines of the opposite angles
is called the Law of Sines and may be expressed as follows :
The sides of a triangle are proportional to the sines of the opposite
angles.
If we multiply - = -; — - by 6, and divide by sin J., we have
h sinJ?
a b
sin J. sinJ3
Similarly, we may obtain the following :
a b c
sin J. sinJ3 sinC
and this is frequently given as the Law of Sines.
It is also apparent that a sin J3 = 6 sin J., a sin C = c sin ^, and 6 sin C = c sin J9,
three relations which are still another form of the Law of Sines.
THE OBLIQUE TRIANGLE 109
106. The Law of Sines extended. There is an interesting extension
of the Law of Sines with respect to the diameter of the circle circum-
scribed about a triangle.
Circumscribe a circle about the triangle ABC and draw the radii
OBy OCy as shown in the figure. Let R denote the radius. Draw
OM perpendicular to EC. Since the angle BOC is a central angle
intercepting the same arc as the angle Ay the angle 50C = 2-4;
hence the angle BOM = A ; then
BM = R siaBOM= R ainA.
Therefore a = 2 iJ sin -4 .
In like manner, b = 2RsmBy
and c = 2 i2 sin C.
a b c
Therefore 2R =
sin A sin 5 sinC
That is, The ratio of any side of a triangle to the sine of the oppo-
site angle is numerically eqtial to the diameter of the circumscribed
circle.
Exercise 47. Law of Sines
a sin^
1. Consider the formula 7 = -: — - when B = 90° ; when A = 90* ;
1. . « i_ , 6 sm5 '
when A=^ B\ when a = 0.
. Prove by the Law of Sines that the bisector of an angle of a
triangle divides the opposite side into parts proportional to the
adjacent sides.
3. Prove Ex. 2 for the bisector of an exterior angle of a triangle.
4. The triangle ABC has A = 78°, B = 72°, and c = 4 in. Find the
diameter of the circumscribed circle.
6. The triangle ABC has ^ = 76° 37', B = 81° 46', and c = 368.4 ft.
Find the diameter of the circumscribed circle.
. What is the diameter of the circle circumscribed about an equi-
lateral triangle of side 7.4 in. ? What is the diameter of the circle
inscribed in the same triangle ?
7. What is the diameter of the circle circumscribed about an isos-
celes triangle of base 4.8 in. and vertical angle 10° ?
8. What is the diameter of the circle circumscribed about an isos-
celes triangle whose vertical angle is 18° and the sum of the two equal
sides 18 in. ?
no PLA2^ TRIGONOMETRY
107. Applications of the Law of Sines. If we have given any side
of a triangle, and any two of the angles, we are able to solve the tri-
angle by means of the Law of Sines. Thus, if we have given «, A ,
and By in this triangle, we can find the remaining parts as follows :
1. C = 180*-(^+5).
^ b siaB
a aiaA
asin^ a . ^
.*. b = — : — 7" = -: — 7 X SinB.
sin^ sin^
<j __ sinC ^ . _ ^ ^^ ^ ^
a BiaA ' * sin^ sin^l
For example, given a = 24.31, ^ = 45* 18', and 5 = 22° 11', solve
the triangle.
The work may be arranged as follows :
a == 24.31 log a = 1.38578 = 1.38578
^ = 45* 18' colog sin^ = 0.14825 = 0.14825
B=: 22*11' log sin.g= 9.57700 log sin C = 9.96556
il + 5 == 67* 29' log b = 1.11103 log c = 1.49959
.-. C = 112* 31' .-. b = 12.913 .-. e = 31.593
When — lOis omitted after a logarithm or cologarithm to which it belongs,
it must still be remembered that the logarithm or cologarithm is 10 too large.
The length of a having.been given only to four significant figures, the values
of b and c are to be dei)ended upon only to the same number of significant
figures in pi*actical measurement. In the above example a is given to only four
significant figures, and hence we say that b = 12.91, and c = 81.50.
Exercise 48. Law of Sines
Solve the triangle ABC^ given the followirig parU :
1. a = 500, A = 10* 12', B = 46* 36'. ^
2. a = 795, A = 79* 59', B = 44* 41'.
3. a = 804, A = 99* 55', B = ^5* 1'.
4. a = 820, A = 12* 49', B = 141* 59^
6. c = 1005, A = 78* 19', B = 54* 27'.
6. 6 = 13.57, B = 13* 57', C = 57* 13'.
7. a = 6412, A = 70* 55', C = 52* 9'.
8. J = 999, A = 37* 58', C = 65* 2\
c
THE OBLIQUE TRIANGLE 111
Solve ExB, 9-14 without using logarithms :
9. Given b = 7.071, A = 30°, and C = 105°, find a and c.
10. Given c = 9.562, A = 45°, and B = 60°, find a and b,
11. The base of a triangle is 600 ft. and the angles at the base
are 30° and 120°. Find the other sides and the altitude.
12. Two angles of a triangle are 20° and 40°. Find the ratio of
the opposite sides.
13. The angles of a triangle are as 5 : 10 : 21, and the side oppo-
site the smallest angle is 3. Find the other sides.
14. Given one side of a triangle 27 in., and the adjacent angles
each equal to 30°, find the radius of the circumscribed circle.
16. The angles B and C of a triangle ABC are 50° 30' and 122° 9'
respectively, and BC is 9 mi. Find AB and A C
16. In a parallelogram, given a diagonal d and the angles x and y
which this diagonal makes with the sides, find the sides. Compute
the results when d = 11.2, x = 19° 1', and y = 42° 54'.
7. A lighthouse was observed from a ship to bear N. 34° E.;
after the ship sailed due south 3 mi. the lighthouse bore N. 23° E.
Find the distance from the lighthouse to the ship in each position. \ '^ io
The phrase to hear N, S4° E. means that the line of sight to the lighthouse is \
in the northeast quarter of the horizon and makes, with a line due north, an
angle of 34°.
*^*^<18. A headland was observed from a ship to bear directly east ;
after the ship had sailed 5 mi. N. 31° E. the headland bore S. 42° E.
Find the distance from the headland to the ship in each position.
19. In a trapezoid, given the parallel sides a and b, and the angles
X and y at the ends of one of the parallel sides, find the nonparallel
sides. Compute the results when a = 15, b = 7, x = 70°, y = 40°.
• ' SJO. Two observers 5 mi. apart on a plain, and facing each other,
find that the angles of elevation of a balloon in the same vertical
plane with themselves are 55° and 58° respectively. Find the dis-
tance from the balloon to each observer, and also the height of the
balloon above the plain.
21. A balloon is directly above a straight road 7 J mi. long, joining
two towns. The balloonist observes that the first town makes an
angle of 42° and the second town an angle of 38° with the perpen-
dicular. Find the distance from the balloon to each town, and also
the height of the balloon above the plain.
112 PLANE TRIGONOMETRY
108. The Ambiguous Case. As mentioned in § 104, if two sides
of a triangle and the angle opposite one of them are given, the solu-
tion will lead, in general, to two triangles. Thus,. if we have the
two sides a and b and the angle A given, we proceed to solve the
triangle as follows ;
C = 180*-(^+5);
hence we can find C if we can find B,
e sinC
Furthermore,
a sin^
asinC
whence c = . . ,
hence we can find e if we can find C, and we can also find c if we
cai^ find B, But to find B we have q
sin B __b
sin A a
h^inA
whence sin B =
a
--.-f.--'"^
Therefore we do not find B directly, but only sin 5. But when an
angle is determined by its sine, it admits of two values which are
supplements of each other (§ 86) ; hence either of the two values
of B may be taken unless one of them is excluded by the conditions
of the problem.
In general, therefore, either of the triangles ABC and AB^C fulfills
the given conditions.
Exercise 49. The Ambiguous Case
In the triangle ABC given a, J, and A, prove that :
1. If a > b, then A> By B \^ acute, and there is one and only one
triangle which will satisfy the given conditions.
2. If a = by both A and B are acute, and there is one and only one
triangle which will satisfy the given conditions, and this triangle is
isosceles.
3. If a < by then A must be acute to have the triangle possible, and
there are in general two triangles which satisfy the given conditions.
4. li a = b sin^, the required triangle is a right triangle.
6. If a <^ sin ^, the triangle is impossible.
6. If ^ = B, there is one, and only one, triangle.
THE OBLIQUE TRIANGLE 118
109. Number of Solutions to be expected. We may summarize the
results found on page 112 as follows :
There are two solutions if A is acute a/nd ths value of a lies he-
tween h and h sin A,
There is no solution if A is OASUte and a<.b sin A; or if A is obtuse
and a<.bf or a=^b.
There is one solution in ea^h of the other cases.
The number of solutions can often be determined by inspection. In case of
doubt, find the value of 6 sin ^.
We can also determine the number of solutions by considering the value of
log sin B, If log sin B = 0, then sin B = 1 and B = 00®. Therefore the triangle
require^ is a right triangle. If log sin B>0, then sinB>l, and hence tiie
triangle is impossible. If log sinB < 0, there is oim sohition when a > 6 ; there
are two solutions when a < 6.
When there are two solutions, let B\ C\ c^, denote the unknown parts of the
second triangle ; then
and c' =
sin J.
110. niustrative Problems. The following may be taken as illus-
trative of the above cases :
1. Given a = 16, ^ = 20, and A = 106*, find the remaining parts.
In this case a < 6 and A > 90°. Since a < 6, it follows that A<B* Hence if
A > 90*^, B must also be greater than 90°. But a triangle cannot have two
obtuse angles. Therefore the triangle is impossible.
2. Given a = 36, 6 = 80, and A = 30®, find the remaining parts.
Here we have dsin^ = 80 x ^ = 40 ; so that a < 6 sin^ and the triangle is
impossible. Draw the figure to illustrate this fact.
3. Given a = 25, 6 = 50, and A = 30®, find the remaining parts.
Here we have 6sin^ = 60x ^ = 25; but a is also equal to 25. Hence B
must be a right angle. ABC is therefore a right triangle and there is only one
solution.
4. Given a = 30, 6 = 30, and A = 60®, find the remaining parts.
Here we have a = 6, and A an acute angle. Hence there is one solution and
only one. It is evident, also, that the triangle is not only isosceles but equilateral.
5. Given a = 3.4, b = 3.4, and A = 45®, find the remaining parts.
Here we have a = 6, and A an acute angle. Hence there is one solution and
only one. It is evident, also, that the triangle is not only isosceles but right.
114
PLANE TRIGONOMETRY
6. Given a = 72,630, h = 117,480, and A = 80^ 0' 50", find B,
C\ and c,
log ft
log sin A
colog a
log sin B = 0.20222
5.06997
9.99337
5.13888
Here log sin J5 > 0.
Therefore sin 5>1, which is impossible.
Therefore there is no solution.
7. Given a = 13.2, h = 15.7, and A = 57^ 13' 15", find B, C, and c.
log h = 1.19590
log sin A = 9.92467
colog a = 8.87943
log sin B = 0.00000
.-. 5=90°
.-. C = 32* 46' 45"
c = 6 cos A
log h = 1.19590
log cos ^ = 9.73352
log c = 0.92942
.-. c = 8.5
Therefore there is one solution.
Since B = 90°, the triangle is a right triangle.
8. Given a = 767, h = 242, and A = 36° 53' 2", find 5, C, and c.
log 6 = 2.38382
log sin A = 9.77830
colog a = 7.11520
log sin B = 9.27732
.-. J5=10°54'58"
.-. C=132°12'0"
Here a > ft, and log sin 5 < 0.
Therefore there is one solution.
loga = 2.88480
log sin C = 9.86970
colog sin A = 0.22170
logc = 2.97620
.-. c = 946.68
= 946.7
9. Given a = 177.01, ft = 216.45,
mr\ A = 35* 36' 20",
find the
other parts.
log ft = 2.33536
log a = 2.24800
2.24800
log sin A = 9.76507
log
sin C = 9.99462
9.23035
colog a = 7.75200
colog
sin A = 0.23493
log c = 2.47755
0.23493
log sin 5 =9.85243
1.71328
.-. B = 45* 23' 28" or
.-. c = 300.29 or 51.675
134* 36' 32"
= 300/29 or 51.68
.-. 0= 99*0' 12" or
9* 47' 8"
Here a <b, and log sin B < 0.
Therefore there are two solutions.
,
•
THE OBLIQUE TRIANGLE 115
Exercise 50. The Oblique Triangle
Find the number of solutions^ given the following :
1. a = 80, ft = 100, A = 30**.
'*^2. a = 50, b = 100, A = 30^
3. a = 40, b = 100, A = 30**.
4. a = 100, b = 100, A = 30**.
5. a = 13.4, b = 11.46, A = IV 20'.
^6. a = 70, ft = 76, ^ = 60^
7. a = 134.16, ft = 84.64, 5 = 62*' 9'.
8. a = 200, ft = 100, ^ = 30**.
Solve the triangles, given the following :
9. a = 840, ft = 486, A = 21*^ 31'.
10. a = 9.399, ft = 9.197, ^ = 120*^ 36'.
11. a i= 91.06, ft = 77.04, ^=61** 9'.
12. a = 66My ft = 66.66, 5 = 77** 44'.
13. a = 309, ft = 360, A = 21** 14'.
14. a = 34, ft = 22, 5 = 30** 20'.
^15. ft = 19, c = 18, C=16**49'.
16. a = 8.716, ft = 9.787, A = 38** 14' 12".
17. a = 4.4, ft = 6.21, A = 67** 37' 17".
18. Given a = 75yb = 29, and B = 16** 16', find the difference be-
tween the areas of the two triangles which meet these conditions.
19. In a parallelogram, given the side a, a diagonal d, and the
angle A made by the two diagonals, find the other diagonal. As a
special case consider the parallelogram in which a = 36, d = 63,
and A = 21** 36'.
20. In a parallelogram ABCD, given AD=S in., BD = 2.5 in., and
A = 47** 20', find AB.
21. In a quadrilateral ABCD, given AC = 4 in., ABAC = 36**,
Z5 = 76** 20', ZD = 38** 30', and ABAD = 70** 40', find the length
of each of the four sides.
22. In a pentagon ABCDE, given Z^ = 110** 60', AB = 106** 30',
ZE = 104** 10', ZJ5^C = 30**, ADAE = W5&, Z ADC = 52'' SO',
and AC = 6 in., find the sides and the remaining angles of the
pentagon.
116
PLANE TRIGONOMETRY
111. Law of Cosines. This law gives the value of one side of
a triangle in terms of the othei two sides and the angle included
between them.
In either figure, a« = A« + BL^,
In the first figure, BD = c — AD.
In the second figure, BD =:AD — c.
In either case, BD^ =^AD^ -^ 2 c x AD -{- <^.
Therefore, in all cases, a* = A* -f- AD + c^ — 2 c x AD,
Now h^ +Xd' = l^,
and AD = b cos A.
Therefore a* = ft^ + c* — 26ccosil.
In like manner it may be proved that
6« = c2 + a^-2cacosJB,
and c^ = a* + 6^ — 2a^ cos C
The three formulas have precisely the same form, and the Law
of Cosines may be stated as follows :
The square on any side of a triangle is equal to the sum of the
squares on the other two sides diminished hy twice their product into
the cosine of the included an^le.
It will be seen that if -4 = 90°, we have
aa = 62 + ca-26ccos90°
= &a + c2.
In other words we have the Pythagorean Theorem as a special case. Hence
this is sometimes called the Generalized Pythagorean Theorem,
It will also be seen that the law includes two other familiar propositions of
geometry, one of which is the following :
In an obtuse triangle the aqiuire on the side opposite the obtuse angle is equivalent
to the sum of the squares on the other two sides increased by tujice the product of
one of those sides by the projection of the other upon thai, side.
This and the analogous proposition are given as exercises on page 117.
%
r
THE OBLIQUE TRIANGLE 117
Exercise 51. Law of Cosines
1. Using the figures on page 116, prove that, whether the angle
B is acute or obtuse, c = a cos B -{- b cos A,
2. What are the two symmetrical formulas obtained by changing
the letters in Ex. 1 ? What does the formula in Ex. 1 become when
5 = 90**?
3. Show that the sum of the squares on the sides of a triangle
is equal to 2(ab cos C -{- he cos^ -f- ca cos B),
4. Consider the Law of Cosines in the case of the triangle a = 5,
^^ = 12, c = 6.
5. Given a = 5, ^ = 6, and C = 60®, find c,
6. Given a = 10, h"^ 10, and C = 45**, find c
Given a = 8, ft = 5, and C = 60®, find c.
8. From the formula a^ =^ h^ -\- <^ — 2 he cos A deduce a formula
for cos^. From this result find the value of A when h^ -{• c^ =z a*.
9. Prove that if — r — = the triangle is either isosceles or
' \.4r ha
.right.
^^ T^ ^1 ^ cos^ . cos B , cos C a^ -{-h^ + (?
^10. Prove that 1 ; 1 = r—-;
a h e 2abe
J2 ^ ^2 a* 4- h^ 4- c*
11. Prove that — cos yl + t cos 5 + — cos C = — ^r— ; .
a o e 2abc
12. From the Law of Cosines prove that the square on the side op-
posite an acute angle of a triangle is equal to the sum of the squares
on the other two sides minus twice the product of either side and
the projection of the other side upon it.
13. As in Ex. 12, consider the geometric proposition relating to
the square on the side opposite an obtuse angle.
44. In the parallelogram ABCD, given ^J5 = 4 in., ^D = 5 in., and
A = 38® 40', find the two diagonals.
15. In the parallelogram ABCD, given ^5 = 7 in., .4C = 10in.,
and /.BAC = 36® 7', find the side BC and the diagonal BD.
16. In the quadrilateral ABCD, given .4^' = 3.6 in., AD = 4: in.,
BC = 2.4 in., ZA CB = 29® 40', and ZCAD=z 71® 20', find the other
two sides and all four angles of the quadrilateral.
17. In the pentagon ABCDE, given AB=i 3.4 in., i4C = 4.1 in.,
^Z> = 3.9in., ^£:=2.2in., ZJ5^C = 38® 7', ZC^Z>= 41® 22', and
ZJ'AE = 32® 6', find the perimeter of the pentagon.
118 PLANE TRIGONOMETRY
112. Law of Tangents. Since - = —. — -> by the Law of Sines, it
follows by the theory of proportion that
a — b sinA — sinB
a-\'b sin^ + sin5
This is easily seen without resorting to the theory of proportion. For, since
asinB = b 8inA (§ 106), we have
asinB — 68in-4 = 6 sin^— asinB
Adding, a sin ^ — 6 sin B = a sin^ — 6 sin B
asin^ + a sin 5 — 6 sin ul — 6 sin 5 = asin-4 — asin^ + 6 sin -4 — 6 sin ^,
or (a — b) (sin A + sin J?) = (a + 6) (sin A — sin B);
a — 6 sin-4— sin5
whence, by division.
a-\-b sinul+sin^
T> i. v ff^Ao sm^ — sm5 ta.nUA — B)
But by §103, -. — 7- — 7—z = 7 — , ;^ . „( -
-^ ' sin^-f-sin^ tanJ(^ + 5)
^■- r«, ^ a— ^ tan|(il — J?)
^^ Therefore = tt ^ •
a+b tan|(il + J?)
By merely changing the letters,
a — c _ tan ^(A — C)
a-^c "" tan i(^ + C) '
and ^^tanK^-C)^
b-^-c tani(5+C)
Hence the Law of Tangents :
Tfie difference between two sides of a triangle is to their sum as
the tangent of half the difference between the opposite angles is to
the tangent of half their sum.
In the case of a triangle, if we know the two sides a and 6 and
the included angle C, we have our choice of two methods of solving.
From the Law of Cosines we can find c, and then, from the Law of
Sines, we can find A and B, Or we can find ^ + J5 by taking C from
180°, and then, since we also know a-\-b and a — b, we can find
A — B. From A -\- B and A — B we can find A and B, This second
method is usually the simpler one.
If 6 > a, then B>A. The formula is still true, but to avoid negative numbers
the formula in this case should be written
6-a _ tan^(g — ^)
6 + a~tan^(5 + u4)'
THE OBLIQUE TRIANGLE 119
Exercise 52. Law of Tangents
Firid the form to which 7 = ?-^— — -~ reduces when :
^ a + b taii^(^ + ^)
1. C=90^ 3. A=B=C,
2. a = 6. 4. ^-5=90*, and 5 = a
Prove the following formulas :
5. |^ = tani(B-C)cotJ(JB + C).
b — c
6. tani(5 — C)=7 cotiA.
^ ^ ^ o -f- c
7.
8.
9.
sin^ + sin^ tan^(^+^)
siiiil — sin5 tan ^(^—5)
sin^-f-sinC 2 sin^(^ + C)cosi(^ — C)
8in5 - sin C ~ 2 cos ^(5 + C)^\xi\{B - C) '
sin A -\- sin 5 , , , , „^ ^ t ^ ^
10. -, — , . „ = tan^(^+Jg)cot4^M-jB).
sm^ — sin5 2V • / 2V y
11. To what does the formula in Ex. 8 reduce when A=B?
12. To what does the formula in Ex. 9 reduce when B = C=: 60® ?
13. To what does the formula in Ex. 10 reduce when the triangle
is equilateral ?
14. To what does the Law of Tangents, in the form stated at the
top of this page, reduce in the case of an isosceles triangle in which
a = b? What does this prove with respect to the angles opposite
the equal sides ?
16. By the help of the Law of Tangents prove that an equilateral
triangle is also equiangular.
16. By the help of the Law of Tangents prove that an equiangular
triangle is also equilateral.
17. Given any three sides and any three angles of a quadrilateral,
show how the fourth side and the fourth angle can be found. Show
also that it is not necessary to have so many parts given, and find
the smallest number of parts that will solve the quadrilateral.
18. What sides, what diagonals, and what angles of a pentagon is it
necessary to know in order, by the aid of the Law of Tangents alone,
to solve the pentagon ?
120 PLANE TRIGONOMETRY
113. Applications to Triangles. The La^ of Cosines and the Law
of Tangents are frequently used in the solution of triangles. This
is particularly the case when we have given two sides, a and ft, and
fche included angle C.
There are two convenient ways of
finding the angles A and 5, the first being
by the Law of Tangents. This law may
be written
tani(^-5)=-— r X tan J(.4 4-iB).
Since ^(^ 4-^) = i(180**- C), the value of i(^ +5) is known, so
that this equation enables us to find the value of \{A—B), We
then have i(^+5) + i(^-5) = ^
and \{A •\.B)-\{A -B)=B.
The second method of finding A and ^ is as follows : In the above
figure let BD be perpendicular to A C,
.^ ^ , BD BD
Then tan^ = •— — =
AD AC -DC
Now BD = a sin C,
and DC = a cos C.
asinC
tan -4 =
h — a cos C
Since A and C are now known, B can be found.
This is not so convenient as the first method, because it is not so well adapted
to work with logarithms.
The side o may now be found by the Law of Sines, thus :
a sin C b sin C
C = — : --} or C = — : — TT'
smA smB
Instead of finding ^4 and jB first, and from these values finding c,
we may first find c and then find A and B, To find c first we may
write the Law of Cosines (§ 111) as follows :
e = Va^ + &2 _ 2 o^ cos C.
Having thus found c, and already knowing a, b, and C, we have
. , asinC . ^ ft sin C
sin^ = 9 siuB =
c c
In general this is not so convenient as the first method given above, because
the formula for c is not so well adapted to work with logarithms.
THE OBLIQUE TRIANGLE 121
114. lUustrative Problems. 1. Given C = 63** 35' 30", a = 748, and
h = 375, find A^ B, and c.
We see that a + Z^ = 1123, a — ft=373, and .4+5 = 180*-C =
116** 24' 30". Hence i(A +B) = 58"" 12' 15".
log (a-h)= 2.57171 log b = 2.67403
colog (a + ft) = 6.94962 log sin C = 9.95214
log tan i (^ + jB) = 0.20766 eolog sin 5 = 0.30073
log tan J (^ - J5) = 9.72899 log c = 2.82690
.-. i(A-B)= 28** 10' 54" .-. c = 671.27
After finding ^(A —B) we combine this with \(A+B) and find
A = 86** 23' 9" and 5 = 30** 1' 21".
«
In the above example, in finding the side c we use the angle B rather than
the angle -4, because A is near 90°. The use of the sine of an angle near 90°
should be avoided, because it varies so slowly that we cannot determine the
angle accurately when the sine is given.
2. Given a = 4, c = 6, and B = 60**, find the third side h.
Here the Law of Cosines may be used to advantage, because the numbers
are so small as to make the computation easy. We have
6 = Va2 + c2 - 2ac cosJ? = Vl6 + 36 - 24 = V28 ;
log 28 = 1.44716, log V28 = 0.72368, V28 = 6.2915 ;
that is, to three significant figures, b = 6.292.
Exercise 53. Solving Triangles
Solve these triangles^ given the following parts :
"iTa = 77.99, h = 83.39, C = 72** 15'.
2. 5 = 872.5, c = 632.7, ^=80**.
3. a =17, Z>=12, C = 59**17'.
4. ^» = Vk, c = V3, ^ = 35*^53'.
^" . / 5. a = 0.917, h = 0.312, (? = 33** V 9".
- ^^ 6. a = 13.715, e = 11.214, B = 16** 22' 36".
"^ 7. 5 = 3000.9, c= 1587.2, yl=86**4'4".
8. a = 4527, h = 3465, C = 66** 6' 27".
9. a = 55.14, b = 33.09, C = 30** 24'.
10. a = 47.99, b = 33.14, C = 175** 19' 10".
11. a = 210, b = 105, C = 36** 52' 12".
12. a = 100, 5 = 900, O = 65^
/
I
122 PLANE TRIGONOMETRY
Solve these triangles^ given the follomng parts :
13. a = 409, b = 169, C = 117.7**.
14. a = 6.26, b = 5.05, C = 106.77^
16. a = 3718, b = 1607, C = 95.86^
16. a = 46.07, b = 22.29, C = 66.36°.
17. ft = 445, c = 624, ^ = 10.88^
18. ft =16.7, c = 43.6, ^ = 57.22°.
19. If two sides of a triangle are each equal to 6, and the in-
cluded angle is 60®, find the third side by two different methods.
20. If two sides of a triangle are each equal to 6, and the in-
cluded angle is 120°, find the third side by three different methods.
21. Apply the first method given on page 120 to the case in which
a is equal to ft ; that is, the case in which the triangle is isosceles.
22. If two sides of a triangle are 10 and 11, and the included
angle is 50°, find the third side.
23. If two sides of a triangle are 43.301 and 25, and the included
angle is 30°, find the third side.
24. In order to find the distance between two objects, A and B,
separated by a swamp, a station C was chosen, and the distances
CA = 3825 yd., CB = 3475.6 yd., together with
the angle ACB= 62° 31', were measured. Find
the distance from A to J5.
25. Two inaccessible objects, A and B, are
each viewed from two stations, C and X), on the
same side oi AB and 562 yd. apart. The angle
ACB is 62° 12', J5C/)41°8', ^i)5 60°49', and
ADC 34° 51'. Required the distance AB.
26. In order to find the distance between two objects, A and B,
separated by a pond, a station C was chosen, and it was found that
CA = 426 yd., CB = 322.4 yd., and ACB= 68° 42'. Required the
distance from A to B.
27. Two trains start at the same time from the same station and
move along straight tracks that form an angle of 30°, one train at
the rate of 30 mi. an hour, the other at the rate of 40 mi. an hour.
How far apart are the trains at the end of half an hour ?
28. In a parallelogram, given the two diagonals 6 and 6 and the
angle which they form 49° 18', find the sides.
'» %
THE OBLIQUE TRIANGLE 123
115. Given the Three Sides. Given the three sides of a triangle, it
is possible to find the angles by the Law of Cosines. Thus, from
a^ = &2 ^ ^ __ 2 ftc coSil,
we have co8^ =
2 he
This formula is not, however, adapted to work with logarithms. In order to
remedy this difficulty we shall now proceed to change its form.
Let s equal the semiperimeter of the triangle ; that is,
let a + ft + c = 25.'
Then ft + c — a = 2s — 2a=:2(s — a),
c + a — i = 2(s — ft),
and a + ft — c = 2 (s — c).
Hence 1 — COS^l =1 rr-i — — = ^r^
2 ftc 2 ftc
^ a^^(l,-cY ^ (a + b- c)(a - b + c)
"" 2ftc "" 2fto
__ 2(5-ft)(5-c)
"" be
In the same way the value of 1 -f- cos A is
"^ 2ftc ■" 2ftc "" 2ftc
(ft + c + a)(b + c--a) _ 2s(s — a)
"" 2ftc " be '
But from § 102 we know that
1 — cosyl = 2 sin^ Jyl, and 1 + cos^ = 2 cos^ J^l.
o • u, ^ 2(s--ft)(s — c) -- „, ^ 28(8 — a)
.*. 2 sm^i^ = -^^ :/^ ^> and 2 cos^^ = — 4 ^'
^ be ^ be
It therefore follows that
, ls(s — a)
and cofllA=^— jj
SlU QC
Furthermore, since tan x = > we have
coscc
* 1 5(5 — a)
124 PLANE TRIGONOMETRY
By merely changing the letters in the formulas given on page 123,
we have the following :
. , _ |(g — a)(g ---c) . 1(8 — a)(s — b)
^ y8(8 — b) ^ y 8(8 — c)
There is then a choice of three different formulas for finding the value of
each angle. If half the angle is very near 0°, the formula for the cosine will
not give a very accurate result, because the cosines of angles near QP differ little
in value ; and the same is true of the formula for the sine when half the angle
is very near 90°. Hence in the first case the formula for the sine, and in the
second that for the cosine, should be used.
But in general the formulas for the tangent are to be preferred, the tangent
as a rule changing more rapidly than the sine or cosine.
It is not necessary to compute by the formulas more than two angles, for
the third may then be found from the equation -4 + B + C = 180°. There is this
advantage, however, in computing all three angles by the formulas, that we
may then use the sum of the angles as a test of the accuracy of the results.
116. Checks on the Angles. In case it is desired to compute all the
angles for the purpose of checking the work, the formulas for the
tangent may be put in a more convenient form.
The formula for tan ^A may be written thus :
^j^.^QZMEi
(8-a)(8-b)(8-c)
8
Hence, if we put
i (5-a)(8-ft)(rr^
we have tan^A =
Likewise, tanJ-B = -9 tanJC =
8 — b 8 —- c
For example, if a = 3, b = 8.5, and c = 4.5, we have 5 = 5.5, s— a = 2.5,
« — 6 = 2, and « — c = 1. ^________
\ 6.6 \5.6 \11
.-. tan J -4 = 0.9634 -^ 2.6 = 0.8814.
.-. A = 4P 46'.
THE OBLIQUE TRIANGLE 125
Exercise 54. Fommlas of the Triangle
1. Given tan J ^4 = -vl^ — -r-^ — r-^ > express the value of log tan \A,
jI 8 18 ^~* Qfj
2. Given sin J ^ = -^^ ^^ ^ > express the value of log sin J A,
3. Given r = -O ^ ^^ '^ ^ > express the value of log r.
4. Given tan J-4 = > express the value of log tan ^A,
8 ~" (Xf
T
5. Given tan J-4 = > express the value of logr.
8 ""~ Cb
6. Of the three values for tan^^,
1 — COSii
1 + COSil
s(s — a)
(§ 102)
(§ 116)
and ^^^ZMEMU, (§116)
which is the easiest to treat by logarithms ? Express the logarithms
of the results and show why your answer is correct.
7. Given a = 4, ft = 6, and o = 6, find the value of tan J^, and
then find the value of A.
8. Deduce the equation
* \ 8(8 — a)
from the equation
tan J ^ = -y T— 7
^ \l + cos-4
9. Discuss the formula
* \ s (5 — a)
1 l (.-«)(«-&)(«^c) ^
« — a > 8
for the case of an equilateral triangle, say when a = 4.
126
PLANE TRIGONOMETRY
117. Illustrative Problems. 1. Given a = 3.41, h = 2.60, c = 1.58,
find the angles.
Since it is given that a = 3.41, h = 2.60, and o = 1.68, it follows
that 25 = 7.59 and « = 3.795. Therefore
5- a = 0.385, «- ft =1.195, 5 - c = 2.215.
Using the formula of § 115 and
tan \By we may arrange the work as
colog s = 9.42079
colog(s — a)= 0.41454
log(s-. ft) =0.07737
log(5-c) = 0.34537
2 )0.25807
logtani^ = 0.12903
.-. i^= 53^ 23' 20"
.-. ^ = 106** 46' 40"
.-. ^ + J5 = 153** 39' 54",
the corresponding formula for
follows :
colog s= 9.42079-10
log(5-a)= 9.58546-10
colog(s-ft)= 9.92263-10
log \s - c) = 0.34537
2 )19.27425-20
logtanJ5= 9.63713-10
.-. ^5 =23** 26' 37"
.-. jB = 46*^ 53' 14"
and C = 26** 20' 6".
2. Solve the above problem by finding all three angles by the use
of the formulas on page 124.
Since it is given that a =3.41, ft =2.60, and c = 1.58, it follows
that 28 = 7.59 and s = 3.795. Therefore
s - a = 0.385,
ft = 1.195, 5 - c = 2.215.
Here the work may be compactly arranged as follows, if we find log tan i^,
etc., by siMrajcimg log (s — a), etc., from log r instead of adding the cologarithm.
log(5- a) =9.58546
log (s - ft) = 0.07737
log (s - c) = 0.34537
colog s = 9.42079
log 1^ = 9.42899
log r = 9.71450
logtanj^ =10.12903
logtanJB= 9.63713
logtan^C= 9.36912
J^= 53** 23' 20"
iB= 23** 26' 37"
^C = 13*^10' 3"
A = 106** 46' 40"
J5= 46** 53' 14"
C= 26** 20' 6"
Cheelc. ^+J5+C=180** 0' 0"
Even if no mistakes are made in the work, the sum of the three angles found
as above may differ very slightly from 180° in consequence of the fact that
computation with logarithms is at best only a method of close approximation .
When a difference of this kind exists, it should be divided among the angles
according to the probable amount of error for each angle.
THE OBLIQUE TRIANGLE 127
Exercise 55. Finding the Angles
Find the three angles of a triangle^ given the three sides as follows:
1. 51, 66, 20. 6. 43, 50, 57. 11. 6, 8, 10.
2. 78, 101, 29. 7. 37, 58, 79. 12. 6, 6, 10.
3. Ill, 145, 40. 8. 73, 82, 91. 13. 6, 6, 6.
4. 21, 26, 31. 9. V5, V6, V7. 14. 6, 9, 12.
5. 19, 34, 49. 10. 21, 28, 35. 16. 3, 4, 5.
16. Given a =14.5, h = 55.4, and c = 66.9, find A, B, and C.
17. Given a = 2, 6 = V6, and c = V3 — 1, find A, B, and C.
18. Given a = 2, ft = V6, and c = Vs + 1, find A, B, and C.
19. The sides of a triangle are 78.9, 65.4, and 97.3 respectively.
Find the largest angle.
20. The sides of a triangle are 487.25, 512.33, and 544.37 respec-
tively. Find the smallest angle. _
_ /o 1^ -4 -v/ Q i
21. Find the angles of a triangle whose sides are j=- > y=- >
rz 2 V 2 2 "v 2
and —^ respectively.
22. Of three towns. A, B, and C, A is found to be 200 mi. from B
and 184 mi. from C, and B is found to be 150 mi. due north from C
How many miles is A north of C ?
23. Under what visual angle is an object 7 ft. long seen by an
observer whose eye is 5 ft. from one end of the object and 8 ft. from
the other end ?
24. The sides of a triangle are 14.6 in., 16.7 in., and 18.8 in.
respectively. Find the length of the perpendicular from the vertex
of the largest angle upon the opposite side.
25. The distances between three cities. A, B, and C, are measured
and found to be as follows: ^J5 = 165 mi., ylC = 72 mi., and
BC =1S5 mi. B is due east from A. In what direction is C from A ?
What two answers are admissible ?
26. In a quadrilateral ABCD, AB=2 in., BC = S in., CZ) = 3 in.,
DA = 4 in., and yl C = 4 in. Find the angles of the quadrilateral.
27. In a parallelogram ABCD, AB=^ 2 in., AC = 3 in., and AD
= 2.5 in. Find ZC5^.
28. In a rectangle ABCD, ^5 = 3.3 in., and ^ C = 5^ in. Find the
angles that each diagonal makes with the sides.
128
PLANE TRIGONOMETRY
118. Area of a Triangle. The area of a triangle may be found if
the following parts are known :
1. Two sides and the included angle ;
2. Two angles and any side ;
3. The three sides.
These cases will now be considered.
Case 1. Given two sides and the included angle.
Lettering the triangle as here shown, and designating CD by h
and the area by S, we have c
But A = a sin B.
Therefore
S=lacsiaB.
Also S=:^aJ) sin Cj and 8= ^bc sin A, a
Exercise 56. Area of a Triangle
Find the areas of the triangles in which it is given that :
1, a = 27, c = 32,
rj = 43,
c = 5.3,
c = 7.6,
b = 19.4,
b = 64.32,
c = 168.6,
c = 417.8,
c = 29.62,
c = 1634,
11. Prove that the area of a parallelogram is equal to the product
of the base, the diagonal, and the sine of the angle included by them.
12. Find the area of the quadrilateral ABCD, given AB = 3 in.,
^ C = 4.2 in., AD = 3.8 in., ZBAD:= 88** 10', ZBAC =: 36** 20'.
13. In a quadrilateral ABCD, BC z= 5.1 in., .4C = 4.8 in., CD =
3.7 in., Z^CJ5 = 123M2', and ZDCA =117*26'. Draw the figure
approximately and find the area.
14. In the pentagon ABODE, AB=z 3.1 in., AC = 4.2 in., ^/) =
3.7 in., AE = 2.9 in., AA= 132** 18', ABAC = 38** 16', and ADAE =
53° 9'. Find the area of the pentagon.
2. a = 35,
3. a = 4.8,
4. a = 9.8,
5. a = 17.3,
6. a = 48.35,
7. & = 127.8,
8. ^» = 423.9,
9. ^» = 32.78,
10. ft = 1487,
B = 40^
B = 37^
B = 39** 27'.
B = 48.6^
C = 56.26**.
C = 62** 37'.
A = 72** 43'.
A = 68** 27'.
A = 57** 32' 20".
A = 61** 30' 30".
THE OBLIQUE TRIANGLE
129
Case 2. Given two angles and any side.
If two angles are known the third can be found, so we may
consider that all three angles are given.
O
it follows that
And since
we have
c =
sini4
S = iacsinB (Case 1),
, asinC . „ a^sinBsinC
^ = ^ a — : — — sin B =
sin^ " 2sin^
Since all three angles are known we may use this formula; or,
since sin (B + C)= sin (180® — ^4) = sin^, we may write it as follows :
a'sin^sinC
S =
2Bin(B+C)
Exercise 57. Area of a Triangle
Find the areas of the triangles in which it is given that :
1. a =17,
2. a =182,
3. a = 298,
4. a =19.8,
6. a =2487,
6. ft = 483.7,
7. ft = 627.4,
8 c = 296.3,
9. c = 17.48,
10. c = 96.37,
B = 48^
B = 63.5^
B = 78.8^
B = 39** 20',
B = 87** 28',
A = 84** 32',
A = 73** 42',
A = 68** 36',
A = 36** 27' 30",
C = b2\
C = 78.4**.
C = 96.6^
C = 88** 40'.
C = 69** 32'.
C = 78** 49'.
C = 63** 37'.
B = 42** 36'.
B = 73** 60'.
B = 69** 52' 60".
A = 42** 23' 36",
11. In a parallelogram ABCD the diagonal AC makes with the
sides the angles 27** 10' and 32** 43' respectively. AB is 2.8 in. long.
What is the area of the parallelogram ?
130 PLANE TRIGONOMETRY
Case 3. Given the three sides.
Since, by § 101, smB=2 sin J 5 cos J 5,
and, by § 115, sin ^5 = ^^JliH^Zfl ,
and cosi5 = -J^i^ ^>
'' y ac
by substituting these values for sin ^B and cos ^B in the above
equation, we have
2 ,
sm5 = — ■y/s(s-a)(s — b)(s-c).
By putting this value for sin B in the formula of Case 1, we have
the following important formula for the area of a triangle :
S = ^s(s — a)(s — b)(s — c).
This is known as Heron's Formula for the area of a triangle, having been
given in the works of this Greek writer. It is often given in geometry, but the
proof by trigonometry is much simpler.
A special case of finding the area of a triangle when the three
sides are given is that in which the radius of the circumscribed
circle or the radius of the inscribed circle is also given.
If R denotes the radius of the circumscribed circle, we have,
from § 106, ^
sin 5 =
2R
By putting this value of sin 5 in the formula
of Case 1, we have ^^ ^
If r denotes the radius of the inscribed circle, we may divide the
triangle into three triangles by lines from the center of this circle to
the vertices ; then the altitude of each of the three triangles is equal
to r. Therefore , ^
S = \r(a + b+c) = rs.
By putting in this formula the value of S from Heron's Formula,
we have
^J (s-a)(8-b)(s^c)
From this formula, r, as given in § 116, is seen to be equal to the
radius of the inscribed circle.
THE OBLIQUE TRIANGLE 131
Exercise 58. Area of a Triangle
Find the area% of the triangles in which it i% given that : •
1. a = 3, ^» = 4, c = 5. 4. a = 1.8, h = 3.7, c = 2.1.
2. a = 15, h^ 20, c = 25. 5. a = 5.3, h = 4.8, c = 4.6.
3. a = 10, b = 10, c = 10. 6. a = 7.1, «» = 5.3, c = 6.4.
7. There is a triangular piece of land with sides 48.5 rd., 52.3 rd.,
and 61.4 rd. Find the area in square rods ; in acres.
Find the areas of the triangles in which it is given that :
8. a = 2.4, b = 3.2, c = 4, R = 2,
9. a = 2.7, b = 3.6, c = 4.5, R = 2.25.
10. a = 3.9, b = 5,2, c = 6.5, i? = 3.25.
1 1. a = 12, ^ = 12, c = 12, R = 6.928.
12. Given a = 60, 5 = 40** 35M2", area =12, find the radius of
the inscribed circle.
Find the areas of the triangles in which it is given that :
13. a = 40, Z^ = 13, c = 37.
14. a = 408, «> = 41, c = 401.
15. a = 624, b = 206, c = 445.
16. ^^ = 8, c = 5, ^ = 60^
17. a = 7, c = 3, ^ = 60^
18. ^ = 21.66, c = 36.94, ^ = 66° 4' 19".
19. a = 215.9, c = 307.7, A = 25** 9' 31".
20. b = U% ^ = 70** 42' 30", i5 = 39^8' 28".
21. a = 4474.5, b = 2164.5, C = 116** 30' 20".
22. a = 510, c = 173, B = 162** 30' 28".
23. If a is the side of an equilateral triangle, show that the area
is J a" V3.
24. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the
included angle is 46** 24'. Find the area.
25. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they
form a right angle. Find the area.
26. Two angles of a triangle are 76** 54' and 57** 33' 12", and the
included side is 9 ch. Find the area.
27. The three sides of a triangle are 49 ch., 50.25 ch., and 25.69 ch.
Find the area.
132
PLANE TRIGONOMETRY
28. The three sides of a triangle are 10.64 ch., 12.28 eh., and
9 eh. Find the area.
29. The sides of a triangular field, of which the area is 14 A.,
are proportional to 3, 5, 7. Find the sides.
30. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first
bears N. 82** 30' W. ; the second S. 24* 15' E. Find the area.
31. The base of an isosceles triangle is 20, and its area is
100 -f- VS ; find its angles.
32. Two sides and the included angle of a triangle are 2416 ft.,
1712 ft., and 30® ; and two sides and the included angle of another
triangle are 1948 ft., 2848 ft., and 150®. Find the sum of their areas.
33. Two adjacent sides of a rectangle are 52.25 ch. and 38.24 ch.
Find the area.
34. Two adjacent sides of a parallelogram are 59.8 ch. and 37.05 ch.,
and the included angle is 72® 10'. Find the area.
35. Two adjacent sides of a parallelogram are 15.36 ch. and
11.46 ch., and the included angle is 47® 30'. Find the area.
36. Show that the area of a quadrilateral is equal to one half the
product of its diagonals into the sine of the included angle.
37. The diagonals of a quadrilateral are 34 ft. and 56 ft., inter-
secting at an angle of 67®. Find the area.
38. The diagonals of a quadrilateral are 75 ft. and 49 ft., inter-
secting at an angle of 42®. Find the area.
39. in the quadrilateral ABCD we have AB, 17.22 ch. ; AD, 7A5 ch. ;
CD, 14.10 ch. ; BC, 5.25 ch. ; and the diagonal AC, 15.04 ch. Find
the area.
40. Show that the area of a regular polygon of n sides, of which
na'
one side is a, is — r- cot
180*
4 n
41. One side of a regular pentagon is 25. Find the area.
42. One side of a regular hexagon is 32. Find the area.
43. One side of a regular decagon is 46. Find the area.
44. If r is the radius of a circle, show that the area of the regular
180®
circumscribed polygon of n sides is nr^ tan > and the area of the
n
360*
n
n
regular inscribed polygon is - r^ sin
45. Obtain a formula for the area of a parallelogram in terms of
two adjacent sides and the included angle.
CHAPTER VIII
MISCELLANEOUS APPLICATIONS
119. Applications of the Right Triangle. Although the fonnulaS
for oblique triangles apply with equal force to right triangles, yet
the formulas developed for the latter in Chapter IV are somewhat
simpler and should be used when possible. It will be remembered that
these formulas depend merely on the definitions of the functions.
Exercise 59. Right Triangles
1. If the sun's altitude is 30°, find the length of the longest
shadow which can be cast on a horizontal plane by a g
stick 10 ft. in length.
2. A flagstaff 90 ft. high, on a horizontal plane, x.^'Ml_^^ ^
casts a shadow of 117 ft. Find the altitude of the sun. ^ ^ tj*
3. If the sun's altitude is 60°, what angle must a stick make with
the horizon in order that its shadow in a horizontal plane may be
the longest possible ?
4. A tower 93.97 ft. high is situated on the bank of a river. The
angle of depression of an object on the opposite
bank is 25° 12'. Find the breadth of the river.
6. The angle of elevation of the top of a tower
is 48° 19', and the distance of the base from the point of obser-
vation is 95 ft. Find the height of the tower and the distance of its
top from the point of observation.
6. From a tower 58 ft. high the angles of depression of two
objects situated in the same horizontal line with
the base of the tower, and on the same side, are ^^'z
30° 13' and 45° 46'. Find the distance between ab
these two objects.
7. From one edge of a ditch 36 ft. wide the angle of elevation
of the top of a wall on the opposite edge is 62° 39'. Find the
length of a ladder that will just reach from the point of observation
to the top of the wall. • .
138
134 , PLANE TRIGONOMETRY
8. The top of a flagstaff has been partly broken off and touches
the ground at a distance of 15 ft. from the foot of the staff. If the
length of the broken part is 39 ft., find the length of the whole staff.
9. From a balloon which is directly above one town the angle
of depression of another town is qbserved to be 10* 14'. The towns
being 8 mi. apart, find the height of the balloon.
10. A ladder^ 40 ft. long reaches a window 33 ft. high, on one
fide of a street. Being turned over upon its foot, the ladder reaches
another window 21 ft. high, on the opposite side of the street. Find
the width of the street.
11. From a mountain 1000 ft. high the angle of depression of a
ship is 27® 35' 11". Find the distance of the ship from the summit
of the mountain.
12. From the top of a mountain 3 mi. high the angle of depres-
sion of the most distant object which is visible on the earth's sur-
face is found to be 2® 13' 50". Find the diameter of the earth.
13. A lighthouse 54 ft. high is situated on a rock. The angle of
elevation of the top of the lighthouse, as observed from a ship, is
4® 52', and the angle of elevation of the top of the rock is 4® 2'.
Find the height of the rock and its distance from the ship.
14. The latitude of Cambridge, Massachusetts, is 42** 22' 49". What
is the length of the radius of that parallel of latitude ?
15. At what latitude is the circumference of the parallel of lati-
tude equal to half the equator ?
16. In a circle with a radius of 6.7 is inscribed a regular polygon
of thirteen sides. Find the length of one of its sides.
17. A regular heptagon, one side of which is 5.73, is inscribed in
a circle. Find the radius of the circle.
18. When the moon is setting at any place, the angle at the moon
subtended by the earth's radius passing through that place is 57' 3".
If the earth's radius is 3956.2 mi., what is the moon's distance from
the eartti's center ?
19. A man in a balloon observes the angle of depression of an
object on the ground, bearing south, to be 35® 30'; the balloon drifts
2^ mi. east at the same height, when the angle of depression of the
same object is 23® 14'. Find the height of the balloon.
20. The angle at the earth's center subtended by the sun's radius
is 16' 2", and the sun's distance is 92,400,000 mi. Find the sun's
diameter in miles.
\
;otJs Aiijn^ioNS 135
21. A man standing south of a tower a^H»n ziA same horizontal
plane observes its angle of elevation to be 54*16'; he goes east
100 yd. and then finds its angle of elevation is 50® 8'. Find the
height of the tower.
22. A regular pyramid, with a square base, has a lateral edge 150 ft.
long, and the side of the base is 200 ft. Pind the inclination of the
face of the pyramid to the base.
2S. The height of a house subtends a right angle at a window on
the other side of the street, and the angle of elevation of the top of
the house from the same point is 60®. The street is 30 ft. wide.
How high is the house ?
24. The perpendicular from the vertex of the right angle of a
right triangle divides the hypotenuse into two segments 364.3 ft.
and 492.8 ft. in length respectively. Find the acute angles of the
triangle.
25. The bisector of the right angle of a right triangle divides the
hypotenuse into two segments 431.9 ft. and 523.8 ft. in length
respectively. Find the acute angles of the triangle.
26. Find the number of degrees, minutes, and seconds in an arc
of a circle, knowing that the chord which subtends it is 238.25 ft.,
and that the radius is 196.27 ft.
27. Calculate to the nearest hundredth of an inch the chord which
subtends an arc of 37® 43' in a circle having a radius of 542.35 in.
28. Calculate to the nearest hundredth of an inch the chord which
subtends an arc of 14® in a circle having a radius of 475.23 in.
29. In an isosceles triangle ABC the base AB is 1235 in., and
AA=Z.B = 64® 22'. Find the radius of the inscribed circle.
30. Find the number of degrees, minutes, and seconds in an arc
of a circle, knowing that the chord which subtends it is two thirds
of the diameter.
31. Find the number of degrees, minutes, and seconds in an arc
of a circle, knowing that the chord which subtends it is three fourths
of the diameter.
32. The radius of a circle being 2548.36 in., and the length of a
chord BC being 3609.02 in., find the angle BAC made by two
tangents drawn at B and C respectively.
33. Find the ratio of a chord to the diameter, knowing that the
chord subtends an arc 27® 48'. If the diameter is 8 in., how long is
the chord ? If the chord is 8 in., how long is the diameter ?
136
PLAXE TRIGONOMETRY
34. Find the length of the diameter of a regular pentagon of
which the side is 1 in., and the length of the side of a r^^ular
pentagon of which the diameter is 1 in.
35. Two circles of radii a and h are externally tangent. The com-
mon tangents APy BPy and the line of centers CC*P are drawn as
shown in the figure. Find sin APC.
36. In Ex. 35 find Z.APC, know-
ing that a = 3 6.
37. In AABC,/LA = 68* 26' 27",
ZB = 75* 8' 23", and the altitude //,
from C, is 148.17 in. Required the
lengths of the three sides.
38. Two axes, OX and OF, form a right angle at O, the center of
a circle of radius 1091 ft Through P, a point on OX 1997 ft from
O, a tangent is drawn, meeting OY at C. Re-
quired OC and the angle CPO,
39. Find the sine of the angle formed by
the intersection of the diagonals of a cube.
40. The angle of elevation of the top of
a tower observed at a place A, south of it, is
30°; and at a place B, west of A, and at a distance of a from it,
the angle of elevation is 18°. Show that the height of the towei
V5-1
a
IS
V2-I-2V5
, the tangent of 18° being
\/lO -I- 2 VS
41. Standing directly in front of one cormn- of a flat-roofed house,
which is 150 ft in length, I observe that the horizontal angle which
the length subtends has for its cosine V^, and that the vertical angle
3
subtended by its height has for its sine 77^ * What is the height
of the house ?
42. At a distance a from the foot of a tower, the angle of eleva-
tion A of the top of the tower is the complement of the angle of
elevation of a flagstaff on top of it. Show that the length of the
staff is 2a cot 2 A,
43. A rectangular solid is 4 in. long, 3 in. wide, and 2 in. high.
Calculate the tangent of the angle formed by the intersection of
any two of the diagonals.
44. Calculate the tangent as in Ex. 43, the solid being I units long,
w units wide, and h units high.
MISCELLAKEOUS APPLICATIONS 187
120. Applications of the Oblique Triangle. As stated in § 119^ when
conditions permit of using a right triangle in making a trigono-
metric observation it is better to do so. Often, however, it is impos-
sible or inconvenient to use the right triangle, as in the case of an
observation on an inclined plane, and in such cases resoi*t to the
oblique triangle is necessary.
Exercise 60. Oblique Triangles
1. Show how to determine the height of an inaccessible object
situated on a horizontal plane by observing its angles of elevation
at two points in the same line with its base and measuring the
distance between these two points.
2. Show how to determine the height of an inaccessible object
standing on an inclined plane.
3. Show how to determine the distance between two inaccessible
objects by observing angles at the ends of a line of known length.
4. The angle of elevation of the top of an inaccessible tower stand-
ing on a horizontal plain is 63° 26' ; at a point 600 ft. farther from
the base of the tower the angle of elevation of the top is 32° 14'.
Find the height of the tower.
5. A tower stands on the bank of a river. From the opposite bank
the angle of elevation of the top of the tower is 60° 13', and from a
point 40 ft. further off the angle of elevation is 50° 19'. Find the
width of the river.
6. At the distance of 40 ft. from the foot of a vertical tower on
an inclined plane, the tower subtends an angle of 41° 19'; at a point
60 ft. farther away the angle subtended by the tower is 23° 45'.
Find the height of the tower.
7. A building makes an angle of 113° 12' with the inclined plane
on which it stands ; at a distance of 89 ft. from its base, measured
down the plane, the angle subtended by the building is 23° 27'. Find
the height of the building.
8. A person goes 70 yd. up a slope of 1 in 3J from the bank of a
river and observes the angle of depression of an object on the oppo-
site bank to be 2J°. Find the width of the river.
9. A tree stands on a declivity inclined 15° to the horizon. A man
ascends the declivity 80 ft. from the foot of the tree and finds the
angle then subtended by the tree to be 30°. Find the height of
the tree.
138 • PLANE TRIGONOMETRY
10. The angle subtended by a tree on an inclined plane is, at a
certain point, 42** 17', and 326 ft. further down it is 21** 47'. The
inclination of the plane is 8° 63'. Find the height of the tree.
11. From a point B at the foot of a mountain, the angle of elevation
of the top A is 60°. After ascending the mountain one mile, at an
inclination of 30° to the horizon, and reaching a point C, an observer
finds that the angle A CB is 135°. Find the number of feet in the
height of the mountain.
12. The length of a lake subtends, at a certain point, an angle of
46° 24', and the distances from this point to the two ends of the
lake are 346 ft. and 290 ft. Find the length of the lake.
13. Along the bank of a river is drawn a base line of 600 ft. The
angular distance of one end of this line from an object on the oppo-
site side of the river, as observed from the other end of the line,
is 63°; that of the second extremity from the same object, observed
at the first, is 79° 12'. Find the width of the river.
14. Two observers, stationed on opposite sides of a cloud, observe
its angles of elevation to be 44° 66' and 36° 4'. Their distance from
each other is 700 ft. What is the height of the cloud ?
15. From the top of a house 42 ft. high the angle of elevation of
the top of a pole is 14° 13'; at the bottom of the house.it is 23° 19'.
Find the height of the pole.
16. From a window on a level with the bottom of a steeple the
angle of elevation of the top of the steeple is 40°, and from a second
window 18 ft. higher the angle of elevation is 37° 30'. Find the
height of the steeple.
17. The sides of a triangle are 17, 21, 28. Prove that the length
of a line bisecting the longest side and drawn from the opposite
angle is 13.
18. The sum of the sides of a triangle is 100. The angle at ^ is
double that at B, and the angle at B is double that at C. Determine
the sides.
19. A ship sailing north sees two lighthouses 8 mi. apart in a line
due west; after an hour^s sailing, one lighthouse bears S.W., and
the other S. 22° 30' W. (22° 30' west of south). Find the ship's rate.
20. A ship, 10 mi. S.W. of a harbor, sees another ship sail from
the harbor in a direction S. 80° E., at a rate of 9 mi. an hour. In what
direction and at what rate must the first ship sail in order to catch
up with the second ship in Ij hr.?
MISCELLANEOUS APPLICATIONS 139
21. Two ships axe a mile apart. The angular distance of the first
ship from a lighthouse on shore, as observed from the second ship,
is 35** 14' 10" ; the angular distance of the second ship from the light-
house, observed from the first ship, is 42® 11' 53". Find the distance
in feet from each ship to the lighthouse.
22. A lighthouse bears N. 11° 15' E., as seen from a ship. The
ship sails northwest 30 mi., and then the lighthouse bears east. How
far is the lighthouse from the second point of observation ?
23. Two rocks are seen in the same straight line with a ship,
bearing N. 15° E. After the ship has sailed N.W. 5 mi., the first rock
bears E., and the second N.E. Find the distance between the rocks.
24. On the side OX of a given angle ZOFa point A is taken such
that OA = d. Deduce a formula for the length .45 of a line from A
to OF that makes a given angle a with OX. From
this formula, aj is a minimum when what sine is
the maximum ? Under those circumstances what
is the sum of and a ? Then what is the size of
Z.B? State the conclusion as to the size of Za
in order that x shall be the minimum.
25. Three points, A, By and C, form the vertices of an equilateral
triangle, AB being 500 ft. Each of the two sides AB and ^C is seen
from a point P under an angle of 120° ; that is, Z ^ P5 = 120* = Z CPA .
Find the length of A P.
26. A lighthouse facing south sends out its rays extending in a
quadrant from S.E. to S.W. A steamer sailing due east first sees
the light when 6 mi. away from the lighthouse and continues to see
it for 45 min. At what rate is the ship sailing ?
27. If two forces, represented in intensity by the lengths a and b,
pull from P in the directions PC and PA, respectively, and if Z.APC
is known, the resultant force is represented in
intensity and direction by /, the diagonal of
the parallelogram ABCP, Show how to find/
and AAPBj given a, h, and Z.APC,
28. Two forces, one of 410 lb. and the other
of 320 lb., make an angle of 51° 37'. Find the intensity and the
direction of their resultant.
29. An unknown force combined with one of 128 lb. produces
a resultant of 200 lb., and this resultant makes an angle of 18°
24' with the known force. Find the intensity and direction of tne
unknown force.
140 PLANE TRIGOKOMETRY
30. Wishing to determine the distance between a church A and a
tower -B, on the opposite side of a river, a man measured a line CD
along the river (C being nearly opposite ^), and observed the angles
ACB, 58* 20'; ACB, 95** 20'; ^DB, 53* 30'; BBC, 98* 45'. CB is
600 ft. What is the distance required ?
31. Wishing to find the height of a summit A, a man measured a
horizontal base line CD^ 440 yd. At C the angle of elevation of A
is 37* 18', and the horizontal angle between D and the summit of
the mountain is 76* 18' ; at D the horizontal angle between C and
the summit is 67* 14'. Find the height.
32. A balloon is observed from two stations 3000 ft. apart. At the
first station the horizontal angle of the balloon and the other station
is 75* 25', and the angle of elevation of the balloon is 18*. The hori-
zontal angle of the first station and the balloon, measured at the
second station, is 64* 30'. Find the height of the balloon.
33. At two stations the height of a kite subtends the same angle A,
The angle which the line joining one station and the kite subtends
at the other station is B ; and the distance between the two stations
is a. Show that the height of the kite is J a sin^ sec B,
34. Two towers on a horizontal plain are 120 ft. apart. A person
standing successively at their bases observes that the angle of eleva-
tion of one is double that of the other ; but when he is halfway be-
tween the towers, the angles of elevation are complementary. Prove
that the heights of the towers are 90 ft. and 40 ft.
35. To find the distance of an inaccessible point C from either
of two points A and 5, having no instruments to measure angles.
Prolong CA to a, and CB to h, and draw AB^ Ab, and Ba. Measure
AB, 500 ft. ; aA, 100 ft. ; aB, 560 ft. ; bB, 100 ft. ; and Ab, 550 ft.
Compute the distances A C and BC.
36. To compute the horizontal distance between two inaccessible
points A and B when no point can be found whence both can be seen.
Take two points C and D, distant 200 yd., so that A can be seen
from C, and B from D. From C measure CF, 200 yd. to F, whence
A can be seen ; and from D measure DE, 200 yd. to E, whence B
can be seen. Measure AFC, 83*; ACD, 53*30'; ACF, 54*31'; BDE,
54* 30'; BDC, 156* 25'; DEB, 88* 30'. Compute the distance AB.
37. A column in the north temperate zone is S. 67* 30' E. of an
observer, and at noon the extremity of its shadow is northeast of him.
The shswiow is 80 ft. in length, and the elevation of the column at
the observer's station is 45*. Find the height of the column.
MISCELLANEOUS APPLICATIONS
141
121. Areas. In finding the areas of rectilinear figures the effort
is made to divide any given figure into rectangles, parallelograms,
triangles, or trapezoids, unless it already has one of these forms.
For example, the dotted lines show how the above figures may be
divided for the purpose of computing the areas. A regular polygon
would be conveniently divided into congruent isosceles triangles
by the radii of the circumscribed circle.
Exercise 61. Miscellaneous Applications
1. In the trapezoid^5Ci>itis known thatZ^ = 90*,Z5=32*25',
AB = 324.35 ft., and CD = 208.16 ft. Find the area.
2. Find the area of a regular pentagon of which each side is 4 in.
8. Find the area of a regular hexagon of which each side is 4 in.
4. The area of a regular polygon inscribed in a circle is to that
of the circumscribed regular polygon of the same number of sides
as 3 to 4. Find the number of sides.
5. The area of a regular polygon inscribed in a circle is the
geometric mean between the areas of the inscribed and circumscribed
regular polygons of half the nimiber of sides.
6. Find the ratio of a square inscribed in a circle to a square cir-
cimiscribed about the same circle. Find the ratio of the perimeters.
7. The square circumscribed about a circle is four thirds the in-
scribed regular dodecagon.
8. In finding the area of a field ABODE a surveyor measured
the lengths of the sides and the angle which each side makes with
the meridian (north and south) line through its
extremities. AD happened to be a meridian line.
Show how he could compute the area.
9. Two sides of a triangle are 3 and 12, and
the included angle is 30®. Find the hypotenuse of
the isosceles right triangle of equal area.
10. In the quadrilateral A B CD we have given A B,
BC,Z.AyZ.B, and Z. C, Show how to find the area of the quadrilateral.
11. In Ex. 10, suppose ^5 = 176 ft., J5C = 198 ft., Z^ = 95*',
Z5 = 92® 15', and Z C = 96® 45'. What is the area ?
142 PLANE TRIGONOMETRY
122. Surveyor's Measures. In measuring city lots snrveyors com-
monly use feet and square feet, with decimal parts of these units.
In measuring larger pieces of land the following measures are used :
16 J feet (ft.) = 1 rod (rd.)
66 feet = 4 rods = 1 chain (ch.)
100 links (li.) = 1 chain
10 square chains (sq. ch.) = 160 square rods(sq.rd.)=lacre(A.)
We may write either 7 ch. 42 li. or 7.42 ch. for 7 chains and 42 links. The
decimal fraction is rapidly replacing the old plan, in which the word link was
used. Similarly, the parts of an acre are now written in the decimal form
instead of, as formerly, in square chains or square rods.
Areas are computed as if the land were flat, or projected on a horizontal
plane, no allowance being made for inequalities of surface.
123. Area of a Field. The areas of fields are found in various
ways, depending upon the shape. In general, however, the work is
reduced to the finding of the areas of triangles
or trapezoids.
For example, in the case here shown we may draw a
north and south line E^A' and then find the sum of the
areas of the trapezoids ABB'A', BCC'B% CDiyC\ and ^
DEE'iy, From this we may subtract the sum of the
trapezoids^ 6?6?'^', GFF'G' and FEWF\ The result will j
be the area of the field. (
Instead of running the imaginary line E'A' outside
the field, it would be quite as convenient to let it pass j
through F^ A^ E^ or C. The method of computing the
area is substantially the same in both cases.
For details concerning surveying, beyond what is here given and is included
in Exercise 60, the student is referred to works upon the subject.
Exercise 62. Area of a Field
1. Find the number of acres in a triangular field of which the
sides are 14 ch., 16 ch., and 20 ch.
2. Find the number of acres in a triangular field having two sides
16 ch. and 30 ch., and the included angle 64° 15'.
3. Find the number of acres in a triangular field having two angles
68.4** and 47.2^ and the included side 20 ch.
4. Required the area of the field described in § 123, knowing that
^^' = 8 ch., BB^ = 12 ch., CC^ = 13 ch., DD^ = 12 ch., EE' = 8 ch.,
FF' = 1 eh., GG' = 2 ch., ^'(y' = 6 ch., G'B' = 1.5 ch., B'F' = 2.3 ch.,
F'C = 3 ch., CD' = 4 ch., D'E' = 2.9 ch.
MISCELLANEOUS Al^PLlCATlOlSfS l4S
6. In a quadrangular field ABCD, AB runs N. 27* E. 12.5 ch.,
EC runs K 30** W. 10 ch., CD runs S. 37** W. 15 ch., and DA runs
S. 47** E. 11.5 ch. Find the area.
That AB ia N. 27°E. means that it makes an angle of
27° east of the line running north through A. ^
6. In a triangular field ABC, AB mns N. 10® E.
30 ch., BC runs S. 30** W. 20 ch., and CA runs S. 22** E. j
13 ch. Find the area.
7. In a field ABCD, AB runs E. 10 ch., BC runs ^
N. 12 ch., CD runs S. 68** 12' W. 10.77 ch., and DA
runs S. 8 ch. Find the area.
8. A field is in the form of a right triangle of which the sides
are 15 ch., 20 ch., and 25 ch. From the vertex of the right angle a
line is run to the hypotenuse, making an angle of 30** with the side
that is 16 ch. long. Find the area of each of the triangles into
which the field is divided.
Using a protractor^ draw to scale the fields referred to in the
following examples^ and find the areas :
9. ^5, N. 72** E. 18 ch., CZ), K 68** W. 21 ch.,
BC, N. 10** E. 12.5 ch., DA, S. 12** E. 26.3 ch.
10. AB, N. 45** E. 10 ch., CD, S. 15** W. 18.21 ch.,
BC, S. 75** E. 11.55 ch., DA, N. 45** W. 19.11 ch.
11. AB, N. 5** 30' W. 6.08 ch., CD, S. 3** E. 5.33 ch.,
BC, S. 82** 30' W. 6.51 ch., DA, E. 6.72 ch.
12. AB, N. 6** 15' W. 6.31 ch., CD, S. 5** E. 5.86 ch.,
BC, S. 81** 50' W. 4.06 ch., DA, K 88** 30' E. 4.12 ch.
13. A farm is bounded and described as follows: Beginning at
the southwest corner of lot No. 13, thence N. IJ** E. 132 rods and
23 links to a stake in the west boundary line of said lot; thence
S. 89** E. 32 rods and 15^ links to a stake ; thence N. 1 J** E. 29 rods
and 15 links to a stake in the north boundary line of said lot ; thence
S. 89** E. 61 rods and 18-^^ links to a stake ; thence S. 32^** W. 54 rods
to a stake ; thence S. 35^** E. 22 rods and 4 links to a stake ; thence
S. 48** E. 33 rods and 2 links to a stake ; thence S. 7 J** W. 76 rods
and 20 links to a stake in the south boundary line of said lot ; thence
N. 89** W. 96 rods and 10 links to the place of beginning. Containing
85.65 acres, more or less. Verify the area given and plot the farm.
This is a common way of describing a farm in a deed or a mortgage.
144 PLANE TRIGONOMETRY
124. The Circle. It is learned in geometry that
= 2 irvj and a = tt/^,
where c = circumference, r = radius, a = area, and ir = 3.14159-1-
= 3.1416— = about 3|. For practical purposes ^ may be taken.
Furthermore, if we have a sector with angle n degrees,
ft
the area of the sector is evidently ^^ of ttt*.
From these formulas we can, by the help of the
formulas relating to triangles, solve numerous prob-
lems relating to the circle.
Exercise 63. The Circle
1. A sector of a circle of radius 8 in. has an angle of 62.5®.
A chord joining the extremities of the radii forming the sector cuts
off a segment. What is the area of this segment ? b
2. A sector of a circle of diameter 9.2 in. has an
angle of 29® 42'. A chord joining the extremities
of the radii forming the sector cuts off a segment.
What is the area of the remainder of the circle ?
3. In a circle of radius 3.5 in., what is the area included between two
parallel chords of 6 in. and 5 in. respectively? (Give two answers.)
4. A regular hexagon is inscribed in a circle of radius 4 in. What
is the area of that part of the circle not covered by the hexagon ?
5. In a circle of radius 10 in. a regular five-pointed
star is inscribed. What is the area of the star ? What
is the area of that part of the circle not covered by
the star ?
6. In a circle of diameter 7.2 in. a regular five-
pointed star is inscribed. The points are joined,
thus forming a regular pentagon. There is also a regular pentagon
formed in the center by the crossing of the lines of the star. The
small pentagon is what fractional part of the large one ?
7. A circular hole is cut in a regular hexagonal plate
of side 8 in. The radius of the circle is 4 in. What is
the area of the rest of the plate ?
8. A regular hexagon is formed by joining the mid-points of the
sides of a regular hexagon. Find the ratio of the smaller hexagon
to the larger.
CHAPTER IX
PLAlfB SAILING-
125. Plane Sailing. A simple and interesting application of plane
trigonometry is found in that branch of navigation in which the
surface of the earth is considered a plane. This can be the case
only when the distance is so small that the curvature of the earth
may be neglected.
This chapter may be omitted if further applications of a practical nature are
not needed.
1SS6. Latitude and Departure. The difference of latitude between
two places is the arc of a meridian between the parallels of latitude
which pass through those places.
Thus the latitude of Cape Cod is 42P 2' 21'' N. and the latitude of Cape Hat-
terajs is 36° 16' 14" N. The difference of latitude is 6° 47' 7".
The departure between two meridians is the length of the arc
of a parallel of latitude cut off by those meridians, measured in
geographic miles.
The geographic mile, or knot, is the length of 1' of the equator. Taking the
equator to be 131,386,456 ft., -^ of *3^ of this length is 0082.66 ft., and this
is generally taken as the standard in the United States. The British Admiralty
knot is a little shorter, being 6080 ft. The term "mile" in this chapter refers
to the geographic mile, and there are 60 mi. in one degree of a great circle.
Calling the course the angle between the track of the ship and the
meridian line, as in the case of N. 20® E., it will be evident by drawing
a figure that the difference in latitude, expressed in distance, equals
the distance sailed multiplied by the cosine of the course. That is
diff. of latitude = distance x cos C.
In the same way we can find the departure. This is evidently
given by the equation
departure = distance x sin C.
For example, if a ship has sailed N. 30® E. 10 mi., the difference
in latitude, expressed in miles, is
10 cos 30® = 10 X 0.8660 = 8.66,
and the departure is 10 sin 30® = 10 x 0.5 = 5-
146
146
PLAlfE TRIGONOMETBY
127. The Compaw. The mariner divides the circle into 32 equal
parts called points. There are therefore 8 points in a right angle,
and a point is 11° 15'. To sail two
points east of north means, therefore,
to sail 22° 30' east of north, or north-
northeast (N.N.E.) as shown on the
compass. Northeast (N.E.) is 45° east
of north. One point east of north is
called north by east (N. by E.) and one
point east of south is called south by
east (S. by E.). The other terms used,
and their significance in angular measure,
will best be understood from the illustration and the following table :
KonxH
0-1
ti
11 10
Polnta
1
n! by E.
N. by W.
S. by E.
S. by W.
N.N.E.
K.N.W.
iii
19 41 IB
22 3»
1-i
S.S.E.
S.8.W.
N.E. by N.
N.W. by N.
v,
3-i
25 lS4fi
28 730
33 45
ti
S.E. by S.
S.W. by S.
N.E.
N.W.
3-1
38 33 4A
39MI0
45
3-1
3-i
3-1
S.E.
8.W.
N.E. by B.
N.W. by W.
47 48 45
B6 16
8.B. by E.
S.W. by W.
E.N.E.
W.N.W.
t!
S4 4118
OT30
ti
E.S.E.
W.S.W.
E. by N.
W. by H.
73 7 30
7B5«]5
78 45
H
E. byS.
W. by 8.
E.
W.
T-1
84 22 30
I'*
E.
W.
The compass varies in dISerent, parts of the earth ; hence, In sidling, the
compass course is not the same as the true course. The true course Is the com-
pass course, with allowances for variation of the needle in difierent parts of the
earth, for deviation caused by the iron in the ship, and for leeway, the angle
which the ship makes with her track.
PLANE SAILING 147
Exercise 64. Plane Sailing
1. A ship sails from latitude 40° N. on a course N.E. 26 mi. Find
the difference of latitude and the departure.
2. A ship sails from latitude 35® N. on a course S.W. 53 mi. Find
the difference of latitude and the departure.
3. A ship sails from a point on the equator on a course N.E. by
N. 62 mi. Find the difference of latitude and the departure.
4. A ship sails from latitude 43° 45' S. on a course N. by E. 38 mi.
Find the difference of latitude and the departure.
5. A ship sails from latitude 1° 45' N. on a course S.E. by E. 25 mi.
Find the difference of latitude and the departure.
6. A ship sails from latitude 13° 17' S. on a course N.E. by E. | E.,
until the departure is 42 mi. Find the difference of latitude and the
latitude reached.
7. A ship sails from latitude 40° 20' N. on a N.N.E. course for
92 mi. Find the departure.
8. If a steamer sails S.W. by W. 20 mi. what is the departure
and the difference of latitude?
9. If a sailboat sails N. 25° W. until the departure is 25 mi., what
distance does it sail ?
10. A ship sails from latitude 37° 40' N. on a N.E. by E. course
for 122 mi. Find the departure.
11. A yacht sails 6 J points west of north, the distance being 12 mi.
What is the departure ?
12. A steamer sails S.W. by W. 28 mi. It then sails N.W. 30 mi.
How far is it then to the west of its starting point ?
13. A ship sails on a course between S. and E. 24 mi., leaving
latitude 2° 52' S. and reaching latitude 2° 58' S. Find the course and
the departure.
14. A ship sails from latitude 32° 18' N., on a course between N.
and W., a distance of 34 mi. and a departure of 10 mi. Find the
course and the latitude reached.
15. A ship sails on a course between S. and E., making a differ-
ence of latitude 13 mi. and a departure of 20 mi. Find the distance
and the course.
16. A ship sails on a course between N. and W., making a differ-
ence of latitude 17 mi. and a departure of 22 mi. Find the distance
and the course.
148 PLANE TBIGONOMETRY
128. Poicllel Sailli^. Sailing due east oi due west, remaining on
the same parallel of latitude, is called parallel sailing.
129. finding XHSereace In Longitude. In parallel sailing the dis-
tance sailed is, by definition (g 126), the departure. From the
departure the difference in longitude is found aa follows ;
Let ^B be the departure. Then in rt. A O^iJ
^AOD=W-hL
Hence ^ = Bin(90= - lat.) = cos lat
The triangles DAB and OEQ are similar, the arcs being (§ 125)
considered straight lines.
„^ , DA AB DA AB
Therefore 5fi = EQ' ''' OA=W
Hence cos lat = -=— ■
Therefore EQ = t--=AB x see lat.
cos lat.
That is, Dlfi. long. = depart, x sec lat.
That is, the number of minutes in the difference in longitude is the product
. of the number of miles in the departure by the aecant of the latitude, tlie
nautical, or geographic, mile being a minute of longitude on the equator.
Exercise 65. Parallel Sailing
1. A ship in latitude 42° 16' N., longitude 72° 16' W., sails due
east a distance of 149 mi. What is the position of the point reached?
2. A ship in latitude 44° 49' S., longitude 119° 42' E., sails due
west until it reaches longitude 117° 16' E. Find the distance made.
3. A ship in latitude 60° 15' N,, longitude 60° 16' W., sails due
west a distance of 60 mi. What is the position of the point reached ?
PLANE SAILING 149
130. Middle Latitude Sailing. Since a ship rarely sails for any
length of time due east or due west, the difference in longitude can-
not ordinarily be found as in parallel sailing (§,§128, 129). Therefore,
in plane sailing the departure between two places is measured gen-
erally on that parallel of latitude which lies midway between the
parallels of the two places. This is called the method of middle
latitude sailing. Hence, in middle latitude sailing,
DifF. long. = depart. X sec mid. lat.
produces no great error, eacept in very high latitudes or
Exercise 66. Middle Latitude Sailing
1. A ship leaves latitude 31° 14' N., longitude 42° 19' W., and sails
KN.E. 32 mi. Find the position reached.
3. Leaving latitude 49''57'N., longitude 15° 16' W., a ship sails
between S. and W. till tlie departure is 38 mi. and the latitude is
49° 38' N. Find the course, distance, and longitude reached.
3. Leaving latitiide 42° 30' N., longitude 58° 51' W., a ship sails
S.E. by S. 48 mi. Find the position reached.
4. Leaving latitude 49° 67' N., longitude 30° W., a ship sails
S. 39° W. and reaches latitude 49° 44' N, Find the distance and
the longitude reached.
5. Leaving latitude 37° N., longitude 32° 16' W., a ship sails be-
tween K. and W. 45 mi. and reaches latitude 37° 10' N. Find the
course and the longitude reached.
6. A ship sails from latitude 40° 28' N., longitude 74° W., on an
E.S.E. course, 62 mi. Find the latitude and longitude reached.
7. A ship sails from latitude 42° 20' N., longitude 71° 4' W., on a
N.N.E. course, 30 mi. Find the latitude and longitude reached.
160 PLANE TEIGONOMETRY
131. Ttwrene Sailing. Id case a ship sails from one point to an-
other on two or more difFerent courses, the depEirtiiie and difference
of longitude are found by reckon-
ing each course separately and com-
bining the results. For example,
two such courses are shown in the
figure. This is called the method
of traverse sailing.
No new principles are involved in
tiaverae Moling, as will be seen in solv-
iDg Ex. 1, given below.
ExercUe 67. Traverse Sailing
1. Leaving latitude 37° 16' S., longitude 18° 42' W., a ship sails
N.E. 104 mi., then N.N.W. 60 mi., then W. by S. 216 mi. Find the
position reached, and its bearing and distance from the point left.
For the first course we have difference of latitude 78.5 K., departure 73.5 E.;
for the second course, difference of latitude 55.4 N., departure 23 W. ; for the
third course, difference of latitude 42.1 S., departure 211.8 W.
On the whole, then, the ship has made 12S.0 mi. of oorth latitude and 12.1 ml.
of south latitude. The place reached is therefore on a parallel of latitude 66.8 mi,
to the north of the parallel left ; that is, in latitude 35° 49.2' S.
In the same way the departure is found to be 101.3 mi. W., and the middle
latitude la 36° 32.6'. With these data we find the difference of longitude to be
801', or 3° ai' W. Hence the longitude reached is 22° 3' W.
With the difference of latitude 66.8 mi. and the departure 161.3 mi., we find
ihe course to be N. 61°43'W. and the distance IS3.2mi. The ship has reached
the same point that it would have reached if it had sailed directly on a course
N. 61° 43' W. for a distance of 183.2 mi,
2. A ship leaves Cape Cod (42° 2' K"., 70° 3' W.) and sails S.E. by S.
114 mi., then N. by E. 94 mi., then W.N.W. 42 mi. Find its position
and the total distance.
3. A ship leaves Cape of Good Hope (34° 22' 8., 18° 30' E.) and
sails N.W. 126 mi., then N. by E. 84 mi., then W.S.W. 217 mi. Find
its position and the total distance.
4. A ship in latitude 40° N. and longitude 67° 4' W. sails N.W.
60 mi., then N. by W, 62 mi., then W.S.W. 83 mi. Find its position.
5. A ship sailed S.S.W. 48 mi., then S.W. by S. 36 mi., and then
N.E. 24 mi. Find the difference in latitude and the departure.
6. A ship sailed S. i E. 18 mi., S.W. i S. 37 mi,, and then S,S,W
^ W. 56 mi. Find the difCui-eace in latitude and the depai'ture.
CHAPTER X
GRAPHS OF FUNCTIONS
132. Circular Measure. Besides the methods of measuring angles
which have been discussed already and are generally used in
practical work, there is another method that is frequently employed
in the theoretical treatment of the subject. This takes for the unit
the angle subtended by an arc which is equal in length to the radius,
and is known as circular measure,
133. Radian. An angle subtended by an arc equal in length to the
radius of the circle is called a radian.
The term " radian " is applied to both the angle and
arc. In the annexed figure we may think of a radius
bent around the arc -4 B so as to coincide with it. Then
^AOB is a radian.
134. Relation of the Radian to Degree Measure.
The number of radians in 360® is equal to the
number of times the length of the radius is contained in the length
of the circumference. It is proved in geometry that this number is
2 TT for all circles, ir being equal to 3.1416, nearly. Therefore the
radian is the same angle in all circles.
The circumference of a circle is 2 tt times the radius.
Hence 2 tt radians = 360°, and tt radians = 180**.
Therefore 1 radian = ^^^^ = 57.29578** = 57** 17' 45",
TT
TT . !
and 1 degree = t^ radian = 0.017453 radian/
135. Number of Radians in an Angle. From the definition of radian
we see that the number of radians in an angle is equal to the length
of the subtending arc divided by the length of the radius.
Thus, if an arc is 6 in. long and the radius of the circle is 4 in., the number
of radians in the angle subtended by the arc is 6 in. -j- 4 in., or 1 J.
This may be reduced to degrees thus :
1 J X 67.29678° = 86.94867°,
or, for practical purposes, 1 J x 67.3° = 86.9° = 86° 64^
161
152 PLANE TRIGONOMETRY
136. Reduction of Radians and Degrees. From the values found in
§ 134 the following methods of reduction are evident :
To reduce radians to degrees^ multiply 57^ 17^ 46^\ or 67 29678^ ^
hy the number of radians.
To reduce degrees to radians^ multiply 0.017453 hy the number
of degrees.
These rules need not be learned, since we do not often have to make these
reductions. It is essential, however, to know clearly the significance of radian
measure, since we shall often use it hereafter. In solving the following problems
the rules may be consulted as necessary.
In particular the student should learn the following :
360° = 2 ?r radians, 60° = J ?r radians,
180° = TT radians, 30° = J ^ radians,
90° = J TT radians, 16° = ^ ?r radians,
46° = J ?r radians, 22.6° = \^l^ radians.
The word radians is usually understood without being written. Thus sin 27r
means the sine of 2 tt radians, or sin 360° ; and tan J ir means the tangent of
\ IT radians, or 46°. Also, sin 2 means the sine of 2 radians, or sin 114.69166°.
Exercise 68. Radians
Express the following in radians :
1. 270°. 3. 56.25°. 6. 196.5°.
2. 11.25°. 4. 7.5°. 6. 1440°.
Express thefoUmving in degree measure :
9. IjTT. 11. l^TT. 13. ^TT.
10. IjTT. 12. IjTT. 14. StT.
State the quadrant in which the following angles lie :
17. f TT. 19. IfTT. 21. 2.5 TT. 23. 1.
18. f TT. 20. IfTT. 22. — 3.4 7r. 24. —2.
Express the following in degrees and also in radians :
25. I of four right angles. 27. § of two right angles.
26. I of four right angles. 28. | of one right angle.
29. What decimal part of a radian is 1°? V?
30. How many minutes in a radian ? How many seconds ?
31. Express in radians the angle of an equilateral triangle.
32. Over what part of a radian does the minute baud of ^ clock
move in 15 min. ?
7.
200°.
8.
3000°
15.
67r.
16.
10 TT.
GRAPHS OF FUNCTIONS 163
137. Functions of Small Angles. Let A OP be any acute angle, and
let X be its circular measure. Describe a (;ircle of unit radius about
O as center and take Z.AOP' = -'Z.AOP. Draw the tangents to
tlie circle at P and P', meeting OA in T. Then we see that
chord PP' < arc PJ*'
<PT-\- P"/\
Dividing by 2, MP < arc A P < PT,
or sin y < x < tan a*.
Dividing by sin a*, 3 < -r- — < sec x,
„_, . sin X
W hence 1 > > cos x.
X
sin X
Therefore the value of *■ lies between cos x and 1.
X
If, now, the angle x is constantly diminished, cos x approaches
the value 1.
sm X
Accordingly, the limit of > as x ap])roa(;hes 0, is 1.
X
Hence when x denotes the circular measure of an angle near 0^ we may
use X instead of sin x and instead of tan x.
For example, required to find the sine and cosine of 1'.
If X is the circular measure of 1',
2^ _ 314159 + _ . 00029088 4-
360x60- 10800 - 0-00029088 +,
X =
the next figure in x being 8.
Now sinx > but < x ; hence sin 1' lies between and 0.000290889.
Again, cos 1' = Vl-sinn' > Vl - (0 0003)^ > 0.9999999.
Hence cos 1' = 0.9999999 +•
But, as above, sin ir > cc cos x.
.-. sinl' > 0.000290888 x 0.9999999
> 0.000290888 (1 - 0.0000001)
> 0.000290888 - 0.0000000000290888
> 0.000290887.
Hence sin 1' lies between 0.000290887 and 0.000290889 ; that is,
t© eight places of decimals,
sin 1'-= 0.00029088+,
the next figure being 7 or 8.
164 PLANE TRIGONOMETRY
138. Angles having the Same Sine. If we let AXOP ^Xy in this
figure, and let P' be symmetric to P with respect to the axis YY\ we
shall have Z XOP' = 180* - cc, or tt - a:. And
since - = sin x = sin (tt — x) we see that x and
r ^
TT — X have the same sine.
Furthermore, sin x = sin (360** -h a;), and
sin (180** - aj) = sin (360** + 180'* - x). That
is, we may increase any angle by 360® without
changing the sine. Hence we have sina; = sin(7i • 360®-f j^), and
sin (180° — ic) = sin(7i . 360° + 180° — x). Using circular measure
we may write these results as follows :
sin X = sin (2 kir + x), and sin (tt — a;) = sin (2 A; + 1 tt — x).
These may be simplified still more, thus :
sin X = sin [titt -h (— l)"ar]
where n is any integer, positive or negative.
Thus if n = we have sin a; = sin (0 • -tt + (— l)^x) = sin x ; if n = 1 we have
sinx = sin(ir— x) ; if n = 2 we have sinx = sin (27r + x); and so on.
Since the sine is the reciprocal of the cosecant, it is evident that x and
nir + (— l)*x have the same cosecant.
To find four angles whose sine is 0.2688, we see by the tables that sin 15°= 0.2588.
Hence we have sin 16° = sin [nw + (— 1)" • 15°] = sin (ir - 16°) = sin (2 ?r + 16°)
= sin (3 IT — 15°) ; and so on.
Exercise 69. Sines and Small Angles
1. Find four angles whose sine is 0.2756.
2. Find six angles whose sine is 0.5000.
3. Find eight angles having the same sine as ^ tt.
4. Find four angles having the same cosecant as ^ tt.
5. Find four angles having the same cosecant as 0.1 tt.
Griven ir = 3,141592653589^ compute to eleven decimal places :
6. cosl'. 7. sinl'. 8. tanl'. 9. sin 2'.
10. From the results of Exs. 6 and 7, and by the aid of the formula
sin 2 aj = 2 sin a? cos x, compute sin 2\ carrying the multiplication to
six decimal places. Compare the result with that of Ex. 9.
11. Compute sin 1° to four decimal places.
X X
12. From the formula cos a; = 1 — 2 sin^ — > show that cos a; > 1 — -x- •
GRAPHS OF FIJNCTIONS
155
139. Angles having the Same Cosine. If we let /.XOP = x, in
this figure, and let P' be symmetric to P with respect to the axis
XX\ we shall have Z.XOP' = 360^ - a;, or - x,
depending on whether we think of it as a
positive or a negative angle. In either case
its cosine is -> the same as cosaj.
r
In either case cos x = cos (n • 360® — x).
In general, cos x = cos (2 nir ± x),
where n is any integer, positive or negative.
Thus if n = 0, we have cos a; = cos (± a;) ; if n = 1, we have cos x = cos (2 ir± x) ;
if n = 2, we have cosx = cos(47r ± x); and so on.
Since the cosine is the reciprocal of the secant, it is evident that x and 2mr±x
have the same secant.
140. Angles having the Same Tangent. Since we have tan x =
a
— a
-a
and tan (180° -f a;) = — r > we see that tan x = tan (180® -f x). In
general we may say that
tan X = tan (2 kir + a;) = tan (2 /ctt -f- tt -|- x).
This may be written more simply thus :
tan X = tan (nir -f- x),
where n is any integer, positive or negative.
Thus if we have tan 20° given, we know that iwr + 20° has the same tangent.
Writing both in degree measure, we may say that n • 180° + 20° has the same
tangent. If n = 1, we have 200° ; if n = 2, we have 380° ; if n = 3, we have 660° ;
and so on. Furthermore, if n = — 1, we have —160° ; and so on.
Since the cotangent is the reciprocal of the tangent, it is evident that x and
nir -{■ x have the same cotangent.
Exercise 70. Angles having the Same Functions
1. Find
2. Find
3. Find
4. Find
5. Find
6. Find
7. Find
8. Find
9. Find
10. Find
two positive angles that have ^ as their cosine.
two negative angles that have ^ as their cosine.
four angles whose cosine is the same as the cosine of 25°
four angles that have 2 as their secant.
two positive angles that have 1 as their tangent.
two negative angles that have 1 as their tangent.
four angles that have V3 as their tangent.
four angles that have Vs as their cotangent.
four angles that have 0.5000 as their tangent.
four negative angles whose cotangent is 0.5000.
156 PLANE TKIGONOMETKY
141. Inverse Trigonometric Functions. If y == sin Xj iihan x is the
angle whose sine is y. This is expressed by the symbols x = sin~^ y,
or X = arc sin y.
In American and English books the symbol sin-^ y is generally used ; on the
continent of Europe the symbol arc sin y is the one that is met.
The symbol sin~^y is read "the inverse sine of y/^ *Hhe antisine
of y/' or " the angle whose sine is y." The symbol arc sin y is read
" the arc whose sine is y" or " the angle whose sine is y."
The sjnnbols cos-i x, tan-i x, cot-i x, and so on are similarly used.
The symbol sin-iy must not be confused with (sin y)~^. The former means
the angle whose sine is y ; the latter means the reciprocal of siny.
We have seen (§ 138) that sin-^ 0.5000 may be 30°, 150^ 390^ 510^
and so on. In other words, there are many values for sin"^ x ; that is,
Inverse trigonometric functions are many-valued.
142. Principal Value of an Inverse Function. The smallest positive
value of an inverse function is called its principal value.
For example, the principal value of sin-i 0.6000 is 30° ; the principal value
of cos-i 0.6000 is 60° ; the principal value of tan-i (— 1) is 136° ; and so on.
Exercise 71. Inverse Functions
Prove the following formulas :
1. sin'^x -4- cos"^x- = ^ TT. 3. sec'^x + csc"^^ = ^tt.
2. tan~^aj -f- cot~^ic = ^tt. 4. sin~^(— x) = — sin~^x.
Find two values of each of the following :
5. sin-^^V3. 7. tan-^|V3. 9. sec-^2.
6. csc-^V2. 8. tan-^oo. 10. cos-^(— ^V2).
11. Find the value of the sine of the angle whose cosine is ^;
that is, the value of sin(cos~^^).
Find the values of the following :
12. sin(cos-^^ Vs). 13. sin(tan-^l). 14. cos(cot-^l).
Prove the following formulas :
15. tan(tan-ix + tan-i2^) = :^-=tiL. 17. tan(2tan-ia;) = --^^.
^ L — xy ^ ^ 1 — ar
GEAPHS OF FUNCTIONS 167
Find four values of each of the following :
19. tan-i 0.5774. 21. sin-^ 0.9613. 23. cot"^ 0.2756.
20. cot-i 0.6249. 22. sin" ^ 0.3256. 24. 008-^0.9455.
25. Solve the equation y — sin"^^.
26. Find the value of sin(tan-^^ + tan"^^).
27. If sin~^x = 2 cos-^cc, find the value of x.
Prove the following formulas :
28. cos (sin-^ x) = Vl — x^,
29. cos (2 sin-^ a;) = 1 — 2 a:^.
30. sin(sin"'^aj)= 05.
31. sin (sin-^cc -|- sin-^y) = x Vl — y^-\-y Vl — aA
32. tan-i 2 + tan"^ i = i ^r-
33. 2tan-iaj = tan-i[2aj:(l-ar^)].
34. 2 sin-^ic = sin-i(2 x Vl - x^),
35. 2 cos-^aj = cos-^(2a5^ — 1).
36. 3 tan-^a; = tan-^ [(3 x - x^)\ (1 -- 3 a^].
37. sin"^ Va; : y = tan""^ Va; : {y — x).
38. sin"^ V(a; — y):(x — z) = tan""^ V(a; — y) : (y — »)
39. sin-^aj = sec""^(l : Vl — ar^).
40. 2 sec-^aj = tan-^ [2 Va:^ - 1 : (2 - x^'].
41. tan-^ J 4- tan-^ i = i w*-
42. tan-^ J + tan-'^ ^ = tan"^ f
43. sin-if + sin-^j| = sin-^f|.
44. 8in-i^V82 + sin-i4^ViT= JTT.
45. sec-^ + sec-i l| = 75** 45'.
46. tan-^(2 + V3)- tan-^(2 - V3)= sec-*2.
47. tan-^J + tan-^J + tan-^;j^-|-tan-^i = Jw.
48. sin-^aj + sin"^ Vl — ar^ = J tt.
49. sin-10.5 + sin-i J V3 = sin-^l.
50. tan-^ i = tan-^ J + tan-^ f
51. tan-^0.5 + tan-10.2 + tan-^0.125 = J w-
52j tan-^1 4- tan-^2 + tan-^3 = tt.
53. tan-^ f + tan-^ J + tan-^ tt = i ^•
54. cos-^^ VlO 4- sin-i i Vs = J tt.
168
PLiNE TRIGONOMETBY
143. Gtaph of 1 Fnnctlos. Ab in algebra, so in trigonometFy, It is
poBsiUe to represent a function graphically. Before taking up the
Bul^ect of graphs in trigonometry a few of the simpler cases from
aJgebra will be considered.
Suppose, for example, we have the expression 3x + 2. Since the
value of this expression depends upon the value of x, it is called a
function of x. This fact is indicated by the equation
/W-3I + 2,
read " function x = 3x •\- 2." But since /(«) is not so easily written
as a single letter, it is customary to replace it by some such letter as
y, writing. the equation
y=3x + 2.
If X = 0, we Bee that y = 2; if a: = 1,
then y = 5; and so on. We may form a
table of such values, thus :
I
y
«
V
2
2
1
6
-1
-1
2
8
_2
-4
8
:
11
-3
-T
11
\zAdlLZZrrz'.
We may then plot the points (0, 2), (1, 6), (2, 8), -..,(— 1, — 1),
(— 2, — 4), ■ - -, as in g 77, and connect them. Then we have the
graph of the function So; + 2.
The graph shows that the function, y or f(x), changes in value much more
rapidly than the variable, z. It also shows that the function does not become
negative at the same time that the variable does, its value being 2 when z = 0,
and ^ when x=— \. This kind oi function in which z is of the flrat degree
only is called a iinear fanetion because Its graph is a straight line.
Exercise 72. Graphs
Plot the graphs of the following functiont :
1. 2x. 6. a; — 1. 9. ~2-x.
2. 1^3!. 6. 23!+l. 10. 23; + 3.
« + L
7. 3-a;.
8. 4-^3;.
-3.
12. 3 — 2a;,
13. 0.5 a;
14. 1.4 X
-i^x-2it.
GEAPHS OF FUNCTIONS 159
144. Qnph of ■ Qnadntic Function. We shall now consider fuao-
tions of the second degree in the variable. Such a function ia
called a quadratie function.
Taking the function a^ + * — 2, we
write
y=:a^ + a:-2.
Preparing a table of values, as on
page 16S, we have
>
s
>
V
-2
-2
1
.-1
-2
2
4
-2
S
10
-3
4
4
18
-4
10
In order to see where the function lies between y =— 2 and y ——2, we
let X, =— \. We find tbat when z =— J, y =— 2}. Similarlj if we give to z
other vaJues between and — 1, we shall find that y In every case lies between
and — 2.
Plotting the points and drawing through them a smooth curve, we
have the graph aB here shown.
Thiscurveisajrarotola. All graphs of fuactionaof the form yssiw* + (w + e
are parabolas.
Graphs of functions of the form x* + y* = i^, oi y = ± Vr' — e^, are circle*
with their center at 0.
Graphs of functiona ot the fonn a^ -t- 6*^' = c' are ellipaet, these becoming
circles if a = b.
Graphs of functions of the form a*ifl — b^j/> = c' are hyperbolas.
There are more general equatlona of all these conic seetiont, but these sufBce
for our present purposes. The graph of every quadratic function in x and ]/ is
always a oonio section.
Exercise 73. Qrapha of Quadratic FnnctionB
Plot the graphs of the follomnff /unctions :
1. ar". 5
3. 23r'. 6
3. ^. 7. x^-
4. 3^ + 1. 8. x'' +
160
PLANE TKIGONOMETRy
145. Graph of the Sine. Since sin x is a function of x, we can plot
the graph of sin x. We may represent x, the arc (or angle), in de-
grees or in radians on the a;-axis. Representing it in degrees, as
more familiar, we may prepare s
I. table of values as follows :
120° 186° 160° 166° 180° ■
If we represent each unit on the y^ixis by J, and each unit on the
a»-azis by 30°, the graph is as follows :
The graph shows very clearly that the sine of an angle x is poutlve between
the Taluee 1 = 0° and x = 180°. and also between the yalues x= — 360° and
I = — 180° ; that it is negative between the TaJnes x = — 180° and z = 0°, and
also between the values x = 180° and z — 360°. It also shows that the sine
changes ftom posiUve to negative as the angle increases and passes through
— 180° and 180°, and that the sine changes from negative to positive as the
angle Increases and passes through the values — 360°, 0°, and 360°. These facts
liave been found analytically (§84), but they are seen more clearly b; studying
the graph.
If we nse radian measure for the arc (angle), and represent each
onit on the a^asis by 0.1 tt, the graph is as follows :
the only diffeienca being that we have
the z-azis, thus elongating the curve in the
146. Periodicity of FunctlooB. This curve shows graphically what
we have already found, that periodically the sine comes back to any
given value.
Thus an z= 1 when x=-270°, 90°, 460P, ■■■, returning to this value for
increase of the angle by every 360°, oi 2ir radians. The j>mod of the sine is
therefore said to be S60° or 2 tt.
GRAPHS OF FUNCTIONS
Exercise 74. Graphs of Trigonometric Fnnctioas
1. Verify the following plot of the graph of cos x :
2. What is the period of cos x ?
S. Verify the following plot of the graph of tail a: :
4. What is the period of tan x ?
5. Verify the following plot of the graph of cotfc:
- '^ \ ih»^ .„y^..
s
mm
6. What is the period of cot a; ?
7. Verify the following plot.of the graph of Bee it:
mmm-
r i f iiiii 'i i
^:y=
-ffct
8. What is the period of sec x ?
9. Plot the graph of esc x, and state the period. Also state at
what values of x the sign of cscx changes.
10. Plot the graphs of sin a; and coax on the same paper. Wiat
does the figure tell as i» the mutual relation of these functions ?
162 PLANE TRIGONOMETRY
Exercise 75. Miscellaneous Exercise
Find the areas of the triangles in which :
1. a = 25, 6 = 25, c = 25. 3. a = 74, i = 75, c = 92.
2. a = 25, i = 33J, c = 41§. 4. a = 2J, 6 = 3^, c = 4J.
5. Consider the area of a triangle with sides 17.2, 26.4, 43.6.
6. Consider the area of a triangle with sides 26.3, 42.4, 73.9.
7. Two inaccessible points A and B are visible from D, but no
other point can be found from which both points are visible. Take
some point C from which both A and D can be seen and measure CD,
200 ft. ; angle ADC, 89**; and angle ACD, 50** 30'. Then take some
point E from which both D and B are visible, and measure DE,
200 ft.; angle BDE, 54** 30'; and angle BED, 88** 30'. At D measure
angle ADB, 72** 30'. Compute the distance AB.
8. Show by aid of the table of natural sines that sin x and x agree
to four places of decimals for all angles less than 4** 40'.
9. If the values of log x and log sin x agree to five decimal places,
find from the table the greatest value x can have.
10. Find four angles whose cosine is the same as the cosine of 175**.
11. Find four angles whose cosine is the same as the cosine of 200**.
12. How many radians in the angle subtended by an arc 7.2 in.
long, the radius being 3.6 in. ? How many degrees ?
13. How many radians in the angle subtended by an arc 1.62 in.
long, the radius being 4.86 in. ? How many degrees ?
Draw the following angles :
14. — TT. 16. -^TT. 18. 2.7 TT. 20. 3 7r-9.
16. -2. 17. -J. 19. 27r-6. 21. 4-7r.
22. Find four angles whose tangent is —p •
23. Find four angles whose cotangent is
V3
24. Plot the graphs of sin x and esc x on the same paper. What
does the figure tell as to the mutual relation of these functions ?
26. Plot the graphs of cos x and sec x on the same paper. What
does the figure tell as to the mutual relation of these functions ?
26. Plot the graphs of tan x and cot x on the same paper. What
does the figure tell as to the mutual relation of these functions ?
CHAPTER XI
TRIGONOMSTRIC IDENTITIES AND EQUATIONS
147. Equation and Identity. An expression of equality which is
true for one or more values of the unknown quantity is called an
equation: An expression of equality which is true for all values of
the literal quantities is called an identity.
For example, in algebra we may have the equation
4x-8 = 7,
which is true only if x = 2.6. Or we may have the identity
(a + 6)2 = a2 + 2a6 + 62,
which is true whatever values we may give to a and 6.
Thus sin X = 1 is a trigonometric equation. It is true for x = 90® or \ ir,
X = 460° or 2iir, x .= 810° or 4j7r, and so on, with a period of 860° or 2ir. In
general, therefore, the equation sin x = 1 is true for x = (2 n + J) ir. It is this
general value that is required in solving a general trigonometric equation.
On the other hand, the equation sin^x = 1 — cos^x is true for all values of x.
It is therefore an identity.
The symbol = is often used instead of = to indicate identity, but the sign of
equality is very commonly employed unless special emphasis is to be laid upon
the fact that the relation is an identity instead'of an ordinary equation.
148. How to prove an Identity. A convenient method of proving
a trigonometric identity is to substitute the proper ratios for the
functions themselves.
CL C
Thus to prove that sin x = 1 : esc x we have only to substitute - for sin x and -
a c * ^ ^
for CSC X. We then see that - = !:-. Similarly, to prove that tan x = sin x sec x,
we may substitute - for tanx, - for sinx, and - for secx. We then have
be b
a _a c
b^c'b'
We can often prove a trigonometric identity by reference to
formulas already proved.
This was done in proving the identity 8in2x = 2 sinx cos x (§ 101), and in
tan X + tan v
proving tan (X + y) = ^^"^^''f"^ (§ 03).
1 — tan X tan y
In some cases it may be better to draw a figure and use a geometric
proofs as was done in § 90.
168
164 PLANE TRIGONOMETRY
£xei:pise 76. Identities
Prove the follovring identities :
2 tan ix « J. o 3 tan x — tan'a?
'•^^'^'"^l-tan'i.:- 6. tan3..= l-3tan'x '
2 tan ix ^ tan 2x -\- tan x sin 3 a;
2. sma;= — ^^^ — 7. -; — 7; -^ = — :
1 + tan-* i X tan 2x — tan x siu as
. _ 2 tan a; ^3 cos x -\- cos 3 a? ,-
3. sin 2 X = Z—-Z — r~ ' S- tt-- • — 77— = cof a;.
1 + tan'^aj 3 sin x — sin 3 x
^ ^ . . . f> 2 sin* a; ^ sin 3 a; + sin 5 x
4. 2 sin a; + sin 2 a; = :;; 9. -— = cota;.
1 — cos X cos 3 a; — cos 5 x
. ^ sin^ 2 a; — sin^ x ^ ^ sin 3 aj + sin 5 a;
6. sin 3 a; = : ^ 10. — : ; — ; — - — = 2cos2ax
sin X sm X + sin 3 x
11. sm a; -I- sin 3 a; H- sm 5 aj =
12. tan 2 x + sec 2 a; =
sma;
cos X H- sin x
cos X — sin aj
sin (x + y)
13. tanar + tany = ^^ — —^'
cos X cos y
, . ^ sin 2x -\- sin 2 v
14. tan(aj + y)= ^ — 7 ^•
^ ' ^^ cos 2 35 + cos 2 y
sin a; + c os y tan [ j- (a; + y) + 45°]
sin a; — cos y tan [J {^ — y)— 45**]
16. sin 2 a; + sin 4 a; = 2 sin 3 x cos aj.
17. sin 4 aj = 4 sin x cos a; — 8 sin^a; cos x,
18. sin 4 aj = 8 cos* a; sin a; — 4 cos x sin x.
19. cos 4 x = 1 — 8 cos^a; + 8 cos*a; =1 — 8 sin^x + 8 sin*x.
20. cos 2 X + cos 4 X = 2 cos 3 x cos x.
21. sin 3 X — sin x = 2 cos 2 x sin x.
22. sin*x sin 3 X + cos*x cos 3 x = cos* 2 x
23. cos*x — sin*x = cos 2 x.
24. cos*x + sin*x = 1 — J sin^ 2 x.
25. cos^x — sin^x = (1 — sin^x cos^x) cos 2 x,
26. cos^x + sin^x = 1 — 3 sin^x cos^x.
27. CSC X — 2 cot 2 x cos x = 2 sin x.
IDENTITIES AKD EQUATIONS 165
Prove the folloyying identities:
28. (sin 2 a; — sin 2 y) tan (x -f ^y) = 2 (sin^.r — sin^y).
3^9. sin 3 a; = 4 sin x sin (60** + x) sin (60** - x).
^^4-30. sin 4 a; = 2 sin x cos 3 a; + sin 2 a;.'
31. sin x + sin (a; — f tt) + sin (^ tt — aj) = 0.
32. cos X sin (y — «) + cos y sin (« — ar) 4- cos z sin (x — y)= 0.
w 33. cos (aj + y) sin y — cos (a; + z) sin «
"^ •. - = sin (x + y) c<^s y — sin (x -\- z) cos «.
34. cos (a; H- 3^ + «) + cos {x -\-y — z)->r cos (x — y -\- z)
+ cos (y -H « — x) = 4 cos a; cos y cos z.
35. sin (x + y) cos (x — y) + sin (y + «) cos (3^ — z)
+ sin (« H- aj) cos (« — a?) = sin 2 aj + sin 2 y 4- sin 2 «.
36. sin (aj + y) + cos (x — y)=2 sin (x -f i tt) sin (3^ 4- i tt).
37. sin (x + y)— cos (x — y) = — 2 sin (a; — i tt) sin (3^ — i tt).
98. cos (x + y)GOS (aj — y) = cos^ x — sin^ y.
39. sin (aj + y) sin (x — y)= sin^ a; — sin^ y.
v 40. sin a; + 2 sin 3 aj 4- sin 6 a; = 4 cos^a; sin 3 x.
If A, B, C are the angles of a triangle^ prove that :
41. sin.2yl + sin 2B -{- sin 2 C = 4 sin A ninB sin C.
42. cos 2A -\- cos 2^4- cos 2C = — 1 — 4: cos A cos B cos C.
4.3. sin SA + sin SB + sin 3 C = — 4 cos § ^ cos f B cos | a
44. cos^^ + cos^jB + cos^ C = 1 — 2 cos A cos B cos C.
ijf ^ + -5 + C= 56^*", ;?rove «Aat ;
45. tan^ tanJB + tan^B tan C + tan C tsmA = 1.
46. sin^^ + sin'jB + sin^ C = 1 — 2 sin^ sin^B sin C
47. sin 2^ + sin 2 JB 4- sin 2 C = 4 cos A cos B cos C,
48. Prove that cot~^ 3 4- csc"^ V5 = J tt.
49. Prove that x + tan~^ (cot 2 x) = tan~^ (cot x).
Prove the following statements :
sin 75° 4- sin 15° «
^0. . -go . . go = tan 60°.
sm 75° — sm 15°
51. sin 60° 4- sin 120° = 2 sin 90° cos 30°.
52. cos 20° 4- cos 100° 4- cos 140° = 0.
63. cos 36° + sin 36° = V2 cos 9°.
54. tan 11° 15' 4- 2 tan 22° 30' 4- 4 tan 45° = cot 11° 16'.
166 PLANE TRIGONOMETRY
149. How to solve a Trigonometric Equation. To solve a trigonometrio
equation is to find for the unknown quantity the general value which
satisfies the equation.
Practically it suffices to find the values between 0° and 860^, since we can
then apply our knowledge of the periodicity of the various functions to give us
the other values if we need them.
There is no general method applicable to all cases, but the follow-
ing suggestions will prove of value :
1. If functionB of the sum or difference of two angles are involved^
reduce su^h functions to functions of a single angle.
Thus, instead of leaving sin {x + y) in an equation, substitute for sin {x + jf)
its equal sin x cos y + cosx sin 2/.
Similarly, replace cos2x by cos^x — sin^x, and replace the functions of ^x
by the functions of x,
2. If several functions are involved^ reduce them to the same
function.
This is not always convenient, but it is frequently possible to reduce the
equation so as to involve only sines and cosines, or tangents and cotangents,
after which the solution can be seen.
3. If possible^ employ the msthod of factoring in solving the
final equation.
4. Check the results hy substituting in the given equation.
For example, solve the equation cosx = sin 2x.
By §101, sin2x = 2sinxcosx.
.*. cosx = 2 sin X cosx.
.*. (1 — 2 sin x) cosx = 0.
.'. cosx = 0, or 1 — 2 sin X = 0.
.-. X = 90° or 270°, 30° or 150°, or these values increased by 2nir.
Each of these values satisfies the given equation.
Exercise 77. Trigonometric Equations
Solve the following equations :
1. sina; = 2sin(^7r + a;). 7. sin aj = cos 2 aj.
2. sin 2 a; = 2 cos x. 8. tan x tan 2 aj = 2.
3. cos 2 a; = 2 sin x. 9. sec aj = 4 cso x,
4. sinaj -fcosa; = 1. 10. cos^ -f cos 2^ = 0.
6. sin aj 4- cos 2 a; = 4 sin^a. 11. cot ^ ^ + esc ^ = 2.
6. 4 cos 2a; + 3 cosx = 1. 12. cot aj tan 2 a; = 3.
A.'
IDENTITIES AND EQUATIONS 187
Solve the following equations:
13. sin X + sin 2 x = sin 3 x, 33. sin x sec 2 x = 1.
14. sin 2 05 = 3 sw?x — cos^aj. 34. sin^a; + sin 2 ar = 1.
15. cot = i tan 0. 35. cos a; sin 2 aj esc a; = 1.
16. 2 sin ^ = cos 0, 36. cot x tan 2 a; = sec 2 aj.
17. 2 sin^aj + 6 sin aj = 3. 37. sin 2 a; = cos 4 x,
18. tan a; sec x = V2. 38. sin 2 « cot ;?; — sin* « = -J-.
19. cos ar — cos 2 a; = 1. 39. tan^a; = sin 2 x.
20. cos 3 aj 4- 8 cos'a; = 0. 40. sec 2 a; 4- 1 = 2 cos x.
21. tan X -\- cot x = tan 2 aj. 41. tan 2 a; 4- tan 3 ar = 0.
22. tan a; H- sec a; = a. 42. esc x = cot x -\- VS. •
23. cos 2 a; = a (1 — cos a;). 43. tan a; tan 3 ar = — §.
24. sin"^ i aj = 30°. 44. cos 5x -\- cos Sx + cos aj =
25. tan"^a; + 2 cot"^aj = 136°. 45. sin* a; — cos*ar = k.
26. sec X — cot x = esc x — tan x. 46. sin a; + 2 cos aj = 1.
27. tan 2 x tan x = 1. 47. sin 4 a; — cos 3 a; = sin 2 a*.
28. tan*ar + cot*a; = ^. 48. sin aj + cos x = sec x.
29. sin a; + sin 2 a; = 1 — cos 2 aj. 49. 2 cos a; cos 3 a; + 1 = 0.
30. 4cos 2a; + 6sinaj = 6. 50. cos3x — 2cos2arH-cosa;=:0
31. sin 4 X — sin 2 ar = sin x. 51. sin (x — 30°) = ^ V3 sin x.
32. 2 sin^x + sin* 2x = 2. 52. sin-^a? + 2 cos'^x = f tt.
53. sin-^x + 3 cos-^x = 210°.
^^ 1 — tanx ^
54. :; : = cos 2 X.
1 + tanx
55. tan(i7r 4-x)4- tan(:i^7r — x)= 4.
56. Vl + sin X — Vl — sin X = 2 cos x.
57. sin(46° + x)H-cos(45°-x)=l.
58. (1 — tan x) cos 2 x = a (1 + tan x).
59. sin'x + cos^x = Y^ sin* 2 x.
60. sec (x + 120°) + sec (x - 120°) = 2 cos x.
61. sin*x cos*x — cos*x — sin*x + 1 = 0.
62. sin X + sin 2x4- sin 3 x = 0.
63. sin ^ 4- 2 sin 2 ^ 4- 3 sin 3 ^ = 0.
64. sin 3 X = cos 2 x — 1.
65. sin (x 4- 12°) 4- sin (x - 8°) = sin 20°.
168 PLANE TRIGONOMETRY
Solve ihefollomng equations:
66. tan (60** + X) tan (60* - «) =- 2.
67. tan X -\- tan 2 x = 0.
68. sin (x + 120°) + sin (x + 60**) = J.
69. sin (aj -h 30**) sin (a- - 30**) = J.
70. sin 2^ = cos 3^.
71. 8in*a; + cos*ic = j.
72. sin*a5 — cos*a; = ^.
73. tan (a; + 30**) = 2 cos aj.
74. sec aj = 2 tan x + ^,
75. sin 11 aj sin 4 a; + sin 5 x sin 2 aj = 0.
76. cos X + cos 3 a; + cos 5x -\- cos 7 x = 0.
77. sin {x + 12**) cos (x - 12**) = cos 33** sin 67
78. sin-^a; + sin'^ J x =120^
79. tan-iaj + tan-i2aj = tan-^3V3.
80. tan- \x+l)-^ tan"^ (a; — 1) = tan"* 2 x.
81. (3 — 4 cos^a;) sin 2 aj = 0.
82. cos 2 ^ sec ^ + sec ^ +1= 0.
83. sin X cos 2 a; tan a; cot 2 a; sec a; esc 2 x = 1.
84. tan(^ + 45**) = 8 tan ft
86. tan(tf + 45**) tan = 2.
86. sin X + sin 3 aj = cos x — cos 3 x.
87. sin Jar(cos 2aj — 2)(1— tan*aj)= 0.
88. tan X + tan 2 x = tan 3 x.
89. cot a; — tan x = sin x + cos x.
Prove the following identities:
^/v /H . i. . i. N/ • N secx cscx
90. (1 + cot X + tan x) (sm x — cos x) = — --t r-*
^ '^^ ^ csc*x sec^x
91.2 CSC 2 X cot X = 1 + cot* X.
92. sin a + sin ft + sin (a + ft) = 4 cos J a cos \ ft sin ^(a + ^)
93. tan(45** + x) - tan (45** - x) = 2 tan 2a.
94. cot*x — cos?x = cot*x cos^x.
96. tan^x — sin^x = tan*x sin'x.
96. cot*x + cot'x = csc*x — csc^x.
97. cos^x + sin^x cos^j^ = cos*?/ "•" sin*?/ cos"aj.
IDENTITIES AND EQUATIONS 169
150. Simultaneous Equations. Simultaneous trigonometric equations
are solved by the same principles as simultaneous algebraic equations.
1. Required to solve for x and y the system
X sin a -H y sin ft = m (1)
X cos a + y cos ft = n (2)
From(l), xsinacosa + y sinftcosa = mcosa. (3)
From (2), x sin a cos a + y cos 6 sin a = n siu a. (4)
From (3) and (4), y sin 6 cos a — y cosft sin a = m cos a — n sin a,
or y sin (6 — a) = m cos a — n sin a ;
m cos a — n sin a
whence y =
Similarly, x =
sin (6 — o)
n sin ft — m cos ft
sin (ft — a)
2. Required to solve for x and y the system
sin X + sin y = a (1)
cos X -f «os y — h (2)
By§108, 28in J(x+y)cosJ(x-y) = a, (3)
and 2 cos i (x + y) cos J (x — y) = ft.
Dividing, tan i (X + y) = I . (4)
.-. sin J (x + y) =
Va2 + ft^
Substituting the value of sin J (x + y) in (3),
cosi(x-y)=iVa2 + ft3. (5)
From (4), x + y = 2 tan-i^ . (6)
From (6), x - y = 2 cos-i jVo« + «?. (7)
From (6) and (7), x = tan-i - + cos-i jVa^ + ft^,
and y = tan-i- — cos-i J Vo* + 6^.
3. Required to solve for x and y the system
y sin a; = a (1)
y cos a; = ft (2)
Dividing, tan x = - .
6
.*. x= tan-*-.
Adding the squares of (1) and (2),
y« (sin«« + cos^x) = ©24. fta.
Therefore y^ = a?'\- ft^,
(tn4 y = ± Va« + 6^.
170 PLANE TRIGONOMETRY
4. Required to solve for x and y the system
y8m(x + a)=m (1)
y cos (x + b) = n (2)
From (1), y8inxcosa + 2/co8X8ina = m.
From (2), y cosxcos6 — y sinx sinft = n.
We may now solve for y sin x and 2/ cosx, and then solve for x and y.
5. Required to solve for r, x, and y the system
r cos X sin y = a (1)
r cos 05 cos y = ^ (2)
r sin a; = c (3)
Dividing (1) by (2),
Squaring (1) and (2) and adding,
Taking the square root,
Dividing (3) by (5),
Squaring (8) and adding to (4),
tany = -;.
f)
.-. y = tan-^i-*
r^cos2x = o2 + 62.
(4)
rco8X = Va2 + 62.
(^)
c
tan X = .
Vo2 + 62
/. X = tan-1 .
Voa + 62
ra = a« + 62 + c2.
.-. r = Va* + 62 + c2.
Exercise 78. Simultaneous Equations
Solve the follovring systems for x and y :
1. sin X -\- sin y = sin a 5. sin^a; 4- 3^ = m
cos ar + cos 3^ = 1 + cos a cos^a; + 3^ = n
2. sin^aj 4- sin* 3^ = a 6. sin a; + sin y = 1
cos^a; — cos'y = h sin a; — sin 3^ = 1
3. sin aj — sin y = 0.7038 7. cos aj + cos y = a,
cos aj — cos 3^ = — 0.7245 cos 2 aj + cos 2y = b
4. a; sin 21® -f 3^ cos 44® = 179.70 8. sin aj -f sin 3^ = 2 m sin a
X cos 21® 4- 2^ sin 44® = 232.30 cos x 4- cos y = 2n cos a
9. Find two angles, x and y, knowing that the sum of their sines
is a and the sum of their cosines is b.
Solve the following systems for r and x :
10. r sin aj = 92.344 11. r sin (x ~ 19® 18') = 69.4034
r cos a? = 205.309 r cos {x - 30® 64') = 147.9347
IDENTITIES AND EQUATIONS ITl
151. Additional Symbols and Functions. It is the custom in advanced
trigonometry and in higher mathematics to represent angles by the
Greek letters, and this custom will be followed in the rest of this
work where it seems desirable.
The Greek letters most commonly used for this purpose are as follows:
a, alpha B^ theta
/?, beta X, lambda
7, gamma Aa, mu
d, delta 0, phi
e, epsilon w, omega
Besides the six trigonometric functions already studied, there are,
as mentioned on page 4, two others that were formerly used and
that are still occasionally found in books on trigonometry. These
two functions are as follows :
versed sine of a: = 1 — cos or, written versin a ;
coversed sine of a: = 1 — sin a, written coversin a.
Exercise 79. Simultaneous Equations
1. Solve for ^ and x : 4. Solve for B and ^ :
versing = x sin B + cos ^ = a
1 — sin <^ = 0.5 sin^ + cos B = h
2. Solve for B and x : 6. Solve for 6 and <^ :
1 — sin B = x a sin*^ — b sin*^ = a
1 + sin B = a a cos*tf — b cos*^ = b
3. Solve for X and /* : 6. Solve for B :
sin \ = V^ sin /* sin^^ 4- 2 cos ^ = 2
tan \ = VS tan /^t cos ^ — cos^tf =
152. Eliminant. The equation resulting from the elimination of
a certain letter, or of certain letters, between two or more given
equations is called the eliminant of the given equations with respect
to the letter or letters.
For example, if c/x = 6 and a'x = b\ it follows by divisiou that a : a' = b : b%
or that ai/ = a\ and this equality, in which x does not appear, is the eliminant
of the given equations with respect to x.
There is no definite rule for discovering the eliminant in trigo-
nometric equations. The study of a few examples and the recalling
of identities already considered will assist in the solutions of the
problems that arisa
172 PLANE TRIGONOMETRY
153. niustrative Examples. The following examples will serve to
illustrate the method of finding the eliminant :
1. Find the eliminant, with respect to ^, of
sin <l> = a
COS <!> = b
Since sin^0 + co8*0 = 1, we have a^ + 6^ = 1, the eliminant.
2. Find the eliminant, with respect to X, of
sec X = m
tanX = n
Since sec^X — tan^X = 1, we have m^ — n^ = 1, the eliminant.
3. Find the eliminant, with respect to fi, of
m sin fi -\- cos /a = 1
n sin fi — cos /^t = 1
Writing the equations 7n sin /* = 1 — cos /ix, n sin /ix = 1 + cos /ix, and multiplying,
we have . „ « . o
mn siu^fji = 1 — cos^/ii = Bin^/i,
Hence mn = 1 is the eliminant.
Exercise 80. Elimination
Find the eliminant with respect to a, ^, X, fi, or <!> of the follmv
ing equations :
1. sin <^ + 1 = a 7. sin <^ + sin 2 <^ = m
cos <^ — 1 = i cos ^ + cos 2 <^ = n
2. tan X — a = 8. a + sin ^ = esc ^
cot X — Z» = h -\- cos ^ = sec ^
3. sin a -f vi = n 9. tan a + ^n a = m
cos a -\- p = q tan a: — sin a = n
4. a + sec <^ = /> 10. ^ sin^ f^ — P cos* /^t = r
j9 -5- cot 4> =^(1 p^ cos* t^—p' sin* fA = r'
5. c sin 2 <^ 4- cos 2 <^ = 1 11. sin 2 ^ + tan 2 ^ = A:
^ sin 2 <^ — cos 2 <^ = 1 sin 2 ^ — tan 2 <^ = Z
6. X =^ r ($ — sin 6) 12. jo = a cos ^ cos <^
^ = r (1 — cos ^) 9 = ^ cos ^ sin ^
e = versine- 1 y/r. r = c sin tf
CHAPTER XII
APPLICATIONS OF TRIGONOMETRY TO ALGEBRA
154. Extent of Applications. Trigonometry has numerous applica-
tions to algebra, particularly in the approximate solutions of equations
and in the interpretation of imaginary roots.
These applications, however, are not essential to the study of spherical trigo-
nometry, and hence this chapter may be omitted without interfering with the
Btudent^s progress.
For example, if we had no better method of sdving quadratic equa-
tions we could proceed by trigonometry, and in some cases it is even
now advantageous to do so. Consider the equation a^ -\- px — q = 0,
Here the roots are
Xi = — iP + h y?^ H- ^ S', »2 = "" i-P r i ^P^+^^
2-y/g r
If we let = tan ^, or ^ = 2 'Sq cot ^, we have
ajj = — V^ cot <^ + -\fq Vcot^<^ + 1
= — V^ cot «^ 4- -v-~ = V^ ( -:— r — cot «^ )
'■ ^ sin^ ^\sm^ /
/- 1 — cos 6 /- .
Similarly,
^2 = — V^ cot \ <l>.
For example, if x« + 1.1102x - 8.86©4 = we have
. „^ 2V8.8694 .
tan 4> = — •;
^ 1.1102 '
whence log tan </> = 0.51876,
and = 78*» d' 2.6''.
Therefore log tan J = 9.87041 - 10.
and log Vq = log V8.8694 = 0.26818.
Hence logx^ = 0.18864,
and x^ = 1.860.
Similarly, «, = - 2.470.
178
174 PLANE TRIGONOMETRY
155. De Moivre't Theorem. Expressions of the form
cos X + i sin x,
where i = V— 1, play an important part in modern analysis.
Since (cos x + isinx) (cos y -j- i sin y)
= cos aj cosy — sin a sin y + i (cob « sin y + siu a; cos y)
= cos(aj 4- y) -h 1 8in(aj + y),
we have (cosaj + tsinaj)*= cos2aj + tsin2«;
and again, (cos x + i sin «)•= (cos a + * sin xy (cos a + i sin ar)
= (cos2a; + t sin 2 a) (cos a + isinx)
= cos 3 a + i sin 3 aj.
Similarly, (cos x -\- I sin aj)"= cos nx + / sin wa!.
To find the nth power of eos x-\- i sin x^ n being a positive integer^
we have only to multiply the angle x by n in the expression.
This is known as De Moivre's Theorem, from the discoverer (c. 1725).
156. De Moivre's Theorem extended. Again, if n is a positive integer
as before, . . ,
/ x ^ , . xy , . .
( cos - + i sm - 1 = cos aj + I sin a.
\ n nj
.'. (cos a; + i sin aj)" = cos - + * sin - •
^ ^ n n
However, x may be increased by any integral multiple of 2 tt with-
out changing the value of cos a; -h t sin x. Therefore the following n
expressions are the nth roots of cos aj + t sin x :
X . , . X x-\-2ir , . . x + 27r
cos- + tsin-> cos f-ism >
n n n n
X + A:*ir . . . aj 4- 4:'7r
cos h *sin > • • •>
n n
ajH-(7i--l)27r , . . aj + (w-l)27r
cos ^^ ^ }- 1 sin ^ ^ •
n n
Hence, if w is a positive integer,
(cos X -^ I sin a;)*
= cos |-*sm (Aj = 0,1,2, . . .,?i— 1).
n n
Similarly, it may be shown that
— 711/ fft
(cos x + i sin a;)** = cos— (aj + 2 kir) -f i sin— (aj + 2 kir\
^ ^ n n
(A; = 0, 1, 2, . • ., n — 1, ?» and n being integers, positive or negative.)
APPLICATIONS TO ALGEBRA 176
157. The Roots of Unity. If we have the binomial equation
a^-l=0,
we see that a^ = 1,
and X =s the nth root of 1,
of which the simplest positive root is Vl or 1. Since the equation
is of the nth degree, there are n roots. In other words, 1 has n nth
roots. These are easily found by De Moivre's Theorem.
There are no other roots than those in § 156. For, letting A; = n, n + 1, and so
on, we have
cos i^ — ^ + t sin ^^ — -
n n
= cos I - + 2 IT ) + i sin ( - + 2 IT I = cos - + i sin - »
\n / \n J n n
, « + (n+l)2w , . . x + (n+l)2w
and cos ^ — + t sin ^^ —
n n
= cos I — ■ + 2wj+ tsini h 2wj
aj + 2ir . . . «+ 2ir
= cos + i sin ,
n n
and 80 on, all of which we found when ik = 0, 1, 2, • • • , n — 1.
For example, required to find the three cube roots of 1.
If COS0 + t sin = 1, the given number,
then = 0, 2 IT, 4 IT, • • • .
Also (cos0 + i sin0)* = 1» = the three cube roots of 1.
But (cos* + i 8in0)* = co8^^^I±^ + 1 8ln ^^±1,
3 8
where ik = 0, 1, or 2, and = 0, 2 w, 4 ir, • • • .
Therefore 1^ = cos 2ir + isin2ir = l,
or li = cosjir + tsin§ir= cosl2(y» + tsinl20»
= - J + jVs . i =- 0.6 + 0.8660 i,
or l* = cosJw + <8in Jw = co8 240°+ tsin240®
= - J- i\/3.i=-0.5-0.8660i.
Th^ three cube roots of 1 are therefore
Thete roots could, of course, be obtained algebraically, tiius :
a^ - 1 = 0,
whencA (x — !)(«« + x + 1) - 0;
and either x — 1 = 0, whence x = 1,
or x« + X + 1 = 0, whence x = — J ± \ V— 8.
Most equations like x" — a ^ ccumot, however, be solved algebraically.
176 PLANE TRIGONOMETRY
Required to find the seven 7th roots of —1; that is, to solve the
equation x' = — 1, or a;' + 1 = 0.
If CO80 + { 8iii0 = ~ 1, the given number,
then = IT, 3 IT, 6 IT, • • • .
Also (COB0 + ifidn0)t = coB— i — ^— -^ + t8in— i — i_L_z:,
7 7
where fc = 0, 1, • • •, 6, and = w, 8 w, • • • .
That is, in this case
(co80 + t sin0)7 = cos^ — h t sin ^ i— .
Hence the seven 7th roots of 1 are
cos- + i sin- = cos 25P 42' 51 f" + i sin 25<^ 42' 51^'',
cos— + i sin — = cos 77° 8' 34f '' + t sin 77° 8' 84^',
, 6w , . . 6v ... 9^ , . . 9ir
and cos H tsm — » cosw + tsinw, cos Htsin — 9
7 7 7 7
IItt . . . llir ISw , . , 18ir
cos +t8in » cos l-tsin
7 7 7 7
All these values may be found from the tables. For example,
cos 25° 42' 51f " + i sin 25° 42' 51 f" = 0.9010 + 0.4839 V^.
Exercise 81. Roots of Unity
1. Knd by De Moivre's Theorem the two square roots of 1.
2. Find by De Moivre's Theorem the four 4th roots of 1.
3. Find three of the nine 9th roots of 1.
4. Find the five 6th roots of 1.
6. Find the six 6th roots of -f 1 and of — 1.
6. Find the four 4th roots of — 1.
7. Show that the sum of the three cube roots of 1 is zero.
8. Show that the sum of the five 6th roots of 1 is zero.
9. From Exs. 7 and 8 infer the law as to the sum of the nth
roots of 1 and prove this law.
10. From Ex. 9 infer the law as ix) the sum of the nth roots of Jc
and prove this law.
11. Show that any power of any one of the three cube .roots of 1
is one of these three roots.
12. Investigate the law implied in the statement of Ex. 11 for the
four 4th roots and the five 6th roots of 1.
APPLICATIONS TO ALGEBRA 177
158. Roots of Numbers. We have seen that the three cube roots
of 1 are
X
X
J = cos 120* -h i sin 120* = - j + j V^,
,2 = cos 240* + i sin 240* = - J - J V- 3,
and x^ = cos 360* + i sin 360* = cos 0* + i sin 0* = 1.
Furthermore, x^ is the square of aj^, because
(cos 120* + i sin 120*)^ = cos (2 . 120*) + i sin (2 . 120*),
by De Moivre's Theorem. We may therefbre represent the three
cube roots by o, o^, and either u? or 1.
In the same way we may represent all n of the nth. roots of 1 by
o), 0)^, (I)*, • • • , (!)*• or 1.
If we have to extract the three cube roots of 8 we can see at once
that they are ^ ^ j o a
•^ 2, 2 <i), and 2 « ,
because 2» = 8, (2a))« = 2»u)»= 8 . 1 = 8,
and (2 a)^» = 2» o)« = 2» {u?f = 2» 1^ = 8.
In general, to find the three cube roots of any number we may
take the arithmetical cube root for one of them and multiply this
by <i) for the second and by a? for the third.
The same is true for any root. For example, if w, w^, «*, w*, and w^ or 1 are
the five 6th roots of 1, the five 5th roots of 32 are 2 w, 2 a;^^ 2 w^, 2 w*, and 2 w^ or 2.
Exercise 82. Roots of Numbers
1. Find the three cube roots of 126.
2. Find the four 4th roots of — 81 and verify the results.
3. Find three of the 6th roots of 729 and verify the results.
4. Find three of the 10th roots of 1024 and verify the results.
6. Find three of the 100th roots of 1.
6. Show that, if 2 w is one of the complex 7th roots of 128, two of
the other roots are 2 a? and 2 w*.
7. Show that either of the two complex cube roots of 1 is at the
same time the square and the square root of the other.
8. Show that a result similar to the one stated in Ex. 7 can be
found with respect to the four 4th roots of 1.
9. Show that the sum of all the 71th roots of 1 is zero.
10. Show that the sum of the products of all the nth. roots of 1,
taken two by two, is zero.
178 PLANE TRIGONOMETRY
199. Properties of Los^arithniB. The properties of logarithms have
already been studied in Chapter III. These properties hold true
whatever base is taken. They are as follows :
1. The logarithm of 1 is 0,
2. The logarithm of the base itself is 1,
3. The logarithm, of the reciprocal of a positive number is the
negative of the logarithm of the number,
Ai, The logarithm, of the product of two or m/rre positive numbers is
found by adding the logarithm's of the several factors,
6. The logarithm, of the quotievt of two positive numbers is found
by subtra^ing the logarithm, of the divisor Jrom the logarithm of the
dividend,
6. The logarithm, of a power of a positive number is found by
m,ultiplying the logarithm, of the number by the exponent of the power,
7. The logarithm of the real positive value of a root of a positive
number is found by dividing the logarithm of the number by the index
of the root.
160. Two Important Systems. Although the number of different
systems of logarithms is unlimited, there are but two systems which
are in common use. These are
1. The common system, also called the Briggs, denary, or decimal
system, of which the base is 10.
2. The natural system, of which the base is the fixed value which
the sum of the series
111 1
1 + 7 + 7^ + 7-4-7; +
1 ■ 1.2 ■ 1.2-3 1.2.3.4 •
approaches as the number of terms is indefinitely increased. This
base, correct to seven places of decimals, is 2.7182818, and is denoted
by the letter e.
Instead of writing 1*2, 1*2*3, 1*2*3*4, and so on, we may write either
2 !, 3 !, 4 !, and so on, or [2, [3, [4, and so on. The expression 2 1 is used on the
continent of Europe, [2 being formerly used in America and England. At pres-
ent the expression 2 1 is coming to be preferred to [2 in these two countries.
The common system of logarithms is used in actual calculation;
the natural system is used in higher mathematics.
The natural logarithms are also known as Naperian logarithms, in
honor of the inventor of logarithms, John Napier (1614), although
these are not the ones used by him. They are also known as hypei
bolic logarithms.
APPLICATIONS TO ALGEBRA
179
161. Exponential Series. By the binomial theorem we may expand
(l + i)"'and have
V^ + ;;j=^+«' + — ^^r- + T, + •••• (1)
2!
3!
This is trae for all values of x and n, provided n > 1. If n is not greater
than 1 the series is not eonvergent ; that is, the sum approaches no definite limit.
The further discussion of convergency belongs to the domain of algebra.
When a; = 1 we have
(^-i)-=
1 + 1 +
n \ n/\ nl
2! "^ 3!
Hence, from (1) and (2),
nx
1 + 1 +
1_1 (l-l)(l-?)
21 "^ 3l
+
4-3 4-i)(«-D
= 1 + a- H TT h
+
2! 3!
If we take n infinitely large, (3) becomes
that is.
In particular, if aj = 1 we have
^ = i + ,+^+|! +
We therefore see that we can compute the value of e
by simply adding 1, 1, ^ of 1, -J- of ^ of 1, and so on,
indefinitely, and that to compute the value to only a few
decimal places is a very simple matter. We have merely
to proceed as here shown.
Here we take 1, 1, ^ of 1, -J- of ^ of 1, J of J of ^ of 1,
and so on, and add them. The result given is correct
to five decimal places. The result to ten decimal places
is 2.7182818284.
(^)
(3)
(4)
2
3
4
6
6
7
8
9
1.000000
1.000000
0.600000
0.166667
0.041667
0.008333
0.001388
0.000198
0.000026
0.000003
e = 2.71828.
180 PLANE TRIGONOMETRY
162. Expansion of sin jr, cos jt, and tan jr. Denote one radian by 1,
and let
cos 1 H- i sm 1 = k.
Then cos a; + i sin a; = (cos 1 -f- * sin 1)* = A;%
and, putting — x for a;,
cos {—x)+i sin(— x) = cos a; — t sin a; = k"^.
That is, cos x + i sin x — kf^y
and cos x ^ i sin a; = A;"'*.
By taking the sum and difference of these two equations, and
dividing the sum by 2 and the difference by 2t, we have
cos a; = h(^"'"^"*)>
and sin aj = — (Af — k"").
But kf = (6^^*)* = e**°«^*, and A;"* = e-*i<«*.
and e-^«>.* = l-a.logA:4-^^'-^^^|f^+....
.-. C03a; = -(fe' + ^-*)=l+ ^n, ' + 4! "* '
and sin a; = T -^ a; log * H ^ ° - ^ H ^ ° ' + • • • i- •
Dividing the last equation by «, we have
sina; If, ,. . ar'qogA:)' . a;«aogA;)» , "I
^- = IV°^^+ 3! + 6! +'"r
But remembering that x represents radians, it is evident that the
smaller x is, the nearer sin x comes to equaling x ; that is, the more
nearly the sine equals the arc.
Therefore the smaller x becomes, the nearer comes to 1, and
* 1
the nearer the second member of the equation comes to t log k.
We therefore say that, as x approaches the limit 0, the limits of
these two members are equal, and
1 = -rlogA;;
whence log A: = t,
and Aj = e'.
APPLICATIONS TO ALGEBRA 181
Therefore, we have
coax =^ (e^ + 6-^0= 1 - 2; + fj - It + • • •'
1 ,^. ^, X* , a;* a;'
From the last two series we obtain, by division,
sinaj . aj« . 2a:* . VI x^
cos a; 3 15 315
By the aid of these series, which rapidly converge, the trigonometric func-
tions of any angle are readily calculated.
In the computation it must be remembered that x is the circular measure of
the given angle.
Thus to compute cosl, that is, the cosine of 1 radian or cos 57.29678°, or
approximately cos 57.8°, we have
COSl = l— — - + - •T-+ r
21 41 61 81
^ 1 _ 0.5 + 0.04167 - 0.00130 + 0.00002
= 0.5403 = cos 57° 18'.
163. Euler'8 Formula. An important formula discovered in the
eighteenth century by the Swiss mathematician Euler will now be
considered. We have, as in § 162,
/i«8 /y.5 ,^7
. Uy JL> JL/
a? X* x^
and cos aj = 1 — 777 + 7-. — ^, 4- • • •.
2! 4! 6!
By multiplying by i in the formula for sin x, we have
. ia^ , ia^ ix\
tsina. = ia.-- + ;5y-^ + ....
Adding,
COS 05 + 1 sm a; = l + ta; — g-f— o7 + T74--»^— ••••
By substituting ix for x in the formula for e*, we see that
^ = 1 + ^^ + - + — + — + — +...
... a? ix^ , x* , ta^
In other words,
e*^ = cos X + i sin x.
182 PLANE TRIGONOMETRY
164. Deductions from Euler's Formula. Euler's formula is one of
the most important formulas in all mathematics. From it several
important deductions will now be made.
Since f^ = cos a; + i sin a;, in which x may have any values, we
may let a; = tt. We then have
6»' = cos 7r4-isin7r = — 1 + 0,
or e'* = — 1.
In this formula we have combined four of the most in teres ting numbers of
mathematics, e(the natural base), {(the imaginary unit, V— l), w(the ratio of
the circumference to the diameter), and — 1 (the negative unit).
Furthermore, we see that a real number (e) may be affected by an imaginary
exponent (iw) and yet have the power real (— 1).
Taking the square root of each side of the equation 6*' = — 1,
we have
e
tir
Taking the logarithm of each side of the equation e'' = — 1,
^« Ji*^® t'7r = log(-l).
Hence we see that — 1 has a logarithm, but that it is an imaginary number
and is, therefore, not suitable for purposes of calculation.
Since cos <^ H- t sin <^ = cos (2 A;7r + <^) -h i sin (2 hir + <^), we see
that e*', which is equal to cos ^ + isin<^, may be written e^^*^+*^,
or we may write
g«< ^ g(2 *r + «)f ^ cog <^ -f I sin «^ = cos (2 A;7r + «^) -f- i sin (2 kir + «^)
Hence (2 A;7r + <^) * = log [cos (2 A;7r 4- ^) -f- * sin (2 kit: + <^)].
If«^ = 0, 2A;7ri = logl.
If ik = 0, this reduces to = log 1.
If fc = 1 we have 2 tt* = log 1 ; if A; = 2, we have 4 tti = log 1, and so on. In
other words, log 1 is multiple-valued, but only one of these values is real.
If «^ = 7r, (2 kir + 7r)i = (2 y5: + l)'7ri = log(- 1).
Hence the logarithms of negative numbers are always imaginary ; for if A; =
we have iri = log(— 1) ; if ik = 1 we have 3 tti = log(— 1) ; and so on.
If we wish to consider the logarithm of some number ivr, we have
^^hni ^ ;vr(cog 2 A^TT -f- i sin 2 kir).
Hence log iV + 2 kiri = log iV + log (cos 2 Ajtt + t sin 2 kif)
= log iV -f- log 1 = log N,
That is, log N = log ^ + 2 kid. Hence the logarithm of a niunber is the
logarithm given by the tables, + 2A;iri. If ik = we have the usual logarithm,
but for other values of ik we have imaginaries.
APPLICATIONS f ALGfifi&A 1 8^
Exercise 83. Properties of Logarithms
Prove the following propertie% of logarithms a% given in § 159^
using h as the base:
1. Properties 1 and 2. 3. Property 4. 6. Property 6.
2. Property 3. 4. Property 5. 6. Property 7.
Find the value of each of the following :
7. 5! 8. 7! 9. 6! 10. 8! 11. 10!
Simplify the following :
,n 10^ ,o 1^^ ,. 7! _ 16! ,^ 20!
^^- -37- ''• -8! • ^"- 5!- ^^- 14!- ^^^ 17!
/ 1 1 \^
17. Find to five decimal places the value of ( 1+1+ 97 + oT + * • *) •
18. Find to five decimal places the value of f 2 + ^ + ^ + . . . 1 .
By the use of the series for cos x find the following :
19. cos^. 20. cos^. 21. cos 2. 22. cosO.
By the use of the series for sin x find the following :
23. sinl. 24. sin ^. 25. sin 2. 26. sinO.
By the use of the series for tan x find the follovring :
27. tanO. 28. tanl. 29. tan^. 30. tan 2.
Prove the following statements :
81. ««''' = 1. 32. e'"^=i^. 33. e^ = V^, 34. e' = V^.
GHven log ^2 = 0.6931, find two logarithms (to the base e') of:
35. 2. 36. 4. 37. V2. 38. - 2.
Given log ^5 = 1.609, find three logarithms (to the base e) of:
39. 5. 40. 25. 41. 125. 42. - 5.
Given logj.0 = 2.302585, find two logarithms (to the base e) of:
43. 100. 44. -10. 45. 1000. 46. VlO.
47. From the series of § 162 show that sin(— <^) = — sin ^.
48. Prove that the ratio of the circumference of a circle to the
diameter equals — 2 log (t*) = — 2 i log i.
184 PLANE TRIGONOMETRY
Bxercise 84. Review Problems
1. The angle of elevation of the top of a vertical cliff at a point
575 ft. from the foot is 32^ 15'. Find the height of the cliff.
2. An aeroplane is above a straight road on which are two observers
1640 ft. apart. At a given signal the observers take the angles of ele-
vation of the aeroplane, finding them to be 58® and 63® respectively.
Find the height of the aeroplane and its distance from each observer.
3. Prove that ( Vcsc a; + cot aj — Vcscaj — cot a?)* = 2 (esc x — 1).
4. Given sin a; = 2 m/(m^ + 1) and sin y = 2 n/(ri? + 1), find the
value of tan (x + y),
6. Find the least value of cos^a; + sec^aj.
6. Prove that 1 — sin^a;/sin^y = C08^a;(l — tan* a/tan* y).
7. Prove this formula, due to Euler : tan*^ J + tan~^J = J tt.
8. Prove that cot J a; — cot aj = esc aj.
9. Prove that (sinaj-ftcosa;)** = cosn(^7r — aj)4-tsinw(j7r — a;).
10. Show that log i= ^7ri and that log (— i) = — ^ wi,
11. Through the excenters of a triangle ABC lines are drawn
parallel to the three sides, thus forming another triangle A'B^C.
Prove that the perimeter of AA'B'C* is 4r cot J ^ cot J 5 cot ^C,
where r is the radius of the circumcircle.
12. Given two sides and the included angle of a triangle, find
the perpendicular drawn to the third side from the opposite vertex.
13. To find the height of a mountain a north-and-south base line is
taken 1000 yd. long. From one end of this line the summit bears
N. 80° E., and has an angle of elevation of 13° 14' ; from the other
end it bears N. 43° 30' E. Find the height of the mountain.
14. The angle of elevation of a wireless telegraph tower is observed
from a point on the horizontal plain on which it stands. At a point a
feet nearer, the angle of elevation is the complement of the former.
At a point b feet nearer still, the angle of elevation is double the first.
Show that the height of the tower is [(a + by — ^ a^] .
Prove the following formulas :
15. 2co8*aj = cos 2 ar -f 1. 17. 8 cos*ar = cos 4 aj + 4 cos 2 a; + 3.
16. 2 sin*a; = — cos 2 a; + 1. 18. 4cos*a; = cos 3ar + 3cos x.
19. 4 sin* a; = — sin 3 a; + 3 sin a;.
20. 8 sin*a; = cos 4 a; — 4 cos 2 a; 4- 3,
FORMULAS
185
THE MOST IMPORTANT FORMULAS OF PLANE
TRIGONOMETRY
Right Triangles (§§ 15-21)
1. y = rain <^. r
4t, X = y cot <^.
2, x = r cos 6. r-
b. r =x sec <^.
3. y = x tan <f>, ^
6, r = y CSC <^.
Relations of Functions (§§ 13, 14, 89)
/
7. sin <^ =
csc<^
./
12. cot <f> =
tan^
1
8. cos <h = rt/ 13. sec 6 =
sec<^ cos<^
9. tan^ = — : — f 14. csc^ = -: — -
cot <f) \ sin ^ ^
10. sin<^ csc<^=l. 16. tan <^ cot <^ = 1. /20. 1 -htan^<^ = sec^<^. >/
11. cos<^ sec<^=l. 16. sin^<^ + cos^<^=l./ 21. 1 + cot^<^ = csc^<^
,„ . . cos«^
17. sm <b = — — ^ •
cot<^
* « i. . sin 6
18. tan<t = -^.
cos<^
. . cos<f>
19. cot <h = -. — - •
sm <f}
J
;y
Functions OYx±y {%% 90-100)
J
26
22. sin (aj -f y) = sin x cos y + cos a; sin y.
23. sin (cc — y) = sin a; cos y — cos x sin y. ^
24. cos (a; + y) = cos a; cos y — sin a; sin y, \i
25. cos (aj — y) = cos aj cos y + sin x sin y. i^
, . , ^ tanar + tany,/ ^^ ./ , x cotacoty — 1
. tan(a;4-y) = q — ^ ^\t 28. cot(ar + v) = — r ^ —
cot y 4- cot x
«« i. / \ tana; — tany - / ^ .. . cotajcoty + l
27. tan(aj — y) = 3-7-;: --^\J29. cot(a; — y) = — : ^-f —
^ ^^ 1 + tanaj tany ^ ^ ^^ cot y — cot aj
Functions of Twice an Angle (§ 101)
30. sin 2 <^ = 2 sin ^ cos ^. 32. cos 2 ^ = cos^^ — sin^<^.
31. tan2i^
1-tanV
2cpt^
Functions of Half an Angle (§ 102)
• , . Il — cos 6
34. sin J ^ = ± >j \ 2' T' "
«« i. , . Il — cosA
3 6: ■ tan J 1^ = ± -^ t^ -^
4- cos<^
35. cos ^ ^
-^
4- cos <^
37. cot^<^
--^
cos<^
cos<^
186 PLANE TRIGONOMETRY
Functions involving Half Angles (§ 101)
X QC
38. sin a; =: 2 sin - cos ^ •
40. COS X = C08*;r — S
2 Un 1
39. tana; =
cot«|-l
i-*-1
2cot|
'2'
Sums and Differences of Functions (§ 103)
42: sin.4 -f sin 5 = 2 sin \{A'\-B) cos J (^ — ^)-
43. sin^ — sin5 = 2 cos \{A + 5) sin ^(.4 — B),
44. C0S.4 + cos5 = 2 cos \{A + 5)cos ^(^ — 5).
45. C0S.4 — cos5 = — 2sin J(^4-5)sini(.4--5).
sin ^4- sin^ _ tan^(^ + ^)
sinX — sin5 tan J (^4 — 5)
Laws of Sines, Cosines, and Tangents (§§ 106, 111, 112)
^.v T u • ^ sin^
47. Law of sines, — = -: — >
h sin5
a ft c
sini4 sin 5 sinC
48. Law of cosines, a* = ft* + c* — 2 6c coSi4.
49. Law of tangents, — -^ = - — f ; ^ "^ J^ > if a > ft ;
^ ' a + ft tani(.4 + 5) '
ft — a tan 1(5— /I) .„ ^-
r"; — = I — ttt; — rr > if a < ft.
ft 4- a tan^(B-f^)
Formulas in Terms op Sides (§§ 115, 116)
aH-M:_£ (5-a)(^-ft)(^-c)
60. ^ = 5. 53. ^-^^ ^^^ ^ ^ = r
.>js^
.,,i„j..^5^ME5. „.^„j,,JS^3.
62. cos-^i4 = xl ^^ ^^ ' 65. tan-J'^ =
s — flt
Areas of Triangles (§ 118)
66. Area of triangle ABC = J ac sin 5 = J r (a + ft + ^J) =» r«
r-, Z-) 77-, r ohc c? sin B sin C
WS (8 — a) (S — ft) U — C) 5B: -r— - = rr— : — .^ , ^. ^
^ ^^ ^^ ' 4iJ 28in(B + (7)
INDEX
AbsciBW 78
Addition fonnulEis .... 07, 101
Algebra, applications to ... . 173
Ambiguous easi' 112
Angle, functions of an .... 3, 4
of depraasion 18
of elevation 18
negative 77, 9a
positive 77
Angles, diflerenoe of 100
diSeriDg by HO" S2
greater tlian aao" 87
liaviDg tbe same f unc.lloiis 164, 166
how measured 2
8umof 97
AntUogarithm 48
Areas 66,128,141,142
Base 40
BrIggH 39
Changes in the functions ... 26
Characteristic 48
negative 44, 61
Circle 144
Circular measure 151
Cologaritbm 64
Compass 146
Complementary angles .... 7
Conversion table SO
Coordinates 78
Cosecant 4, 22
Cosine 4, 16, 116, 180
Codnes, law of . 116
Cotangent 4, 20
ConrM 146
Covereed rine 1 71
Decimal table 30
De Molvre'a Theorem . . , , 174
DepoTtura 146
Depression, angle of 1
Difference of two angles , . . . U
of two functions IC
Divldon by logarithms . . . 42, C
Elevation, angle of 1
Ellmlnant It
Equation .... 163, 166, 169, 11
Elder 18
Elder's Formula 18
Expansion in series 18
Exponential equation £
series 17
Formulas, important 18
Fractional exponent £
Functions as lines S
changes in the S
graphs of II
inveree II
line values of E
logarithms of 6
of a negative angle .... C
of an angle 3, 1
of any angle 8
of half an angle ... 104, 1£
of small angles IG
oftheditferenceof two angles 10
of the sum of two angles . , i
of 30", 46°, flO"
of twice an angle 10
reciprocal 1
relations of 13, 1
variations in 8
Graphs of functions IC
Half angles 104, IS;
Identity It
Interpolation SI, 32, 4
188
INDEX
PAGE
Inverse functions 156
Isosceles triangle 70
Latitude 145
Laws of the characteristic ... 44
of cosines 116
of sines 108
of tangents 118
Logarithm 40
Logarithms 39
of functions 60
properties of 178
systems of 178
use of tables of .... 46, 61
Mantissa 43
Middle latitude sailing .... 149
Multiplication by logarithms . 42, 50
Napier 39
Negative angle 77, 92
characteristic 44, 51
lines 77
Oblique angles 77
triangle 107
Ordinate 78
Origin 78
Parallel sailing 148
Plane sailing 145
trigonometry 1
Polygon, regular 72
Positive angle 77
Power, logarithm of .... 43, 56
Practical use of the cosecant . . 22
of the cosine 16
of the cotangent 20
of the secant 21
of the sine 14
of the tangent ...... 18
PAGE
Quadrant 78
Radian 151
Reciprocal functions 12
Reduction of functions to first
quadrant 90
Regular polygon 72
Relations of the functions . 12, 13, 94
Right triangle 34, 63, 133
Root, logarithm of 48, 57
Roots of numbers 177
of unity 176
Secant 4, *21
Series, exponential 179
Sexagesimal table 28
Signs of functions 86
Simultaneous equations .... 169
Sine 4, 14, 108, 180
Sines, law of 108
Sum of two angles 97
of two functions 105
Surveyor's measures 142
Symbols 3, 4, 40, 171
Tables explained 10, 28, 30, 46. 48, 61
Tangent 4, 18
Tangents, law of 118
Traverse sailing 150'
Trigonometric equation .... 163
identity 163
Trigonometry, nature of ... . 1
plane 1
Unity, roots of 175
Variations in the functions . , 86
Versed sine 171
ANSWERS
ANSWERS
PLANE TRIGONOMETRY
1. COfiBrs
3. tan id.
7. sin A =
8. sin -4 =
9. sin -4 =
10. sin^ =
11. sin id =
12. 8in^ =
csc-4 =
13. a3 + &^
14. sin^ =
cotw4
2n
secw4 =
16. sin^ =
sec^ =
16. sin^ =
sec -4. =
17. sin^ =
secul =
19. sin^
20. sin^
C8C^
21. sin^
C8C^
22. sinB
cscB
28. sinB
oecB
24. EdnB
C8CB
625.5
11
Hi
271 ^ n' — 1 , . 2n
— : 0,0% A = ; tan A =. —
n2 4- 1 n2 + 1 n^-1
n2 + 1 ^ n2 + 1
-— ; c8cA = .
n2-l* 2n
2n ^ w2-l ^ . 2n . ^ n« - 1
— ; cos^ = ; tan^ = — ; cot-d. = — ;
n2 + l' n^ + l n2-l* 2n
n2 + 1 . n2 + 1
; csc^ = .
n^-l 2n
2 mn . m^ —n^ ^ . 2 mn ^ . m^ -^n*
—- -; co8^ = — -; tan^ = — -; cot-A = ;
m* + 71^ m^ + n^ m^ — v? 2 mn
m^-^n^ . m2 + n^
— ; csc^ = .
m^ -^n^ 2 mn
2 mn . m^ -- n^ ^ . 2 m>n ^ . m^ — n^
— ; C08-4 = — ; tan^ = — ; cot A = ;
m^ + n^ 7n* + 71^ tti^ — n' 2 ttiti
7^2 + 712 . 7^2 + 7l2
: — ; CSC-A = .
7n2 -- 7i2 2 mn
: ^ V2 = cos^ ; tan-4 = 1 = cot A ; sec-4 = Vi = c«c-4._
: f VS; cosul = J VS; tan^ = 2 ; cot^ = J ; 8ec^ = V6;
; J ; COS A = J Vs ; tan^ = § Vs ; cotA = ^ VB ; secvl = J Vs ;
■■•?»%
cosB=-^; tanB = Jjy 1 cotB =
C06B =
T%;8ecB = V4^;
co8B = |ft; tanB=^\; cot 5 = ^3^; secB=:f|f;
1
2 PLANE TRIGONOMETRY
P + g P-^Q P-Q 2pq
26. 8in^ = ^^"*'^^ = co8g; cot^ = , ^ =tan^;
P + ^ Vp'^ + ^a
cos^ = — = sinB ; sec-4 = ^, = csc5 ;
P + Q •V2pq
tan A — = cotB ; csc^ = ^ = secB,
27. sin^ = ^^ ."t^ = cos^; cotA = — ^ = tan ^;
P + 1 P
cos^ = -- ■ = sin.B; sec -4 = Vp + 1 = esc 7?;
Vp + 1
tan J. = Vp = cotB: csc-4 = — — = secJB.
P
28. 12.3. 87. 2.5 ; 1.6. 47. a = 4.501 ; b = 5.862.
29. 1.54. 88. 1.5 mi. ; 2 mi. 48. a = 6.8801 ; b = 8.1962.
80. 9. 40. a = 0.342 ; b = 0.94. 49. a = 160.75 ; b = 191.5.
81. 6800. 41. a = 1.368 ; b = 3.76. 60. a = 1.88 ; b = 0.684.
82. 4000. 42. a = 1.197 ; b = 3.29. 51. c = 2.128 ; b = 0.728.
88. 227.84. 43. a = 1.6416 ; b = 4.512. 52. c = 5.848 ; a = 5.494.
84. 3 Vl3 ; 9. 44. a = 2.565 ; b = 7.05. 63. c = 26.6 ; 6 = 9.1.
85. 5 Vs ; J V3. 46. a = 0.643 ; b = 0.766. 54. a = 412.05 ; c = 438.6.
36. 6 ; 8. 46. a = 1.929 ; b = 2.298. 66. 142.926 yd.
66. 1^; 24 ft.
Exercise 2. Page 7
1. cos 60®. 6. cos 40°. 9. cos30^ 13. cos 14^30'. 17. cos25^ 21. tan29».
2. sin 70°. 6. cot 30°. 10. sin 30°. 14. cot 7° 15'. 18. cot 10°. 22. sec 12°.
3. cot 50°. 7. esc 15°. 11. cot 45°. 16. esc 21° 45'. 19. esc 13°. 23. cosl°.
4. esc 65°. 8. sec 5°. 12. esc 45°. 16. sin 1° 50^. 20. sin 38°. 24. sin 4°.
26. esc 2°. 27. sin 7 J°. 29.45°. 31.30°.
26. cosl2j°. 28. cot 1.4°. 80. 46°. 82. 30°.
Exercise 3. Page 9
1. 0.5. 6. 1.1547. 9. 1.7320. 13. Vi. 17. J VS. 21. J.
2. 0.8660. 6. 2. 10. 0.5773. 14. ^Vo. 18. |V2. 22; 3.
3. 0.6773. 7. 0.8660. 11. ^: 15. V3. 19. |V3. 28. jVa
4. 1.7320. 8. 0.5. 12. 1.1547. 16. jVS. 20. VS. 34. Vs.
26. co8 27«42'20'':" 27. csc2°27'0''. 29. oosl4.2°. 31. cot21.18°.
26. cot 14° 31' 25". 28. sin 1° 59^ 33". 80. sin 7.25°. 82. dsc 4.06°.
88. 90°. „ J90^ 40. 22° 30^. 48. J VS. 47. 2 VS. 61. 1.
84.60°. w + l* 41.18°. 44. V2. 48.2. 62. jVa
86. 22° 30^. 88. 90°. . ^ 90° 46. Vo. 49. J VS.
86. 18°. 89. 60°. * n + 1* 46. §V3. 60. jVS;
ANSWERS 8
Exercise 4. Page 10
1. 0.0872. 7. 0.3584. 13. 0.9135. 19. 5.1446. 26. 1.0000. 31. 1.4396.
2. 0.2419. 8. 0.5000. 14. 0.9135. 20. 5.1446. 26. 1.0000. 32. 1.4396.
3. 0.3584. 9. 0.9945. 16. 0.8192. 21. 0.3839. 27. 1.0353. 33. 0.0038.
4. 0.5000. 10. 0.9945. 16. 0.8192. 22. 0.3839. 28. 1.0353. 34. 0.0054.
6. 0.0872. 11. 0.9703. 17. 11.4301. 23. 1.0000. 29. 4.8097. 36. 2 sec 10°.
6. 0.2419. 12. 0.9703. 18. 11.4301. 24. 1.0000. 30. 4.8097. 36. 2 esc 10°.
37. 2 cos 15°.
38. 3 sin 20° > sin (3 x 20°) and > sin (2 x 20°).
39. 3 tan 10° < tan(3 x 10°) and > tan (2 x 10°).
40. 3 cos 10° > cos (3 x 10°) and > cos (2 x.lO°). •*'^ ' '"^
41. No.
42. The sin, tan, sec increase and the cos, cot, esc decrease.
Exercise 5. Page 12
12. 37.6.
13. 1.
14. 100. 16. 60. 16. 12.86.
17. 22.64.
Exercise 6. Page 15
1. 1.736.
4. 57.45
7. 39°. 10. 54 ft.
13. 449.9 ft
2. 3.882.
6. 12°.
8. 43°. 11. 4.326 ft.
3. 41.01.
6. 20°.
9. 30°. 12. 479.9 ft.
Exercise 7. Page 16
1. 10.83.
8. 5.935.
16. 63°. 22. 411.4 ft. 29.
6 in.
2. 13.46.
9. 4.884.
16. 70°. 23. 383 ft. 30.
28.19 ft.; 21.21ft.;
3. 25.58.
10. 7.311.
17. 54°. 24. 43°.
12.68ft.; 30ft.; Oft
4. 31.86.
11. 10°.
18. 60°. 26. 7.794 in. 31.
60°;
0°.
5. 55.73.
12. 17°.
19. 70°. 26. 166.272 sq. in. 82.
25°;
65°.
6. 1.873.
13. 26°.
20. 84°. 27. 5.667. 33.
30° and 60°;
7. 5.972.
14. 60°.
21. 60°. 28. 27.71 ft.
31° and 59°.
34.
749.9 ft.
Exercise 8. Page 19
1. 12.02.
6. 5.928.
11. 45°. 16. 64°.
20.
159.7 ft.
2. 11.04.
7. 14.78.
12. 8°. 17. 148 ft. 8 in.
21.
45°; 90°; 45°
3. 28.84.
8. 44.01.
13. 9°. 18. 29°.
22.
15.76 ft.
4. 45.04.
9. 107.1.
14. 19°. 19. 2.517 mi;
23.
6.14 ft.
6. 98.
10. 453.8.
16. 22°. 3.916 mi.
Exercise 9. Page 20
24.
1.03 in.
1. 26.11.
4. 85.81.
7. 26.60. 10. 25°.
13. 113 ft.
2. 12.35.
6. 544.0.
8. 68.80. 11. 28.87 ft.
14. 123^6 ft
3. 162.6.
6. 26.84.
9. 45°. 12. 428.4 ft.
Exercise 10. Page 21
1. 40.40.
4. 33.63.
7. 41°. 10. 67.74 ft.
13. 26.11 ft
2. 61.77.
6. 66.50.
8. 60°. 11. 1369 ft
<.
8. 101.2.
6. S39.4.
9. 22.66 ft. 12. 91.64 ft.
4 PLANE TRIGONOMETRY
Szerciae 11. Page 22
1. 49.60. 8. 80.62. 5. 81.19. 7. 64^ 9. 66<». 11. 1113 ft.
2. 64.87. 4. 64.80. 6. 162.8. 8. 28^ 10. 46<>. 12. 13.69 mi
18. 19.82 mi. 14.267.0 ft. 16.67.61ft. 16. 17.23 in.
Exercise 12. Page 23
8. tanx. 4. secx. 6. secx. 6. csc x. 7. cotx. 8. cscx. 16. 18°. 8 5. rsinx.
86. a = cm; 6 = c Vl — m^. 37. a = 6m ; c = 6 Vm^ + 1.
Exercise 13. Page 26
2. 0. 8. No. 18. 2.3109. 19. 37°. 26. 19°. 31. 16^
3. 1. 9. 46°. 14. 0.6873. 20. 46°. 26. 48*^. 32. 37°.
4. ». 10. 0.6462; 15. 6°. 21. 6°. 27. 34°. 88. J.
5. 0. 0.7631. 16. 24°. 22. 13°. 28. 40°.
6. The tangent. 11. 0.3680. 17. 44°. 28. 22°. 29. 64°.
7. No. 12. 2.7173. 18. 26°. 24. 14°. 30. 30°.
Exercise 14. Page 29
1. 0.7647. 7. 0.7428. 18. 0.8708. 19. 63.47. 25. 69.38. 81. 19.70 ft.;
2. 0.9004. 8. 0.6663. 14. 0.8708. 20. 20.90. 26. 49.83. 22.62 ft.
3. 0.7646. 9. 0.6693. 15. 1.1483. 21. 26.27. 27. 94.36. 32. 19.72 ft.;
4. 0.9016. 10. 0.6667. 16. 17.73. 22. 48.29. 28. 74.93. 22.61ft.
5. 0.7638. 11. 0.6700. 17. 32.16. 28. 66.48. 29. 88.35. 83. 120,5 ft.
6. 0.7646. 12. 0.6700. 18. 46.01. 24. 64.84. 80. 47° 66'. 84. 71.77 ft.
Exercise 15. Page 30
1.0.0087. 6.0.0716. 11.0.9972. 16.1.0000. 21. 12.66 in. ;
2. 0.0070. 7. 0.9972. 12. 0.9974. 17. 0.0716. 0.9970 in.
8. 0.0698. 8. 0.0769. 13. 0.0767. 18. 143.2. 22. 390 ft.
4. 0.9973. 9. 12.71. 14. 13.96. 19. 0.0062. 28. 0.7477 in.;
5. 0.0787. 10. 13.62. 15. 0.0769. 20. 0.0734. 9.630 in.
Exercise 16. Page 33
1. 0.4667. 14. 12.1624. 24. 70° 46' 30'; 35. 10.7389. 48. 44°38'30".
2. 0.6726. 15. 16.3140. 0.3490. '^' 36. 0.9808. 49. 69° 16'.
8.0.8338. 16.10.4662. 25. 79° 30' 15"; 37.4.6787. 50. 78° 8' 30".
4. 0.9099. 17. 8.7149. 0.1852. 88. 4.1626. 51. 78° 8' 15".
5. 0.8066. 18. 7.2246. 26. 0.4306. 39. 3.6108. 52. 14° 45.
6. 0.7289. 19. 6.6686. 27. 0.4313. 40. 3.3602. 53. 0.7658.
7. 0.4335. 20. 6.0826. 28. 0.5410. 41. 31° 30'. 54. 0.6438.
8. 0.6438. 21, 39° 43' 30"; 29. 0.6646. 42. 36° 16'. 56. 0.6639.
9.0.6418. 0.7691. 80.0.9046. 48. 41° 18' 30". 56. 33° 10' 16^
10. 0.9209. 22. 60° 16' 30"; 81. 0.1990. / 44. 44° 36' 30". 1.6298.
11. 1.2882. 0.6391. ' 32. 4.9560.-^ 45. 38° 16'. 57. 31° 8' 30";
12. 2.6018. 28. 71° 29' 40"; 83. 0.1490. 46. 39° 30'. 0.6042.
18. 3.1266. 0.9483. 84. 7.8279. 47. 17° 46'
ANSWERS 6
Kxercise 17. Page 87
1. A = 860 62', B= 68«8', c = 6. 8. ^ = 43°33', B= 4e«27', a = 08.14.
2. ^ = 32° 36', B = 67'' 26', b = 10.96. 9. B = 67° 46', a = 26.73, c = 60.12.
3. B = 77° 43', b = 24.34, c = 24.98. 10. ^ = 43° 49', a = 191.9, c = 277.2.
4. ^ = 46° 42', b = 9.801, c = 14.29. 11. ^ = 68° 43', B = 21° 17', c = 102.0.
6. B = 62° 18', a = 15.90, 6 = 20.67. 12. .4 = 3° 20', B= 86°40', b = 102.8.
e.A = 66° 48', a = 127.7, 6 = 67.39. 18. A = 84° 62', b = 0.2802, c = 3.138.
7. -4 = 34° 18', B = 66° 42', a = 12.96. 14. ^ = 70° 48', B = 19° 12', 6 = 6.916.
15. B = 61° 31', a = 36.47, 6 = 44.62.
16. ^ = 22° 37', B = 67° 23', a = 6, c = 13.
17. ^ = 63°8', B = 36°62',a = 40, c = 60.
18. ^ = 22° 37', B= 67° 23', a = 12.6, c = 32.6.
19. B = 54° 49' 30", 6=3.647, c= 4.340. 21. ^ = 60° 41' 30", 6=3.693, c= 7.339
80. B = 47° 47' 30", 6=6.284, c=8.486. 22. ^ = 63° 39' 30", 6=6.812, c=9.808.
23. B = 60° 17' 30", a = 3.370, 6 = 6.906.
24. B = 66° 39' 30", a = 203.08, b = 297.26.
26. B = 48° 49' 20", a = 218.68, c = 332.14.
26. B = 64.6°, b = 100.6, c = 111.6.
27. B = 66.6°, a = 10.37, b = 22.76. 80. B = 26.64°, a = 67.10, b = 38.61.
28. B = 67.46°, a = 21.62, b = 33.72. 81. A = 39.41°, 6 = 64.77, r = 70.88.
29. B = 34.49°, a = 65.94, b = 46.80. 82. B = 21.76°, a = 226.6, c = 242.8.
38. 29.20 in. 87. 43.30 in. 41. 13.26 ft.
34. 23.73 in. 88. 60.05 in. 42. 16.82 in.; 18.60 in.
35. 42.26 in. 89. 66° 18' 36", 33° 41' 24". 48. 12.42 ft.
86. 64.26 in. 40. A = 41° 24' 30", B = 48° 36' 30". 44. 66.89 in.
45. 9° 36' 40".
Bzercise 18. Page 41
1. 5. 8. 4. 5. 6. 7. 8. 9. 6. 11. 3. 18. 3. 15. 4. 17. 3. 19. 6.
2. 2. 4. 4. 6. 7. 8. 6. 10. 4. 12. 2. 14. 3. 16. 2. 18. 5. 20. - 1.
21. -2; -3; -4. 24. 1; 2; 3; 6; 9; 10; -2; -4;
22. 1 and 2 ; 2 and 3 ; 3 and 4; -6;— 6;— 7;- 8.
4 and 6 ; 5 and 6 ; 8 and 9. 25. l;4;6;7;8;-l;-2;— 3;
23. — 2 and - 1; - 8 and — 2 ; - 4 ; — 5 ; - 6 ; - 7.
- 4 and - 3 ; - 1 and ; 26. ; — 4 ; - 6 ; 7 ; 8.
— 2 and — 1 ; — 8 and — 2.
27. 1 and 2. 81. 2 and 3. 85. 3 and 4. 89. 6 and 6.
28. 1 and 2. 82. 2 and 3. 86. 3 and 4. 40. 6 and 7.
29. 1 and 2. 33. 2 and 3. 87. 3 and 4. 41. 6 and 7.
30. 1 and 2. 84. 2 and 3. 88. 3 and 4. 42. 7 and 8.
Exercise 19. Page 45
1. 1.
6. 3.
11. -1.
16. - 4.
21. 1.68681.
2. 1.
7. 2.
12. - 2.
11 - 3.
22. 0.68681.
8. 2.
8. 1.
18. -1.
18. - 6.
28. 2.68681.
4. 0.
9. 0.
14. - 1.
19. - 1.
24. 4.68681.
5. 3.
10. 4.
15. - 3.
20. -2.
25. 6.6868L
PLANE TRIGONOMETRY
36. 7.68681.
27. 1.68681.
28. 2.58681.
29. 4.68681.
30. 8.67724.
81. 0.67724.
82. 4.67724.
88. 7.67724.
84. 2.67724.
85. 6.67724.
86. 0.40603.
87. 1.40603.
88. 1.40603.
39. 3.40603.
40. 4.40603.
41. 7.40603.
42. 0.39794.
48. 1.39794.
44. 1.39794.
46. 2.39794.
4a 4.39794.
47. 7.39794.
Exercise 20. Page 47
1. 0.30103.
2. 1.30103.
8. 2.30103.
4. 3.30103.
6. 3.32222.
6. 3.83244.
7. 3.33365.
8. 0.33365.
9. 3.64220.
10. 3.64963.
11. 3.74671.
12. 3.84663.
18. 3.72304.
14. 1.83566.
15. 0.89906.
16. 2.92168.
17. 1.84610.
18. 1.87606.
19. 1.87862.
20. 1.87892*.
21. 2.40654.
22. 3.66630.
23. 4.95424.
24. 2.26042.
26. 4.09132.
26. 4.09160.
27. 4.09167.
28. 2.09167.
29. 2.37037.
30. 1.61624.
31. 1.75037.
82. 1.61576.
88. 5.61409.
84. 2.56155.
36. 7.82948.
86. 17.72562.
37. 9.19606.
88. 5.26893.
39. 2.51989.
40. 3.20732.
41. 4.86198.
42. 0.48124.
48. 0.95424.
44. 0.90309.
46. 4.22472.
46. 2.87595.
47. 6.32328.
48. 12.70040.
49. 19.68460.
60. 0.15062.
51. 1.66062.
62. 1.17969.
63. 0.46458.
54. 0.64167.
66. 1.08030.
56. 2.16224.
67. 0.79034.
68. 1.14477.
69. 0.54254.
60. 0.99155.
61. 2.00072.
62. 0.75343.
68. 1.19855.
Exercise 21. Page 49
1. 3.
2. 3000.
3. 0.003.
4. 304.6.
6. 37,020.
6. 46.
7. 467.6.
8. 0.000056.
9. 6505.
10. 0.06796.
11. 0.0006095.
12. 0.66.
13. 6.695.
14. 7.6.
15. 7,806,000,000.
16. 79,950,000.
17. 1.7102.
18. 27.005.
19. 370.15.
20. 0.38065.
21. 0.0043142.
22. 43,144.
23. 4.3646.
24. 0.049074.
26. 694,640,000.
26. 0.00067555.
27. 6846.5.
28. 685.55.
29. 77,553.
80. 785.65.
81. 7917.3.
32. 8.5652.
83. 875.18.
84. 2.
86. 3.46591 ;
3.45864. .
36. 2955.
37. 0.0066062.
38. 0.65163.
89. 91.226.
40. 53,159,000.
41. 0.000010746.
42. 5.72784;
534,360.
43. 353,780.
44. 7.2388.
46. 107.
46. 26,459.
47. 16,693,000.
48. 129.66.
49. 4.9341.
Exercise 22. Page 50
1. 10.
9. 66.
17. 12,000.
25. 603.9.
83. 210.
2. 24.
10. 18.
18, 18,000.
26. 1282.8.
34. 946.
3. 16.
11. 100.
19. 660,000.
27. 184,670.
85. 5005.
4. 36.
12. 2400.
20. 180,000.
28. 11,099.
86. 38,645.
6. 8.
18. 1600.
21. 1034.6.
29. 1609.9.
87. 627,400
6. 21.
14. 3500.
22. 2192.3.
80. 17,468.
88. 276.67,
7. 12.
16. 8000.
23. 13.31.
81. 18.212 in.
8. 18.
16. 21,000.
24. 20.266.
82. 113.04 ft.
ANSWERS
1. 7.68964.
2. 3.68964.
3. 7,68964.
4. 3.09497.
6. 0.00000.
6. 1.99999.
Exercise 23. Page 51
7. 4.03939.
8. 2.00010.
9. 1.99999.
10. 0.00000.
13. 0.1248.
14. 0.0001248.
16. 0.0043707.
16. 0.11422.
11. 1,248,000. 17. 0.0000003125. 23. 3309.6.
12. 124.8. 18. 0.25121. 24. 452.27
19. 0.02240.
20. 0.00015725.
21. 1.3020.
22. 38.079.
25. 22.936.
26. 34.108.
27. 16-51.
Exercise 24. Page 53
1. 1.97519.
13. 3.89100.
26. 5.
37. 0.00999.
49. 60.87.
2. 3.66078.
14. 2.00000.
26. 84.
88. 0.0709.
60. 0.6527.
3. 1.68618.
16. 2.11220.
27. 82.002.
39. 0.0204.
51. 20.
4. 3.70404.
16. 2.00286.
28. 76.
40. 0.065.
62. 50.
6. 5.00000.
17. 1.71172.
29. 35.6.
41. 0.48001.
63. 700.
6. 9.70000.
18. 5.
30. 73.002.
42. 2.143.
M. 800.
7. 7.00000.
19. 5.
31. 92.
43. 0.4667.
66. 9000.
8. 7.00000.
20. 3.
32. 105.
44. 0.004667.
66. 11,000.
9. 3.76439.
21. 4.
38. 63.
45. 1.913.
67. 120,000
10. 2.00000.
22. 3.
84. 77.
46. 1.123.
68. 0.01.
11. 2.90000.
23. 5.
36. 0.0129.
47. 12.86.
69. 871.1 ; 2
12. 6.90000.
24. 3.
86. 1290.
48. 5.184.
Exercise 25. Page 54
1.
2.60206.
5. 4.42585.
9. 0.30103.
13. 1.62187-
17.
1.
2.
3.88606.
6. 3.36927.
10. 0.14267.
14. 2.20698.
18.
0.1
3.
2.56225.
7. 2.28727.
11. 1.08092.
16. 3.22185.
19.
0.
4.
1.23433.
8. 1.14188.
12. 2.13906.
16. 4.15490.
20.
1.
Exercise 26. Page 55
J. 1.
8. 0.44272.
16. 6.1649.
22. 105.47.
2. 6.
9. 1.7833.
16. 0.42742.
23. 3,013,400.
3. 3.
10. 1000.
17. 1.4179.
24. 0.081528.
4. 0.5.
11. 0.092.
18. 0.031169.
25. 232.24.
6. 1.
12. 1.8.
19. 40.464.
26. 0.0000007237
6. 2.
13. 0.01.
20. 0.14621.
27. 103.33.
7. 0.11111.
14. 0.21.
21. 2893.2.
Exercise 27. Page 56
1. 4.
6. 728.98.
11. 4,782,800.
16. 83,522.
2. 8.
7. 64.
12. 16,777,000.
17. 15,625.
3. 32.
8. 125.
18. 19,486,000.
18. 6,103,600,000
4. 1024.
9. 1.
14. 11,391,000.
19. 15,625.
5. 80.998.
10. 40,365,000.
15. 11.391.
20. 244,140,000.
8
PLANE TRIGONOMETRY
21. 16,413,000,000,000,000.
82. 7,700,600.
28. 31,137,000,000.
24. 292,360,000,000,000.
25. 2.1436.
26. 180.11.
27. 0.000000000001.
28. 0.00000002048.
29. 0.06766.
80. 0.00000011766.
81. 0.018741.
82. 164.86.
88. 167.6.
84. 41,961.
86. 2.0727.
86. 0.0019720.
87. 0.023651.
88. 0.00016228.
89. 0.0000076624.
40. 0.00000012603.
41. 9.8696; 31.006,
42. 21.991 ; 163.94
3063.6.
Exercise 28. Page 57
1. 1.4142.
2. 1.71.
8. 1.3206.
4. 1.2394.
5. 1.1487.
6. 2.2796.
7. 6.6669.
8. 3.0403.
9. 3.3166.
10. 1.4422.
11. 2.802.
12. 1.2023.
18. 0.64773.
14. 0.3684.
15. 0.067406.
16. 0.064491.
17. 20.729.
18. 1.9733.
19. 3.9096.
20. 0.0028827.
21. 1.7726; 1.4646.
22. 1.3313; 2.1460;
6.6684; 0.42378;
0.40020; 0.79637.
Exercise 29. Page 59
1. X =
2. x =
8. x =
4. X =
5. X =
21. x =
22. x =
23. x =
24. x =
25. x =
26. X =
3.
4.
4.
4.
3.
6. X = 4.2479.
7. X = 3.9300.
8. X = 4:2920.
9. x = 6.6610.
10. X = 3.0499.
2, 2/ = 2.
log a — logp
log(l + r) '
log r + log I — log a
logr
1, -3.
logo— logp
log(l+r«) *
log[8(r-l) + a]-logq
logr
11. X = 3.
12. X = 3.3219.
18, x=- 0.087616.
14. X = 4.4190.
15. X = - 0.047964.
27. X = 2, - 1.
28. 0.062467.
29. 3.1389.
80. 0.036161.
81. 0.03476.
82. 6.
log 6
88.
84.
log a
logn
log 6
16. X = 3, y = 1.
17. X = 6, 2/ = 1.
18. X = 1, 2/ = 1.
19. X = 2, y = 2.
20. X = 3, 2/ = 2.
85. 2 ; 7.2730 ;
2.0009 ; 2.0043.
log a
86. 1;
87. x =
88. - 1.
log 6'
log 6
1;3;4
log a — log b
Exercise 30. Page 62
1. 9.66706 - 10.
2. 9.97015 - 10.
8. 9.90796 - 10.
4. 9.82561 - 10.
5. 10.67196-10.
6. 9.32747 - 10.
7. 10.67196-10.
8. 9.32747 - 10.
9. 9.20613 - 10.
10. 9.99626 - 10.
11. 9.14412-10.
12. 9.14412 - 10.
18. 8.89464 - 10.
14. 9.99651 - 10.
16. 9.23510 - 10.
16. 9.87099 - 10.
17. 9.68826 - 10.
18. 10.10706-10.
19. 9.65763 - 10.
20. 9.96966 - 10.
21. 9.98436 - 10.
22. 9.42096 - 10.
28. 9.48632 - 10.
24. 9.68916 - 10.
25. 9.96340 - 10.
26. 11.13737-10.
27. 9.74766-10.
28. 9.66368 - 10.
29. 10.17675-10.
80. 9.82332 - 10.
81. 6.51166 - 10.
82. 8.25667-10.
83. 6.79257-10.
84. 8.66813 - 10.
85. 7.45643-10.
86. 8.16611-10.
87. 8.11603-10.
88. 8.00469 - 10.
89. 8.24916 - 10.
40. 8.24916 - 10.
41. 8.63264-10.
42. 8.63206 - 10.
43. 9.32607-10.
44. 9.32607 - 10.
45. 10.39604-10
46. 7°30'.
47. 32<»2r.
48. 68° 27.
ANSWERS
9
49. 85*=>80'.
60. 4°30'.
51. 3P38'.
52. 68° 36'.
53. 60^82'.
54. 3»° 2'.
55. 63*=>4r23".
56. 77«»6'.
57. 79°.
58. 70°.
59. 20° 13' 30''.
60. 32° 22' 16".
61. 49° 34' 12".
62. 61° 47' 36".
68. 37° 8' 48".
64. 60° 48' 16".
65. 8° 49' 30".
66. 8° 46' 30".
I//
67. 67*»4f
68. 49° 26' 7".
69. 38° 22' 80".
70. 2° 3' 30".
71. 89° 49' 10".
£xerci8e 31. Page 67
1. u4 = 30=',
B = 60°,
b = 10.39,
5=31.18.
2. 5 = 30°,
a = 6.928,
c = 8,
S = 13.86.
8. 1^=60°,
6 = 6.196,
c=6.
5=7.794.
4. ^ = 46°,
B = 46°,
c = 6.667,
S = S.
5. /f = 43° 47',
B= 46° 13',
6 - 2.086,
S = 2.086.
6. B = 66° 30'.
a = 260,
6 = 676,
S = 71,880.
7. B = 61° 66',
a = 1073,
6 = 2012,
S = 1,079,600.
8. B = 60° 26',
a = 46.96,
b = 66.62,
S = 1278.
9. B=64°,
a = 0.6878,
6 = 0.8090,
S = 0.2378.
10. A = 68° 13',
a = 186.7,
b = 74.22,
S = 6892.
11. ^ = 13° 36',
a = 21.94,
b = 90.79,
5 = 996.8.
12. B = 86° 25',
b = 7946,
c = 7972,
S = 2,631,000.
13. 5 = 63° 16',
b = 66.03,
c = 81.14,
iS = 1678.
14. B = 4°,
b = 0.0006694,
c = 0.00802,
S = 0.000002238.
15. A =_ig° 12',
a = 53.12,
c = 73.60,
S = 1363.
16. ^ = 86° 22',
a = 31.60,
c = 31.66,
5=31.60.
17. A = 13° 41',
b = 4075,
c = 4194,
5 = 2,021,000.
18. A = 21° 8',
b = 188.9,
c = 202.6,
S = 6893.
19. -4 = 44° 36',
6 = 2.221,
c = 3.119,
5=2.431.
20. B = 62° 4',
a = 3.118,
c = 6.071,
S = 6.236.
21. A = 31° 24',
B = 68° 36',
6 = 7333,
5 = 16,410,000.
22. ^ = 66° 3',
B = 33° 67',
b = 48.32,
5=1734.
23. ^ = 66° 14',
B = 24° 46',
6 = 3.917,
5 = 16.63.
24. ^ = 63° 16',
B = 36° 46',
a = 1758,
5 = 1,164,000.
25. ^ = 63° 31',
B - 36° 29',
a = 24.68,
5 = 226.2.
26. ^ = 63°,
B = 27°,
c = 43.
5 = 373.9.
27. A = 4° ,42',
B = 86° 18',
c = 16.
5=9.187.
28. A = 81° 30',
B = 8° 30',
c = 419.9,
5 = 12,890.
29. ^ = 38° 69',
B = 61° 1',
c = 21.76,
5=116.8.
80. ^ = 1° 22',
B = 88° 38',
6 = 91.89,
5 = 100.6.
31. A = 39° 48',
E = 60° 12',
c = 7.811,
5=16.
82. ^ = 30' 12",
2? = 89° 29' 48"
,6=70,
5=21.63.
88. A = 43° 25',
B = 46° 40',
a = 1.189,
5=0.7488.
34. i^ = 71° 46',
85. B = 60° 62',
b = 21^25,
a = 6.^88,
c = 22.37,
c = 13.74,
5 = 74.37.
5=49.13.
86. B = 20"? 6',
a = 63.86,-
6 = 23.37,
5 = 746.16.
87. A = 46° 66',
a = 1,9.40,
6 = 18.78,
5 = 182.15.
38. A = 41° 11',
b = 63.72,
c = 71.38,
5 = 1262.4
39. A = 65° 16',
a = 12.98,
c = 16.80,
5 = 58.42.
40. ^ = 3° 66',
a = 0.6806,
6 = 8.442,
5 = 2.460.
10 PLANE TRIGONOMETRY
41. 8=^c^AnAcoaA. 48. 8= jlj^ t&nA.
48. 5= Ja^cot^. 44. S^^aVc^-a^.
45. ^ = 40° 46' 48", B = 49° 14' 12", 6 = 11.6, c = 16.315.
46. ^ = 66° 13' 20", B = 34° 46' 40", a = 7.2, c = 8.766.
47. B = 61°, a = 3.647, 6 = 6.68, c = 7.623.
48. A = 27° 2' 30", B = 62° 67' 30", a = 10.002, b = 19.696.
49. 19° 28' 17" ; 70° 31' 43". 61. 16.498 mi.
60. 3112 mi. ; 19,653 mi. , 52. Between 1° 16' 30" and 1°19' 10".
63.212.1ft. 68. 69° 44' 35". 63. 7.071 mi.; 67.686.9ft.
54. 732.2 ft. 59. 95.34 ft. 7.071 mi. 68. 6.667 ft
55. 3270 ft. 60. 23° 60' 40". 64. 19.05 ft. 69. 136.6 ft.
56. 37.8 ft. 61. 36°r42". 65. 20.88 ft. 70. 140 ft.
67. 1°26'66". 62. 69° 26' 38". 66. 66.66 ft. 71. 84.74 ft.
Exercise 32. Page 71
1. C = 2(90°— J.), c = 2acos^, ^ = asin^.
2. A = 90°— ^C, c =2acos^, h = a^TiA.
3. = 2(90°-^), a = — - — , h = aanA.
^ " 2 COS A
4.^ = 90°-JC, a = — - — , /i = asin^.
^ 2C08^
5. C = 2(90°-^), a = , c = 2aco8A,
sin A
6. -4 = 90° — i C, a = , c =2 a cos A,
J sin -4
7. sin^ = -, = 2(90°-^), c = 2aco8A.
8. tan^=— , (7 = 2(90°-^), a = -^
c anA
9. ^ = 67° 22' 60", C - 46° 14' 20", h = 13.2.
10. c = 0.21943, h = 0.27384, S = 0.03004.
11. a = 2.066, h = 1.6862, S - 1.9819.
12. a = 7.706, c = 3.6676, S = 13.726.
13. A = 26° 27' 47", C = 129° 4' 26", a = 81.41, h = 36.
14. A = 81° 12' 9", G = 17° 36' 42", a = 17, c = 6.2.
16. c = 14.049, h = 26.649, S = 187.2.
16. /S = a2 sin ^C cos JO. 19. 28.284 ft. ; 21. 94° 20'. 24. 37.699 sq. in.
17. S = a^ sin A cos A, 4626.44 sq. ft. 22. 2.7261. 26. 0.8776.
18. S=h^ tan J C. 20. 0.76536. 23. 38° 56' 33".
Exercise 33. Page 72
1. r = 1.618, h = 1.6388, S = 7.694. 4. r = 1.0824, c = 0.82842, S = 3.3137.
2. A = 0.9848, p = 6.2614, 8 = 3.0782. 6. r = 2.6942, h = 2.4891, c = 1.461.
8. ^ = 19.754, c = 6.267, 8 = 1236. 6. r = 1.6994, h = 1.441, p = 9.716.
7. 0.51764 in. 9. 0.2238 sq. in. 13. 6.283.
' ~ 90^* 11. 1.0236 in.
^^ n 12. 0.062821 ; 6.2821.
ANSWERS
11
Exercise 34. Page 73
2. 29.76 sq. in.
3. 104.07 sq.ft.
4. 36.463 sq. in.
6. 20.284 in.
7. 37.319 ft.
8. 342.67 ft.
9. 36.602 ft.;
86.602 ft.
10. 120.03 ft.
11. 2.9101 mi.;
3.631 mi.
12. 11° 47'';
49.206 ft. •
13. 62° 36' 42".
14. 60° 36' 68".
16. 6.3609 in.
16. 20 in.
17. 7.7942 in.
18. 40° 7' 6".
19. 77° 8' 31".
20. 94.368 ft.;
26° 42' 68".
21. 24.662 ft.
22. 196.93 ft.
28. 220.8 ft.
24. 1916.3 ft.
2fi. 362.09 ft.
26. 69° 2' 10".
27. 14.772 in. ;
16.696 in.
28. 73.21ft.
29. 26° 36' 9".
30. 26.613 in.
31. 7.6 ft.
32. 69° 68' 64";
173.08 ft.
33. 7.2917 ft.
34. 19.061.
35. 1.732 in.
86. 2676.8 mi.
37. 26.776 ft.;
19.46 ft.
88. 10.941ft.;
20.141 ft.
39. 65.406 ft.
40. Between isr
and 132'.
41. 43° 18' 48".
42. 2.6068 in.
43. 14.642 in. ;
26.87 in.
44. 6471.7 ft.
Exercise 35. Page 80
29. 10.
83. IJ.
37. 0.
41. 6.10.
46. 28 J in.
49. fVs.
30. 16.
34. 3J.
38. 7.
42. 6.10.
46. 9.43 in.
60. Yes.
31. 13.
36. 3.
39. 5.
43. 8.24.
47. 2.
61. Octagon,
82. 2}.
86. 5.
40. 16.
44. 4.24.
48. 3V3.
2.829.
Exercise 36. Page 84
16. I. 18. II. 20. III.
17. I. 19. II. 21. IV.
30. On OX,
61. jV6; jV3; ^/'S; ^VS.
62. 90°.
63. 60*.
22. I. 24. III. 26. I. 28. III.
23. II. 26. IV. 27. II. 29. On 0Y\
64. sin = ^V2; cos =— ^V2; tan=— 1;
CSC = V2 ; sec = — V2 ; cot = — 1.
66. sin = ; cos = — 1 ; -tan = ;
CSC = 00 ; sec = — 1 ; cot = oo.
Exercise 37. Page 88
62. 2 ; one in Quadrant I, one in Quadrant II.
63. 4 ; two in Quadrant I, two in Quadi-ant IV.
64. 2; 1; 1; 1; 1.
66. Between 90° and 270° ; between 0° and 90° or between
between 0° and 90° or between 270° and 360° ; between
67. 1 ; ; ; oo ;
1 ; 00 ; 1 ; 0.
69. Ill; II.
60. 40 ; 20.
61. 0.
62. 0.
68. 0.
64. a3-62 4.4fl55j
66. -2(a2 + 62).
66. 0.
67. (♦.
76. 30° ; 160° ; 390°
77. 30° ; 330° ; 390°
78. 60° ; 120° ; 420°
79. 60°; 300°; 420°
80. 30° ; 210° ; 390«>
180° and 270» ;
180° and 360°.
81. 60° ; 240° ;
510°.
690°.
480°.
660°.
670°,
420°
82. 210°
88. 120°
84. 225°
86. 135°
86. 135°
87. 185°
600°.
330°.
240^.
315°
226°
315°
316°
12 PLANE TRIGONOMETRY
Exercise 38. Page 91
1. sin IQP. 9. tan 78°. 17. - cot 66*>. 25. - sin 7° W 3".
2. -cos 20°. 10. cot 82°. 18. -cot 13°. 26. cos 86° 54' 46''.
8. -tan 32°. 11. -sin 86°. 19. -sinO°. 27. -tan 37° 61' 45"
4. - cot 24°. 12. - sin 16°. 20. cos 0°. 28. cot 16° lO' 3".
5. sin 0°. 18. - tan 78°. 21. sin 31° 60'. 29. sin 32.26°.
6. - tan 0°. 14. - tan 36° 22. - cos 12° 20'. 80. - cos 62.25S
7. - sin 20°. 16. cos 70°. 23. tan 86° 30'.
8. - cos 46°. 16. cos 10°. 24. - cot 72° 20'.
Exercise 39. Page 93
1. cos 10®. 10. — cot 0°. 19. - sin 86°. 28. - cot 9.1°.
2. cos 30°. 11. -cot 29°. 20. cos 76°. 29. 1^.0262.
8. cos 20°. 12. -cot 39°. 21. cos 87°. 80: -0.6483.
4. cos 40°. 13. - tan 4° 1'. 22. - sin 6°. 81. - 0.7729.
6. - sin 6°. 14. - tan 7° 2'. 23. tan 80°. 32. 0.6040.
6. - sin 7°. 16. - tan 8° 3'. 24. tan 30°. 88. - 0.1304.
7. - sin 21°. 16, - tan 9° 9'. 26. - tan 20°. 34. 0.8686.
8. - sin 37°. 17, - sin 3°. 26. - cot 1.6°. 35. 0.1367.
9. - cot 1°. 18. - sin 9°. 27. - cot 7.8°. 86. - 0.1364.
87. 9.89947 - 10. 40, - (10.62286 - 10). 43. 10.14763 - 10.
88. - (9.83861 - 10) . 41. - (9.91969 - 10) . 44. - (9.82489 - 10) .
39. - (9.79916 - 10). 42, 9.92401 - 10. 46. 226°; 316°; 686°; 676
:o
6. sin X = ±
7. cos X = ±
8. secx = ±
Exercise 40. Page 95
1 19. 46°. 27. 60°.
Vcot^x + l' 20. 30°. 28. 60° or 180°.
1 21. 60°. 29. 46°.
Vtan2x + 1 22. 46°. 80. 30°.
1 23. 46°. 81. 46°.
Vl-sin2x 24. 46°. 32. ;}V6; f V5.
9. cscx=4- — I— . ^'^' 88. iVl6;Vl6.
Vl - C082 X 26. 46°. 84. J ; 6.
35. sinx = § V6, cosx = ^ VB, tan x = 2 ; cscx = J VB, secx = V6, cot x = J.
86. vV "^ ; tV ^/17. 41. 45° or 226°. 45. 270° or 30°.
37. ^^ ; J^. 42. 45°, 136°, 226°, 46. 30° or 160°.
38. When x = 0°. or 316°. 47. 46°, 136°, 226°
39. 0° or 180°. 43. 46° or 225°. or 316°.
40. 38° 10'. 44. 0° or 60°. 48. 60°,
53. cos^=jV6, tan-4=jV5, csc^=|, sec-4=fV6, cot-4=jV6.
54. sin -4 = J V7, tan-4=jV7, csc-4= ^V7j_ sec-4= |, cotA—^Vl,
55. ^nAz=^^VlS,coBA=:^rf-\/Ts, cacA=^V^, sec^ = iVl3, cotA=%,
56. sin-4.= j, _ cos-4=|, tan-4=J, csc^ = |, secA=:^.
57. sirrA = I V6, cos^=§, tan4 = ^V5, C8C^ = §V5, cot^=|V6
58. cos^ = ^^, tan A = J^, esc A = •}- J, sec -4 = -^/, cot -4 = ^.
59. cos-4 = §, tan -4 = j^, csc-4 = |, sec -4 =- J, cot-4 = |.
ao. sin^ = if{, tan-4 = |j, csc^ = f |, secul = f ^, cot-4 = f J.
ANSWERS
13
61. sin -A = § J, tan^ = V, csgA = |f,
62. sin -4 = J, co8-4 = J, csc^ = |,
63. sin -4 = J V2, cos J. = J V2, tan^ =
64. sin ^ = § Vs, cos -4 = J Vs, tan A =
65. sin ^ = J Vs, cos -4 = J, tan -4 =
66. sin -4 = J V2, co8-4 = J V2, tan 4 =
67. cos^ =Vl— m^, tan-4 =
m
sec ^ = y , cot -4 = ^;5.
sec ^ = J, cot -4 = |.
1, csc^ = V2,_ 8ecA = ^/2.
2, csc^ = J Vs, sec -4 = V6._
VS, csc^ = ^Vs, cot A = J Vs.
1, sec^ = V2, cot-4 = l.
2m
vrr
68.
m^
CSC -4 = — , sec^ =
m
Vi=
;, COt^
Vi-:
m-'
69.
l-m2
m2 + n2
m'-
70. cos 0° = 1, tan 0° = 0, esc 0^ = oo,
71. cos 90® = 0, tan 90° = oo, esc 90° = 1,
72. sin 90° = 1, cos 90° = 0, esc 90° = 1,
73. sin 22° 30' = — ^ , cos 22° 30'
m 2mn
sec 0° = 1, cot 0° = 00.
sec 90° = 00, cot 90° = 0.
sec 90° = 00, cot 90°= 0.
1
\/4 - 2 V2
\ /4-f 2V2
CSC 22° 30' = V4 + 2V2, sec 22° 30' = V^ - 2 V2.
tan 22° 30' = V2 - 1,
^^ l-cos^^
74. — +
cos -4
cos*^
1 - cos2 A
Exercise 41. Page 98
1. 0.26875.
5. 1.
9. 0.866.
18. 0.6.
2. 0.96676.
6. 0.
10. - 0.5.
14. - 0.866.
3. 0.96576.
7. 0.96676.
11. 0.707.
16. 0.26876.
4, 0.26876.
8. - 0.26876.
12. - 0.707.
16. - 0.96676
Exercise 42.
Page 99
1. 0.268.
5. 00.
9. - 1.732.
13. - 0.677.
2. 3.732.
6. 0.
10. - 0.677.
14. -1.732.
3. 3.732.
7. -3.732.
11, -1.
16. - 0.268.
4. 0.268.
8. - 0.268.
12. -].
16. - 3.732.
1.
a- ih .
7. cosy.
8. amy.
9. coty.
10. cosy.
11. siny.
12. — sin y.
13. — cosy.
Exercise 43. Page 102
14. — cos y.
16. — sin y
16. siny.
17. sinx.
18. — cosx.
19. — sinx.
20. — cotx.
21. tanx.
22. — tanx.
28. cotx.
24. — sin y.
26. J V2 (cos y — sin y).
26. jV2{cosy + siny).
27.
28.
29.
1— tany
1 + tan y
V3 cot y — 1
cot y + V3
jV3coty + 1
cot y — ^ V3
30. tany.
31. 0.8671 ; 0.2222.
82. 3.732 ; 0.268.
33. 1; H ; ^5°-
34. X + y = 90°, 270° in
the three cases.
87. 136° 406°.
14 I'hANE TRIGONOMETRY
Exercise 44. Page 103
5.1. 7. -J. 9.0.8492. 11. - 1.1776. 18. ^JJ. 15. 3 sin « - 4 8in«aj
6. J Vs. 8. /y. 10. 0.6827. 12. 1.7161. 14. ^fj. 16. 4 cos»x - 8 cos a;
Exercise 45. Page 104
1. 0.2688. 8. 0.2679. 6. 7.6928. 7. 0.9239. 9. 2.4142.
2. 0.9669. 4. 3.7321. 6. 0.3827. 8. 0.4142. 10. 5.0280.
11. 0.10061; 0.99493. 12. 0.38730; 0.92196; 0.42009; 2.3806
Exercise 46. Page 105
8. 0. jg cos (X + y) 22^ co8(x>~y)
9. Jv3. * sin X cosy * sin x sin y
- g 2 ^ 19. tan^x. cos (x + y)
* sin2x ^ cos(x — y) ' sin x sin y '
18. 2 cot 2 X. ' cos X cosy 24. tan x tan y.
^^ cos(x-y) 2j^ cos(x + y) 27. j.
sin X cos y ' cos x cos y
Exercise 47. Page 109
1, a = 6sinu4; 6=:asinB; a = h; 8inA = sinB. 6. 8.6460in.; 4.2728 in
4. 8 in. 7. 27.6498 in.
6, 1000 ft. 8. 9.1121 in.
Exercise 48. Page 110
1. C = 123° 12', 6 = 2061.6, c = 2362.6. 11. Sides, 600 ft. and 1039.2 ft.
2. C = 66° 20', 6 = 667.69, c = 663.99. altitude, 619.6 ft.
3. C = 36° 4', b = 677.31, c = 468.93. 12. 866 : 1607.
4. C = 26° 12', b = 2276.6, c = 1673.9. 18. 6.438 ; 6.857.
5. C = 47° 14', a = 1340.6, b = 1113.8. 14. 16.688 in.
6. ^ = 108° 60^, a = 63.276, c = 47.324. 16. AB=: 59.664 mi. ;
7. B = 66° 66', b = 6685.9, c = 5357.5. AC= 64.286 mi.
S. B= 77°, a = 630.77, c = 929.48. 18. 4.1366 and 8.6416.
9. a = 6 ; c = 9.669. 17. 6.1433 mi. and 8.7918 mi.
10. a = 7 ; b = 8.673. 18. 6.4343 mi. and 5.7673 mi .
19. 8 and 6.4723.
20. 4.6064 mi. ; 4.4494 mi. ; 3.7733 mi.
21. 6.4709 mi.; 5.8018 mi. ; 4.8111 mi.
Exercise 50. Page 115
1. Two. 3. No solution. 6. One. 7. No solution
2. One. 4. One. 6. Two. 8. One.
9. B= 12° 13' 34", C = 146° 16' 26", c = 1272.1.
10. B = 67° 23' 40", C = 2° 1' 20", c = 0.38626.
11. B = 41° 12' 66", C = 87° 38' 4", c = 116.83.
12. A = 64° 31', C = 47° 46'. c = 60.496.
18, B = 24° 67' 26", C = 133° 48' 34", c = 616.7 ;
R= 156° 2' 34", C'= 3° 43' 26", c'= 65.414.
ANSWERS
15
14. A = 51<' 18' 27", C = 98° 21' 88",
A'= 128° 41' 83", 0'= 20° 58' 27",
16. ^ = 147° 27' 47", B = 16° 43' 13",
A' = 0° 54' 18", i^' = 168° 16' 47",
C = 97° 44' 20",
, C'=5°47'16",
c = 48.098;
c'= 15.598.
a = 35.519 ;
«'= 1.0415.
c = 13.954 ;
c'= 1.4202.
c= 2.7901.
16. i^=44°l'28",
B'= 185° 58' 32",
17. B = 90°, C = 82° 22' 43",
18. 420. 19. 124.62. 20. 8.2096 in.
21. ^B= 8.8771 in.; BO = 2.8716 in. ; CD = 8.7465 in. ; ^D = 6.1817 in.
22. C=:125°6',D=93°24';^B = 4.3075in.;BO=8.1288in.; Ci)=5.431in
DE = 4.4186 in. ; AE = 5.0522 in.
Exercise 51. Page 117
2. 6 = acosC+ccos^; Qg^ _ ?!!_±_£!zi^- 90° !«• ^^ = 1-9249 im ;
a = ftcosC+ccosJ?; * ~ 26c ' ' CD = 4.4431 in. ;
c = b cos-4,
4. Impossible.
6. 5.
6. 7.655.
7. 7.
14. ^0 = 8.499 in.;
BD = 8.1254 in.
16. BC = 5.9924 in. ;
BD= 8.3556 in.
Sxercise 52.
1. - — ^ = tan(^-.45°).
a + 6 ^ '
2. tan J(-4 — B) = 0.
8. a = 6.
4. a + 6 = (a - &)(2 + Vs).
2 sin ^ tan A
Page 119,
V3 V^
18.
^ = 109° 26' ;
J? = 112° 13' 40";
C = 88° 11' 40";
D = 50° 8' 40".
17. 13.3157 in.
^^' "6" -
^=00 V3.
11.
9 or CO = GO.
14. tan J(^- J?) = 0; A=B.
17. 5.
18. Side8AB,BC,AE; diagonal ^ 7) ;
angles B, CAD, DAE.
Exercise 53. Page 121
I. A
= 51° 15', B :
= 56° 30',
c
= 95.24.
2. B
= 60° 45' 2", C :
= 39° 14' 58",
a
= 984.83.
8. A
= 77° 12' 53", B
= 43° 30' 7",
c
= 14.987.
4. B
= 93° 28' 36", C
= 50° 38' 24",
a
= 1.3131.
6. A
= 182° 18' 27", B :
= 14° 34' 24",
c
= 0.6775.
6. ^
= 118° 55' 49", C:
= 45° 41' 35",
5
= 4.1554.
7.
B = 65° 13' 51",
C= 28° 42' 5",
a = 3297.2.
19.
6.
8.
A = 68° 29' 15",
B = 45° 24' 18",
c = 4449.
20.
10.392.
9.
^=117° 24' 32",
J?=32°ir28",
c = 31.431.
21.
^ = ^=90°- JC,
10.
^ = 2° 46' 8",
B = 1° 54' 42",
c = 81.066.
asinC
11.
JL = 116°33'54",
B = 26° 33' 54",
c = 140.87.
c ^^ ■ •
sin^
12.
^ = 6° 1' 55",
B = 108° 58' 5",
c =862.5.
22.
8.9212.
13.
A = 45° 14' 20",
B=17°8'40",
c = 510.02.
28.
25.
14.
A = 41° 42' 33",
B = 32° 31' 15",
c = 9.0398.
24.
3800 yd.
16.
A = 62° 58' 26",
5 = 21° 9' 58",
c = 4151.7.
26.
729.67 yd.
16.
A = 84° 49' 58",
B = 28° 48' 26",
c = 42.374.
26.
480.85 yd.
17.
B= 24° 11' 20",
C = 144° 55' 52",
a = 206.
27.
10.266 mi.
18.
J3 = 20® 86' 84",
C = 102° 10' 14",
a = 37.5.
28.
2.3386 and 6.0032
16
PLANE TRIGONOMETRY
Bxerciae 54. Page 125
1. ^ [log(8 — 6) + log(8 — c) + colog 8 + colog(8 — «)] . 4. log v + colog (8 — a) .
8. ^ [log (8 — 6) + log (8 — c) + colog b + colog c] . 6. log(8— a) + log tan ^A.
8. ^ [log (8— a) + log (8 — 6) + log (8 — e) + colog 8] . 6. The second.
7. V|, or 0.87796 ; 41<' 24' 34^'.
Bxercise 55. Page 127
9. A=:W>.
18. eO°; 60°; eO°.
14. 28° 57' 18''
16. 36° 52' 12"
126° 62' 12"; 14° 15'.
136° 23' 50"; 11° 25' 15
143° 7' 48" ; 9° 31' 40".
56° 6' 36"; 81° 47' 11".
30° 24'; 133° W 24".
57° 59' 44"; 75° 10' 41"
1. 38° 62' 48"
2. 32° 10' 55"
8. 27° 20' 32"
4. 42° 6' 13" ;
6. 16° 25' 36"
6. 46° 49' 35"
7. 26° 29" ; 43° 25' 20" ; 110° 34' 11".
8. 49° 84' 58"
9. 51° 53' 12"
10. 36° 52' 12"
11. 36° 52' 12"
12. 33° 33' 27"
//
//
58° 46' 58"; 71° 38' 4
59° 81' 48"; 68° 35'.
58° 7' 48"; 90°.
53° 7' 48" ; 90°.
33° 83' 27"; 112° 53' 6
46° 34' 6"-; 104° 28' 36
53° 7' 48"; 90°.
16. 8° 19' 9" ; 33° 33' 36" ; 138° 7' 15".
17. 45° ; 120° ; 16*».
18. 45° ; 60° ; 75°.
19. 84° 14' 34".
20. 54° 48' 54".
21. 105° ; 15° ; 60».
22. 54.516.
28. 60°.
24. 12.434 in.
25. 4° 23' 2" W. of N. or W. of &
26. ^ = 90° 37' 3";
B = 104° 28' 41" ;
a= 96° 55' 44";
D = 67° 58' 32".
27. 82° 49' 10".
28. 36° 52: 11 ^5
53° 7' 49".
Exercise 56. Page 128
1. 277.68.
4. 27.891. 7.
10,280.9.
10. 1,067,760.
2. 452.87.
6. 139.53. 8.
82,362.
12. 10.0067 sq.ia
8. 8.0824.
6. 1380.7. 9.
409.63.
18. 18.064 sq. in.
14. 13.41 sq. in.
Exercise 57. Page 129
1. 85.926.
8. 436,540. 6. 7,408,200.
7. 176,384.
9. 92.963.
2. 28,531.
4. 157.63. 6. 398,710.
8. 25,848.
10. 8176.7.
11. 5.729 sq. in.
Exercise 58. Page 131
1. 6.
14. 8160.
29. 13.93 ch., 23.21 ch., 32.60 ch
2. 150.
15. 26,208.
80. 14 A. 5.54 sq. ch.
9. 43.301.
/ 16. 17.3206.
81. 30° ; 30° ; 120°.
4. 1.1367.
17. 10.392.
82. 2,421,000 sq. ft.
6. 10.279.
18. 365.68.
88. 199 A.
8 sq. ch.
6. 16.307;
19. 29,450 ; 6982.8.
84. 210 A.
9.1 sq. ch.
7. 1224.8 sq.
rd.; 20. 15,540.
86. 12 A. 9.78 sq.ch.
7.655 A.
21. 4,333,600.
87. 876.84
sq. ft.
8. 3.84.
22. 13,260.
88. 1229.5
sq. ft.
9. 4.8599.
24. 3 A. 0.392 sq. ch.
39. 9 A. 0.055 sq. eh.
10. 101.4.
25. 12 A. 3.45sq. ch.
41. 1075.3.
11. 62.354.
26. 4 A. 6.634 sq.ch.
42. 2660.4.
12. 0.19076.
27. 61 A. 4.97 sq.ch.
43. 16,281.
18. 240.
28. 4 A. 6.638 sq.ch.
46. Ai«a =
B 06 sin.^.
ANSWERS
17
Bzerciae 59. Page 13S
1. 20 ft.
2. 3r84'6".
8. 30°.
4. 199.70 ft.
6. 106.69 ft.;
142.85 ft.
6. 43.12 ft.
7. 78.36 ft.
8. 75 ft.
9. 1.4442 mi.
10. 56.649 ft.
11. 2159.0 ft.
18. 7912.8 mi.
18. 260.21 ft. ;
3690.3 ft.
14. 2922.4 mi.
16. 60°.
16. 3.2068.
17. 6.6031.
18. 238,410 mi.
19. 1.3438 mi.
30. 861,860 mi.
21. 235.81 yd.
22. 26° 34'.
28. 69.282 ft.
24. 49° 18' 42'' :
.//
40° 41' 18
//
25. 50° 29' 35
39° 30' 25".
26. 74° 44' 14".
27. 360.61 in.
28. 115.83 in.
29. 388.62 in.
80. 83° 37' 40".
81. 97° 11'.
82. 89° 50' 18".
88. 0.2402 ;
1.9216 in. ;
33.306 in.
84. 1.7 in.;
0.588 in.
86.
a — 6
a + 6
86. 30°.
87. 97.86 m.;
153.3 in. ;
159.31 in.
88. 1802.5 ft.;
33° 6' 51".
89. 0.9428.
41. 45 ft.
48. 0.9524^
44.
A* - P - w*
Exercise 60. Page 137
4. 460.46 ft.
8.
422.11yd. 12. 265.78 ft
•
16. 210.44 ft.
6. 88.936 ft.
9.
41.411ft. 18. 529.49 ft
•
18. 19.8; 35.7;
6. 56.664 ft.
10.
234.51ft. 14. 294.69 ft
•
44.5.
7. 51.595 ft.
11.
12,492.6 ft. 16. 101.892 ft.
19. 13.657 mi. per
hour.
OBainO
24. a = : ;
sma
28.
658.361b.; 22° 23' 47'
20. N.76°56'E.;
with first force.
13.938 mi. per
hour.
sin a ;
29.
88.3261b.; 45° 37' 16'*
21. 3121.1ft.;
90°; B=90°;
with known force.
3633.6 ft.
Za = 90°-O.
80.
757.60 ft.
22. 25.433 mi.
86. 288.67 ft.
81.
520.01 yd.
88. 6.3397 mi.
26. 11.314 mi. per hour.
82.
1366.4 ft.
86. 536.28 ft. ; 600.16 ft. 86. 345.46 yd.
87. 61.23 ft.
1. 19,647 sq. ft.
2. 27.527 sq. in.
1. 11.124A.
2. 21.617A.
8. 15.129A.
Exercise 61. Page 141
8. 41.569 sq. in.
4. 6.
Exercise 62. Page 142
4. 14 A.
5. 13.77A.
6. 10.026A.
7. lOA.
8. 4.5348A. ;
10.4652 A.
Exercise 63. Page 144
1. 6.5223 sq. in.
2. 66.2343 sq. in.
8. 3.583 sq. in. ; 27.6565 sq. in.
4. 8.6965 sq. in.
6. 112.26 sq. in.;
201 .9 sq. in.
6. J; iV2.
9. 6.
11. 40,320 sq. ft.
9. 36.38A.
10. 20.07 A.
11. 3.766A.
18. 2.485A.
6. 0.14270.
7. 116.012 sq. in.
8. }.
18 PLANE TRIGONOMETRY
Exercise 64. Page 147
1. 18' 23'';
6. 13' 63";
10. 101.44 mi.
18.385 mi.
20.787 mi.
11. 11.483 mi.
2. 37' 29^' ;
6. 19' 62";
12. 44.6 mi.
37.4776 mi.
12° 67' 8" S.
13. S. 76° 31' 20" E.;
8. 61' 33'';
7. 36.207 mi.
23.2374 mi.
34.446 mi.
8. 16.6296 mi.;
14. N.17°6'14"W.;
4. 37' 16" ;
11' 6.7".
32° 60' 30" N.
7.4136 mi.
9. 69.166 mi.
16. 23.864 mi.;
6. 27.803 mi.; :
N.62<^18'21"W.
S. 66° 68' 34" E.
Exercise 65. Page 148
1. 42^16'N.; 68° 64' 39" W. 2. 103.67 mi. 8. 60°16'N.; 62° 16' 65" W.
Exercise 66. Page 149
1. 31° 26' 16" N. ; 8. 41° 60' 6" N. ; 6. 40° 4' 16" N. ;
41° 44' 23" W. 58° 16' 1" W. 72° 44' 66" W.
2. S. 63° 26' W. ; 4. 16.727 mi. ; 7. 42° 47' 43" N. ;
42.486 mi.; 30° 16' 19" W. 70° 48' 26" W.
16° 14' 62" W. 6. N. 77° 9' 38" W. ;
32° 28' 32" W.
Exercise 67. Page 150
1. 86° 49' 10" S. ; 22° 2' 44" W. ; N. 61<^ 42' W. ; 183.16 mi.
2. 42° 16' 29" N. ; 69° 6' 11" W.; 44.939 mi.
8. 32° 63' 34" S.; 13°1'63"E; 287.16 mi.
4. 41° r 40" N. ; 69° 64' 1" W.
6. 67' 19"; 21.4 mi.
6. 1°37'8"; 46.662 mi.
Exercise 68. Page 152
1. |ir.
6. \%\ir.
9. 270°.
18.
7^30'.
17. II.
21. n.
2. T^ir.
6. 8 IT.
10. 240°.
14.
640°.
18. II.
22. II.
8. /yw.
n.^ir.
11. 210°.
16.
1080°.
19. III.
28. I.
4. ^V"*
8. Jyiir.
12. 225°.
16.
1800°.
20. IV.
24. III.
26. 216°, fir.
28. 33° 46', ^
«".
80. 3437.76';
206,266".
26. 300°, \ft.
29. 0.017453;
81. ' ^ TT radiaiiR.
27. 120°, §ir.
0.0002909.
82. ^ IT radians.
Exercise 69. Page 154
1. 16°, 164% 876°, 624°. 6. 18°, 162°, 878°, 522°.
2. 30°, 150°, 890°, 610°, 750°, 870°. 6. 0.99999996769.
8. 30°, 160°, 890°, 610°, 760°, 870°, 1110°, 1230°. 7. 0.00029088820.
4. 67° 80', 112° 80', 427° 30', 472° 30'. 8. 0.00029088821.
9. 0.00068177632. 10. 0.000682. 11. 0.0176.
ANSWERS
19
1. 60°, 300°.
2. - 60°, - 800°.
8. 26°, 336°,
385°, 696°.
4. 60°, 300°,
420°, 660°.
Exercise 70. Page 155
6. 46°, 225°.
e. - 135°, - 315°.
7. 60°, 240°,
420°, 600°.
8. 30°, 210°,
390°, 670°.
9. 26° 34', 206° 34',
386° 34', 666° 34'.
10. -116° 34', -296° 34-,
- 476° 84', - 666° 84'.
6. 60°, 120°.
6. 45°, 135°.
19. 60°, 240°,
420°, 600°.
20. 68°, 238°,
418°, 698^.
21. 74°, 106°,
434°, 466°.
2. 360° or 2 v,
4. 180°or7r.
18. iV2.
Exercise 71. Page 156
7. 30°, 210°. 9. 60°, 300°. 11. J VS.
8. 90°, 270°. 10. 136°, 226°. 12. J. 14. | V2.
22. 19°, 161°, 25. 19° 28' 17",
379°, 621°. 160° 31' 43".
23. 15° 24' 30", 195° 24' 30", 26. ± ■\^/2
376° 24' 30", 666° 24' 30". 27. ± A Vs or 0.
24. 19°, 341°,
379°, 701°.
1. 270.63.
2. 416.66.
8. 2696.8.
4. 4.168.
5. Impossible.
6. Impossible.
7. 346.48 ft.
Exercise 74. Page 161
6. 180°or7r.
8. 360° or 2 ir.
Exercise 75. Page 162
9. 40' 9".
10. - 176°, 186°,
636°, 646°.
11. - 200°, 160°,
660°, 620°.
12. 2 radians ;
114° 35' 30".
Exercise 77. Page 166
9. 180° and 360°.
10. Complements.
13. ^radian;
19° 6' 66".
22. 30°, 210°,
390°, 670°.
28. 60°, 240°,
420°, 600°.
1. J TT or ^ TT.
2. 90° or 270°.
8. 21° 28' or 168° 32'.
4. 0° or 90°.
6. 30^, 160°, 199° 28', or 340° 32'.
6. 61° 19', 180°, or 308° 41'.
7. 30^, 160^, or 270°.
8. 36° 16', 144° 44', 216° 16', or 324° 44'.
9. 76° 68' or 256° 68'.
10. 60°, 180°, or 300°.
11. 90° or 143° 8'.
12. 30°, 160P, 210°, or 330^.
18. 0°, 120°, 180°, or 240°.
14. 46°, 161° 34', 225°, or 341° 34'.
16. 60°, 120°, 240°, or 300°.
16. 26° 34' or 206° 34'.
17. 30° or 160°.
18. 46° or 136°.
19. 60°, 90°, 270°, or 300°.
20. 60°, 90°, 120°, 240°, 270°, oi
300°.
21. 32°46',147°14',212°46',or327°14'.
a2-l
22. tan-i
2a
28. 0087^ ( = -i— ).
24. 1.
25. 1.
26. 0°, 46P, 90°, 180°, 226°, or 270^.
27. 30°, 150^, 210^, or 330^.
20 PLANE TRIGONOMETRY
38. 80°, eOP, 120°, 16(F, 210°, 240°, 800^, 80. 60°, 90°, 120°, 240°, 270°, or 800°.
or 830°. 61. 0^, 90°, 180°, or 270°.
29. 0°, 66° 42^, 180°, or 204° 18'. 62. 0°, 90°, 120°, 180^, 240°, or 270^.
80. 14° 29', 30°, 160°, or 166° 81'. 68. 0°, 74° 6', 127° 26', 180°, 232° 86'.
81. 0°, 20^, 100°, 140°, 180°, 220°, 260°, or 286° 65'.
or 840°. 64. 0°, 180°, 220° 89', or 819° 21'.
88. 46°, 90°, 136°, 226°, 270°, or 316°. 65. 8° or 168°.
88. 30^, 160°, or 270°. 66. 40°12', 139° 48', 220°12', or 819°48'.
84. 26° 34', 90°, 206° 34', or 270°. 67. 0^, 60°, 120^, 180°, 240°, or 300°.
86. 46°, 136°, 226°, or 316°. 68. 30° or 3**"°.
86. 46°, 186°, 226°, or 316°. 69. 60°, 120°, 240°, or 300°.
87. 16°, 76°, 136°, 196°, 265°, or 316°. 70. 18°, 90°, 162°, 234°, 270^, or 306°.
88. 45°, 136°, 226°, or 316°. 71. 30°, 60°, 120°, 160°, 210°, 240°, 300°,
89. 0° 46°, 180°, or 226°. or 330°.
40. 0°, 90°, 120^, 240°, or 270°. 72. 53° 8', 126° 62', 233° 8', or 306° 52'.
41. 0°, 36°, 72°, 108°, 144°, 180^, 216\ 78. 30°.
262°, 288°, or 324°. 74. 22° 37' or 143° 8'.
42. 120°. 75. 0°, 20°, 30°, 40^, 60°, 80°, 90°, 100^,
48. 64° 44', 126° 16', 234° 44', 306° 16'. 120°, 140^, 160°, 160°, 180°, 200°,
44. 30^, 60°, 90°, 120°, 160°, 210°, 240°, 210°, 220°, 240°, 260°, 270^, 280^,
270^, 30 0°, or 330^. 300^, 320^, 330^, or 340°.
4JI Rin-i 4. E ''®- 22i°. 45°, 67i°, 90°, 112^°, 136°,
=^\ 2 ' 157 J°, 202 J°, 226°, 247^°, 270^,
46. 90°, 216° 62', or 323° 8'. 292 J° 316°, or 337 J°.
47. 30°, 90°, 160°, 210°, 270°, or 330°. 77. 46° or 225°.
48. 0^, 45°, 180°, or 225°. 78. ± 1 or db | Vn.
49. 46°, 60°, 120^, 136°, 226°, 240°, 300°, 79. J VS or - J VS.
or 316°. 80. or ± 1.
50. 0°, 46°, 135°, 226°, or 316°. 81. 0^, 80°, 90°, 150°, 180°, 210°, 270°,
51. 90° or 270°. or 330^.
52. J Vs. 82. 120° or 240°.
58. i. 88. 60°, 120°, 240°, or 300°.
54. 0°, 46°, 90°, 180°, 226°, or 270°. 84. 10° 12', 34° 48', 190° 12', or 214° 48'.
55. 30^, 160^, 210°, or 330°. 85. 29°19', 105° 41', 209°19', or 286°41'.
56. 60°. 86. 0^, 45°, 90°, 180°, 226°, or 270°.
67. 105° or 345°. 87. 0°, 45°, 135°, 225°, or 316°.
58. 136°, 315°, or i8in-i(l - a). 88. 0°, 60°, 120°, 180°, 240°, or 300°.
59. 30°, 60°, 120^, 160^, 210°, 240^, 300°, 89. 27° 68', 136°, 242° 2', or 316°.
or 330°.
Exercise 78. Page 170
1. x = a,y = 0; oTZ = 0,y = a, 4. x = 100, y = 200.
V:; — h K • 1 . /wi— n+1
m + n— 1
y =
^ 2
6. as = 90°,
8. a; = 76° 10', y = l&'9(r. y = 0° or 180°,
y = sln-i ±
laTb
ANSWERS 21
7. X = oofl-i j(a ± V6 - a« + 2) ; y = cos-i \{a ± V6 - a^ + 2).
m
8. X = tan- 1 — tan a + J cos- 1 [2 m« - (2 m« - 2 n^) cos^ a - 1] ;
n
y = tan-i — tana— Jcos-i[2m2 — (2m2 — 2n2)cos2a— 1].
n
9. X = tan-if + cos-i J Va2 + &«; y = tan-i ^ - cos-i A Vo^ + ft*,
10. X = 24° 18', r = 226.12 ; x = 204° IS', r = - 226.12.
11. X = 42° 28', r = 161 ; X = 222° 28', r = - 161.
Exercise 79. Page 171
1. = 30° or 160°; x = 0.184 or 1.866.
2, = sin-i (a — 1) ; x = 2 — a.
8. X = 46°, 136°, 226°, or 816°; /i = 80°, 160°, 210°, or 830°.
^ = iain-i(-^--l)-i8in-i^^.
6. ^ = 0°.
Exercise 80. Page 172
1. fl^ + 6» - 2(a - 6) = - 1. 7. (m8 + n8 - l)^ = (n + l)^ + m^.
2-«^ = l- 8. afcU afet = 1.
^ 1 ^ 11 9. (m + n)V4-(m-n)2 = 2(m-n)
p '^ ^ 10. p'r =— r'p.
^- ^ = ^- • II4 ik* + i* = 2 W(JfeZ - 2).
6. x = ±V2r2/-2^3 + rver8in-iJ. 12. a%%-2 + a^c^ + ^^V* = a^ft^c".
Exercise 81. Page 176
1. 1 ; — 1.
2. 1 ; V^ ; - 1 ; - V^.
8. 1 ; 0.7660 + 0.6428 i ; 0.1736 + 0.9848 i.
4. 1; j(V6-l + iVlO + 2V6); j(- V6- 1 + i Vlo32V6) ;
j(- V5«.l_fVlO-2V6); j(V5 - 1 - i VlO + 2 Ve).
6. 1; i+ JVITS; - J+ jVZrS; -1; -J- jV-S; J-iVZ^.
jV3 + iV3T; V3T; -^V8 + jVZT; - jVs-iV^; -V-i;
6. iV2+iV-2; - jV2+iV=^; _iV2-iV:r2; jV2- JvCi;
22 PLANE TRIGONOMETRY
Exercise 82. Page 177
1. -f +fV-3; -f*. jV:r3;_5^
2. |V2+^V^; - 3^+ 3V-2; -|V2- 3V-2; ^\/2- |\/^.
8. J+lVTs; -l + ^VTs; -8.
4. 2 (cos 86° + i sin 86°) ; 2 (cos 72° + i sin 72°) ; 2 (cos 108° + t sin 108°).
5. 0.9980 + 0.0628 i ; 0.9921 + 0.1253 i ; 0.9823 + 0.1874 1.
Exercise 83. Page 183
»//
7.120. 18.1.64871. 28. tan 66° 40^ 12'
8. 5040. 19. cos 28° 39'. 29. tan 28° 38' 20".
9. 720. 20. cos 7° lO'. 80. tan 86° 23' 16".
10. 40,320. 21. cos 114° 26' 32". 86. 0.6931 + 2 Tri ; 0.6931 -\- 4 m.
11. 3,628,800. 22. cosO°. 86. 1.3862 + 27ri ; 1.3862 -\- ^iri.
12. 604,800. 23. sin 57° 17' 48". 87. 0.3465 + 2 th ; 0.3466 + 4 th.
18. 90. 24. sin 28° 38' 40". 38. 0.6931 + tti ; 0.6931 + 37ri.
14. 42. 26. sin 65° 24' 46" or 89. 1.609 + 2 tti ; 1.609 + 4 Tri ;
16. 15. sin 114° 35' 16". 1.609 + Qiri.
16. 6840. 26. sin 0° or sin 180°. 40. 3.218 + 2 Tri ; 3.218 + 4 Tri ;
17. 7.38883. 27. tanO°. 3.218+ 6 tti.
41. 4.827 + 2Tri ; 4.827 + 4Tri ; 4.827 + 6Tri.
42. 1.609 + Tri; 1.609+ Stti; 1.609 +5Tri.
48. 4.605170 + 2Tri ; 4.606170 + 4Tri.
44. 2.302585 + Tri ; 2.302585 + 3Tri.
46. 6.907765 + 2 Tri ; 6.907766 + 4 iri.
46. 1.161292 + 2Tri ; 1.151292 + 4Tri.
Exercise 84. Page 184
1. 362.8ft.. 2m(ng-l)+2n(mg-l) 12. 6sinC7.
2. 1445.67ft.; 1704.7ft.; ' . (m^ - 1)V - 1) - 4 mn ' 18. 794.73ft.
1622.5 ft. 6. 2.
TRIGONOMETRIC AND
LOGARITHMIC TABLES
BY
GEORGE WENTWORTH
AND
DAVID EUGENE SMITH
GINN AND COMPANY
BOSTON • NEW TORK • CHICAGO • LONDON
ATLANTA • DALLAS • COLUMBUS • BAN FRANCISCO
COPYBIGHT, 1914, BY OEOBGE WENTWOBTH
AND DAVID EUGENE SMITH
ALL KIQHTS BBSBBTED
S25.4
GINN AND COMPANY • PRO-
PRIETORS • BOSTON • U.S.A.
PREFACE
In preparing this new set of tables for the use of students of
trigonometry care has been taken to meet the modern requirements
in every respect, while preserving the best features to be found in
those tables that have stood the test of long use. In our country
the large majority of teachers prefer five-place logarithmic tables,
and for this preference they have cogent reasons. While a five-place
table gives the results to a degree of approximation closer than is
ordinarily required, nevertheless if a student can use such a table it
is a simple matter to use one with four or six places. One who has
been brought up to use a table with only four places, however, finds
it less easy to adapt himself to a table having a larger number of
places. On this account the basal tables of logarithms given in this
book have five decimal places. For the natural functions, however,
four decimal places are quite sufficient for the kind of applications
that the student will meet in his work in trigonometry, and the gen-
eral custom of using four places has been followed in this respect.
Following the usage found in the best tables, unnecessary figures
have been omitted, thus relieving the eye strain. Where, as on
page 28, the first two figures of a mantissa are the same for several
logarithms, these figures are given only in the line in which they
first occur and in the lines corresponding to multiples of five. Where,
however, a table is to be read from the foot of the page upwards, as
well as from the top downwards, the first two figures are given both
at the bottom and at the top of the vacant space, as on page 51, so
that the computer may have no difficulty in seeing them in what-
ever direction the eye is moving over the table.
It will also be seen that great care has been bestowed upon the
selection of a type that will relieve the eye from fatigue as far as
possible, and upon an arrangement of figures that will assist the
computer in every way. It is believed that this care, together with
the attention given to spacing and to the general appearance of
the- page, has resulted in the most usable set of trigonometric and
logarithmic tables that has thus far been printed.
• • •
111
iv PEEFACE
In recognition of the tendency at the present time to use four-place
tables in certain lines of work, Table I has been prepared. Teachers
are advised, however, for the reasons already stated, to use the five-
place table first and until it is clearly understood, taking Table I
for the work that requires only a low degree of approximation.
The tendency to use decimal parts of a degree instead of minutes
and seconds is one that will undoubtedly increase. This tendency
is therefore recognized by the introduction of a conversion table.
By its use the student can instantly adapt the common tables to
the decimal plan. At the same time it is apparent that students
will be called upon to use the sexagesimal division of the degree
almost exclusively for years to come, and for this reason the emphar
sis should be placed, as it is in the authors' Plane and Spherical
Trigonometry, upon the sexagesimal instead of the decimal division.
It is confidently believed that teachers and students will find in
these tables all that they need for the purposes of the computation
required in every line of work in trigonometry.
GEORGE WENTWORTH
DAVID EUGENE SMITH
CONTENTS
PAOB
Introduction 1
Table I. Foub-Place Mantissas of Logarithms of
Integers and Trigonometric Functions . 17
Table II. Circumferences and Areas of Circles . . 24
Table III. Five-Place Mantissas op Logarithms of
Integers from 1 to 10,000 27
Table IV. Proportional Parts 46
Table V. Logarithms op Constants 48
Table VI. Logarithms of Trigonometric Functions . 49
Table VII. Corrections for Small Angles 78
Table VIII. Natural Functions , 79
Table IX. Conversion op Degrees to Eadians . . . 102
Table X. Conversion of Minutes and Seconds to
Decimals of a Degree, and op Decimals
OF A Degree to Minutes and Seconds . . 104
INTKODUCTION
1. Logarithm. The power to which a given number, called the
bdse, must be raised to equal another given number is called the
logarithm of this second given number.
For example, since 10* = 1000,
therefore, to the base 10, 3 = the logarithm of 1000.
In this case IQOO is called the ar Uilogarithm ^ol 3^ this being the number
corresponding to the logarithm.
In this Introduction only the most important facts relating to logarithms
are given. For a more complete treatment see the Wentworth-Smith Plane
and Spherical Trigonometry, Chapter III.
2. Symbolism. For "logarithm of iV" it is customary to write
logiV, If we wish to specify logiV to the base b we write log^iV,
reading this " logarithm of iV to the base 5."
For example, since 2^ = 8, we see that logjS = 8 ; and since 6^ = 25, log526 = 2.
3. Base. We may take various bases for systems of logarithms,
but for practical calculation in trigonometry, 10 is taken as the base.
Logarithms are due chiefly to John Napier of Scotland (1614), but the
base 10 was suggested by Henry Briggs of Oxford. Hence logarithms to the
base 10 are often called Briggs logarithms.
4. Logarithm of a Product. The logarithm of the product of several
nuTtibers is equal to the sum of the logarithms of the numbers.
For if -4 = 10*, then x = log-4;
and if J5 = 10i', then y = \ogB.
Therefore -45 = 10*+", and x-^y = logAB.
For example, log (247 x 7.21) = log 247 + log 7.21.
5. Logarithm of a Quotient. The logarithm of the quotient of two
numbers is equal to the logarithm of the dividend m,inus the logarithm
of the divisor.
For if A = 10*, then x = log-4.;
and if B=zlQff, then y = \ogB.
A A
Therefore — = lO*"", and x — y = log—*
B B
For example, log (9.2 -*- 6.7) = log 9.2 — log 6.7.
1
2 TABLES
6. Log^arithm of a Power. The logarithm of a power of a number
is equal to the logarithm of the number multiplied by the exponent.
For if x = logA, then ^ = 10*.
Raising to the pih power, Ap = 10^.
Hence log AP = px=:plogA,
For example, log 7.2* = 5 log 7.2.
7. Logarithm of a Root. The logarithm of a root of a number is equal
to the logarithm, of the number divided by the index of the root.
For if aj = logJ., then ^ = 10*.
1 X
Taking the rth root, AT = 10^.
Hence logJ = ? = i^.
° r r
For example, log 4^9.36 = \ log 9.86.
8. Characteristic and Mantissa. Usually a logarithm consists of an
integer plus a decimal fraction.
The integral part of a logarithm is called the characteristic.
The decimal part of a logarithm is called the mantissa.
Thus, if log 2353 = 3.37162, the characteristic is 3 and the mantissa is
0.37162. This means that lOS-s^i® = 2363, or that the 100,000th root of the
837,162d power of 10 is approximately 2353.
The logarithms of integral powers of 10 are, of course, integers, the mantissa
in every such case being zero. For example, since 1000 = 10", log 1000 = 3.
9. Finding the Characteristic. The characteristic is not usually
given in a table of logarithms, because it is easily found mentally.
The characteristic of the logarithm^ of a number greater than 1 is
positive and is one less than the number of integral places in the
number,
9
The characteristic of the logarithm, of a number between and 1 is
negative and is one greater than the number of zeros between the deci-
mal point and the first significant figure in the number.
For example, since lO^ = 1000 and 10* = 10,000, it is evident that log 7260
lies between 3 and 4.
For further explanation see the Wentworth-Smith Plane Trigonometry, § 46.
10. The Negative Characteristic. The mantissa is always consid-
ered as positive. If log 0.02 = — 2 + 0.30103, we cannot write it
— 2.30103 because this would mean that both mantissa and character-
istic are negative. Hence the form 2.30103 has been chosen, which
means that only the characteristic 2 is negative.
In practical computation it is more often written 0.30103 — 2, or 8.80108—10,
but when written by itself the form 2.80103 is convenient.
INTRODUCTION 3
11. Mantissa independent of Decimal Point. The mantissa of the
logarithm of a number is unchanged by any change in the position
of the decimal point of the number.
For if 108.87107 = 2350, then log 2360 = 8.37107.
Dividing by 10, 10287107 = 236, and log 286 =2.37107.
That is, the mantissa of log 2360 is the same as that of log2S6.0, and so on,
wherever the decimal point is placed.
This is of great importance, for if the table gives the mantissa for only 236,
we know that this is the mantissa for 0.236, 2.86, 23.6, 236,000, and so on.
12. Logarithms Approximate. Logarithms are, in general, only
approximate. Although log 1000 is exactly 3, log 7 is only approxi-
mately 0.84510.
To four decimal places, log 7 = 0.8461 ; to five decimal places, 0.84610 ; to
six decimal places, 0.846098, and so on.
In a four-place table there is a possible error of J of 0.0001 ; in a five-place
table, of ^ of 0.00001, and so on, but in each case the probable error is much less.
If several logarithms are added the possible error is correspondingly increased.
In findjn g^ antiloegirithm s the firgt figure found by interpolation is usually
accurate, the second is doubtful, and the third is rarely trustworthy.
13. Cologarithm. The logarithm of the recipicocal of a number is
called the cologarithm of the number.
The cologarithm of x is expressed thus : colog x.
Since colog x = log - = log 1 — log x = — logo;, we have
X
colog X =— logx.
That is, colog 2 = — log 2.
To avoid a negative mantissa this may be written
colog X = 10 — log X — 10.
For example, colog 2 = — log 2 = 10 — 0.30103 — 10
= 9.69897-10.
14. Use of the Cologarithm. Since to divide by a number is the
same as to multiply by its reciprocal, instead of subtracting the
logarithm of a divisor we may add its cologarithm.
The cologarithm of a number is easily written by looking at the logarithm
in the table. Thus, since log 20 = 1.30103, we find colog 20 by mentally sub-
tracting this from 10.00000 — 10. This is done by beginning at the left and
subtracting the number represented by each figure from 9, except the right-
hand figure, which we subtract from 10.
For example, if we have to simplify
625 X 7.51
2.73x14.8'
it is easier to add log 626, log 7.61, colog 2.73, and colog 14.8, than to add log
626 and log 7.61, and then to add log 2.73 and log 14.8, and finally to subtract.
4 TABLES
15. General Use of the Tables. In writing down a logarithm always
write the characteristic before looking for the mantissa. Otherwise
the characteristic may be forgotten.
Some computers find it convenient to paste paper tabs so that
they project from the side of the first page of each table, thus allow-
ing the book to be opened quickly at the desired table.
Although a table of proportional parts is given, it is best to ac-
custom the eye to interpolate quickly from the regular table.
TABLE I
16. Kature of Table I. This is a table of logarithms of integers
from 1 to 1000, and of the sine, cosine, tangent, and cotangent, the
mantissas extending to four decimal places and the characteristics
being 10 too large, as in Table VI. For the ordinary computations
of physics and mensuration this is sufficient, the results in general
being correct to four figures.
There is a growing disposition to use the convenient four-place table for
ordinary work. Most teachers prefer, however, to use a five-place table, since
the student who can use this will have no trouble with the simpler four-place
table. For this reason the computations in the Wentworth-Smith Plane and
Spherical Trigonometry are based upon the five-place tabl3.
17. Arrangement of the Table. The vertical columns headed N con-
tain the numbers, and the other columns the logarithms. On page 17
the characteristics as well as the mantissas are given, but on pages 18
and 19 only the mantissas are given, the characteristics being deter-
mined by § 9. To find the mantissa for 16, look on the line to the right
of 16 and in the column marked O. This mantissa, 0.2041, is the same
as that for 1.6, 160, 1600, and so on. To find the mantissa for 167, look
on the line to the right of 16 and in the column marked 7. This man-
tissa, 0.2227, is the same as that for 0.167, 16.7, 167,000, and so on.
The table of trigonometric functions is arranged for every 10', this
being sufficient for many practical purposes.
18. To find a Logarithm or Antilogarithm. The method of finding
the logarithm of a number or the antilogarithm of a logarithm is
the same as that employed with a five-place table (§§ 21-24).
TABLE II
19. Kature of Table II. This table (pages 24 and 26) contains the
circumferences and areas of circles of given radii, and the diam-
eters of circles of given circumference or given area. It often saves
a considerable amount of computation in problems involving circles,
cylinders, spheres, and cones.
INTRODUCTION 5
TABLE III
20. Arrangement of Table in. In this table (pages 27-45) the ver-
tical columns headed N contain the numbers, and the other columns
the logarithms. On page 27 both the characteristic and the mantissa
are printed. On pages 28-46 the mantissa only is printed, and the
decimal point and unnecessary figures are omitted so as to relieve
the eye from strain.
The fractional part of a logarithm is only approximate, and in a
five-place table all figures that follow the fifth are rejected.
Thus, if the mantissa of a logarithm written to seven places is 5326143 it is
written in this table (a five-place table) 63261. If it is 5329788 it is written
53298. If it is 5328461 or 5328499 it is written in this table 53285. If the man- ^
tissa is 5325506 it is written 53255 ; and if it is 5324486 it is vmtten 53245. ^
21. To find the Logarithm of a Kumber. If the given number con-
sists of one or two significant figures, the logarithm is given on
page 27. If zeros follow the significant figures, or if the number
is a proper decimal fraction, the characteristic must be determined.
If the given number has three significant figures, it will be found
in the column headed N (pages 28-45) and the mantissa of its loga-
rithm will be found in the next column to the right.
For example, on page 28, log 145 = 2.16137, and log 14500 = 4.16137.
If the given number has four significant figures, the first three
will be found in the column headed N, and the fourth will be found
at the top of the page in the line containing the figures 1, 2, 3, etc.
The mantissa will be found in the column headed by the fourth figure.
For example, on pages 41 and 44 we find the following :
log 7682 =3.88547,. log 76.85 =1.88564;
log 93280 = 4.96979, log 0.9468 = 9.97626 - 10.
22. Interpolation for Logarithms. If the given number has five
or more significant figures, a process called interpolation is required.
Interpolation is based on the assumption that between two consecutive man-
tissas of the table the change in the mantissa is directly proportional to the
change in the number. This assumption is not exact, but the error does not,
in general, affect the first figure found in this manner.
For example, required the logarithm of 34237.
The required mantissa is (§ 11) the same as the mantissa for 3423.7 ; therefore
it will be found by adding to the mantissa of 3423 seven tenths of the difference
between the mantissas for 3423 and 3424.
The mantissa for 3423 is 53441, and the mantissa for 3424 is 53453.
The difference between these mantissas (tabular difference) is 12.
Hence the mtotissa for 3423.7 is 53441 + (0.7 of 12) = 53449.
Therefore the required logarithm of 34237 is 4.53449.
6 TABLES
23. To find the Antilogarithm. If the given mantissa can be found
in the table, the first three significant figures of the required number
will be found in the column headed N in the same line with the
mantissa, and the fourth figure at the top of the column containing
the mantissa. The position of the decimal point is determined by
the characteristic (§ 9).
1. Find the antilogarithm of 0.92002.
The number for the mantissa 92002 is 8318. (Page 42.)
The characteristic is ; therefore the required number is 8.318.
2. Find the antilogarithm of 6.09167.
The number for the mantissa 09167 is 1235. (Page 28.)
The characteristic is 6 ; therefore the required number is 1,236,000.
3. Find the antilogarithm of 7.60326 - 10.
The number for the mantissa 50325 is 3186. (Page 32.)
The characteristic is — 3 ; therefore the required number is 0.003186.
SS4. Interpolation for Antilogarithms. If the given mantissa cannot
be found in the table, find in the table the two adjacent mantissas
between which the given mantissa lies, and the four figures corre-
sponding to the smaller of these two mantissas will be the first four
significant figures of the required number. If more than four figures
are desired, they may be found by interpolation, as in the following
examples :
1. Find the antilogarithm of 1.48762.
Here the two adjacent mantissas of the table, between which the given man-
tissa 48762 lies, are found to be (page 32) 48756 and 48770. The antilogarithms
are 3073 and 3074. The smaller of these, 3073, contains the first four significant
figures of the required number.
The difference between the two adjacent^ mantissas is 14, and the difference
between the corresponding numbers is 1.
The difference between the smaller of the two adjacent mantissas, 48766, and
the given mantissa, 48762, is 6. Therefore the number to be annexed to 3073
is ^ of 1, which is 6.43, and the fifth significant figure of the required anti-
logarithm is 4.
Hence the required antilogarithm is 30. 734.
2. Find the antilogarithm of 7.82326 - 10.
The two adjacent mantissas between which 82326 lies are (page 39) 82321
and 82328. The antilogarithm having the mantissa 82321 is 6656.
The difference between the two adjacent mantissas is 7, and the difference
between the corresponding numbers is 1.
The difference between the smaller mantissa, 82321, and the given mantissa,
82326, is 5. Therefore the number to be annexed to 6656 is f of 1, which is
0.7, and the fifth significant figure of the required antilogarithm is 7.
Hence the required antilogarithm is 0.0066667.
INTRODUCTION 7
TABLE IV
25. Proportional Parts. In interpolating (§§ 22, 24) we hare to
find fractional parts of the difference between two numbers or two
logarithms.
For example, in finding log 73.637 we see that
log 73.54 = 1.86652
log 73.53 = 1.86646
Tabular difference = 6
Yq tab¥ilar difference = 4
Adding 1.86646 and 0.00004, we have
log 73.637 = 1.86650
These fractional parts of a tabular difference are called jprapor-
tional parts,
26. Nature of Table IV. In Table IV the proportional parts of all
differences from 1 to 100 are given, so that by turning to the table
we can make any ordinary interpolation at a glance.
For example, if the difference (D) is 6, as in the first case considered in § 24,
the table shows that -j^ of this difference is 4.2, the last figure being rejected
because it is less than 6. In such a simple case, however, we would make the
interpolation mentally, without reference to the table.
If the difference were 87, and we wished -^ of this difference, the table
shows at once that this is 78.3, from which we would reject the last figure
as before.
In some sets of tables the proportional parts are printed beside the loga-
rithms themselves, but this necessitates the use of a small type that is trying
to the eyes. It is usually easier to make the interpolation mentally than to use
the table of proportional parts, but where a large number of interpolations are
to be made at the same time the table is helpful.
27. Table IV for Multiplication. By ignoring the decimal points
Table IV may be used as a multiplication table, the column marked
D containing the multiplicands, the multipliers 1-9 appearing at
the top, and the products being given below.
For example, 8 x 79 = 632, as is seen by looking to the right of 79 and
under 8.
TABLE V
28. Logarithms of Constants. There are certain constants, such as
TT, w^, 2 TT, V2, and so on, that enter frequently into the computations
of trigonometry. To save the trouble of looking for the logarithms
of these numbers in the regular table, or of computing their loga-
rithmSy Table Y has been prepared.
8 TABLES
TABLE VI
29. Nature of Table VI. This table (pages 49-77) contains the
logarithms of the trigonometric functions of angles. In order to
avoid negative characteristics, the characteristic of every logarithm
is printed 10 too large. Therefore — 10 is to be annexed to each
logarithm.
On pages 49-55 the characteristic remains the same throughout each column
and is printed at the top and the bottom of the column ; but on pages 56-77
when the characteristic changes one unit in value the place of each change is
marked with a bar. Above each bar the proper characteristic is printed at the
top of the column ; below each bar the characteristic is printed at the bottom.
On pages 56-77 the log sin, log cos, log tan, and log cot are given
for every minute from 1° to 89°. Conversely, this part of the table
gives the value of the angle to the nearest minute when log sin,
log COS, log tan, or log cot is known, provided log sin or log cos lies
between 8.24186 and 9.99993, and log tan or log cot lies between
8.24192 and 11.75808.
If the exact value of the given logarithm of a function is not found in the
table, the value nearest to it is to be taken unless interpolation is employed
as explained in § 30.
If the angle is less than 45° the number of degrees is printed at
the top of the page, and the number of minutes in the column to
the left of the columns containing the logarithms. If the angle is
greater than 45° the number of degrees is printed at the bottom
of the page, and the number of minutes in the column to the right
of the columns containing the logarithms.
If the angle is less than 45P the names of its functions are printed at the
top of the page ; if greater than 46°, at the bottom of the page. Thus,
log sin 21° 37' = 9.66631-10. Page 66
log cot 36° 63' = 10.12473 - 10 = 0.12473. Page 73
log cos 69° 14' = 9.64969-10. Page 65
log tan 45° 69' = 10.01491 - 10 = 0.01491. Page 77
log tan 75° 12' = 10.57806 - 10. Page 62
log cos 82° 17'= 9.12799-10. Page 59
If log cos X = 9.87468 - 10, x = 41° 28^. Page 76
If log cot X = 9.39353 - 10, x = 76° 6'. Page 62
If log sin X = 9.99579 - 10, x = 82° 2'. Page 59
Iflogtanx = 9.02162- 10, x = 6°. Page 58
If logsin = 9.47760 — 10, the nearest log sin in the table is 9.47774 — 10
(page 64), and the angle corresponding to this value is 17° 29'.
If log tan = 0.76520 = 10.76520 — 10, the nearest log tan in the table is
10.76490 — 10 (page 60), and the angle corresponding to this value is 80° 15'.
For the method of interpolating, see § 30.
INTRODUCTION 9
30. Interpolation. If it is desired to obtain the logarithm of the
function of an angle that contains seconds, or to obtain the value of
an angle in degrees, minutes, and seconds from a logarithm of a func-
tion, interpolation must be employed. The theory of interpolation
has already been given in §§ 22 and 24.
Here it must be remembered that the difference between two consecutive
angles in the table is 1^, and that therefore a proportional part of 60^' must be
taken. It must also, be remembered that log sin and log tan increase as the
angle increases, but log cos and log cot diminish as the angle inc^ases.
1. Find log tan 70* 46' 8".
Log tan 70° 46' = 0.45731. (Page 65.)
The difference between the mantissas of log tan 70° 46' and log tan 70° 47'
is 41, and ^ of 41 = 5.
As the function is increasing, the 5 must be added to the figure in the fifth
place of the mantissa 45731 ; therefore log tan 70° 46' 8" = 0.45736.
2. Find log cos 47° 36' 4".
Log cos 47° 35' = 9.82899 - 10. (Page 76.) '
The difference between this mantissa and the mantissa of log cos 47° 36'
is 14, and -^ of 14 = 1.
As the function is decreasing, the 1 must be subtracted from the figure in
the fifth place of the mantissa 82899 ; therefore log cos 47° 35' 4" = 9.82898 - 10.
3. Find X when log sin x = 9.45359 — 10.
The mantissa of the nearest smaller log sin in the table is 45334. (Page 63.)
The angle corresponding to this value is 16° 30'.
The difference between 45334 and the given mantissa, 45359, is 25.
The difference between 45334 and the next following mantissa, 45377, is 43
(the tabular difference) and H of 60"= 35".
As the function is increasing, the 35" must be added to 16° 30' ; therefore
the required angle is 16° 30' 35".
4. Find x when log cot x = 0.73478.
The mantissa of the nearest smaller log cot in the table is 73415. (Page 60.)
The angle corresponding to this value is 10° 27'.
The difference between 73415 and the given mantissa is 63.
The difference between 73415 and the next larger mantissa is 71 (the
tabular difference) and |-| of 60"= 63".
As the function is decreasing, the 53" must be subtracted from 10° 27';
therefore the required angle is 10° 26' 7".
5. Find x when log cos x = 0.83584.
The mantissa of the nearest smaller log cos in the table is 83446. (Page 57.)
The angle corresponding to this value is 86° 5'.
The difference between 83446 and the given mantissa is 138.
The tabular difference is 184, and -ffl of 60" is 45".
As the function is decreasing, 45" must be subtracted from 86° 5' ; therefore
X = 86° 5' - 45", or 86° 4' 15".
10 TABLES
31. The Secant and Cosecant. In working with logarithms we very
rarely use either the secant or the cosecant ; for sec x = 1/cos x, and
log sec X = colog cos a;. If, however, log sec or log esc of an angle
is desired, it may be found from the table by the formulas,
sec -4 = , hence log sec A = colog cos A ;
GoaA
CSC -4 = , hence log esc A = colog sin^l.
« sin^
For example,
log sec 8° 28' = colog cos 8° 28' =0.00476. Page 59
log CSC 18° 86' = colog sin 18° 86' =0.49626. Page 64
log sec 62° 27' = colog cos 62° 27' = 0.83487. Page 69
log CSC 69° 36' 44" = colog sin 69° 36' 44" = 0.06418. Page 70
32. Functions of Small Angles. If a given angle is between 0^ and
1®, or between 89® and 90°; or, conversely, if a given log sin or
log cos does not lie between the limits 8.24186 and 9.99993 in the
table ; or if a given log tan or log cot does not lie between the
limits 8.24192 and 11.7*5808 in the table,— then pages 49-66 of
Table VI must be used.
On page 49, log sin of angles between 0° and 0° 3', and log cos of
the complementary angles between 89° 57' and 90°, are given to
every second ; for the angles between 0° and 0° 3', log tan = log sin,
and log cos = 0.00000 ; for the angles between 89° 67' and 90°,
log cot = log cos, and log sin = 0.00000.
On pages 50-62, log sin, log tan, and log cos of angles between
0° and 1°, or log cos, log cot, and log sin of the complementary
angles between 89° and 90°, are given to every 10".
When log tan and log cot are not given, they may be found by the formulas,
log tan = colog cot. log cot = colog tan.
Conversely, if a given log tan or log cot is not contained in the table, then
the colog must be found ; this will be the log cot or log tan, as the case may be,
and will be contained in the table.
On pages 63-55 the logarithms of the functions of angles
between 1° and 2°, or between 88° and 89°, are given in the manner
employed on pages 50-52. These pages should be used if the angle
lies between these limits, and if not only degrees and minutes but
degrees, minutes, and multiples of 10" are given or required.
When the angle is between 0° and 2°, or 88° and 90°, and a greater degree
of accuracy is desired than that given by the table, interpolation may be em-
ployed with some degree of safety ; but for these angles interpolation does not
always give true results, and it is better to use Table YII.
INTRODUCTION 11
33. Illustrative Problems. The following problems illustrate the
use of Table VI for small angles :
1. Find log tan 0° 2' 47", and log cos 89° 37' 20".
log tan OP 2' 47'' = log sin 0* 2' 47'' = 6.90829 - 10. Page 49
log cos 89° 37' 20" = 7.81911 - 10. Page 51
2. Find log cot 0° 2' 15".
10 - 10
log tan 0° 2' 16"= 6.81691-10 Page 49
Therefore log cot 0° 2' 16"= 3.18409
3. Find log tan 89° 38' 30".
10 - 10
log cot 89° 38' 30" = 7.79617-10 Page 61
Therefore log tan 89° 38' 30" = 2.20883
4. Find x when log tan aj = 6.92090 - 10.
The nearest log tan is 6.92110 — 10 (page 49), and the angle corresponding
to this value of log tan is 0° 2' 62".
5. Find x when log cos x = 7.70240 — 10.
The nearest log cos is 7.70261 — 10. Page 60
The corresponding angle for this value is 89° 42' 40".
6. Find X when log cot x = 2.37368.
This log cot is not contained in the table.
The colog cot = 7.62632 — 10 = log tan.
The log tan in the table nearest to this is (page 60) 7.62610 — 10, and the
angle corresponding to this value of log tan ia 0° 14' 30".
34. Angles between 90° and 360°. If an angle x is between 90° and
360°, it follows, from formulas established in trigonometry, that,
Between Off" and 180'' Between ISff" and 27 ff*
log sin X = log sin (180° — x) log sin x = log sin (x — 180°)„
log cos X = log cos (180° — x)^ log cos x = log cos (x — 180°)„
log tan X = log tan (180° — «)„ log tan x = log tan (x — 180°)
log cot X = log cot (180° — x)^ log cot x = log cot (x — 180°)
Between 270'' and SeO''
log sin X ^ log sin (360° — x\
log cos X = log cos (360° — x)
log tan X = log tan (360° — x\
log cot X = log cot (360° — x\
In these formulas the subscript n means that the function is negative.
The logarithm of a negative number is imaginary, so we have to take the loga-
rithm of the number as if it were positive ; but when we find the function itself
we must treat it as negative.
12 TABLES
TABLE VII
35. Nature of Table VII. This table (page 78) must be used when
great accuracy is desired in working with angles between 0** and 2**
or bet^/e^n 88* and 90^
The values of S and T are such that when the angle a is
expressed in seconds, ^ , . , „
5 = log sm a — log a'\
T === log tan a — log a".
Hence follow the formulas given on page 78.
The values of S and T are printed with the characteristic 10 too
large, and in using them — 10 must always be annexed.
36. niustratiye Problems. The following problems illustrate the
use of Table VII for angles near 0** or 90® :
1. Find log sin (f 58' 17". 3. Find log tan 0' b2' 47.6".
:'/
0° 68' IT' = 3497'' 0° 62' 47.6" = 3167.6'
log 3497 = 3.64370 log 3167.6 = 3.60072
8 = 4.68666 - 10 T = 4.68661 -- 10
log sin 0° 68' 17" = 8.22926 - 10 log tan QP 62' 47.6" = 8.18633 - 10
2. Find log cos 88® 26' 41.2". 4. Find log tan 89® 64' 37.362".
90° - 88° 26' 41.2" = 1° 33' 18.8" 90° - 89° 64' 37.362" = 0° 6' 22.638"
= 6698.8" = 322.638"
log 6698.8 = 3.74809 log 322.638 = 2.60871
8 = 4.68662 - 10 T = 4.68668 -- 10
log cos 88° 26' 41.2" = 8.43361 - 10 log cot 89° 64' 37.362" = 7.19429 - 10
This is nearer than by page 64. log tan 89° 64' 37.362" = 2.80671
5. Find x when log sin aj = 6.72306 - 10.
6.72306-10
8 = 4.68667 - 10
Subtracting, 2.03749 = log 109.016
and 109.016" = 0° 1' 49.016"
6. Find x when log cot x = 1.67604.
colog cotx = 8.32396 — 10
T = 4.68664 - 10
Subtracting, 3.63832 = log 4348.3
and 4348.3" = 1° 12' 28.3"
7. Find x when log tan x = 1.66407.
colog tan X = 8.44693 — 10
• T = 4.68669 - 10
Subtracting, 3.76024 = log 6767.6
6767.6" = 1° 36' 67.6"
and 90° - 1° 36' 67.6" = 88° 24' 2.4"
Therefore the angle required is 88° 24' 2.4".
INTRODUCTION 18
TABLE VIII
37. Nature of Table Vni. This table (pages 79-101) contains the
natural sines, cosines, tangents, and cotangents of angles frcm 0** to
90®, et intervals of 1'. If greater accuracy is desired, interpolation
may be employed.
The table is arranged on a plan similar to that used in Table VL
Angles from 0° to 44*^ are listed at the top of the pages, the minutes being
read downwards in the left-hand column. Angles from 46° to 89° are listed at
the bottom, the minutes being read upwards in the right-hand column.
The names of the functions at the top of the columns are to be used in read-
ing downwards, and those at the bottom are to be used in reading upwards.
38. niustrative Problems. The following problems illustrate the
use of Table VIII :
1. Find sin 6** 29'.
We find directly from the table (page 82) that
sin 6° 29^ = 0.0966
2. Find cot 78° 18'.
We find directly from the table (page 86) that
cot 78° 18' = 0.2071
3. Find cos 42° V 30".
From the table (page 100), cos 42° T = 0.7418
Tabular difference = 0.0002.
1^ of this difference = 0.0001
Since the cosine is decreasing, we subtract.
.-. cos 42° T 30'' = 0.7417
4. Find tan 76° 36' 26".
From the tiable (page 86), tan 76° 35' = 8.8900
Tabular difference = 0.0047.
II of this difference = 0.00196 = 0.0020
Since the tangent is increasing, we add.
.-. tan 75° 35' 25" = 3.8920
TABLE IX
39. Katuie of Table IX. TMs table converts degrees to radians,
and also degrees and parts of a degree indicated by 10', 20', 30', 40',
and 60'.
40. Illustrative Problems. The following problems illustrate the
use of Table IX :
1. Express 62° as radians.
From the table, 62° = 1.0821 radians.
2. Expreas 82° 40' as radians.
From the table, 82° 4^ = 1.4428 radians.
14 TABLES
TABLE X
41. Kataie of Table X. This table converts minutes to thousandths
of a degree, and seconds to ten-thousandths of a degree, this being
accurate enough for all the purposes of elementary trigonometry.
It also converts thousandths of a degree, from 0.001" to 0.009**, to
seconds; and hundredths of a degree to minutes and seconds, so
that a computer who has the decimal divisions of an angle given can
easily find the sexagesimal equivalent.
Table X thus provides for using the decimal divisions of the
degree instead of the ancient sexagesimal division into minutes
and seconds.
There seems to be little doubt that the cumbersome division of the degree
into 60 minutes, and the minute into 60 seconds, will disappear in due time, by the
introduction either of the grade (0.01 of a right angle) divided decimally or of
decimal divisions of the degree. At present, however, it must be remembered
that our instruments for the measure of angles are generally arranged upon
the sexagesimal scale, and that we can serve the new system best by making
the change gradually. It is of first importance that the student shall learn how
to use the common sexagesimal system.
42. Illustrative Problems. The following problems illustrate the
use of the table:
1. Find sin 21.34^
By Table X, 0.34° = 20^ 24''
Hence we have to find sin 21° 20^ 24''.
By Table VIII, sin 21° 2(y 24" = 0.36390
2. Find log tan 16.963^
By Table X, 0.96° = 67' 36"
and 0.003°= 11
//
.-. 15.963° = 15° 57' 47"
By Table V, log tan 15° 57' 47" = 9.45644- 10
3. Find COS 63.72^
By Table X, 0.72° = 43' 12"
Hence we have to find cos 63° 43' 12".
By Table VIII, cos 63° 43' 12" = 0.4427
4. Find tan 68.661^
By Table X, 0.661° = 39' 4"
Hence we have to find tan 68° 39' 4".
By Table VIII, tan 68° 39' 4" = 2.5638
6. Find log cot 66.388^
By Table X, 0.388° = 23' 17"
Hence we have to find log cot 66° 23' 17".
By Table VIII, log cot 56° 23' 17" = 9.82262
INTRODUCTION 16
EXERCISE
Umig Table /, find the logarithms of thefollomng :
1. 75.
7.
67.8. 13. 0.726.
19. 8.
25.
140.
2. 96.
8.
42.6. 14. 7.260.
20. 0.8.
26.
141.
3. '37.
9.
93.9. 15. 72.60.
21. 0.08.
27.
14.2.
4. 423.
10.
4.27. 16. 24.3.
22. 0.008.
28.
1.43.
5. 668.
11.
6.42. 17. 2.43.
23. 8.08.
29.
0.144.
6. 647.
12.
7.63. 18. 0.243.
24. 8.80.
30.
0.146.
Using Table L
\ find the antihgarithms
of the following :
31. 1.4771.
37. 2.6988.
43.
1.9610.
49.
1.9618.
32. 0.9031.
38. 1.6690.
44.
0.9607.
50.
2.8978.
33. 1.7076.
39. 4.6749.
45.
3.9763.
51.
0.9336.
34. 1.9031.
40. 3.9696.
46.
2.6196.
52.
4.8460.
35. 1.9346.
41. 0.9681.
47.
0.6360.
53.
1.3714.
36. 0.8461.
42. 2.8494.
48.
2.6640.
54.
2.4448.
Using Table ij find the logarithms of ih^ following :
55. logsin29^ 61. log sin 6* 10'. 67. log sin 20^0'.
56. log cos 42^ 62. log cos 7** 20'. 68. log cos 42** 20'.
57. logtan61^ 63. log tan 6** 30'. 69. log tan 37* 60'.
68. log cot 20°. 64. log cot 8* 60'. 70. log cot 82* 40'.
69. log sin 46*. 65. log sin 46* 10'. 71. log sin 22* 30'.
60. log cos 46*. 66. log cos 44* 80'. 72. log tan 81* 10'.
Using Table I, find the valine of x in the following :
73. log sin X = 9.7861. 79. log sin x = 9.8068.
74. log sin X = 9.9116. 80. log cos x = 9.9262.
76. log tan a; = 9.9772. 81. log cos x = 9.9101.
76. log tan a = 9.8771. 82. log tan a = 8.9118.
77. log cos X = 9.9089. 83. log tan aj = 9.0093.
78. log cot X = 10.0711. 84. log cot X = 10.1944.
Using Table Illy find the logarithms of tJiefollomng :
85. 1476. 88. 664.8. 91. 29.37. 94. 0.4236.
86. 2836. 89. 392.7. 92. 42.86. 96. 0.09873.
87. 4293. 90. 686.4. 93. 63.91.' 96. 487.48.
Using Table HI, find the antilogarithms of the following :
97. 2.02078. 100. 0.82766. 103. 2.96873. 106. 0.70804.
98. 3.66967. 101. 1.82988. 104. 3.81792. 107. 2.34404.
99. 1.76686. 102. 2.96062. 105. 1.82726. 108. 3.36064.
16 TABLES
Umig Table VZfind thefoUowing hgarithms:
109. log sin 10**. 116. log sin 1' SI*'. 123. log sin 10' 37".
110. log sin 30**. 117. log tan 37' 60". 124. log cot 67** 42'.
111. log sin 60°. 118. log cos nO'. 126. log cos 32** 36' 10".
112. log sin 79**. 119. log cot 88** 24'. 126. log tan 73** 42' 16".
113. log cos 87**. 120. log sin 19** 37'. 127. log sin 16** 16' 16".
114. log tan 33**. 121. log cos 72** 43'. 128. log cos 29** 32' 40".
116. log cot 72**. 122. log cot 88** 18'. 129. log cot 78** 33' 26".
Using Table V% find the valtie of x in the follotving :
130. log sin X = 9.62663. 133. log sin x = 9.93386.
131. log cot X = 9.67668. 134. log cot a; = 9.76837.
132. log cos X = 9.73436. 136. log cos x = 9.99843.
U»ing Table IV, find th£ follotving :
136. 0.8 of 37. 137. 0.6 of 79. 138. 0.7 of 68. 139. 0.9 of 29.
U»ing Table F", find the follotving :
140. log 4 TT. 141. log ^. 142. log67.2968^ 143. log ^.
TlBing Table VII,find the follotving :
144. log sin 67". 146. log sin 48". 146. log tan 89** 68' 10".
Using Table V^ find the follotving :
147. 27r . 87. 148. tt • 761 149. ^. 160. — .
27r 47r
Using Table VUl^find the follotving :
151. sin 10** 17'. 155. cos 46** 38'. 169. cot 1** 62'.
162. sin 37** 40'. 166. cos 78** 19'. 160. cot 63** 48'.
153. sin 68** 10'. 157. tan 16° 29'. 161. cot 10** 9^ 10".
164. cos 10** 39'. 168. tan 88** 8'. 162. cot 6** 17' 8".
163. The angles whose sines are 0.6113 and 0.7801.
Using Table IX^ express the follotving :
164. b2^ 40' as radians. 166. 0.8116 radians as degrees.
Using Table Xy express the follotving :
166. 31' as a decimal of a degree. 167. 0.96** as minutes and seconds.
17
TABLE I
FOUR- PLACE MANTISSAS
OF THE COMMON LOGARITHMS OF
INTEGERS FROM 1 TO 1000
AND OF THE TRIGONOMETRIC FUNCTIONS
On this page the logarithms of integers from 1 to 100 are given in
full, with characteristics as well as mantissas. On account of the great
differences between the successive mantissas, interpolation cannot safely be
e mployed on -t hi s pag e. On pages 18 and 19 are given the mantissas of
numbers &om 100 to 1000, and on pages 20-23 the logarithms of trigono-
metric functions.
1-100
N
log;
N
log
N
log
N
log
N
log
1
0.0000
21
1. 3222
41
1. 6128
61
1. 7853
81
1. 9085 ,
2
0. 3010
22
1. 3424
42
1. 6232
62
1. 7924
82
1. 9138
3
0. 4771
23
1. 3617
43
1.6335
63
1.7993
83
1. 9191
4
0.6021
24
1.3802
44
1.6435
64
1.8062
84
1. 9243
5
0.6990
25
1. 3979
45
1. 6532
65
1. 8129
85
1.9294
6
0. 7782
26
1. 4150
46
1.6628
66
1.8195
86
1. 9345
7
0. 8451
27
1. 4314
47
1. 6721
67
1. 8261
87
1. 9395
8
0.9031
28
1.4472
48
1. 6812
68
1. 8325
88
1.9445
9
0. 9542
29
1.4624
49
1.6902
69
1. 8388
89
1.9494
10
1.0000
30
1. 4771
50
1.6990
70
1.8451
90
1. 9542
11
1.0414
31
1. 4914
51
1. 7076
71
1. 8513
91
1.9590
12
1. 0792
32
1. 5051
52
1.7160
72
1. 8573
92
1.9638
13
1. 1139
33
1. 5185
53
1. 7243
73
1.8633
93
1.9685
14
1.1461
34
1. 5315
54
1. 7324
74
1.8692
94
1. 9731
15
1. 1761
35
1.5441
55
1.7404
75
1. 8751
95
1.9777
16
1.2041
36
1. 5563
56
1. 7482
76
1.8808
96
1. 9823
17
1.2304
37
1.5682
57
1. 7559
77
1.8865
97
1.9868
18
1. 2553
38
1. 5798
58
1.7634
78
1.8921
98
1.9912
19
1. 2788
39
1.5911
59
1.7709
79
1. 8976
99
1.9956
20
1. 3010
40
1.6021
60
1. 7782
80
1.9031
100
2.0000
N
log
N
log
N
log
N
log
N
log
1-100
18
100-500
Each mantissa should be preceded by a decimal point, and the i
proper
characteristic should be written.
On account of the great differences between the successive mantissas |
in the first ten rows, interpolation should not be employed in
that part of 1
the table. Table III should be used in
this case. In
general, an error of 1
one unit may appear in the last figure of any interpolated value.
N
O
0000
1
0043
2
0086
3
0128
4
0170
5
0212
6
0253
7
0294
8
0334
9
0374
lO
11
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1106
13
1139
1173
1206
1239
1271
1303
1335
1367
1399
1430
14
1461
1492
1523
1553
1584
1614
1644
1673
1703
1732
15
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
19
2788
2810
2833
2856
2878
2900
2923
2945
2%7
2989
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
29
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
30
4771
4786
4800
4814
4829
4843
4857
4871
4886
4900
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
32
5051
5065
5079
5092
5105
5119
5132
5145
5159
5172
33
5185
5198
5211
5224
5237
5250
5263
5276
5289
5302
34
5315
5328
5340
5353
5366
5378
5391
5403
5416
5428
35
5441
5453
5465
5478
5490
5502
5514
5527
5539
5551
36
5563
5575
5587
5599
5611
5623
5635
5647
5658
5670
37
5682
5694
5705
5717
5729
5740
5752
5763
5775
5786
38
5798
5809
5821
5832
5843
5855
5866
5877
5888
5899
39
5911
5922
5933
5944
5955
5966
5977
5988
5999
6010
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
44
6435
6444
6454
6464
6474
6484
6493
6503
6513
6522
45
6532
6542
6551
6561
6571
6580
6590
6599
6609
6618
46
6628
6637
6646
6656
6665
6675
6684
6693
6702
6712
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
48
6812
6821
6830
6839
6848
6857
6866
6875
6884
6893
49
6902
6911
6920
6928
6937
6946
6955
6964
6972
6981
50
6990
O
6998
1
7007
2
7016
3
7024
4
7033
5
7042
6
7050
7
7059
8
7067
9
If
100-500
500-1000
19
N
O
1
2
3
4
5
6
7
8
9
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
51
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
53
7243
7251
7259
7267
7275
7284
7292
7300
7308
7316
54
7324
7332
7340
7348
7356
7364
7372
7380
7388
7396
55
7404
7412
7419
7427
7435
7443
7451
7459
7466
7474
56
7482
7490
7497
7505
7513
7520
7528
7536
7543
7551
57
7559
7566
7574
7582
7589
7597
7604
7612
7619
7627
58
7634
7642
7649
7657
7664
7672
7679
7686
7694
7701
59
7709
7716
7723
7731
7738
7745
7752
7760
7767
7774
eo
7782
7789
7796
7803
7810
7818
7825
7832
7839
7846
61
78^"?
7860
7931
7868
7938
7875
7945
7882
7952
7889
7896
7966
7903
7973
7910
7980
7917
7987
62
7924
1 ^\jy
7959
63
7993
8000
8007
8014
8021
8028
8035
8041
8048
8055
64
8062
8069
8075
8082
8089
8096
8102
8109
8116
8122
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
66
8195
8202
8209
8215
8222
8228
8235
8241
8248
8254
67
8261
8267
8274
8280
8287
8293
8299
8306
8312
8319
68
8325
8331
8338
8344
8351
8357
8363
8370
8376
8382
69
8388
8395
8401
8407
8414
8420
8426
8432
8439
8445
70
8451
8457
8463
8470
8476
8482
8488
8494
8500
8506
71
8513
8519
8525
8531
8537
8543
8549
8555
8561
8567
72
8573
8579
8585
8591
8597
8603
8609
8615
8621
8627
73
8633
8639
8645
8651
8657
8663
8669
8675
8681
8686
74
8692
8698
8704
8710
8716
8722
8727
8733
8739
8745
75
8751
8756
8762
8768
8774
8779
8785
8791
8797
8802
76
8808
8814
8820
8825
8831
8837
8842
8848
8854
8859
77
8865
8871
8876
.8882
8887
8893
8899
8904
8910
8915
78
8921
8927
8932
8938
8943
8949
8954
8960
8%5
8971
79
8976
8982
8987
8993
8998
9004
9009
9015
9020
9025
80
9031
9036
9042
9047
9053
9058
9063
9069
9074
9079
81
9085
9090
9096
9101
9106
9112
9117
9122
9128
9133
82
9138
9143
9149
9154
9159
9165
9170
9175
9180
9186
83
9191
9196
9201
9206
9212
9217
9222
9227
9232
9238
84
9243
9248
9253
9258
9263
9269
9274
9279
9284
9289
85
9294
9299
9304
9309
9315
9320
9325
9330
9335
9340
86
9345
9350
9355
9360
9365
9370
9375
9380
9385
9390
87
9395
9400
9405
9410
9415
9420
9425,
9430
9435
9440
88
9445
9450
9455
9460
9465
9469
9474
9479
9484
9489
89
9494
9499
9504
9509
9513
.9518
9523
9528
9533
9538
90
9542
9547
9552
9557
9562
9566
9571
9576
9581
9586
91
9590
9595
9600
9605
9609
9614
9619
9624
9628
9633
92
9638
9643
9647
%52
9657
9661
9666
9671
9675
9680
93
9685
9689
9694
9699
9703
9708
9713
9717
9722
9727
94
9731
9736
9741
9745
9750
9754
9759
9763
9768
9773
95
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
96
^823
9827
9832
9836
9841
9845
9850
9854
9859
9863
97
9868
9872
9877
9881
9886
9890
9894
9899
9903
9908
98
9912
9917
9921
9926
9930
9934
9939
9943
9948
9952
99
9956
9961
9965
9%9
9974
9978
9983
9987
9991
9996
lOO
0000
0004
0009
0013
0017
0022
0026
0030
0035
0039
N
O
1
2
3
4
5
6
7
8
9
500-l(
)00
20
LOGAEITHMS OF SINES
o
C
lO'
20'
30'
40'
50'
60'
o
o
— 00
7.4637
7.7648
7.9408
8.0658
8.1627
8.2419
89
1
8.2419
8.3088
8.3668
8.4179
4637
5050
5428
88
2
5428
5776
6097
6397
6677
6940
7188
87
3
7188
7423
7645
7857
8059
8251
8436
86
4
8436
8613
8783
8946
9104
8.9256
8.9403
85
5
8.9403
8.9545
8.9682
8.9816
8.9945
9.0070
9.0192
84
6
9.0192
9.0311
9.0426
9.0539
9.0648
0755
0859
83
7
0859
0961
1060
1157
1252
1345
1436-
82
8
1436
1525
1612
1697
1781
1863
1943
81
9
1943
2022
2100
2176
2251
2324
2397
80
lO
9.2397
9.2468
9.2538
9.2606
9.2674
9.2740
9.2806
79
11
2806.
2870
2934
2997
3058
3119
3179
78
12
3179
3238
3296
3353
3410
3466
3521
77
13
3521
3575
3629
3682
3734
3786
3837
76
14
3837
3887
3937
3986
4035
. 4083
4130
75
15
9.4130
9.4177
9.4223
9.4269
9.4314
9.4359
9.4403
74
16
4403
4447
4491
4533
4576
4618
4659
73
17
4659
4700
4741
4781
4821
4861
4900
72
18
4900
4939
4977
5015
5052
5090
5126
71
19
5126
5163
5199
5235
5270
5306
5341
70
20
9.5341
9.5375
9.5409
9.5443
9.5477
9.5510
9.5543
69
21
5543
5576
5609
5641
5673
5704
5736
68
22
5736
5767
5798
5828
5859
5889
5919
67
23
5919
5948
5978
6007
6036
6065
6093
66
24
6093
6121
6149
6177
6205
6232
6259
65
25
9.6259
9.6286
9.6313
9.6340
9.6366
9.6392
9.6418
64
26
6418
6444
6470
6495
6521
6546
6570
63
27
6570
6595
6620
6644
6668
6692
6716
62
28
6716
6740
6763
6787
6810
6833
6856
61
29
6856
6878
6901
6923
6946
6968
6990
eo
30
9.6990
9.7012
9.7033
9.7055
9.7076
9.7097
9.7118
59
31
7118
7139
7160
7181
7201
7222
7242
58
32
7242
726?
7282
7302
7322
7342
7361
57
33
7361
7380
7400
7419
7438
7457
7476
56
34
7476
7494
7513 .
7531
7550
7568
7586
55
35
9.7586
9.7604
9.7622
9.7640
9.7657
9.7675
9.7692
54
36
7692
7710
7727
7744
7761
7778
7795
53
37
7795
7811
7828
7844
7861
7877
7893
52
38
7893
7910
7926
7941
7957
7973
7989
51
39
7989
8004
8020
8035
8050
8066
8081
50
40
9.8081
9.8096
9.8111
9.8125
9.8140
9.8155
9.8169
49
41
8169
8184
8198
8213
8227
8241
8255
48
42
8255
8269
8283
8297
8311
8324
8338
47
43
8338
8351
8365
8378
8391
8405
8418
46
44
■ 9.8418
9.8431
9.8444
9.8457
9.8469
9.8482
9.8495
45
o
60'
50'
40'
30'
20'
10'
O'
o
LOGARITHMS OF COSINES
LOGARITHMS OF COSINES
21
o
O'
lO'
20'
30'
40'
50'
60'
o
o
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
9.9999
89
1
9.9999
9.9999
9.9999
9.9999
9.9998
9.9998
9997
88
2
9997
9997
99%
9996
9995
9995
9994
87
3
9994
9993
9993
9992
9991
9990
9989
86
4
9989
9989
9988
9987
9986
9985
9983
85
5
9.9983
9.9982
9.9981
9.9980
9.9979
9.9977
9.9976
84
6
9976
9975
9973
9972
9971
9969
9968
83
7
9968
9966
9964
9963
9961
9959
9958
82
8
9958
9956
9954
9952
9950
9948
9946
81
9
9946
9944
9942
9940
9938
9936
9934
80
10
9.9934
9.9931
9.9929
9.9927
9.9924
9.9922
9.9919
79
11
9919
9917
9914
9912
9909
9907
9904
78
12
9904
9901
9899
9896
9893
9890
9887
77
13
9887
9884
9881
9878
9875
9872
9869
76
14
9869
9866
9863
9859
9856
9853
9849
75
16
9.9849
9.9846
9.9843
9.9839
9.9836
9.9832
9.9828
74
16
9828
9825
9821
9817
9814
9810
9806
73
17
9806
9802
9798
9794
9790
9786
9782
72
18
9782
9778
9774
9770
9765
9761
9757
71
19
9757
9752
9V48
9743
9739
9734
9730
70
20
9.9730
9.9725
9.9721
9.9716
9.9711
9.9706
9.9702
69
21
9702
9697
9692
9687
9682
9677
%72
68
22
9672
9667
9661
9656
9651
9646
9640
67
23
9640
9635
9629
9624
9618
9613
9607
66
24
9607
9602
9596
9590
•
9584
9579
9573
65
25
9.9573
9.9567
9.9561
9.9555
9.9549
9.9543
9.9537
64
26
9537
9530
9524
9518
9512
9505
9499
63
27
9499
9492
9486
9479
9473
9466
9459
62
28
9459
9453
9446
9439
9432
9425
9418
61
29
9418
9411
9404
9397
9390
9383
9375
eo
30
9.9375
0.9368
9.9361
9.9353
9.9346
9.9338
9.9331
59
31
9331
9323
9315
9308
9300
9292
9284
58
32
9284
9276
9268
9260
9252
9244
9236
57
33
9236
9228
9219
9211
9203
9194
9186
56
34
9186
9177
9169
9160
9151
• 9142
9134
55
35
9.9134
9.9125
9.9116
9.9107
9.9098
9.9089
9.9080
54
36
9080
9070
9061
9052
9042
9033
9023
53
37
9023
9014
9004
8995
8985
8975
8965
52
38
8965
8955
8945
8935
8925
8915
8905
51
39
8905
8895
8884
8874
8864
8853
8843
50
40
9.8843
9.8832
9.8821
9.8810
9.8800
9.8789
9.8778
49
41
8778
8767
8756
8745
8733
8722
8711
48
42
8711
8699
8688
8676
8665
8653
8641
47
43
8641
8629
8618
8606
8594
8582
8569
46
44
9.8569
9.8557
9.8545
9.8532
9.8520
9.8507
9.8495
45
o
60'
60'
40'
30'
20'
10'
O'
o
LOGARITHMS OF SINES
22
LOGABITHMS OF TANGENTS
o
O'
lO'
20'
30'
40'
50'
60'
o
o
— 00
7.4637
7.7648
7.9409
8.0658
8.1627
8.2419
89
1
8.2419
8.3089
8.3669
8.4181
4638
5053
5431
88
2
5431
5779
6101
6401
6682
6945
7194
87
3
7194
7429
7652
7865
8067
8261
8446
86
4
8446
8624
8795
8960
9118
8.9272
8.9420
85
5
8.9420
8.9563
8.9701
8.9836
8.9966
9.0093
9.0216
84
6
9.0216
9.0336
9.0453
9.0567
9.0678
0786
0891
83
7
0891
0995
1096
1194
1291
1385
1478
82
8
1478
1569
1658
1745
1831
1915
1997
81
9
1997
2078
2158
2236
2313
2389
2463
80
lO
9.2463
9.2536
9.2609
9.2680
9.2750
9.2819
9.2887
79
11
2887
2953
3020
3085
3149
3212
3275
78
12
3275
3336
3397
3458
3517
3576
3634
77
13
3634
3691
3748
3804
3859
3914
3968
76
14
3968
4021
4074
4127
4178
4230
4281
75
16
9.4281
9.4331
9.4381
9.4430
9.4479
9.4527
9.4575
74
16
4575
4622
4669
4716
4762
4808
4853
73
17
4853
4898
4943
4987
5031
5075
5118
72
18
5118
5161
5203
5245
5287
5329
5370
71
19
5370
5411
5451
5491
5331
5571
5611
70
20
9.5611
9^650
9.5689
9.5727
9.5766
9.5804
9.5842
69
21
5842
5879
5917
5954
5991
6028
6064
68
22
6064
6100
6136
6172
6208
6243
6279
67
23
6279
6314
6348
6383
6417
6452
6486
66
24
6486
6520
6553
6587
6620
6654
6687
65
25
9.6687
9.6720
9.6752
9.6785
9.6817
9.6850
9.6882
64
26
6882
6914
6946
6977
7009
7040
7072
63
27
7072
7103
7134
7165
71%
7226
7257
62
28
7257
7287
7317
7348
7378
7408
7438
61
29
7438
7467
7497
7526
7556
7585
7614
eo
30
9.7614
9.7644
9.7673
9.7701
9.7730
9.7759
9.7788
59
31
7788
7816
7845
7873
7902
7930
7958
58
32
7958
7986
8014
8042
8070
8097
8125
57
33
8125
8153
8180
8208
8235
8263
8290
56
34
8290
8317
8344
8371
8398
8425
8452
55
35
9.8452
9.8479
9.8506
9.8533
9.8559
9.8586
9.8613
54
36
8613
8639
8666
8692
8718
8745
8771
53
37
8771
8797
8824
8850
8876
8902
8928
52
38
8928
8954
8980
9006
9032
9058
9084
51
39
9084
9110
9135
9161
9187
9212
9238
50
40
9.9238
9.9264
9.9289
9.9315
9.9341
9.9366
9.9392
49
41
9392
9417
9443
9468
9494
9519
9544
48
42
9544
9570
9595
9621
9646
9671
9697
47
43
9697
9722
9747
9772
9798
9823
9.9848
46
44
9.9848
9.9874
9.9899
9.9924
9.9949
9.9975
10.0000
45
o
60'
50'
40'
30'
20'
lO'
O'
o
LOGARITHMS OF COTANGENTS
LOGARITHMS OF COTANGENTS
28
o
O'
10'
20'
30'
40'
50'
60'
o
o
00
12.5363
12.2352
12.0591
11.9342
11.8373
11.7581
89
1
11.7581
11.6911
11.6331
11.5819
5362
4947
4569
88
2
4569
4221
3899
3599
3318
3055
2806
87
3
2806
2571
2348
2135
1933
1739
1554
86
4
1554
1376
1205
1040
0882
11.0728
11.0580
85
5
11.0580
11.0437
11.0299
11.0164
11-0034
10.9907
10.9784
84
6
10.9784
10.9664
10.9547
10.9433
10.9322
9214
9109
83
7
9109
9005
8904
8806
8709
8615
8522
82
8
8522
8431
8342
8255
8169
8085
8003
81
9
8003
7922
7842
7764
7687
7611
7537
80
lO
10.7537
10.7464
10.7391
10.7320
10.7250
10.7181
10.7113
79
11
7113
7047
6980
6915
6851
6788
6725
78
12
6725
6664
6603
6542
6483
6424
6366
77
13
6366
6309
6252
61%
6141
6086
6032
76
14
6032
5979
5926
5873
5822
5770
5719
75
15
10.5719
10.5669
10.5619
10.5570
10.5521
10.5473
10.5425
74
16
5425
5378
5331
5284
5238
5192
5147
73
17
5147
5102
5057.
5013
4969
4925
4882
72
18
4882
4839
4797
4755
4713
4671
4630
71
19
4630
4589
4549
4509
4469
4429
4389
70
20
10.4389
10.4350
10.4311
10.4273
10.4234
10.4196
10.4158
69
21
4158
4121
4083
4046
4009
3972
3936
68
22
3936
3900
3864
3828
3792
3757
3721
67
23
3721
3686
3652
3617
3583
3548
3514
66
24
3514
3480
3447
3413
3380
3346
3313
65
25
10.3313
10.3280
10.3248
10.3215
10.3183
10.3150
10.3118
64
26
3118
3086
3054
3023
2991
2960
2928
63
27
2928
2897
2866
2835
2804
2774
2743
62
28
2743
2713
2683
2652
2622
2592
2562
61
29
2562
2533
2503
2474
2444
2415
2386
60
30
10.2386
10.2356
10.2327
10.2299
10.2270
10.2241
10.2212
59
31
2212
2184
2155
2127
2098
2070
2042
58
32
■ 2042
2014
1986
1958
1930
1903
1875
57
33
1875
1847
1820
1792
1765
1737
1710
56
34
1710
1683
1656
1629
1602
1575
1548
55
35
10.1548
10.1521
10.1494
10.1467
10.1441
10.1414
10.1387
54
36
1387
1361
1334
1308
1282
1255
1229
53
37
1229
1203
1176
1150
1124
1098
1072
52
38
1072
1046
1020
0994
0968
0942
0916
51
39
0916
0890
0865
0839
0813
0788
0762
50
40
10.0762
10.0736
10.0711
10.0685
10.0659
10.0634
10.0608
49
41
0608
0583
0557
0532
0506
0481
0456
48
42
0456
0430
0405
0379
0354
0329
0303
47
43
0303
0278
0253
0228
0202
0177
0152
46
44
10.0152
10.0126
10.0101
10.0076
10.0051
10.0025
10.0000
45
o
60r
50'
40'
30'
20'
lO'
O'
o
LOGARITHMS OF TANGENTS
24
CIRCLES, POWERS, AND ROOTS
TABLE II
a
icd
\7Cd^
d^
^
Vd
o
0.0000
0.0000
0.0000
0.0000
1
3.1416
0.7854
1
1
1.0000
1.0000
2
6.2832
3.1416
4
8
4142
2599
3
9.4248
7.0686
9
27
1.7321
4422
4
12.5664
12.5664
16
64
2.0000
5874
5
15.7080
19.6350
25
125
2.2361
1.7100
6
18.8496
28.2743
36
216
4495
8171
7
21.9911
38.4845
49
343
6458
1.9129
8
25.1327
50.2655
64
512
2.8284
2.0000
9
28.2743
63.6173
81
729
3.0000
0801
lO
31.4159
78.5398
100
1,000
3.1623
2.1544
11
34.5575
95.0332
121
1,331
3166
2240
12
37.6991
113.0973
144
1,728
4641
2894
13
40.8407
132.7323
169
2,197
6056
3513
14
43.9823
153.9380
196
2,744
7417
4101
15
47.1239
176.7146
225
3,375
3.8730
2.4662
16
50.2655
201.0619
256
4,096
4.0000
5198
17
53.4071
226.9801
289
4,913
1231
5713
18
56.5487
254.4690
324
5,832
2426
6207
19
59.6903
283.5287
361
6,859
3589
6684
20
62.8319
314.1593
400
8,000
4.4721
2.7144
21
65.9734
346.3606
441
9,261
5826
7589
22
69.1150
380.1327
484
10,648
6904
8020
23
72.2566
415.4756
529
12,167
7958
8439
24
75.3982
452.3893
576
13,824
4.8990
8845
25
78.5398
490.8739
625
15,625
5.0000
2.9240
26
81.6814
530.9292
676
17,576
0990
2.9625
27
84.8230
572.5553
729
19,683
1962
3.0000
28
87.9646
615.7522
784
21,952
2915
0366
29
91.1062
660.5199
841
24,389
3852
0723
30
94.2478
706.8583
900
27,000
5.4772
3.1072
31
97.3894
754.7676
961
29,791
5678
1414
32
100.5310
804.2477
1024
32,768
6569
1748
33
103.6726
855.2986
1089
35,937
7446
2075
34
106.8142
907.9203
1156
39,304
8310
2396
35
109.9557
962.1128
1225
42,875
5.9161
3.2711
36
113.0973
1017.8760
12%
46,656
6.0000
3019
37
116.2389
1075.2101
1369
50,653
0828
3322
38
119.3805
1134.1149
1444
54,872
1644
3620
39
122.5221
1194.5906
1521
59,319
2450
3912-
40
125.6637
1256.6371
1600
64,000
6.3246
3.4200
41
128.8053
1320.2543
1681
68,921
4031
4482
42
131.9469
1385.4424
1764
74,088
4807
4760
43
135.0885
1452.2012
1849
79,507
5574
5034
44
138.2301
1520.5308
1936
85,184
6332
5303
45
141.3717
1590.4313
2025
91,125
6.7082
3.5569
46
144.5133
1661.9025
2116
97,336
7823
5830
47
147.6549
1734.9445
2209
103,823
8557
6088
48
150.7964
1809.5574
2304
110,592
6.9282
6342
49
153.9380
1885.7410
2401
117,649
7.0000
6593
50
157.0796
1963.4954
2500
125,000
7.0711
3.6840
CIRCLES, POWERS, AND ROOTS 25
CIRCUMFERENCES AND AREAS OF CIRCLES
SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS
d
ird
lird^
d^
2500
d^
Vd
Vd
60
157.0796
1963.4954
125,000
7.0711
3.6840
51
160.2212
2042.8206
2601
132,651
1414
7084
52
163.3628
2123.7166
2704
140,608
2111
7325
53
166.5044
2206.1834
2809
148,877
2801
7563
54
169.6460
2290.2210
2916
157,464
3485
7798
55
172.7876
2375.8294
3025
166,375
7.4162
3.8030
56
175.9292
2463.0086
3136
175,616
4833
8259
57
179.0708
2551.7586
3249
185,193
5498
8485
58
182.2124
2642.0794
3364
195,112
6158
8709
59
185.3540
2733.9710
3481
205,379
6811
8930
eo
188.4956
2827.4334
3600
216,000
7.7460
3.9149
61
191.6372
2922.4666
3721
226,981
8102
9365
62
194.7787
3019.0705
3844
238328
8740
9579
63
197.9203
3117.2453
3969
250,047-
7.9373
3.9791
64
201.0619
3216.9909
4096
262,144
8.0000
4.0000
65
204.2035
3318.3072
4225
274,625
8.0623
4.0207
66
207.3451
3421.1944
4356
287,496
1240
0412
67
210.4867
3525.6524
4489
300,763
1854
0615
68
213.6283
3631.6811
4624
314,432
2462
0817
69
216.7699
3739.2807
4761
328,509
3066
1016
70
219.9115
3848.4510
4900
343,000
8.3666
4.1213
71
223.0531
3959.1921
5041
357,911
4261
1408
72
226.1947
4071.5041
5184
373,248
4853
1602
73
229.3363
4185.3868
5329
389,017
5440
1793
74
232.4779
4300.8403
5476
405,224
6023
1983
75
235.6194
4417.8647
5625
421,875
8.6603
4.2172
76
238.7610
4536.4598
5776
438,976
7178
2358
77
241.9026
4656.6257
5929
456,533
7750
2543
78
245.0442
4778.3624 ,
. 6084
474,552
8318
2727
79
248.1858
4901.6699
6241
493,039
8882
2908
80
251.3274
5026.5482
6400
512,000
8.9443
43089
81
254.4690
5152.9974
6561
531,441
9.0000
3267
82
257.6106
5281.0173
6724
551,368
0554
3445
83
260.7522
5410.6079
6889
571,787
1104
3621
84
263.8938
5541.7694
7056
592,704
1652
3795
85
267.0354
5674.5017
7225
614,125
9.2195
4.3968
86
270.1770
5808.8048
7396
636,056
2736
4140
87
273.3186
5944.6787
7569
658,503
3274
4310
88
276.4602
6082.1234
7744
681,472
3808
4480
89
279.6017
6221.1389
7921
704,%9
4340
4647
90
282.7433
6361.7251
8100
729,000
9.4868
4.4814
91
285.8849
6503.8822
8281
753,571
5394
4979
92
289.0265
6647.6101
8464
778,688
5917
5144
93
292.1681
6792.9087
8649
804,357
6437
5307
94
295.3097
6939.7782
8836
830,584
6954
5468
95
298.4513
7088.2184
9025
857,375
9.7468
4.5629
96
301.5929
7238.2295
9216
884,73^
7980
5789
97
304.7345
7389.8113
9409
912,673
8489
5947
98
307.8761
7542.9640
9604
941,192
8995
6104
99
311.0177
7697.6874
9801
970,299
9.9499
6261
lOO
314.1593
7853.9816
10000
1,000,000
10.0000
4.6416
26
CIRCUMFEEENCES AND AREAS OF CIRCLES
If n s= the radius of the circle, the circumference = 2 im.
If n = the radius of the circle, the area
= im!'
•
If n = the circninference of the circle, the radius = —
2ir
n.
If n = the circumference of the circle, the area = —
4ir
n*.
n
2wn
0.00
0.0
1
0.000
0.00
n
2wn
314. 16
7 854
1
2ir'*
7.96
4ir
198.94
O
50
1
6.28
3.1
0.159
0.08
51
320. 44
8171
8.12
206.98
2
12.57
12.6
0.318
0.32
52
326. 73
8495
8.28
215. 18
3
18.85
28.3
0.477
0.72
53
333. 01
8 825
8.44
223. 53
4
25.13
50.3
0.637
1.27
54
339. 29
9161
8.59
232. 05
5
31.42
78.5
0.796
1.99
55
345. 58
9 503
8.75
240.72
6
37.70
113.1
0.955
2.86
56
351. 86
9852
8.91
249. 55
7
43.98
153.9
1.114
3.90
57
358. 14
10 207
9.07
258. 55
8
50.27
201.1
1.273
5.09
58
364.42
10 568
9.23
267.70
9
56.55
254.5
1.432
6.45
59
370. 71
10936
9.39
277. 01
lO
62.83
314.2
1.592
7.96
60
376.99
11310
9.55
286.48
11
69.12
380.1
1.751
9.63
61
383.27
11690
9.71
296.11
12
75.40
452.4
1.910
11.46
62
389. 56
12 076
9.87
305.90
13
81.68
530.9
2.069
13.45
63
395. 84
12 469
10.03
315.84
14
87.96
615.8
2.228
15.60
64
402.12
12 868
10.19
325. 95
16
94.25
706.9
2.387
17.90
65
408.41
13 273
10.35
336. 21
16
100.53
80+. 2
2.546
20.37
66
414. 69
13 685
10.50
346.64
17
106.81
907.9
2.706
23.00
67
420. 97
14103
10.66
357. 22
18
113. 10
1 017. 9
2. 865
25.78
68
427. 26
14 527
10.82
367. 97
19
119. 38
1 134. 1
3.024
28.73
69
433. 54
14957
10.98
378. 87
20
125.66
1 256. 6
3.183
31.83
70
439. 82
15 394
11.14
389. 93
21
131. 95
1 385. 4
3.342
35.09
71
446.11
15 837
11.30
401.15
22
138. 23
1 520. 5
3.501
38.52'
72
452. 39
16 286
11.46
412. 53
23
144.51
1 661. 9.
3.661
42.10
73
458. 67
16 742
11.62
424. 07
24
150.80
1809.6
3.820
45.84
74
464.96
17203
11.78
435. 77
25
157. 08
1963.5
3.979
49.74
75
471. 24
17 671
11.94
447.62
26
163.36
2123.7
4.138
53. 79
76
477. 52
18146
12.10
459.64
27
169. 65
2 290.2
4.297
58.01
77
483.81
18627
12.25
471. 81
28
175. 93
2463.0
4.456
62.39
78
. 490.09
19113
12.41
484.15
29
182. 21
2 642. 1
4.615
66.92
79
4%. 37
. 19607
12.57
496.64
30
188.50
2 827. 4
4.775
71.62
80
502. 65
20106
12.73
509.30
31
194. 78
3 019. 1
4.934
76.47
81
508. 94
20612
12.89
522. 11
32
201.06
3 217.
5.093
81.49
82
515. 22
21124
13.05
535. 08
33
207.35
3 421. 2
5.252
86.66
83
521. 50
21642
13.21
548. 21
34
213.63
3 631. 7
5.411
91.99
84
527. 79
22167
13.37
561. 50
35
219. 91
3 848. 5
5.570
97.48
85
534. 07
22698
13.53
574. 95
36
226. 19
4 071. 5
5.730
103. 13
86
540. 35
23 235
13.69
58a 55
37
232.48
4 300.8
5.889
108. 94
87
546.64
23 779
13.85
602.32
38
238.76
4 536. 5
6.048
114.91
88
552. 92
24 328
14.01
616. 25
39
245. 04
4 778. 4
6.207
121.04
89
559. 20
24885
14.16
630.33
40
251. 33
5 026. 5
6.366
127. 32
90
565. 49
25 447
14.32
644.58
41
257. 61
5 281.0
6.525
133. 77
91
571. 77
26016
14.48
658. 98
42
263.89
5 541. 8
6.685
140.37
92
578. 05
26 590
14.64
673. 54
43
270. 18
5 808.8
6.844
147. 14
93
584. 34
27172
14.80
688.27
44
276.46
6 082. 1
7.003
154.06
94
590.62
27 759
14.96
703. 15
45
282.74
6 361. 7
7.162
161. 14
95
596.90
28353
15.12
718. 19
46
289. 03
6647.6
7.321
168.39
96
603.19
28 953
15.28
733. 39
47
295. 31
6 939. 8
7.480
175. 79
97
609.47
29 559
15.44
748. 74
48
301. 59
7 238.2
7.639
183. 35
98
615. 75
30172
15.60
764.26
49
307.88
7 543.
7.799
191. 07
99
622.04
30 791
15.76
779.94
50
314. 16
2im
7 854.
7.958
1
198.94
4ir
lOO
628. 32
2im
31416
15.92
795. 77
4v
n
n
1
2w*
$T
TABLF. Ill ^{0^^
FIVE-PLACE MANTISSAS
OF THE COMMON LOGARITHMS OF
INTEGERS FROM 1 TO 10,000
On this page the logarithms of integers from 1 to 100 are given in full,
with characteristics as well as mantissas. On account of the great dif-
ferences between the successive mantissas, interpolation cannot safely be
employed on this page.
In the remainder of the table only the mantissas are given.
In general, an error of one unit may appear in the last figure of any
interpolated value.
Table III is to be used when accuracy is required to more than four
figures in the results. In general, the results will be accurate to five figures.
1-100
N io«r
1 0.00000
2 0.30103
3 0.47 712
4 0.60 206
5 0.69897
6 0.77815
7 0.84 510
8 0.90309
9 0.95 424
10 1.00000
11 1.04139
12 1.07918
13 1.11394
14 1.14613
15 1.17609
16 1.20412
17 1.23045
18 1.25 527
19 1.27875
20 1.30103
N log
21 1.32 222
22 1.34 242
23 1.36173
24 1.38021
25 1.39 794 .
26 1.41497
27 1.43136
28 1.44 716
29 1.46240
30 1.47 712
31 1.49136
32 1.50 515
33 1.51851
34 1.53148
' 35 1. 54 407
36 1.55 630
37 1.56 820
38 1.57978
39 1.59106
40 1.60206
N log
41 1.61278
42 1.62 325
43 1.63 347
44 1.64345
45 1.65 321
46 1.66276
47 1.67 210
48 1.68124
49 1.69 020
50 1.69897
61 1. 70 757
52 1. 71 600
53 1.72 428
54 1. 73 239 ,
55 1.74036
56 1.74819
57 1.75 587
58 1.76343
59 1.77 085
60 1.77815
N log
61 1. 78 533
62 1.79 239
63 1.79934
64 1.80618
65. 1.81291
66 L 81 954
67 1.82 607
68 1.83 251
69 1.83 885
70 1.84 510
71 1. 85 126
' 72 1.85 733
73 1. 86 332
74 1.86923
75 1. 87 506
76 1.88081
77 1.88 649
78 r. 89209
79 1.89763
80 1.90309
N log
81 1.90849
82 1.91381
83 1.91908
84 1.92428
85 1. 92 942
86 1.93 450
87 1.93 952
88 1.94448
89 1.94939
90 1.95 424
91 1.95 904
92 l.%379
93 1.96848
94 1.97313
95 1. 97 772
96 1.98227
97 1.98677
98 1.99123
99 1.99564
100 2.00000
N log
N log
N log
N log
N log
1-100
88
• • •
100-160
K
1
a
8
4
5
6
7
8
9
100
00000
00043
00087
00130
00173
00217
00260
00303
00346
00389
101
432
475
518
561
604
647
689
732
775
817
102
860
903
945
988
01030
01072 01115
01157
01 199 01 242 1
103
01284
01326
01368
01410
452
494
536
578
620
662
104
703
745
787
828
870
912
953
995
02036
02078
105
02119
02160
02 202
02 243
02 284
02325
02 366
02407
02 449
02490
106
531
572
612
653
694
735
776
816
857
898
107
938
979
03 019
03 060
03100
03141
03181
03 222
03 262
03 302
108
03342
03 383
423
463
503
543
583
623
663
703
109
743
782
822
862
902
941
981
04021
04060
04100
no
04139
04179
04 218
04 258
04297
04336
04376 04415
04454
04493
111
532
571
610
650
689
727
766
805
844
883
112
922
%1
999
05038
05 077
05115
05154
05192
05 231
05 269
113
05308 05 346 05 385
423
461
500
538
576
614
652
114
690
729
767
805
843
881
918
956
994
06032
115
06070
06108
06145
06183
06 221
06258
06296
06333
06371
06408
116
446
483
521
558
595
633
670
707
744
781
117
819
856
893
930
967
07004
07041
07078 07115
07151
118
07188
07225
07 262
07 298
07335
372
408
445
482
518
119
555
591
628
664
700
737
773
809
846
882
120
07918
07954
07990
08027
08063
08099 08135
08171
08 207
08243
121
08 279
08 314
08350
386
422
458
493
529
565
600
122
636
672
707
743
778
814
849
884
920
95S
123
991
09026
09061
09096
09L32
JB^
09202
09237
09272
09307
124
09342
377
412
447
482
552
587
621
656
125
09691
09 726
09 760
09 795
09830
09 864
09899
09934
09968
10003
126
10037
10072
10106
10140
10175
10209
10243
10278
10312
346
127
380
415
449
483
517
551
585
619
653
687
128
721
755
789
823
857
890
924
958
992
11025
129
11059
11093
11126
11160
11193
11227
11261
11294
11327
361
130
11394
11428
11461
11494
11528
11561
11594
11628
11661
11694
131
727
760
793
826
860
893
926
959
992
12024
132
12 057
12 090
12123
12156
12189
12 222
12 254
12 287
12320
352
133
385
418
450
483
516
548
581
613
646
678
134
710
743
775
808
840
872
905
937
969
13001
135
13033
13 066
13 098
13130
13162
13194
13 226
13 258
13 290
13 322
136
354
386
418
450
481
513
545
577
609
640
137
672
704
735
767
799
830
862
893
925
956
138
988
14019
14 051
14082
14114
14145
14176
14 208
14239
14270
139
14301
333
364
395
426
457
489
520
551
582
140
14613
14644
14 675
14 706
14 737
14 768
14 799
14829
14860
14 891
141
922
953
983
15 014
15 045
15 076
^1
15106
15137
15168
15198
142
15 229
15 259
15 290
320
351
412
442
473
503
143
534
564
594
625
655
685
715
746
776
806
144
836
866
897
927
957
987
16017
16047
16077
16107
145
16137
16167
16197
16 227
16 256
16 286
16316
16346
16376
16406
146
435
465
495
524
554
584
613
643
673
702
147
732
761
791
820
850
879
909
938
%7
997
148
17026
17056
17085
17114
17143
17173
17202
17 231
17 260
17289
149
319
348
377
406
435
46*
493
522
551
580
150
17609
17638
17667
17696
17 725
17 754
17 782
17811
17840
17869
K
1
9
8
4 '
5
6
7
8
9
100-160
160-200
29
K
o
1 2
8
4
5
6
7
8
9
150
17 609
17638 17667
17696
17 725
17 754
17782
17811
17 840
17869
151
898
926 955
984
18013
18041
18 070
18099
18127
18156
152
18184
18213 18 241
18270
298
327
355
384
412
441
153
469
498 526
554
583
611
639
667
696
724
154
752
780 808
837
865
893
921
949
977
19005
155
19033
19061 19089
19117
19145
19173
19201
19229
19257
19285
156
312
340 368
396
424
451
479
507
535
562
157
590
618 645
673
700
728
756
783
811
838
158
866
893 921
948
976
20003
20030
20058
20085
20112
159
20140
20167 20194
20222
20249
276
303
330
358
385
160
20412
20439 20466
20493
20520
20548 20 575
20602
20629
20656
161
683
710 737
763
790
817
844
871
898
925
162
952
978 21005
21032
21059
21085
21112
21139
21165
21192
163
21219
21 245 272
299
325
352.
378
405
431
458
164
484
511 537
564
590
617
643
669
696
722
165
21748 21775 21801
21827
21854
21880
21906
21932
21958 21985 I
166
22011
22037 22063
22089
22115
22141
22167
22194
22 220
22 246
167
272
298 324
350
376
401
427
453
479
505
168
531
557 583
608
634
660
686
712
737
763
169
789
814 840
866
891
917
943
968
994
23019
170
23 045
23070 23096
23121
23147
23172
23198
23 223
23 249
23 274
171
300
325 350
376
401
426
452
477
502
528
172
553
578 603
629
654
679
704
729
754
779
173
805
830 855
880
905
930
955
980 24005
24030
174
24055
24080 24105
24130 24155
24180
24204
24 229
254
279
175
24304
24329 24353
24378
24403
24428
24452
24477
24 502
24 527
176
551
576 601
625
650
674
699
724
748
773
177
797
822 846
871
895
920
944
%9
993
25018
178
25042
25 066 25 091
25115
25139
25164
25188
25 212
25 237
261
179
285
310 334
358
382
406
431
455
479
503
180
25 527
25 551 25 575
25 600
25 624
25 648
25 672
25 696
25 720
25 744
181
768
792 816
840
864
888
912
935
959
983
182
26007
26031 26055
26079
26102
26126
26150
26174
26198
26221
183
245
269 293
316
340
364
387
411
435
458
184
482
505 529
553
576
600
623
647
670
694
185
26717
26741 26764
26788
26811
26834
26858
26881
26905 26928 1
186
951
975 998
27021
27045
27068
27091
27114
27138
27161
187
27184
27207 27 231
254
277
300
323
346
370
393
188
416
439 462
485
508
531
554
577
600
623
189
646
669 692
715
738
761
784
807
830
852
100
27875
27898 27921
27944
27967
27989
28012
28035
28058
28061
191
28103
28126 28149
28171
28194
28 217
240
262
285
307
192
330
353 375
398
421
443
466
488
511
533
193
556
578 601
623
646
668
691
713
735
758
194
780
803 825
847
870
892
914
937
959
981
195
29003
29026 29048
29070
29092
29115
29137
29159
29181
29203
196
226
248 270
292
314
336
358
380
403
425
197
447
469 491
513
535
557
579
601
623
645
198
667
688 710
732
754
776
798
820
842
863
199
885
907 929
951
973
994
30016
30038
30060 30081 1
2QSL
30103
O
30125 30146
30168
30 190_
30211 30233 30255 30276 30 29a
5 6 7 8
.- n: . .
1 2
— - •. •••• . ••
8
4
160-200
80
200-260
N-
O
1
2
8
4
5
6 7
8
» 1
200
30103 3012i
30146
30168
30190
30211
30233 30255 30276 30296 |
201
320
341
363
384
406
428
449 471
492
514
202
535
557
578
600
621
643
664 685
707
728
203
7i0
771
792
814
835
856
878 899
920
942
204
963
984
31006 31027 31048
31069
31091 31112
31133
31154
205
31175
31197
31218
31239
31260
31281
31302 31323
31345 31366 |
206
387
408
429
450
471
492
513 534
555
576
207
597
618
639
660
681
702
723 744
765
785
206
806
827
848
869
890
911
931 952
973
994
209
32015
32035
32056
32 077
32098
32118
32139 32160
32181
32 201
210
32222
32 243
32 263
32 284
32 305
32 325
32346 3234^
32387
32408
211
428
449
469
490
510
531
552 572
593
613
212
634
654
675
695
715
736
756 777
797
818
213
838
858
879
899
919
94&
960 980
33 001
33021
214
33041
33 062
33 082
33102
33122
33143
33163 33183
203
224
215
33 244
33 264
33 284
33 304
33 325
33 345
33365 33385
33 405
33425
216
445
465
486
506
526
546
566 586
606
626
217
646
666
686
706
726
746
766 786
806
826
218
846
866
885
905
925
945
965 985
34005
34025
219
34044
34064
34084
34104
34124
34143
34163 34183
203
223
220
34242
34 262
34 282
34301
34321
34341
34361 34380
34400
34420
221
439
459
479
498
518
537
557 ' 577
5%
616
222
635
655
674
694
713
733
753 772
792
811
223
830
850
869
889
908
928
947 967
986
35 005
224
35 025
35 044
35 064
35083
35102
35122
35 141 35 160 35 180
199
225
35 218
35 238
35 257
35 276
35 295
35 315
35 334 35 353
35 372
35392
226
411
430
449
468
488
507
526 545
564
583
227
603
622
641
660
679
698
717 736
755
774
228
793
813
832
851
870
889
908 927
946
965
229
984
36003
36021
36040
36059
36078
36097 36116 36135
36154
2d0
36173
36192
36211
36229
36248
36267 36286 36305 36324 36342 |
231
361
380
399
418
436
455
474 493
511
530
232
549
568
586
605
624
642
661 680
698
717
233
736
754
773
791
810
829
847 866
884
903
234
922
940
959
977
996
37014
37033 37051
37070
37088
235
37107
37125
37144
37162
37181
37199
37 218 37236
37254
37273
236
291
310
328
346
365
383
401 420
438
457
237
475
493
511
530
548
566
585 603
621
639
238
658
676
694
712
731
749
767 785
803
822
239
840
858
876
894
912
931
949 967
985 38003 1
246
38021
38039
38057
38075
38093
38112
38130 38148
38166 38184 |
241
202
220
238
256
274
292
310 328
346
364
242
382
399
417
435
453
471
489 507
525
543
243
561
578
5%
614
632
650
668 686
703
721
244
739
757
775
792
810
828
846 863
881
899
245
38917
38934
38952
38970 38987
39005
39023 39041
39058
39076
246
39094
39111
39129
39146
39164
182
199 217
235
253
247
270
287
305
322
340
358
375 393
410
428
248
445
463
480
498
515
533
550 568
585
602
249
620
637
655
672
690
707
724 742
759
777
•
250
39794
39811
39829
39846 39863
39881
39898 39915
39933
39950
N
O
1
2
3
4
5
6 7
8
9
200-260
260-800
81
N
O
1
2
3
4
5
6
7
8
9
250
39794
39811
39829
39846
39863
39881
39898
39915
39933
39950
251
967
985
40002
40019
40037
40054
40071
40088
40106
40123
252
40140
40157
175
192
209
226
243
261
278
295
253
312
329
346
364
381
398
415
432
449
466
254
483
500
518
535
552
569
586
603
620
637
255
40654
40671
40688
40 705
40 722
40739 40 756
40 773
40790 40807 I
256
824
841
858
875
892
909
926
943
960
976
257
993
41010
41027
41044
41061
41078 41095
41111
41128
41145
258
41162
179
1%
212
229
246
263
280
296
313
259
330
347
363
380
397
414
430
447
464
481
260
41497
41514
41531
41547
41564
41581
41597
41614
41631
41647
261
664
681
697
714
731
747
764
780
797
814
262
830
847
863
880
8%
913
929
946
963
979
263
996
42012
42029
42045
42062
42078
42095
42111
42127
42144
264
42160
177
193
210
226
243
259
275
292
308
265
42325
42 341
42357
42 374
42 390
42 406
42423
42439
42 455
42472
266
488
504
521
537
553
570
586
602
619
635
267
651
667
684
700
716
732
749
765
781
797
268
813
830
846
862
878
894
911
927
943
959
269
975
991
43008
43024
43 040
43056
43072
43088
43104
43120
270
43136 43152 43169 43185
43 201
43 217
43 233
43 249 43 265
43 281
271
297
313
329
345
361
377
393
409
425
441
272
457
473
489
505
521
S37
553
569
584
600
273
616
632
648
664
680
696
712
727
743
759
274
775
791
807
823
838
854
870
886
902
917
275
43933
43949 43%5
43 981
43996
44012
44028
44 044
44059 44075 |
276
44091
44107
44122
44138
44154
170
185
201
217
232
277
248
264
279
295
311
326
342
358
373
389
278
404
420
436
451
467
483
498
514
529
545
279
560
576
592
607
623
638
654
669
685
700
280
44 716 44 731
44 747
44 762
44 778
44 793
44809
44 824
44840
44 855
281
871
886
902
917
932
948
963
979
994
45 010
282
45 025
45 040
45 056
45 071
45 086
45102
45117
45133
45148
163
283
179
194
209
225
240
255
271
286
301
317
284
332
347
362
378
393
408
423
439
454
469
285
45 484 45 500 45 515
45 530
45 545
45 561
45 576
45 591
45 606
45 621
286
637
652
667
682
697
712
728
743
758
773
287
788
803
818
834
849
864
879
894
909
924
288
939
954
%9
984
46000
46015
46030 46045
46060 46075 1
289
46090 46105
46120 46135
150
165
180
195
210
225
290
46240 46255
46270 46285
46300
46315
46330 46345
46359
46374
291
389
404
419
434
449
464
479
494
509
523
292
538
553
568
583
598
613
627
642
657
672
293
687
702
716
731
746
761
776
790
805
820
294
835
850
864
879
894
909
923
938
953
967
295
46982
46 997
47012
47026
47 041
47056
47 070
47085
47100
47114
2%
47129
47144
159
173
188
202
217
232
246
261
297
276
290
305
319
334
349
363
378
392
407
298
422
436
451
465
480
494
509
524
538
553
299
567
582
5%
611
625
640
654
669
683
698
300
47 712
47 727
47 741
47 756
47 770
47 784
47 799
47813
4782S
47842
If
O
1
2
3
4
5
6
7
8
9
360-800
82
800-860
N
1
2
8
4
5
6 7
8 9
800
47 712
47 727
47 741
47 756
47 770
47 784
47 799 47813
47828 47842
301
857
871
885
900
914
929
943 958
972 966
302
48001
48015
48029
48044
48058
48073
48087 48101
48116 48130
303
144
159
173
187
202
216
230 244
259 273
304
287
302
316
330
344
359
373 387
401 416
d05
48430
48444
48458
48473
48487
48501
48515 48530
48544 48558
306
572
586
601
615
629
643
657 671
686 700
307
714
728
742
756
770
785
799 813
827 841
306
855
869
883
897
911
926
940 954
968 982
309
996
49010
49024
49038
49052
49066
49080 49094
49106 49122
aio
49136
49150
49164
49178
49192
49 206
49220 49234
49248 49262
311
276
290
304
318
332
346
360 374
388 402
312
415
429
443
457
471
485
499 513
527 541
313
554
568
582
596
610
624
638 651
665 679
314
693
707
721
734
748
762
776 790
803 817
315
49831
49845
49859
49872
49 88^
49900
49914 49927
49941 49955
316
969
982
996
50010
50024
50037
50051 50065
50079 50092
317
50106
50120
50133
147
161
174
188 202
215 229
318
243
256
270
284
297
311
325 338
352 365
319
379
393
406
420
433
447
461 474
488 501
820
5051i
50529
50542
50 556
50569
50583
50596 50610
50623 50637
321
651
664
678
691
705
718
732 745
759 772
322
786
799
813
826
840
853
866 880
893 907
323
920
934
947
%1
974
987
51001 51014
51028 51041
324
5105i
51068
51081
51095
51108
51121
135 148
162 175
825
51188
51202
51215
51228
51242
51255
51268 51282
51295 51308
326
322
335
348
362
375
388
402 415
428 441
327
455
468
481
495
508
521
534 548
561 574
328
587
601
614
627
640
654
667 680
693 706
329
720
733
746
759
772
•
786
799 812
825 838
880
51851
51865
51878
51891
51904
51917
51930 51943
51957 51970
331
983
996
52009
52022
52035
52048
52061 52075
52088 52101
332
52114
52127
140
153
166
179
192 205
218 231
333
244
257
270
284
297
310
323 336
349 362
334
375
388
401
414
427
440
453 466
479 492
885
52 504
52 517
52 530
52 543
52 556
52 569
52 582 52 595
52 608 52621
336
634
647
660
673
686
699
711 724
737 750
337
763
776
789
802
815
827
840 853
866 879
338
892
905
917
930
943
956
969 982
994 53007
339
53 020
53033
53046
53 058
53 071
53084
53097 53110
53 122 135
840
53148
53161
53173
53186
53199
53 212
53 224 53 237
53 250 53 263
341
275
288
301
314
326
339
352 364
377 390
342
403
415
428
441
453
466
479 491
504 517
343
529
542
555
567
580
593
605 618
631 643
344
656
668
681
694
706
719
732 744
757 769
845
53 782
53 794
53 807
53 820
53 832
53 845
53 857 53 870
53 882 53 895
346
908
920
933
945
958
970
983 995
54008 54020
347
54033
54045
54058
54070
54083
54095
54108 54120
133 145
348
158
170
183
195
208
220
233 245
258 270
349
283
295
307
320
332
345
357 370
382 394
850
54407
54419
54432
54 444
54456
54469
54481 54494
54 506 54518
N
1
2
8
4
5
6 7
8 9
800-860
860-400
88
N
12
8
4
5
6
7
• 8
O
350
54407 54419 54432
54 444
54456
54 469
54481
54494
54 506
54518
351
531 543 555
568
580
593
605
617
630
642
352
654 667 679
691
704
716
728
741
753
765
353
777 790 802
814
827
839
851
864
876
888
354
900 913 925
937
949
962
974
986
998
55011
855
55 023 55 035 55 047
55 060
55 072
55084
55096
55108
55121
55133
356
145 157 169
182
194
206
218
230
242
255
357
267 279 291
303
315
328
340
352
364
376
358
388 400 413
425
437
449
461
473
485
497
359
509 522 534
546
558
570
582
594
606
618
860
55 630 55 642 55 654
55 666
55 678
55 691
55 703
55 715
55 727
55 739
361
751 763 775
787
799
811
823
835
847
859
362
871 883 895
907
919
931
943
955
967
979
363
991 56003 56015
56027
56038
56050
56062
56074
56086
56098
364
56 110 122 134
146
158
170
182
194
205
217
865
56229 56 241 56253
56265
56277
56289
56301
56312
56324
56336
366
348 360 372
384
3%
407
419
431
443
455
367
467 478 490
502
514
526
538
549
561
573
368
585 597 608
620
632
644
656
667
679
691
369
703 714 726
738
750
761
773
785
797
808
870
56820 56832 56844
56855
56867
56879
56891
56902
56914
56926
371
937 949 961
972
984
996
57008
57 019
57031
57043
372
57054 57066 57078
57089
57101
57113
124
136
148
159
373
171 183 194
206
217
229
241
252
264
276
374
287 299 310
322
334
345
357
368
380
392
875
57403 57415 57426
57438
57449
57461
57473
57 484
57496
57507
376
519 530 542
553
565
576
588
600
611
623
377
634 646 657
669
680
692
703
715
726
738
37B
749 761 772
784
795
807
818
830
841
852
379
864 875 887
898
910
921
933
944
955
967
880
57978 57990 58001
58013
58024
58035
58047
58058
58070
58081
381
58092 58104 115
127
138
149
161
172
184
195
382
206 218 229
240
252
263
274
286
297
309
383
320 331 343
354
365
377
388
399
410
422
384
433 444 456
467
478
490
501
512
524
535
885
58546 58557 58569
58 580
58591
58602
58614
58625
58636
58647
386
659 670 681
692
704
715
726
737
749
760
387
771 782 794
805
816
827
838
850
861
872
388
883 894 906
917
928
939
950
961
973
984
389
995 59006 59017
59028
59040
59051
59062
59073
59084
59095
890
59106 59118 59129
59140
59151
59162
59173
59184
59195
59207
391
218 229 240
251
262
273
284
295
306
318
392
329 340 351
362
373
384
395
406
417
428
393
439 450 461
472
483
494
506
517
528
539
394
550 561 572
583
594
605
616
627
638
649
805
59660 59671 59682
59693
59704
59715
59726
59737
59 748 "59 759 |
3%
770 780 791
802
813
824
835
846
857
868
397
879 890 901
912
923
934
945
956
966
977
398
988 999 60010
60021
60032
60043
60054 60065
60076
60086
399
60097 60108 119
130
141
152
163
173
184
195
400
60206 60217 60228
60239
60249
60260
60271
60282
60293
60304
N
12
8
4
5
6
7
8
O
860-400
84
400-460
N
1
2
3
4
5
6
7
8
400
60206
60217
60228
60239
60249
60260
60271
60282
60293
60304
401
314
325
336
347
358
369
379
390
401
412
402
423
433
444
455
466
477
487
498
509
520
403
531
541
552
563
574
584
595
606
617
627
404
638
649
660
670
681
692
703
713
724
735
405
60 746
60 756
60767
60 778
60788
60 799
60810
60821
60831
60842
406
853
863
874
885
895
906
917
927
938
949
407
959
970
981
991
61002
61013
61023
61034 61045
61055
406
61066
61077
61087
61098
109
119
130
140
151
162
409
172
183
194
204
215
225
236
247
257
268
410
61278
61289
61300
61310
61321
61331
61342
61352
6136^*374 1
411
384
395
405
416
426
437
448
458
469
479
412
490
500
511
521
532
542
553
563
574
584
413
595
606
616
627
637
648
658
669
679
690
414
700
711
721
731
742
752
763
773
784
794
415
61805
61815
61826
61836 61847
61857
61868
61878
61888
61899
416
909
920
930
941
951
962
972
982
993
62 003
417
62014 62024 62034 62045
62055
62066
62076
62086
62097
107
418
118
128
138
149
159
170
180
190
201
211
419
221
232
242
252
263
273
284
294
304
315
420
62325
62335
62346
62356
62 366
62 377
62 387
62 397
62 408
62 418
421
428
439
449
459
469
480
490
500
511
521
422
531
542
552
562
572
583
593
603
613
624
423
634
644
655
665
675
685
6%
706
716
726
424
737
• 747
757
767
778
788
798
808
818
829
425
62 839
62849
62859
62870
62880
62 890
62900
62910
62921
62 931
426
941
951
961
972
982
992
63 002
63 012
63 022
63 033
427
63 043
63 053
63063
63073
63 083
63 094
104
114
124
134
428
144
155
165
175
185
195
205
215
225
236
429
246
256
266
276
286
296
- 306
317
327
337
430
63 347
63 357
63 367
63 377
63 387
63 397
63 407
63 417
63 428
63438
431
448
458
468
478
488
498
508
518
528
538
432
548
558
568
579
589
599
609
619
629
639
433
649
659
669
679
689
699
709
719
729
739
434
749
759
769
779
789
799
809
819
829
839
435
63 849
63 859
63 869
63 879
63 889
63 899
63909
63 919
63 929
63 939
436
949
959
969
979
988
998
64008
64018
64028
64038
437
64048
64058
64068
64078
64088
64098
108
118
128
137
438
147
157
167
177
187
197
207
217
227
237
439
246
256
266
276
286
2%
306
316
326
335
440
64345
64355
64 365
64 375 64385
64 395
64404
64414
64 424
64 434
441
444
454
464
473
483
493
503
513
523
532
442
542
552
562
572
582
591
601
611
621
631
443
640
650
660
670
680
689
699
709
719
729
444
738
748
758
768
777
787
797
807
816
826
445
• 64836
64 846
64 856
64865
64 875
64885
64895
64904
64914
64924
446
933
943
953
963
972
982
992
65 002
65 011
65 021
447
65 031
65 040
65 050
65060
65 070
65 079
65 089
099
108
118
448
128
137
147
157
167
176
"^186
1%
205
215
449
22i
234
244
254
263
273
283
292
302
312
450
65 321
65331
65 341
65 350
65 360
65 369
65 379
65389
65398 65408 |
N
1
2
3
4
5
6
7
8 '
• 1
400-460
450-500 »6
N
O
1
2 3
4
5 6 7 8
450
65 321
65331
65 341 65 350
65 360
. 65369 65379 65389 65398 65408
451
418
427
437 447
456
466 475 485 495 504
452
514
523
533 543
552
562 571 581 591 600
453
610
619
629 639
648
658 667 677 686 696
454
706
715
725 734
744
753 763 772 782 792
455
65 801
65 811
65 820 65 830
65 839
65 849 65 858 65868 65 877 65 887
456
896
906
916 25
935
944 954 963 973 982
457
992
66001
66011 66020
66030
66039 66049 66058 66068 66077
458
66087
096
106 115
124
134 143 153 162 172
459
181
191
200 210
219
229 238 247 257 266
460
66276
66285
66295 66304
66314
66323 66332 66342 66351 66361
461
370
380
389 398
408
417 427 436 445 455
462
464
474
483 492
502
511 521 530 539 549
463
558
567
577 586
596
605 614 624 633 642
464
652
661
671 680
689
699 708 717 727 736
465
66745 66755 66764 66773 66783
66792 66801 66811 66820 66829
466
839
848
857 867
876
885 894 904 913 922
467
932
941
950 960
969
978 987 997 67006 67015
468
67025
67034
67043 67052
67062
67071 67080 67089 099 108
469
117
127
136 145
154
164 173 182 191 201
470
67210
67219 67228 67237
67247
67256 67265 67 274 67284 67293
471
302
311
321 330
339
348 357 367 376 385
472
394
403
413 422
431
440 449 459 468 477
473
486
495
504 514
523
532 541 550 560 569
474
578
587
596 605
614
624 633 642 651 660
475
67669
67679 67688 67697
67 706
67 715 67 724 67733 67 742 67752
476
761
770
779 788
797
806 815 825 834 843
477
852
861
870 879
888
897 906 916 925 934
478
943
952
961 970
979
988 997 68006 68015 68024
479
68034
68043
68052- 68061
68070
68079 68088 097 106 115
480
68124
68133
68142 68151
68160
68169 68178 68187 68196 68205
481
215
224
233 242
251
260 269 278 287 296
482
305
314
323 332
341
350 359 368 377 386
483
395
404
413 422
431
440 449 458 467 476
484
485
494
502 511
520
529 538 547 556 565
485
68 574
68583
68592 68601
68610
68619 68628 68637 68646 68655
486
664
673
681 690
699
708 717 726 735 744
487
753
762
771 780
789
797 806 815 ' 824 833
488
842
851
860 869
878
886 895 904 913 922
489
931
940
949 958
966
.975 984 993 69002 69011
490
69020
69028
69037 69046
69055
69064 69073 69082 69090 69099
491
108
117
126 135
144
152 161 170 179 188
492
197
205
214 223
232
241 249 258 267 276
493
285
294
302 311
320
329 338 346 355 364
494
373
381
390 399
408
417 425 434 443 452
495
69461
69469
69478 69487
69496
69 504 69 513 69522 69 531 69539
496
548
557
566 574
583
592 601 609 618 627
497
636
644
653 662
671
679 688 697 705 714
498
723
732
740 749
758
767 775 784 793 801
499
810
819
827 836
845
854 862 871 880 888
500
69897
69906
69914 69923
69932
69940 69949 69958 69966 69975
N
1
2 3
4
5 6 7 8V ^"
460-600
86
600-660
N
1
2
3 4
5
6
7
8
500
69897
69906
69914
69923 69932
69940
69949
69958
69966
69975
501
984
992
70001
70010 70018
70027
70036
70044
70053
70062
502
70070
70079
088
096 105
114
122
131
140
148
503
157
165
174
183 191
200
209
217
226
234
504
243
252
260
269 278
286
295
303
312
321
505
70329
70338
70346
70355 70364
70372
70381
70389
70398
70406
506
415
424
432
441 449
458
467
475
484
492
507
501
509
518
526 535
544
552
561
569
578
508
) 586
59i
603
612 621
629
638
646
655
. 663
509
' 672
680
689
697 706
714
723
731
740
749
510
70757
70 766
70774
70783 70 791
70800
70808
70817
70825
70834
511
842
851
859
868 876
885
893
902
910
919
512
927
935
944
952 961
969
978
986
995
71003
513
71012
71020
71029
71037 71046
71054
71063
71071
71079
088
514
096
105
113
122 130
139
147
155
164
172
515
71181
71189
71198
71206 71214
71223
71231
71240
71248
71257
516
265
273
282
290 299
307
315
324
332
341
517
349
357
366
374 383
391
399
408
416
425
518
433
441
450
458 466
475
483
492
500
508
519
517
525
533
542 550
559
567
575
584
592
520
71600
71609
71617
71625.71634
71642
71650
71659
71667
71675
521
684
692
700
709 717
725
734
742
750
759
522
767
775
784
792 800
809
817
825
834
842
523
850
858
867
875 883
892
900
908
917
925
524
933
941
950
958 966
975
983
991
999
72008
525
72016
72 024
72032
72041 72 049
72 057
72 066
72 074
72082
72090
526
099
107
115
123 132
140
148
156
165
173
527
181
189
198
206 214
222
230
239
247
255
528
^ 263
272
280
288 296
304
313
321
329
337
529
346
354
362
370 378
387
395
403
411
419
580
72428
72436
72444
72 452 72 460
72469
72477
72485
72493
72 501
531
509
518
526
534 542
550
558
567
575
583
532
591
599
607
616 624
632
640
648
656
665
533
673
681
689
697 705
713
722
730
738
746
534
754
762
770
779 787
795
803
811
819
827
535
72 835
72 843
72852
72860 72868
72 876
72 884
72 892
72 900
72908
536
916
925
933
941 949
957
965
973
981
989
537
997
73 006
73014
73 022 73 030
73038
73 046
73054
73062
73070
538
73 078
086
094
102 111
119
127
135
143
151
539
159
167
175
183 191
199
207
215
223
231
540
73 239
73 247
73 255
73 263 73 272
73 280
73 288
73 296
73 304
73312
541
320
328
336
344 352
360
368
376
384
392
542
400
408
416
424 432
440
448
456
464
472
543
480
488
496
504 512
520
528
536
544
552
544
560
568
576
584 592
600
608
616
624
632
545
73 640
73 648
73 656
73 664 73 672
73 679
73 687
73 695
73 703
73 711
546
719
727
735
743 751
759
767
775
783
791
547
799
807
815
823 830
838
846
854
862
870
548
878
886
894
902 910
918
926
933
941
949
549
957
965
973
981 989
997
74005
74013
74020
74028
550
74036
74044
74052
74060 74068
74076
74084
74092
74099
74107
N
1
2
3 4
5
6
7
8
600-650
660-e
^1^
87
^^^^^
N
1
2 8
4
5
6
7
8
550
74036
74044
74052 74 060
74068
74076
74084
74092
74099
74107
551
115
123
131 139
147
155
162
170
178
186
552
194
202
210 218
225
233
241
249
257
265
553
273
280
288 296
304
312
320
327
335
343
554
351
359
367 374
382
390
398
406
414
421
555
74429
74437
74445 74453
74461
74468
74476
74484
74492
74 500
556
507
515
523 531
539
547
554
562
570
578
557
586
593
601 609
617
624
632
640
648
656
558
663
671
679 687
695
702
710
718
726
733
559
741
749
757 764
772
780
788
796
803
811
560
74819
74827
74834 74842
74850
74858
74865
74873
74 881
74889
561
896
904
912 920
927
935
943
950
958
966
562
974
981
.989 997
75 005
75 012
75020
75028
75035
75043
563
75 051
75 059
75 066 75074
082
089
097
105
113
120
564
128
136
143 151
159
166
174
182
189
197
565
75 20i
75 213
75 220 75 228
75 236
75 243
75 251
75 259
75 266
75 274
566
282
289
297 305
312
320
328
'335
343
351
567
358
366
374 381
389
397
404
412
420
427
568
435
442
450 458
465
473
481
488
4%
504
569
511
519
526 534
542
549
557
565
572
580
570
75 587
75 595
75 603 75610
75 618
75 626
75633
75641
75648
75656
571
664
671
679 686
694
702
709
717
724
732
572
740
747
755 762
770
778
785
793
800
808
573
815
823
831 838
846
853
861
868
876
884
574
891
899
906 914
921
929
937
944
952
959
575
75 967
75974
75982 75989
75997
76005
76012
76020
76027
76035
576
76042
76050
76057 76065
76072
080
087
095
103
110
577
118
125
133 140
148
155
163
170
178
185
578
193
200
208 215
223
230
238
245
253
260
579
268
275
283 290
298
305
313
320
328
335
580
76343
76350
76358 76365
76373
76380
76388
76395
76403
76410
581
418
425
433 440
448
455
462
470
477
485
582
492
500
507 515
522
530
537
545
552
559
583
567
574
582 589
597
604
612
619
626
634
584
641
649
656 664
671
678
686
693
701
708
585
76716
76 723
76 730 76738
76745
76 753
76760
76768
76775
76782
586
790
797
805 812
819
827
834
842
849
856
587
864
871
879 886
893
901
908
916
923
930
588
938
945
953 960
967
975
982
989
997
77004
589
77012
77019
77026 77034
77041
77048
77056
77063
77070
078
590
77085
77093
77100 77107 77115
77122
77129
77137
77144
77151
591
159
166
173 181
188
195
203
210
217
225
592
232
240
247 254
262
269
276
283
291
298
593
305
313
320 327
335
342
349
357
364
371
594
379
386
393 401
408
415
422
430
437
444
505
77452
77459
77466 77474
77481
77488
77495
77 503
77510
77517
596
525
532
539 546
554
561
568
576
583
590
597
597
605
612 619
627
634
641
648
656
663
598
670
677
685 692
699
706
714
721
728
735
599
743
750
757 764
772
779
786
793
801
808
600
77815
77822
77 aw 77837
77844
77851
77859
77866
77873
77880
N
■ 1
2 8
4
5
6
7
8
9
660-600
8S
;(• •
-650
N
1
2
3
4
5
6
7
8
600
77 815
77822
77830
77837
77844
77851
77 859
77866
77873
77880
601
887
895
902
909
916
924
931
938
945
952
602
960
967
974
981
988
996
78003
78010
78017
78025
603
78032
78039
78046
78053
78061
78068
075
082
089
097
604
104
HI
118
125
132
140
147
154
161
168
605
78176
78183
78190
78197
78204
78 211
78219
78 226
78 233
78240
606
247
254
262
269
276
283
290
297
305
312
607
319
326
333
340
347
355
362
369
376
383
608
390
398
405
412
419
426
433
440
447
455
609
462
469
476
483
490
497
504
512
519
526
610
78 533
78 540
78 547
78554
78 561
78 569
78 576
78 583
78 590
78 597
611
604
611
618
625
633
640
647
654
661
668
612
675
682
689
696
704
711
718
725
732
739
613
746
753
760
767
774
781
789
7%
803
810
614
817
824
831
838
845
852
859
866
•
873
880
615
78888
78895
78902
78909
78916
78923
78930
78937
78944
78951
616
958
965
972
979
986
993
79000
79007
79014
79021
617
79029
79036
79043
79 050
79057
79064
071
078
085
092
618
099
106
113
120
127
134
141
148
155
162
619
169
176
183
190
197
204
211
218
225
232
620
79 239
79246
79 253
79260
79 267
79 274
79 281
79288
79295
79302
621
309
316
323
330
337
344
351
358
365
372
^ 622
379
386
393
400
407
414
421
428
435
442
623
449
456
463
470
477
484
491
498
505
511
624
518
525
532
539
546
553
560
567
574
581
625
79588 79 595
79602
79609
79616
79623
79630
79637
79644
79650
626
657
664
671
678
685
692
699
706
713
720
627
727
734
741
748
754
761
768
775
782
789
628
796
803
810
817
824
831
837
844
851
858
629
865
872
879
886
893
900
906
913
920
927
630
79934
79941
79948
79955
79962
79969
79975
79982
79989
79996
631
80003
80010
80017
80024
80030
80037
80044
80051
80058 80065 |
632
072
079
085
092
099
106
113
120
127
134
633
140
147
154
161
168
175
182
188
195
202
634
209
216
223
229
236
243
250
257
264
271
635
80277
80284
80 291
80298 80305
80312
80318
80325
80332
80339
636
346
353
359
366
373
380
387
393
400
407
637
414
421
428
434
441
448
455
462
468
475
638
482
489
4%
502
509-
516
523
530
536
543
639
550
557
564
570
577
584
591
598
604
611
640
80618
80625
80632
80638
80645
80652
80659
80665
80672
80679
641
686
693
699
706
713
720
726
733
740
747
642
754
760
767
774
781
787
794
801
808
814
643
821
828
835
841
848
855
862
868
875
882
644
889
895
902
909
916
922
929
936
943
949
645
80956
80963
80969
80976
80983
80990
80996
81003
81010
81017
646
81 023
81030
81037
81043
81050
81057
81064
070
077
084
647
090
097
104
111
117
124
131
137
144
151
648
158
164
171
178
184
191
198
204
211
218
649
224
231
238
245
251
258
265
271
278
285
650
81291
81298
81305
81311
81318
81325
81331
81338
81345
81351
,1.
1
2
3
4
5
6
7
8
;(• I
-660
650-700
89
N
O
1
2
8
4
5
6
7
8 9
650
81291 81298 81305
81311
81318
81325
81331
81338
81345 81351
651
358
365
371
378
385
391
398
405
411 418
652
425
431
438
445
451
458
465
471
478 485
653
491
498
505
511
518
525
531
538
544 551
654
558
564
571
578
584
591
598
604
611 617
655
81624
81631
81637
81644
81651
81657
81664
81671
81677 31684
656
690
697
704
710
717
723
730
737
743 750
657
757
763
770
776
783
790
7%
803
809 816
658
823
829
836
842
849
856
862
869
875 882
659
889
895
902
908
915
921
928
935
941 948
660
81954
81961
81968
81974
81981
81987
81994
82000
82007 82014
661
82020
82027
82033
82040
82046
82053
82060
066
073 079
662
086
092
099
105
112
119
125
132
138 145
663
151
158
164
171
178
184
191
197
204 210
664
217
223
230
236
243
249
256
263
269 276
665
82282
82 289
82 295
82302
82 308
82315
82 321
82328
82334 82341
666
347
354
360
367
373
380
387
393
400 406
667
413
419
426
432
439
445
452
458
465 471
668
478
484
491
497
504
510
517
523
530 536
669
543
549
556
562
569
575
582
588
59^ 601
670
82607
82614
82 620
82 627
82633
82640
82646
82653
82659 82666
671
672
679
685
692
698
705
711
718
724 730
672
737
743
750
756
763
769
776
782
789 795
673
802
808
814
821
827
834
840
847
853 860
674
866
872
879
885
892
898
905
911
918 924
675
82930
82937
82943
82950 82956
82963
82969 82975
82982 82988
676
995
83 001
83 008
83 014
83020
83 027
83033
83040
83046 83052
677
83059
065
072
078
085
091
097
104
110 117
678
123
129
136
142
149
155
161
168
174 181
679
187
193
200
206
213
219
225
232
238 245
680
83 251
83 257
83 264
83 270
83 276
83 283
83 289
83 296
83302 83308
681
315
321
327
334
340
347
353
359
366 372
682
378
385
391
398
404
410
417
423
429 436
683
442
448
455
461
467
474
480
487
493 499
684
506
512
518
525
531
537
544
550
556 563
e85
83 569
83 575
83 582
83 588
83 594
83601
83 607
83 613
83620 83626
686
632
639
645
651
658
664
670
677
683 689
687
696
702
708
715
721
727
734
740
746 753
688
759
765
771
778
784
790
797
803
809 816
689
822
828
835
841
847
853
860
866
872 879
690
83 885
83 891
83 897
83 904
83 910
83 916
83923
83 929
83 935 83942
691
948
954
960
%7
973
979
985
992
998 84004
692
84011
84017
84023
84029
84036
84042
84048
84055
84061 067
693
073
080
086
092
098
105
111
117
123 130
694
136
142
148
155
161
167
173
180
186 192
605
84198 84 205 84 211
84 217
84 223
84230
84 236
84 242
84 248 84 255
696
261
267
273
280
286
292
298
305
311 • 317
697
323
330
336
342
348
354
361
367
373 379
698
386
392
398
404
410
417
423
429
435 442
699
448
454
460
466
473
479
485
491
497 504
700
84 510
84 516
84 522
84 528
84 535
84 541
84 547
84553
84559 84 566
N
1
2
3
4
5
6
7
8 9
660-700
40
700-760
K
12 8 4
5 6
7
8 9
700
84 510 84516 84522 84 528 84535
84541 84 547
84553
84 559 84566
701
572 578 584 590 597
603 609
615
621 628
702
634 640 646 652 658
665 671
677
683 689
703
696 702 708 714 720
726 733
739
745 751
704
757 763 770 776 782
788 794
800
807 813
705
84819 84825 84831 84837 84844
84850 84856
84 862
84868 84874
706
880 887 893 899 905
911 917
924
930 936
707
942 948 954 960 967
973 979
985
991 997
708
85 003 85 009 85016 85022 85 028
85 034 85 040
85 046
85 052 85 058
709
065 071 077 083 089
095 101
107
114 120
710
85126 85132 85138 85144 85150
85 156 85 163 85 169 85 175 85 181 |
711
187 193 199 205 211
217 224
230
236 242
712
248 254 260 266 272
278 285
291
297 303
713
309 315 321 327 333
339 345
352
358 364
714
370 376 382 388 394
400 406
412
418 425
715
85 431 85437 85 443 85 449 85 455
85 461 85 467
85 473
85479 85 485
716
491 497 503 509 516
522 528
534
540 546
717
552 558 564 570 576
582 588
594
600 606
718
612 618 625 631 637
643 649
655
661 667
719
673 679 685 691 697
703 709
715
721 727
720
85 733 85 739 85 745 85 751 85 757
85 763 85 769
85 775
85 781 85 788
721
794 800 806 812 818
824 830
836
842 848
722
854 860 866 872 878
884 890
896
902 908
723
914 920 926 932 938
944 950
956
962 968
724
974 980 986 992 998
86004 86010
86016
86022 86028
725
86034 86040 86046 86052 86058
86064 86070
86076
86082 86088
726
094 100 106 112 118
124 130
136
141 147
727
153 159 165 171 177
183 189
195
201 207
728
213 219 225 231 237
243 249
255
261 267
729
273 279 285 291 297
303 308
314
320 326
780
86332 86338 86344 86350 86356
86362 86368
86374
86380 86386
731
392 398 404 410 415
421 427
433
439 445
732
451 457 463 469 475
481 487
493
499 504
733
510 516 522 528 534
540 546
552
558 564
734
570 576 581 587 593
599 605
611
617 623
785
86629 86635 86641 86646 86652
86658 86664 86670 86676 86682 |
736
688 694 700 705 711
717 723
729
735 741
737
747 753 759 764 770
776 782
788
794 800
738
806 812 817 823 829
835 841
847
853 859
739
864 870 876 882 888
894 900
906
911 917
740
86923 86929 86935 86941 86947
86953 86958
86964
86970 86976
741
982 988 994 999 87005
87011 87017
87023
87029 87035
742
87040 87046 87052 870^8 064
070 075
081
087 093
743
099 105 111 116 122
128 134
140
146 151
744
157 163 169 175 181
186 192
198
204 210
745
87216 87221 87227 87233 87239
87245 87251
87256
87262 87268
746
274 280 286 291 297
303 309
315
320 326
747
332 338 344 349 355
361 367
373
379 384
748
390 396 402 408 413
419 425
431
437 442
749
448 454 460 466 471
477 483
489
495 500
750
87506 87512 87518 87523 87529
87535 87541
87 547
87 552 87 558
K
12 8 4
5 O
7
8
700-760
•
760-800
41
N
O
1
2
a
4
5
6
7
8
750
87506
87 512
87 518
87 523
87 529
87 535
87 541
87547
87552
87556
751
564
570
576
581
587
593
599
604
610
616
752
622
628
633
639
645
651
656
662
668
674
753
679
685
691
697
703
708
714
720
726
731
754
737
743
•
749
754
760
766
772
777
783
789
755
87 795
87800
87806
87812
87818
87823
87829 87835
87841
87846
756
852
858
864
869
875
881
887
892
898
904
757
910
915
921
927
933
938
944
950
955
961
758
967
973
978
984
990
996
88001
88007
88013
88016
759
88024
88030
88036
88041
88047
88053
058
064
070
076
760
88061
88067
88093
88098
88104
88110
88116
88121
88127
88133
761
138
144
150
156
161
167
173
178
184
190
762
195
201
207
213
218
224
230
235
241
247
763
252
258
264
270
275
281
287
292
298
304
764
309
315
321
326
332
338
343
349
355
360
765
88366
88372
88377
88383
88389
88395
88400
88406
88412
88417
766
423
429
434
440
446
451
457
463
468
474
767
480
485
491
497
502
508
513
519
525
530
768
536
542
547
553
559
564
570
576
581
587
769
593
598
604
610
615
621
627
632
638
643
770
88649 88655
88660
88666
88672
88677
88683
88689
88694
88 700
771
705
711
717
722
728
734
739
745
750
756
772
762
767
773
779
784
790
795
801
807
612
773
818
824
829
835
840
846
852
857
863
868
774
874
880
885
891
897
902
908
913
919
925
775
88930
88936
88941
88947
88953
88958 88964 88969 88975
88981
776
986
992
997
89003
89009
89014
89020
89025
89031
89037
777
89042
89048
89053
059
064
070
076
081
087
092
778
098
104
109
115
120
126
131
137
143
148
779
154
159
165
170
176
182
187
193
198
204
780
89209
89215
89221
89226
89232
89237
89243
89248
89254
89260
781
265
271
276
282
287
293
298
304
310
315
782
321
326
332
337
343
348
354
360
365
371
783
376
382
387
393
398
404
409
415
421
426
784
432
437
443
448
454
459
465
470
476
481
785
89487
89492
89498
89504
89509
89515
89520
89526
89531
89537
786
542
548
553
559
564
570
575
581
586
592
787
597
603
609
614
620
625
631
636
642
647
788
653
658
664
669
675
680
686
691
697
702
789
708
713
719
724
730
735
741
746
752
757
790
89 763
89 768
89 774
89779 89785
89 790
89 796
89801
89807
89612
791
818
823
829
834
840
845
851
856
862
867
792
873
878
883
889
894
900
905
911
916
922
793
927
933
938
944
949
955
960
966
971
977
794
982
988
993
998
90004
90009 90015
90020 90026 90031 1
795
90037
90042
90048
90053
90059
90064 90069 90075
90080 90066 |
796
091
097
102
108
113
119
124
129
135
140
797
146
151
157
162
168
in
179
184
189
195
798
200
206
211
217
222
227
233
238
244
249
799
255
260
266
271
276
282
287
293
296
304
800
90309 90314
90320
90325
90331
90336 90342 90347 90352
90358
N
1
2
3
4
5
6
7
8
760-800
.42
800-860
N
O 1 2 3 4
5
6
7 8 9
800
90309 90314 90320 90325 90331
90336
90342
90347 90352 90358
801
363 369 374 380 38i
390
3%
401 407 412
802
417 423 428 434 439
445
450
455 461 466
803
472 477 482 488 493
499
504
509 515 520
804
526 531 536 542 547
553
558
563 569 574
805
90580 9058i 90590 90596 90601
90607
90612
90617 90623 90628
806
634 639 644 6i0 655
660
666
671 677 682
807
687 693 698 703 709
714
720
725 730 736
806
741 747 752 757 763
768
773
779 784 789
809
79i 800 806 811 816
822
827
832 838 843
810
90849 90854 90859 9086i 90870
90875
90881
90886 90891 90897
811
902 907 913 918 924
929
934
940 945 950
812
956 961 966 972 977
982
988
993 998 91004
813
91009 91014 91020 91025 91030
91036
91041
91046 91052 057
814
062 068 073 078 084
089
094
100 105 110
815
91 116 91 121 91 126 91 132 91 137
91142
91 148
91 153 91 158 91 164
816
169 174 180 185 190
196
201
206 212 217
817
222 228 233 238 243
249
254
259 265 270
818
275 281 286 291 297
302
307
312 318 323
819
328 334 339 344 350
355
360
365 371 376
820
91381 91387 91392 91397 91403
91408
91413
91418 91424 91429
821
434 440 445 450 455
461
466
471 477 482
822
487 492 498 503 508
514
519
524 529 535
823
540 545 551 556 561
566
572
577 582 587
824
593 598 603 609 614
619
624
630 635 640
825
91645 91651 91656 91661 91666
91672
91677
91682 91687 91693
826
698 703 709 714 719
724
730
735 740 745
827
751 756 761 766 772
777
782
787 793 798
828
803 808 814 819 824
829
834
840 845 850
829
855 861 866 871 876
882
887
892 897 903
830
91908 91913 91918 91924 91929
91934 91939 91944 91950 91955 |
831
960 965 971 976 981
986
991
997 92002 92007
832
92012 92 018 92 023 92 028 92033
92038
92 044
92 049 054 059
833
065 070 075 080 085
091
0%
101 106 111
834
117 122 127 132 137
143
148
153 158 163
835
92169 92074 92179 92184 92189
92195
92 200
92 205 92 210 92 215
836
221 226 231 236 241
247
252
257 262 267
837
273 278 283 288 293
298
304
309 314 319
838
324 330 335 340 345
350
355
361 366 371
839
376 381 387 392 397
402
407
412 418 423 •
840
92428 92 433 92 438 92 443 92 449
92 454
92 459
92464 92469 92474
841
480 485 490 495 500
505
511
516 521 526
842
531 .' 536 542 547 552
557
562
567 572 578
843
583 588 593 598 603
609
614
619 624 629 ^
844
634 639 645 650 655
660
665
670 675 681
845
92686 92691 92696 92 701 92706
92 711
92 716
92 722 92 727 92 732
846
737 742 747 752 758
763
768
773 778 783
847
788 793 799 804 809
814
819
824 829 834
848
840 845 850 855 860
865
870
875 881 886
849
891 896 901 906 911
916
921
927 932 937
850
92 942 92 947 92 952 92 957 92 962
92967
92973
92978 92983 92988
N
12 3 4
5
6
7 8 9
800-850
850-900
48
N
O
1 2
8
4
5
6
7
8
9
850
92 942
92 947 92 952
92 957
92 962
92 %7
92 973
92 978
92 983
92 988
851
993
998 93 003
93 008
93 013
93 018
93 024
93 029
93 034
93 039
852
93 044
93 049 054
059
064
069
075
080
085
090
853
095
100 105
110
115
120
125
131
136
141
854
146
151 156
161
166
171
176
181
186
192
855
93197
93 202 93 207
93 212
93 217
93 222
93 227
93 232
93 237
93 242
856
247
252 258
263
268
273
278
283
288
293
857
298
303 308
313
318
323
328
334
339
344
858
349
354 359
364
369
374
379
384
389
394
859 .
Z9^
404 409
414
420
425
430
435
440
445
860
93 450 93 455 93 460 93 465
93 470
93 475
93 480 93 485
93 490 93 495 |
861
500
505 510
515
520
526
531
536
541
546
862
551
556 561
566
571
576
581
586
591
596
863
601
606 611
616
621
626
631
636
641
646
864
. 651
656 661
666
671
676
682
687
692
697
86^
93 702
93 707 93 712
93 717
93 722
93 727
93 732
93 737
93 742
93 747
866
752
757 762
767
772
777
782
787
792
797
867
802
807 812
817
822
827
832
837
842
847
868
852
857 862
867
872
877
882
887
892
897
869
902
907 912
917
922
927
932
937
942
947
870
93 952
93 957 93 962
93 967
93 972
93977
93 982
93 987
93 992
93 997
871
94002
94007 94 012
94 017
94 022
94 027
94 032
94 037
94 042
94 047
872
052
057 062
067
072
077
082
086
091
096
873
101
106 111
116
121
126
131
136
141
146
87^
151
156 161
166
171
176
181
186
191
196
875
94 201
94 206 94 211
94 216
94 221
94 226
94 231
94 236
94 240
94 245
876
250
255 260
265
270
275
280
285
290
295
877
300
305 310
315
320
325
330
335
340
345
878
349
354 359
364
369
374
379
384
389
394
879
399
404 409
414
419
424
429
433
438
443
880
94 448
94 453 94 458
94 463
94 468
94 473
94 478
94 483
94488
94 493
881
498
503 507
512
517
522
527
532
537
542
882
547
552 557
562
567
571
576
581
586
591
883
5%
601 606
611
616
621
626
630
635
640
884
645
650 655
660
665
670
675
680
685
689
885
94 694
94699 94 704
94 709
94 714
94 719
94 724
94 729
94 734
94 738
886
743
748 753
758
763
768
773
778
783
787
887
792
797 802
807
812
817
822
827
832
836
888
841
846 851
856
861
866
871
876
880
885
889
890
895 900
905
910
915
919
924
929
934
890
94939
94944 94949
94 954
94 959
94963
94 968
94 973
94 978
94 983
891
988
993 998
95 002
95 007
95 012
95 017
95 022
95 027
95 032
892
95 036
95 041 95 046
051
056
061
066
071
075
080
893
085
090 095
100
105
109
114
119
124
129
894
134
139 143
148
153
158
163
168
173
177
895
95182
95 187 95 192
95197
95 202
95 207
95 211
95 216
95 221
95 226
896
231
236 240
245
250
255
260
265
270
274
897
279
284 289
294
299
303
308
313
318
323
898
328
332 337
342
347
352
357
361
366
371
899
376
381 386
390
395
400
405
410
415
419
900
95 424
95 429 95 434
95 439
95 444
95 448
95 453
95 458
95 463
95 468
N
1 2
3
4
5
6
7
8
9
860-900
44
900-960
N
O
1
2
3
4
5
6
7
8
900
95 424
95 429
95 434
95 439
95 444
95 448
95 453
95 458
95463 95468
901
472
477
482
487
492
497
501
506
511 516
902
521
525
530
535
540
545
550
554
559 564
903
569
574
578
583
588
593
598
602
607 612
904
617
622
626
631
636
641
646
650
655 660
905
95 665
95 670
95 674
95 679
95 684
95 689
95 694
95 698
95 703 95 708
906
713
718
722
727
732
737
742
746
751 756
907
761
766
770
775
780
785
789
794
799 804
908
809
813
818
823
828
832
837
842
847 852
909
856
861
866
871
875
880
885
890
895 899
010
95 904
95 909
95 914
95 918
95 923
95 928
95 933
95 938
95 942 95 947
911
952
957
961
966
971
976
980
985
990 995
912
999
96004
96009
96014
96019
96 023
96 028
96033
96038 96042
913
96047
052
057
061
066
071
076
080
085 090
914
095
099
lOf
109
114
118
123
128
133 137
015
96142
96147
96152
96156
96161
96166
96171
96175
%180 96185
916
190
194
199
204
209
213
218
223
227 232
917
237
242
246
251
256
261
265
270
275 280
918
284
289
294
298
303
308
313
317
322 327
919
332
336
341
346
350
355
360
365
369 374
020
96379
96384
96388
96393
96398
96402
96407
96412
96417 96421
921
426
431
435
440
445
450
454
459
464 468
922
473
478
483
487
492
497
501
506
511 515
923
520
525
530
534
539
544
548
553
558 562
924
567
572
577
581
586
591
595
600
605 609
025
96614
96619
96624
96628
96633
96638
96642
96647
96652 96656
926
661
666
670
675
680
685
689
694
699 703
927
708
713
717
722
727
731
736
741
745 750
928
755
759
764
769
774
778
783
788
792 797
929
802
806
811
816
820
825
830
834
839 844
030
96848
96853
96 858
96 862
96867
96872
96876
96 881
96886 96890
931
895
900
904
909
914
918
923
928
932 937
932
942
946
951
956
960
965
970
974
979 984
933
988
993
997
97 002
97 007
97011
97 016
97021
97025 97030
934
97035
97039
97044
049
053
058
063
067
072 077
035
97081
97086
97090
97095
97100
97104
97109
97114
97118 97123
936
128
132
137
142
146
151
155
160
165 169
937
174
179
183
188
192
197
202
206
211 216
938
220
225
230
234
239
243
248
253
257 262
939
267
271
276
280
285 .
290
294
299
304 308
040
97313
97 317
97322
97 327
97 331
97336
97340
97 345
97350 97354
941
359
364
368
373
377
382
387
391
396 400
942
405
410
414
419
424
428
433
437
442 447
943
451
456
460
465
470
474
479
483
488 493
944
497
502
506
511
516
520
525
529
534 539
045
97 543
97 548
97 552
97 557
97 562
97 566
97 571
97575
97 580 97585
946
589
594
598
603
607
612
617
621
626 630
947
635
640
644
649
653
658
663
667
672 676
948
681
685
690
695
699
704
708
713
717 722
949
727
731
736
740
745
749
754
759
763 768
050
97 772
97 777
97782
97 786
97 791
97 795
97800
97804
97809 97813
N
1
2
3
4
5
6
7
8
900-950
9
950-1'
• If
45
N
1 2 3 4
5 6 7 8
950
97 772 97 777 97 782 97 786 97 791
97 795 97 800 97 804 97 809 97 813
951
818 823 827 832 836
841 845 850 855 859
952
864 868 873 877 882
886 891 8% 900 905
953
909 914 918 923 928
932 937 941 946 950
954
955 959 964 968 973
978 982 987 991 996
055
98000 98005 98009 98014 98019
98023 98028 98032 98037 98 041
956
046 050 055 059 064
068 073 078 082 087
957
091 0% 100 105 109
114 118 123 127 132
958
137 141 146 150 155
159 164 168 173 177
959
182 186 191 195 200
204 209 214 218 223
960
98 227 98 232 98 236 98 241 98 245
98250 98 254 98 259 98 263 98268
961
272 277 281 286 290
295 299 30f 308 313
962
318 322 327 331 336
340 345 349 354 358
963
363 367 372 376 381
385 390 394 399 403
964
408 412 417 421 426
430 435 439 444 448
965
98453 98457 98462 98466 98471
98475 98480 98484 98489 98493
966
498 502 507 511 516
520 525 529 534 538
%7
543 547 552 556 561
565 570 574 579 583
968
588 592 597 601 605
610 614 619 623 628
%9
632 637 641 646 650
655 659 664 668 673
070
98677 98682 98686 98691 98695
98700 98704 98709 98713 98 717
971
722 726 731 735 740
744 749 753 758 762
972
767 771 776 780 784
789 793 798 802 807
973
811 816 820 825 829
834 838 843 847 851
974
856 860 865 869 874
878 883 887 892 896
975
98900 98905 98909 98914 98918
98923 98927 98932 98936 98941
976
945 949 954 958 963
967 972 976 981 985
977
989 994 998 99003 99007
99012 99016 99021 99025 99029
978
99034 99038 99043 047 052
056 061 065 069 074
979
078 083 087 092 096
100 105 109 114 118
980
99123 99127 99131 99136 99140
99145 99149 99154 99158 99162
981
167 171 176 180 185
189 193 198 202 207
982
211 216 220 224 229
233 238 242 247 251
983
255 260 264 269 273
277 282 286 291 295
984
300 304 308 313 317
322 326 330 335 339
085
99344 99348 99352 99357 99361
99366 99370 99374 99379 99383
986
388 392 3% 401 405
410 414 419 423 427
987
432 436 441 445 449
454 458 463 467 471
003
476 480 484 489 493
498 502 506 511 515
989
520 524 528 533 537
542 546 550 555 559
000
99 564 99568 99572 99 577 99 581
99 585 99 590 99 594 99599 99603
991
607 612 616 621 625
629 634 638 642 647
992
651 656 660 664 669
673 677 682 686 691
993
695 699 704 708 712
717 721 726 730 734
994
739 743 747 752 756
760 765 769 774 778
005
99782 99787 99791 99795 99800
99804 99808 99813 99817 99822
996
826 830 835 839 843
848 852 856 861 865
997
870 874 878 883 887
891 896 900 904 909
998
913 917 922 926 930
935 939 944 948 952
999
957 961 965 970 974
978 983 987 991 996
1000
00000 00004 00009 00013 00017
00022 00026 00030 00035 00039
N
12 3 4
5 6 7 8
960-K
III
46
TABTiE
IV
PROPORTIONAL
PARTS OF DIFFERENCES
D
1
0.1
2
0.2
3
0.3
4
0.4
5
0.5
6
0.6
7
0.7
8
0.8
0.9
1
2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
3
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
4
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
9
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
10
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
11
1.1
2.2
3.3
4.4
5.5
6.6
7.7
8.8
9.9
12
1.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
13
1.3
2.6
3.9
5.2
6.5
7.8
9.1
10.4
11.7
14
1.4
2.8
4.2
5.6
7.0
8.4
9.8
11.2
12.6
15
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
16
1.6
3.2
4.8
6.4
8.0
9.6
11.2
12.8
14.4
17
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
15.3
18
1.8
3.6
5.4
7.2
9.0
10.8
12.6
14.4
16.2
19
1.9
3.8
5.7
7.6
9.5
11.4
13.3
15.2
17.1
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
21
2.1
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
22
2.2
4.4
6.6
8.8
11.0
13.2
15.4
17.6
19.8
23
2.3
4.6
6.9
9.2
11.5
13.8
16.1
18.4
20.7
24
2.4
4.8
7.2
9.6
12.0
14.4
16.8
19.2
21.6
25
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
26
2.6
5.2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
27
2.7
5.4
8.1
10.8
13.5
16.2
18.9
21.6
24.3
28
2.8
5.6
8.4
11.2
14.0
16.8
19.6
22.4
25.2
29
2.9
5.8
8.7
11.6
14.5
17.4
20.3
23.2
26.1
30
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
31
3.1
6.2
9.3
12.4
15.5
18.6
21.7
24.8
27.9
32
3.2
6.4
9.6
12.8
16.0
19.2
22.4
25.6
28.8
33
3.3
6.6
9.9
13.2
16.5
19.8
23.1
26.4
29.7
34
3.4
6.8
10.2
13.6
17.0
20.4
23.8
27.2
30.6
35
3.5
7.0
10.5
14.0
17.5
21.0
24.5
28.0
31.5
36
3.6
7.2
10.8
14.4
18.0
21.6
25.2
28.8
32.4
37
3.7
7.4
11.1
14.8
18.5
22.2
25.9
29.6
33.3
38
3.8
7.6
11.4
15.2
19.0
22.8
26.6
30.4
34.2
39
3.9
7.8
11.7
15.6
19.5
23.4
27.3
31.2
35.1
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
41
4.1
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
42
4.2
8.4
12.6
16.8
21.0
25.2
29.4
33.6
37.8
43
4.3
8.6
12.9 •
17.2
21.5
25.8
30.1
34.4
38.7
44
4.4
8.8
13.2
17.6
22.0
26.4
30.8
35.2
39.6
45
4.5
9.0
13.5
18.0
22.5
27.0
31.5
36.0
40.5
46
4.6
9.2
13.8
18.4
23.0
27.6
32.2
36.8
41.4
47
4.7
9.4
14.1
18.8
23.5
28.2
32.9
37.6
42.3
48
4.8
9.6
14.4
19.2
24.0
28.8
33.6
38.4
43.2
49
4.9
9.8
14.7
19.6
24.5
29.4
34.3
39.2
44.1
50
5.0
1
10.0
2
15.0
3
20.0
4
25.0
5
30.0
6
35.0
7
40.0
8
45.0
9
«
•
PROPORTION A Ti PARTS OF DIFFERENCES
47
This table contains the proportional parts
of differences from 1 to 100.
i^'or exam
pie, if the difference between two numbers is
73, 0.7 of this |
difference is
\ 51.1.
D
1
5.1
2
10.2
3
15.3
4
20.4
5
25.5
6
30.6
7
35.7
8
40.8
9
45.9
51
52
5.2
10.4
15.6
20.8
26.0
31.2
36.4
41.6
46.8
53
5.3
10.6
15.9
21.2
26.5
31.8
37.1
42.4
47.7
54
5.4
10.8
16.2
21.6
27.0
32.4
37.8
43.2
48.6
55
5.5
11.0
16.5
22.0
27.5
33.0
38.5
44.0
49.5
56
5.6
11.2
16.8
22.4
28.0
33.6
39.2
44.8
50.4
57
5.7
11.4
17.1
22.8
28.5
34.2
39.9
45.6
51.3
58
5.8
11.6
17.4
23.2
29.0
34.8
40.6
46.4
52.2
59
5.9
11.8
17.7
23.6
29.5
35.4
41.3
47.2
53.1
60
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
54.0
61
6.1
12.2
18.3
24.4
30.5
36.6
42.7
48.8
54.9
62
6.2
12.4
18.6
24.8
31.0
37.2
43.4
49.6
55.8
63
6.3
12.6
18.9
25.2
31.5
37.8
44.1
50.4
56.7
64
6.4
12.8
19.2
25.6
32.0
38.4
44.8
51.2
57.6
65
6.5
13.0
19.5
26.0
32.5
39.0
45.5
52.0
58.5
66
6.6
13.2
19.8
26.4
33.0
39.6
46.2
52.8
59.4
67
6.7
13.4
20.1
26.8
33.5
40.2
46.9
53.6
60.3
68
6.8
13.6
20.4
27.2
34.0
40.8
47.6
54.4
61.2
69
6.9
13.8
20.7
27.6
34.5
41.4
48.3
55.2
62.1
70
7.0
14.0
21.0
28.0
35.0
42.0
49.0
56.0
63.0
71
7.1
14.2
21.3
28.4
35.5
42.6
49.7
56.8
63.9
72
^2
14.4
21.6
28.8
36.0
43.2
50.4
57.6
64.8
73
7.3
14.6
21.9
29.2
36.5
43.8
51.1
58.4
65.7
74
7.4
14.8
22.2
29.6
37.0
44.4
51.8
59.2
66.6
75
7.5
15.0
22.5
30.0
37.5
45.0
52.5
60.0
67.5
76
7.6
15.2
22.8
30.4
38.0
45.6
53.2
60.8
68.4
77
7.7
15.4
23.1
30.8
38.5
46.2
53.9
61.6
69.3
78
7.8
15.6
23.4
31.2
39.0
46.8
54.6
62.4
70.2
79
7.9
15.8
23.7
31.6
39.5
47.4
55.3
63.2
71.1
80
8.0
16.0
24.0
32.0
40.0
48.0
56.0
64.0
72.0
81
8.1
16.2
24.3
32.4
40.5
48.6
56.7
64.8
72.9
82
8.2
16.4
24.6
32.8
41.0
49.2
57.4
65.6
73.8
83
8.3
16.6
24.9
33.2
41.5
49.8
58.1
66.4
74.7
84
8.4
16.8
25.2
33.6
42.0
50.4
58.8
67.2
75.6
85
8.5
17.0
25.5
34.0
42.5
51.0
59.5
68.0
76.5
86
8.6
17.2
25.8
34.4
43.0
51.6
60.2
68.8
77.4
87
8.7
17.4
26.1
34.8
43.5
52.2
60.9
69.6
78.3
88
8.8
17.6
26.4
35.2
44.0
52.8
61.6
70.4
79.2
89
8.9
17.8
26.7
35.6
44.5
53.4
62.3
71.2
80.1
90
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
91
9.1
18.2
27.3
36.4
45.5
54.6
63.7
72.8
81.9
92
9.2
18.4
27.6
36.8
46.0
55.2
64.4
73.6
82.8
93
9.3
18.6
27.9
37.2
46.5
55.8
65.1
74.4
83.7
94
9.4
18.8
28.2
37.6
47.0
56.4
65.8
75.2
84.6
95
9.5
19.0
28.5
38.0
47.5
57.0
66.5
76.0
85.5
96
9.6
19.2
28.8
38.4
48.0
57.6
67.2
76.8
86.4
97
9.7
19.4
29.1
38.8
48.5
58.2
67.9
77.6
87.3
98
9.8
19.6
29.4
39.2
49.0
58.8
68.6
78.4
88.2
99
9.9
19.8
29.7
39.6
49.5
59.4
69.3
79.2
89.1
lOO
10.0
1
20.0
2
30.0
3
40.0
4
50.0
5
60.0
6
70.0
7
80.0
8
90.0
9
48
TABLE V. LOGAEITHMS OF CONSTANTS
NUMBES
Loo
Number
Loo
Circle = 360°
2.55630
w^ = 9.86960
0.99430
= 21,600^
= 1,2%,000''
4.33445
6.11261
\ = 0.10132
ir8
9.00570 - 10
IT = 3.14159
0.49715
V^ = 1.77245
0.24857
27r = 6.28319
4 7r= 12.56637
0.79818
1.09921
^ = 0.56419
9.75143 - 10
^'*' = 4.18879
3
0.62209
-J^ = 1.12838
0.05246
^ = 0.78540
4
9.89509 - 10
■v^ = 1.46459
0.16572
^ = 0.52360
6
9.71900 - 10
}^ = 0.68278
9.83428 - 10
- = 0.31831
TT
9.50285 - 10
-Ir =0.62035
\47r
9.79264 - 10
^ =0.15915
2v
9.20182 - 10
-^^ = 0.80600
9.90633 - 10
V2 = 1.41421
0.15052
</2 = 1.25992
0.10034
VS = 1.73205
0.23856
VS = 1.44225
0.15904
VS = 2.23606
0.34949
Vb = 1.70997
0.23299
Ve = 2.44948
0.38908
</6 = 1.81712
0.25938
1 radian =
TT
1° = — radians
180
= 57.2958*^
1.75812
1° = 0.01745 radians
8.24188 - 10
= 3437.75'
3.53627
r= 0.00029 radians
6.46373 - 10
= 206,264.81''
5.31443
1" = 0.000005 radians
4.68557 - 10
Base of natural logs., e
logio« = logio 2.71828
0.43429
e = 2.71828
0.43429
lilogio 6 = 2.302585
0.36222
1 m. = 39.3708 in.
1.59517
1 knot = 6080.27 ft.
3.78392
= 1.0936 yd.
0.03886
= 1.1516 mi.
0.06130
= 3.2809 ft.
0.51599
1 lb. Av. = 7000 gr.
3.84510
1 km. = 0.6214 mi.
9.79336 - 10
1 bu. = 2150.42 cu. in.
3.33252
1 mi. = 1.6093 km.
0.20664
1 U.S. gal. = 231 cu. in.
2.36361
1 oz. Av. = 28.3495 g.
1.45254
1 Brit. gal. = 277.463 cu. in.
2.44320
1 lb. Av. = 453.5927 g.
2.65666
Earth's radii
1 kg. = 2.2046 lb.
0.34333
= 3963 mi.
3.59802
11. = 1.0567 liq.qt.
0.02396
and : 1950 mi.
3.59660
1 liq. qt. = 0.9463 1.
9.97603 - 10
lft./lb. = 0.1383 kg./m.
9.14082 - 10
TABLE
VI
THE LOGARITHMS
OF THE TRIGONOMETRIC FUNCTIONS
From 0= to 0'
J', and from 89*57'
to 90°, for every second
From 0" to 2°,
and from 88° to 90
, for every ten seconds
From 1° to 89
, for every minute
To each logaritl
m - 10 ia to tie appeoded
log sin
0°
log Un- tog Bin
log DM = 10.00 000
"
0'
1'
3'
"
"
0'
1'
2'
"
o
_
6.46373
6. 76 476
60
30
6. 16 270
6.63 982
6. 86 167
30
1
4. 68 557
6,47 090
6.76 836
59
31
6.17 694
6.64462
6.S64SS
29
2
4.98 660
6. 47 797
6. 77 193
58
32
6,19072
6. 64 936
6. 86 742
23
3
5.15270
6.18492
6. 77 548
57
33
6.20409
6.65 406
6. 87 027
27
4
5. 28 763
6.49175
6.77 900
S6
34
6. 21 705
6.65 870
6.87 310
26
6
5.38454
6, 49 849
6. 78 248
55
35
6,22964
6.66330
6. 87 591
35
6
S. 46 373
6.50512
6, 78 595
54
36
6. 24 188
6. 66 785
6. 87 870
24
7
5. S3 067
6. 51 165
6.78938
SJ
37
6. 25 378
6. 67 23i
6.88147
23
8
S. 58 866
6.51808
6. 79 273
52
38
6. 26 536
6.67 680
6.88 423
22
9
5.63 982
6. 52 142
6-79616
51
39
6.27 664
6. 63 121
6.63 697
21
10
S. 68 557
6. 53 067
6.79952
BO
40
6.28 763
6, 63 557
0.83969
30
11
S. 72 697
6. 53 683
6. 80 285
49
41
6.29836
6.68990
6. 89 240
19
12
5.76476
6. 54 291
6- 80 615
48
42
6.30832
6.69 418
6.89 509
18
13
5.79952
6. 54 890
6.30943
47
43
6.31904
6.69841
6. 89 776
17
14
5. 83 170
6. 55 481
6. 81 268
46
44
6,32 903
6. 70 261
6.90042
16
15
5.86167
6.56064
6.81591
45
45
6. 33 879
6.70676
6 90306
16
16
S. 88 969
6.56639
6.31911
44
46
6. 34 833
6. 71 038
6.90508
14
17
S. 91 002
6. 57 207
6. 82 230
43
47
6. 35 767
6. 71 496
6.90829
13
18
5. 94 OSS,
6. 57 767
6. 82 545
42
48
6-36632
6.71900
6.91088
12
19
5.96433
6.58320
6.82859
■51
49
6.37 577
6.72300
6. 91 346
11
20
5.93 660
6.58 866
6. 83 170
40
50
6,38454
6.72697
6, 91 602
10
21
6. 00 779
6.59406
6- 83 479
39
SI
0.39 315
6.73090
6. 91 857
9
22
6.02 800
6.59939
6. 83 786
38
52
6. 40 158
6. 73 479
6.92110
8
23
6. 04 730
6.60465
6.84091
37
53
6-409SS
6. 73 865
6.92362
7
24
6.06 579
6.60985
6.84 394
36
54
6. 41 797
6. 74 248
6.92 612
6
25
6,0S3.n
6. 61 499
6. 34 694
35
55
6. 42 594
6.74627
6. 92 861
5
26
6,10055
6. 62 007
6.84 993
34
56
6. 43 376
6. 75 003
6- 93 109
4
27
6.11694
6. 62 509
6. 35 2S9
33
57
6.44 145
6. 75 376
6. 93 355
3
23
6, 13 273
6.63 006
6, 85 584
32
SS
6.44 900
6 75 746
6. 93 599
2
29
6. 14 797
6.63 496
6. 85 S76
31
59
6, 45 643
6. 76 112
6. 93 843
1
30
6.16270
6.63982
6. 86 167
30
60
6.46373
6.76476
6.94 085
O
^^
59'
58'
67'
""
^
69'
68'
57'
^
89"
1<^C0S
50
/ 99
log Bin
log COB
10.00000
log tan
9 n
9 99
log Bin
log COB
10.00000
log tan
/ //
6O0
lOO
7. 46 373
7. 46 373
60
10
5. 68 557
30.00000
5. 68 557
50
10
7
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8. 12 172
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8. 21 189
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8. 13 117
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8. 13 121
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8. 21 576
9.99994
8. 21 581
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306
193
807
51
9
295
583 711
289
51
10
83S13
86 295
97 219
02 781
50
10
8+308
85 571 98 737
01263
50
11
5Z7
233
244
756
49
11
321
559 762
238
49
12
540
271
269
731
48
12
33+
547 787
213
48
13
554
259
295
705
47
13
3+7
534 812
188
47
14
567
247
320
680
46
14
360
522 838
162
+6
15
83 581
86235
97345
02 655
45
15
8+373
85 510 98863
01137
45
16
594
Z23
371
629
4+
16
3SS
497 888
112
17
60S
211
396
60+
43
17
398
485 913
087
+3
13
621
200
+21
579
42
18
411
473 939
061
42
19
634
188
447
553
41
19
424
460 964
036
41
20
83 643
86176
97472
02 528
40
20
84 437
85 448 98989
01011
40
21
661
16+
497
503
39
21
450
+36 99015
00985
39
22
674
152
523
+77
38
22
463
+23 0+0
960
38
23
1+0
543
452
37
23
476
411 065
935
37
24
701
123
573
427
36
24
+89
399 090
910
36
25
83 715
86116
97 593
02 402
35
25
8+502
85 386 99116
00 884
35
26
723
104
624
376
3+
26
SIS
37+ 1+1
859
34
27
741
092
6+9
351
33
27
528
361 166
83+
33
23
755
674
326
32
28
5+0
3+9 191
809
32
Z9
768
068
700
300
31
29
553
337 217
733
31
30
83 78!
86056
97 725
02Z7S
30
30
84 566
85 324 99 242
00 758
30
31
795
04+
750
250
29
31
579
312 267
733
29
32
032
776
224
28
32
592
299 293
707
23
33
821
020
801
199
27
33
605
287 318
27
34
834
86008
826
174
26
34
618
274 343
657
Z6
35
83848
85 996
97851
021+9
25
35
84 630
85 262 99368
00632
25
36
861
98+
877
123
24
36
6+3
250 394
606
2+
37
874
972
902
098
23
37
656
237 419
581
23
3S
837
960
927
073
23
38
669
2Zi +4+
556
22
39
901
9+8
953
0+7
21
39
682
212 469
531
21
40
83 914
85 936
97978
02022
20
40
84694
85 200 99+95
00505
20
41
927
924
98003
01997
19
41
707
187 520
480
19
42
940
912
029
971
18
42
720
175 545
455
13
43
95+
900
05+
946
17
43
733
162 570
430
17
44
%7
883
079
921
16
44
7+5
150 596
404
16
45
83 980
85 876
98104
01896
15
45
8+753
85137 99621
00379
15
46
83993
864
130
870
1+
+6
771
125 646
35+
14
47
84 006
851
155
845
13
47
78+
112 672
328
13
43
020
839
180
820
12
+8
796
100 697
303
12
49
033
827
206
794
U
+9
309
087 722
278
U
50
84 016
85 815
98 231
01769
lo
50
84 822
85074 99 7+7
00253
10
51
0S9
803
256
74+
9
51
83S
062 773
227
9
072
791
281
719
8
52
847
0+9 798
202
8
J3
085
779
307
693
7
53
860
037 823
177
7
Si
098
766
332
663
6
5+
873
02+ 8+8
152
6
55
84112
85 754
98357
01643
5
55
8+SS5
85 012 99874
00126
6
56
12S
742
333
617
4
56
84 999 899
101
4
57
133
730
403
592
3
57
911
986 924
076
3
53
151
718
433
567
2
58
923
974 9+9
051
2
59
164
706
453
542
1
59
936
961 975
OZS
1
OO
84177
85 693
98484
01516
O
60
849+9
8+9+9 C!JTO
00000
9
9
9
10
9
9 10
10
'
log 00.
logiin
logoot
lugtaa
'
'
log 00,
logEin logout
logtal
'
46°
45°
78
TABLE YII
FOR DETERMINING THE FOLLOWING WITH GREATER
ACCURACY THAN CAN BE DONE BY MEANS OF TABLE VI
1. log sin, log tan, and log eoty when the angle is between 0® and 2® ;
2. log coSy log tan, and log coty when the angle is between 88® and 90® ;
3. The value of the angle when the logarithm of the function does not
lie between the limits 8.54 684 and 11. 45 316.
FORMULAS FOR THE USE OF THE NUMBERS S AND T
I. When the angle a is between 0® and 2° :
log sin a = log a"' + S.
log tan a = log a"' + T.
log cot a = colog tan a.
log o^' = log sin a — 5
= log tan a — r
= colog cot a— T.
II. When the angle a is between 88® and 90® :
log cos a = log (90° — ay + S,
log cot a = log (90° — aY' + T.
log tan a = colog cot a.
log (90°— ay' = logcosa — iS
= log cot a — r
= colog tan a — T;
a = 90°-(90°-a).
Values of S and T
a"
2409
3 417
3 823
4190
4840
5 414
5 932
6408
6633
6851
7 267
8
4. 68 557
4. 68 556
4. 68 555
4. 68 55i
4. 68 554
4. 68 553
4. 68 552
4. 68 551
4. 68 550
4. 68 550
4. 68 549
8
log Bin a
8.06 740
8. 21 920
8. 26 795
8. 30 776
8. 37 038
8.41904
8. 45 872
8. 49 223
8. 50 721
8. 52 125
8. 54 684
logsiaa
in
log tan a
200
1726
2432
2976
3 434
3 838
4 204
4 540
4 699
4 853
5146
4. 68 557
4. 68 558
4. 68 559
4.68 560
4. 68 561
4. 68 562
4.68 563
4.68 564
4. 68 565
4. 68 565
4.68 566
6.98 660
7. 92 263
8. 07 156
8. 15 924
8. 22 142
8. 26 973
8.30930
a 34 270
8. 35 766
8. 37 167
8. 39 713
log tan a
log tan a
5146
5 424
5 689
5 941
6184
6417
6642
6859
7070
7173
7274
4. 68 567
4.68568
4. 68 569
4. 68 570
4. 68 571
4. 68 572
4. 68 573
4. 68 574
4. 68 575
4. 68 575
8. 39 713
8.41999
8. 44 072
8. 45 955
8. 47 697
8. 49 305
8. 50 802
8. 52 200
8. 53 516
8. 54 145
8. 54 753
logtaaa
TABLE VIII
NATURAL FUNCTIONS
Owing to tlie rapid change in the functions, interpolation is not
aMurate for the cotangents from 0° to 3°, nor for the tangents from 87°
to 90°. For the same functions interpolation is not accmute, in general,
in the last figure from 3= to 6° and from 84° to 87", respectively.
0" 0"
'
Bin COB tan cot
'
sin cos tan cot
'
1
z
3
4
B
6
7
8
9
lO
ii
H
15
16
17
18
19
20
21
22
Z3
24
35
26
21
23
29
30
0.0000 1.0000 0.0000 Infinite
03 00 03 3437.75
06 00 06 1718.87
09 00 09 1145-92
12 00 12 859.436
0.0015 1.0000 0.OO15 687.549"
17 00 17 572,957
20 00 20 491.106
23 00 23 429.718
26 00 26 3S1-971
0-0029 1-0000 0.0029 343.774
32 00 32 312.521
35 00 35 286.478
38 00 3S 264.441
41 00 41 245.552
0.0044 1.0000 0.0044 229.132
47 00 47 214.858
49 00 49 202-219
52 00 52 190.984
55 00 =5 180,932
0.0058 1.0000 0.0058 171-885
61 00 61 163-700
64 00 64 156-259
67 00 67 149.465
70 00 70 143-237
0-0073 1.0000 0.0073 137.507
76 00 76 132,219
79 00 79 127.321
81 00 81 122.774
8+ 00 84 118,540
0,00S7 1.0000 0.00S7 1H.5S9
60
S9
58
57
56
55
54
53
52
il
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
fi
32
31
30
=1?
32
33
34
35
36
37
3$
39
40
41
42
43
44
45
46
47
43
49
50
51
52
53
54
55
56
57
58
59
60
0.0087 1.0000 0.0087 114.589
90 00 90 110.S92
93 00 93 107.426
96 00 96 104.171
99 1.0000 99 101.107
0.0102 0.9999 0.0102 93.2179
05 99 05 95.4895
08 99 08 92.9085
1! 99 11 90.4633
13 99 13 8S.1436
0.0116 0,9999 0-0116 85.9398
19 99 19 83.8435
22 99 22 81.8470
25 99 25 79,9434
28 99 28 78.1263
0,0131 0.9999 0.0131 76,3900
3+ 99 34 74.7292
37 99 37 73-1390
40 99 40 71.6151
43 99 43 70.1533
0.0145 0.9999 0.0145 68.7501
48 99 48 67.4019
51 99 51 66.1055
54 99 54 64.8580
57 99 57 63.6567
0.0160 0-9999 0.0160 62,4992
63 99 63 61.3829
66 99 66 60.3058
69 99 69 59.2659
72 99 72 58.2612
0.0175 0.9998 0.0175 57.2900
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
fi
5
4
3
2
1
'
poa Bin cot tan
'
'
cos sin cot tan
'
89°
89°
M
/
o
sin cos
tan cot
/
60
0.0175 0.9998
0.0175 57.2900
1
77 98
77 56.3506
59
2
80 98
80 55.4415
58
3
83 98
83 54.5613
57
4
86 98
86 53.7086
56
5
0.0189 0.9998
0.0189 52.8821
55
6
92 98
92 52.0807
54
7
95 98
95 51.3032
53
8
0198 98
0198 50.5485
52
9
0201 98
0201 49.8157
51
10
0.0204 0.9998
0.0204 49.1039
50
11
07 98
07 48.4121
49
12
09 98
09 47.7395
48
13
12 98
12 47.0853
47
14
15 98
15 46.4489
46
15
0.0218 0.9998
0.0218 45.8294
45
16
21 98
21 45.2261
44
17
24 97
24 44.6386
43
18
27 97
27 44.0661
42
19
30 97
30 43.5081
41
20
0.0233 0.9997
0.0233 42.9641
40
21
36 97
36 42.4335
39
22
39 97
39 41.9158
38
23
41 97
41 41.4106
37
24
44 97
44 40.9174
36
25
0.0247 0.9997
0.0247 40.4358
35
26
50 97
5P 39.9655
34
27
53 97
53 39.5059
33
28
56 97
56 39.0568
32
29
59 97
59 38.6177
31
SO
0.0262 0.9997
0.0262 38.1885
30
31
65 96
65 37.7686
29
32
68 96
68 37.3579
28
33
70 96
71 36.9560
27
34
73 96
74 36.5627
26
35
0.0276 0.9996
0.0276 36.1776
25
36
79 %
79 35.8006
24
37
82 96
82 35.4313
23
38
85 96
85 35.0695
22
39
88 96
88 34.7151
21
40
0.0291 0.99%
0.0291 34.3678
20
41
94 96
94 34.0273
19
42
0297 %
0297 33.6935
18
43
0300 96
0300 33.3662
17
44
02 95
03 33.0452
16
45
0.0305 0.9995
0.0306 32.7303
15
46
08 95
08 32.4213
14
47
11 95
11 32.1181
13
48
14 95
14 31.8205
12
49
17 95
17 31.5284
11
50
0.0320 0.9995
0.0320 31.2416
10
51
23 95
23 30.9599
9
52
26 95
26 30.6833
8
53
29 95
29 30.4116
7
54
32 95
32 30.1446
6
55
0.0334 0.9994
0.0335 29.8823
5
56
37 94
38 29.6245
4
57
40 94
40 29.3711
3
58
43 94
43 29.1220
2
59
46 94
46 28.8771
1
60
/
0.0349 0.9994
0.0349 28.6363
cos sin
cot tan
/
o
sin cos
tan cot
/
0.0349 0.9994
0.0349 28.6363
60
1
52 94
52 28.3994
59
2
55 94
55 28.1664
58
3
58 94
58 27.9372
57
4
61 93
61 27.7117
56
5
0.0364 0.9993
0.0364 27.4899
55
6
66 93
67 27.2715
54
7
69 93
70 27.0566
53
8
72 93
73 26.8450
52
9
75 93
75 26.6367
51
10
0.0378 0.9993
0.0378 26.4316
50
11
81 93
81 26.2296
49
12
84 93
84 26.0307
48
13
87 93
87 25.8348
47
14
90 92
90 25.6418
46
15
0.0393 0.9992
0.0393 25.4517
45
16
96 92
96 25.2644
44
17
0398 92
0399 25.0798
43
18
0401 92
0402 24.8978
42
19
04 92
05 24.7185
41
20
0.0407 0.9992
0.0407 24.5418
40
21
10 92
10 24.3675
39
22
13 91
13 24.1957
38
23
16 91
16 24.0263
37
24
19 91
19 23.8593
36
25
0.0422 0.9991
0.0422 23.6945
35
26
25 91
25 23.5321
34
27
27 91
28 23.3718
33
28
30 91
31 23.2137
32
29
33 91
34 23.0577
31
30
0.0436 0.9990
0.0437 22.9038
30
31
39 90
40 22.7519
29
32
42 90
42 22.6020
28
33
45 90
45 22.4541
27
34
48 90
48 22.3081
26
35
0.0451 0.9990
0.0451 22.1640
25
36
54 90
54 22.0217
24
37
57 90
57 21.8813
23
38
59 89
60 21.7426
22
39
62 89
63 21.6056
21
40
0.0465 0.9989
0.0466 21.4704
20
41
68 89
69 21.3369
19
42
71 89
72 21.2049
18
43
74 89
75 21.0747
17
44
77 89
77 20.9460
16
45
0.0480 0.9988
0.0480 20.8188
15
46
83 88
83 20.6932
14
47
86 88
86 20.5691
13
48
88 88
89 20.4465
12
49
91 88
92 20.3253
11
50
0.0494 0.9988
0.0495 20.2056
10
51
0497 88
0498 20.0872
9
52
0500 87
0501 19.9702
8
53
03 87
04 19.8546
7
54
06 87
07 19.7403
6
55
0.0509 0.9987
0.0509 19.6273
5
56
12 87
12 19.5156
4
57
15 87
15 19.4051
3
58
18 87
18 19.2959
2
59
20 86
21 19.1879
1
60
0.0523 0.9986
0.0524 19.0811
/
cos sin
cot tan
88'
87
3*
'
8lD coa
tan cot
'
o
0.0523 0.99S6
0.0524 19.0811
00
1
26 86
27 18,9755
59
2
29 86
30 18.8711
5S
3
32 S6
33 I8.767S
57
35 86
36 18.6656
56
5
0.0538 0.9986
0,0539 18.5645
55
e
41 85
42 18.4645
54
7
44 85
44 18.3655
53
8
47 85
47 13-2677
52
9
SO 85
50 18.1708
51
10
0.0552 0.9935
0,0553 18.0750
50
11
55 85
56 17.9802
49
12
58 84
59 17,8863
48
13
61 84
62 17.7934
47
14
64 84
65 17.7015
46
16
0,0567 0.9984
0-0568 17.6106
45
16
70 84
71 17.5205
44
17
73 84
74 17.4314
43
18
76 83
77 17.3432
42
19
79 83
SO 17.2558
41
20
0.0581 0.9983
0.0SS2 17.1693
4(^
21
84 83
85 17.0837
39
22
87 83
88 16.9990
3S
Z3
90 83
91 16.9150
37
24
93 82
94 16,8319
36
35
0.05% 0.9982
0.0597 16-7496
35
26
0599 82
0600 16.6681
34
27
0602 82
03 16,5874
33
28
05 82
06 16.5075
32
29
08 82
09 16.4283
31
30
0,0610 0,9981
0,0612 16,3499
30
31
13 81
15 16.2722
29
32
16 81
17 16.1952
28
33
19 81
20 16.1190
27
34
22 81
23 16.0435
26
35
0.0625 0.9980
0.0626 15.9687
25
36
28 80
29 15.8945
24
37
31 80
32 15.8211
23
38
34 80
ZS 15.7483
22
39
37 80
38 15.6762
21
40
0.0640 0.9980
0.0641 15.6048
20
41
42 79
44 15.5340
19
+2
45 79
47 15.4638
IS
43
48 79
50 15,3943
J7
44
51 79
S3 15.3254
16
45
0-065+ 0.9979
0.0655 15.2571
15
46
57 78
58 15.1893
14
47
60 78
61 15.1222
13
48
63 78
64 15.0557
12
49
66 78
67 14.9898
n
50
0.0669 0.9978
0.0670 14.9244
lO
51
71 77
73 14.8596
9
52
74 77
76 14.7954
8
53
77 77
79 14,7317
7
54
80 77
82 14.6685
6
55
0.0683 0.9977
0.0685 14.6059
5
56
86 76
88 14.5438
4
57
89 76
90 14-4823
3
58
92 76
93 14.4212
2
59
95 76
96 14.3607
1
00
0.0698 0.9976
0.0699 14.3007
^
cos sin
cot tao
4-
81
'
sin cos
tan cot
~
~o
0.0698 0.9976
0,0699 14.3007
60
1
0700 75
0702 2411
59
2
03 75
OS 1821
58
3
06 75
08 1235
57
4
09 75
11 0655
56
5
0.0712 0.9975
0.0714 14.0079
55
6
15 74
17 13.9507
54
?
IS 74
20 8940
S3
8
21 74
23 8378
52
9
24 74
26 7821
51
10
0.0727 0.9974
0.07Z9 13.7267
50
1
29 73
3i 6719
49
2
32 73
34 6174
48
3
35 73
37 5634
47
4
38 73
40 5098
46
15
0,0741 0.9973
00743 13,4566
46
6
44 72
46 4039
7
47 72
49 3515
43
S
SO 72
52 2996
42
9
S3 72
55 2480
41
20
0.07560.9971
0.0758 13.1969
40
21
58 71
61 1461
39
22
61 71
64 09S8
38
23
64 71
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139
43
18
7+
43
IS
106
42
19
77
47
18
073
41
30
0.2979
0.9546
0.3121
3.2041
40
Zl
S2
45
24
3,2003
39
22
SS
44
27
3.1975
38
23
83
43
31
943
37
24
90
42
34
910
26
35
0.2993
0,9542
0,3137
3.1378
35
26
96
41
40
845
34
27
2999
40
43
313
33
28
3002
39
47
730
32
29
0+
33
50
74S
31
30
0.3007
0.9537
0.3153
3,1716
30
31
10
36
56
63+
29
32
13
35
59
652
33
15
35
63
620
27
3+
IS
34
65
53S
35
0.3021
0.9533
o.31c;d
3.1556
25
36
2+
32
72
52+
24
37
26
31
75
492
23
33
29
30
79
460
22
39
32
29
83
429
21
40
0.303S
0.9523
0.3 1S5
3.1397
20
41
38
27
88
366
19
42
40
27
91
334
13
43
43
26
95
303
17
44
46
25
3198
271
16
45
0.3{M9
0.9524
0.3201
3.1240
15
46
SI
23
04
209
14
47
54
23
07
173
13
43
57
21
11
146
12
49
60
20
14
115
li
50
0.3062
0.9520
0.3217
3.1084
10
51
65
19
20
053
9
52
6S
18
23
3.1022
8
S3
71
17
27
3.0991
7
54
74
16
30
961
6
55
0.3076
0.9515
0.3233
3.0930
5
56
79
14
36
899
4
57
13
40
868
3
58
85
12
43
833
2
59
87
11
46
807
1
60
0J090
0.9511
0J249
3.0777
~
COB sin cot tan
~^
'
Bin
COB
tun
cot
'
0.3090
0.9511
0,3249
3.0777
60
1
93
10
52
746
59
2
96
09
56
716
S3
3
3098
08
59
686
57
4
3101
07
62
655
56
5
0.3104
0.9506
0,3265
3.0625
55
6
07
05
69
595
54
7
10
04
72
565
53
8
12
03
75
535
52
9
15
02
78
505
51
10
0.3118
0.9502
0,3281
3.0475
50
11
21
01
85
445
49
12
23
9500
415
48
13
26
9499
91
385
47
14
29
94
356
46
15
0.3132
0.9497
0,3298
3.0326
45
16
3+
96
3301
296
44
17
37
95
04
267
43
18
40
9+
07
237
42
19
43
93
10
208
41
20
0.3145
0.9492
0.331+
3.0173
40
21
48
92
17
149
39
22
51
91
20
120
38
23
5+
90
23
090
37
24
56
27
061
36
25
0.3159
0.9488
0.3330
3.0032
35
62
87
33
3.0003
34
27
65
86
36
2.9974
33
28
68
85
39
945
32
29
70
84
43
916
31
SO
0,3173
0,9483
0,3346
2,9887
30
31
76
82
49
853
29
32
79
81
52
829
28
33
81
80
56
800
27
3+
84
SO
59
772
26
35
0.31S7
0,9479
0J362
2,9743
35
36
90
73
65
714
2+
37
92
77
69
636
23
38
95
76
72
657
22
39
3 193
75
75
629
21
40
O.3201
0.9474
0.337S
2,9600
20
41
03
73
82
572
19
42
06
73
35
544
18
43
09
71
515
17
12
70
91
437
16
45
0.3214
0.9469
0J395
2,9459
15
46
17
63
3398
431
1+
47
20
67
3401
403
13
4S
23
65
04
375
12
49
25
66
08
347
11
SO
0.3228
0.9465
0J411
2,9319
lO
51
31
64
14
291
9
52
34
6.3
17
263
8
S3
36
62
21
235
7
5+
39
61
24
208
6
55
0.3242
0.9460
0.3427
2.9130
5
56
45
59
30
152
4
57
47
58
34
125
3
SS
50
57
37
097
2
59
53
56
40
070
1
CO
0J2S6
0,9455
0.3443
2,9042
CCS
Bin
cot
tan
~
72"
71°
19°
ac
'
Bin COS
tan
cot
'
o
OJ256 0.9455
0.3443
2,90+2
60
1
53 54
47
2.9015
59
2
61 53
50
2.8987
58
3
64 52
53
960
57
4
67 51
56
933
56
5
0.3269 0.9450
0.3460
2.8905
55
6
72 49
63
878
54
7
75 49
66
851
53
S
78 48
69
324
52
9
80 47
73
797
51
10
0.3283 0.9t46
0.3476
2-8770
50
11
86 45
79
743
49
12
89 ^4
82
716
48
13
91 43
86
639
47
14
9+ 42
662
46
10
0.3297 0.9t41
0J492
2.8636
45
16
3300 W
95
609
44
17
02 39
3499
582
43
18
OS 3S
3502
556
19
08 37
OS
529
41
20
0.3311 0.9436
0.350S
2.S502
40
21
13 35
12
476
39
22
16 34
15
449
33
23
19 33
18
423
S7
24
22 32
22
397
36
25
0.3324 0.9431
0,3525
2.8370
35
27 30
28
3+4
34
27
30 29
31
318
33
28
35
291
32
29
33 27
38
265
31
30
0.3333 0.9426
0.3541
2.8239
30
31
41 25
4+
213
29
32
44 24
43
187
28
33
46 23
51
161
27
34
49 23
54
135
26
36
0.3352 0.9422
0JS58
2.8109
35
36
55 21
61
083
24
37
57 20
6+
057
23
33
60 19
67
032
22
39
63 IS
71
2.S006
21
40
0J36S 0.9417
0-3S74
2.7980
20
41
63 16
77
955
19
42
71 :s
81
929
IS
43
74 14-
84
903
17
44
76 13
37
878
16
45
0,3379 0.9412
0,3S90
2.7352
IS
46
82 U
94
827
14
47
85 10
3597
801
13
43
87 09
3600
776
12
49
90 08
04
751
11
50
0.3393 0-9407
0,3607
2.7725
lO
51
96 06
10
700
9
52
3398 OS
13
675
8
53
3401 04
17
650
7
54
04 03
20
625
6
55
0.3407 0.9402
0.3623
2,7600
S
56
09 01
27
575
4
57
12 9400
30
550
3
58
)S 9399
33
525
2
59
17 98
36
500
1
60
OJ420 0.9397
0.3640
2.7475
O
Z!
cos sin cot tan
'
'
Bin c^os
tan cot
'
0.3420 0.9397
0.3640 2.7475
60
1
23 96
43 450
59
2
26 95
46 425
53
3
28 9+
50 400
57
4
31 93
53 376
56
S
0.3+34 0.9392
0.36.';6 2.7351
55
6
37 91
59 326
54
7
39 90
63 302
53
8
42 89
66 277
52
9
4S 88
69 253
51
lO
0.3443 0.9387
0.3673 2.7228
50
]1
50 86
76 204
49
12
S3 85
79 179
43
13
56 84
83 155
47
14
53 83
86 130
46
dS
0.3461 0,9382
0,36S9 2.7106
45
16
64 31
93 082
44
17
67 80
96 053
43
18
69 79
3699 034
42
19
72 78
3702 2.7009
41
20
0.3475 0.9377
0.3706 2,6985
40
21
78 76
09 961
39
22
80 75
12 937
33
23
83 74
16 913
37
24
86 73
19 889
36
25
0,3488 0.9372
0.3722 2.6865
35
26
91 71
26 841
34
27
9+ 70
29 318
33
28
97 69
32 794
32
29
3499 68
36 770
31
ftO
0.3502 0.9367
0.3739 2.6746
30
31
05 66
42 723
29
32
08 65
45 699
28
33
10 64
49 675
27
34
13 63
52 652
26
35
0,3516 0.9362
03755 2.6628
25
36
13 61
59 60S
24
37
21 60
62 531
23
3S
24 59
65 553
22
39
27 58
69 534
21
40
0J529 0.9356
0.3772 2.6511
30
41
32 55
75 483
19
35 54
79 4G4
18
43
37 53
82 441
17
44
40 S2
85 413
16
4S
0.3543 0.9351
0.3789 2.6395
15
46
46 SO
92 371
14
47
48 49
9S 343
13
43
51 48
3799 323
12
49
54 47
3802 302
11
50
0.3557 0.9346
0.3S0S 2.6279
lO
SI
59 45
09 256
9
52
62 44
12 233
8
53
65 43
15 210
7
54
67 42
19 187
6
55
03570 0.9341
0.3822 2.6165
5
56
73 40
25 142
4
57
76 39
29 119
3
58
78 38
32 096
2
59
81 37
35 074
1
60
0.3584 0.9336
0,3839 2.6051
O
~
cos sin
cot tan
~
70°
68"
90
21'
sin COB
tan
cot
'
o
0.3S84 0.9336
0.3839
2.6051
60
1
86 35
42
028
S9
S9 34
45
2.6006
58
3
92 33
49
2.5983
57
4
95 32
52
961
56
5
0.3597 0.9331
0.3855
2.5938
55
6
3G00 30
59
916
54
7
03 28
62
893
S3
3
OS 27
65
871
52
9
08 26
69
848
51
10
0.3611 0.9325
0.3872
2.5826
60
14 24
75
804
49
12
16 23
79
782
4S
13
19 22
759
47
H
22 21
85
737
46
15
0.362+ 0.9320
0.3S89
2-5715
45
16
27 19
92
693
44
17
30 18
95
671
43
18
33 17
3899
649
42
19
35 16
3902
627
41
20
0.3638 0.931S
0.3906
2.5605
40
21
41 14
09
583
39
22
43 13
12
561
38
23
46 12
16
539
37
24
49 11
19
517
36
25
0.3651 0.9309
0.3922
2.5495
30
26
54 08
26
473
34
27
57 07
29
452
33
28
60 06
32
430
32
29
62 05
36
408
31
30
0.3665 0.9304
0.3939
2.5386
30
31
68 03
42
365
29
32
70 02
46
343
28
33
73 01
49
322
27
34
76 9300
53
300
26
35
0.3679 0.9299
0.3956
2.5279
25
36
81 98
59
257
24
37
84 97
63
236
23
3S
87 96
66
214
22
39
89 95
69
193
21
40
0.3692 0.9293
0.3973
2.5172
30
41
95 92
76
150
19
42
3697 91
79
129
18
43
3700 90
83
108
17
44
03 89
86
086
16
45
0.3706 0.9288
0.3990
2.5065
15
46
08 87
93
044
14
47
11 86
3996
023
13
48
14 85
4000
2.5002
12
49
16 84
03
2.4981
11
50
0.3719 0.9283
0.4006
2.4960
10
51
22 82
10
939
9
52
24 81
13
918
8
S3
27 79
17
897
7
54
30 78
20
876
6
55
0.3733 0.9277
0.4023
2.4355
a
56
35 76
27
834
4
57
38 75
30
813
3
58
41 74
33
792
2
59
43 73
37
772
1
60
0.3746 0.9272
0.4O40
2,4751
~
COS Eln
cot
tan
~
aa-
'
(riu oofl
tan
cot
0.3746 0.9272
0.4040
2.47S1
60
1
49 71
44
730
59
2
51 70
47
709
53
3
54 69
50
689
57
4
57 67
54
668
56
5
0.3760 0.9266
0.4OS7
2.4648
55
6
62 65
61
627
54
7
65 64
6+
606
53
8
68 63
67
586
52
9
70 62
71
566
51
lO
0.3773 0.9261
0.4074
2.4545
50
n
.1776 60
78
525
49
12
^78 59
81
504
48
13
81 58
84
484
47
14
84 57
88
464
46
15
0.3786 0.9255
0.4091
2.4443
45
16
89 5+
95
423
44
17
92 S3
4098
403
43
95 53
4101
383
42
19
3797 51
05
362
41
30
0.3800 0.9250
0.4108
2.4342
40
21
03 49
11
322
39
22
OS 48
IS
302
38
23
08 47
18
282
37
24
11 4S
22
262
36
25
0.3813 0.9244
0.412S
2.4242
35
26
16 43
29
222
34
27
19 42
32
202
33
28
21 41
35
182
32
29
24 40
39
162
31
30
03827 0.9239
0.4142
2.4142
30
31
30 38
46
122
29
32
32 37
49
102
28
33
35 35
52
083
27
34
38 34
56
063
26
35
0.3840 0.9233
0.4159
2.4043
35
36
43 32
63
023
24
37
46 31
66
2.4004
23
33
48 30
69
2.3984
22
39
51 29
73
964
21
40
0.3854 0.9228
0.4176
2.394S
20
41
56 27
80
925
19
42
59 25
83
906
18
43
62 24
87
886
17
44
64 23
90
867
16
45
0.3867 0.9222
0.4193
2.3847
15
46
70 2!
4197
828
14
47
72 20
4200
808
13
4S
75 19
04
789
12
49
78 IS
07
770
11
50
0.3881 0.9216
0.4210
2.3750
lO
51
83 15
14
731
9
52
86 34
17
712
a
53
89 13
21
693
7
54
91 12
24
673
6
55
0.3894 0.9211
0.4228
2.3654
5
56
97 10
31
635
4
57
3899 08
3+
616
3
SS
3902 D7
33
597
2
59
OS 06
41
578
1
60
0.3907 0.9205
0.4245
2.3559
O
~^
cos Bin
cot
tan
~
68°
67"
23°
o
Bin
COB
taD
cot
'
0.3907
0.9205
0.4245
2.3559
60
1
10
04
48
539
59
13
03
52
520
58
3
IS
02
55
501
57
18
9200
58
483
56
5
0.3921
0.9199
0.4262
2.3464
55
6
23
6S
445
54
7
26
97
69
426
53
8
29
96
72
407
S2
9
31
9S
76
388
51
10
0.3934
0,9194
0.4279
2.3369
50
11
37
92
S3
351
49
12
39
91
36
332
48
13
42
90
89
313
47
14
45
93
294
46
16
0.3947
0.9188
0.4296
2.3276
45
16
50
87
4300
257
44
17
53
36
03
238
43
18
55
84
07
220
42
19
S3
83
10
201
41
20
0.3%1
0.9182
0.4314
2.3183
40
21
63
81
17
164
39
22
66
80
20
146
38
23
79
24
127
37
24
71
78
27
109
36
25
0.3974
0.9176
0,4331
2.3090
35
26
77
75
34
072
34
27
79
74
053
33
23
82
73
41
035
32
29
83
72
45
2.3017
31
30
0.3987
0.9171
0.4348
2.2998
30
31
90
69
52
980
29
32
93
55
962
2S
33
95
67
59
944
27
34
3998
66
62
925
26
35
0.4O01
0.9I6S
0.4365
2.2907
25
36
03
64
69
889
24
37
06
62
72
871
23
38
09
61
76
853
39
11
60
79
S35
21
40
0.4014
0.9159
0.4383
2.2817
20
41
17
58
86
799
19
19
57
90
781
18
43
55
93
763
17
44
25
54
4397
745
16
4S
0.4OZ7
0.9153
0.4400
2.2727
15
46
30
52
04
709
14
47
33
51
07
691
13
48
35
50
11
673
12
49
38
48
14
655
11
50
0.4041
0.9147
0.4417
2.2637
lO
51
43
46
21
620
9
52
46
45
24
602
8
53
49
44
584
7
5+
51
43
31
566
6
55
0.4054
0.9141
0.4435
2-2549
5
56
S7
40
38
531
4
S7
59
39
42
513
3
58
62
38
45
496
2
59
65
37
49
478
1
60
0.4067
0.9135
0.4452
2.24GO
~^
cos sId cot tan
I
24°
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60°
59°
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96
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1
2
3
4
5
6
7
8
9
lO
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
do
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
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0.5446
49
51
54
56
0.5459
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63
66
68
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76
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12
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24
27
29
0.5531
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36
39
41
0.5544
46
48
51
53
0.5556
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61
63
65
0.5568
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73
75
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82
85
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76
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60
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52
50
48
0.8347
45
44
42
40
0.8339
37
36
34
32
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29
28
26
24
0.8323
21
20
18
16
0.8315
13
11
10
08
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95
94
92
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23
27
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44
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52
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85
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10
15
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52
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11
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41
0.6745
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1.5399
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369
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1.5350
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330
320
311
1.5301
291
282
272
262
1.5253
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233
224
214
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185
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166
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60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
lO
9
8
7
6
5
4
3
2
1
O
84'
/
o
sin
cos
tan
cot
/
0.5592
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0.6745
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59
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53
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26
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27
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29
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31
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32
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33
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28
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26
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53
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f
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sin
cot
tan
66*
56*
35°
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Hln COH
tan
cot
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0.5736 0,8192
0.7002
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1
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9
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10
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49
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176
13
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14
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15
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16
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44
17
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80
124
42
19
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21
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23
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26
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26
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27
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20
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037
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29
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30
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31
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32
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42
1.4002
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33
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46
1.3994
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34
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35
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40
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44
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45
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46
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47
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12
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51
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781
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59
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cot
tan
~
86*
97
Bin coi
tan
cot
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05878 0.8090
0.726S
1.3764
60
1
80 88
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19
663
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13
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14
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15
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1.3638
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16
15 63
37
630
17
18 61
41
622
43
18
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42
19
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50
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21
27 54
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22
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64
580
38
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572
37
24
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73
564
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26
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26
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82
547
34
27
41 44
539
33
23
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91
531
32
29
46 40
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31
30
0.5948 0.8039
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1.3514
30
31
51 37
04
506
29
32
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498
28
33
55 33
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35
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36
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27
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31
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23
38
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21
40
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41
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42
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54
416
18
43
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58
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17
44
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63
400
16
46
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59
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54°
63°
ST
38°
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tan
cot
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0.6S73
0.7264
0.9462
1.0569
3J5
26
75
62
68
562
34
27
77
60
73
556
33
28
79
53
79
550
2Z
29
81
56
544
31
30
0.6884
0.7254
0,9490
1.0538
30
31
E6
52
9495
S32
29
32
88
50
9501
526
2S
33
90
48
06
519
27
34
92
46
12
513
26
36
0.6894
0.7241
0,9517
1-0507
36
36
96
^?.
23
501
24
37
6898
40
28
495
23
38
6900
38
34
439
22
39
03
36
40
483
21
40
0.6905
0.7234
0,9545
1.0477
20
41
07
32
51
470
19
42
09
30
56
464
IS
43
U
28
62
45B
17
13
26
67
452
16
45
0.6915
0.7224
0.9573
1.0446
15
46
17
22
78
440
14
47
19
20
84
434
13
4S
ZI
IS
90
428
12
49
24
16
9595
422
U
50
0.6926
0.7214
0.9601
1.0416
10
SI
28
12
06
410
9
52
30
10
12
404
8
£3
32
08
IS
398
7
S4
34
06
23
392
6
66
0.6936
0.7203
0.9629
1.0385
5
56
38
7201
34
379
♦
57
40
7199
40
373
3
58
42
97
46
367
2
59
44
95
51
361
1
60
0.6W7
0.7193
0.9657
1.0355
O
CCS Bin cot te.
~
44.
101
'
SlB
coa
tan
cot
"^
~0
0.6947
0.7193
0.9657
1.0355
60
1
49
91
63
349
S9
2
51
S9
68
343
58
3
53
87
74
337
57
4
fS
85
79
331
56
r.
0.6957
0.7183
0.968S
1.0325
65
6
59
81
91
319
54
7
61
79
9696
313
53
8
63
77
9702
307
52
9
65
75
08
301
51
ID
0.6967
0.7173
0.9713
1.0295
60
11
70
71
19
289
49
12
72
'' 69
2£
283
48
13
74
67
30
277
47
i4
76
65
36
271
46
IS
0.6978
0.7163
0.9742
1.0265
45
16
80
61
47
Z59
44
17
82
59
S3
253
43
18
84
57
59
247
42
19
86
55
64
Z41
41
20
0.6988
0.7153
0,9770
1.0Z3S
40
21
90
51
76
230
39
11
92
49
SI
Z24
38
23
95
47
87
218
37
24
97
45
93
212
36
25
0.6999
0.7143
0.9798
1.0206
36
26
7001
41
9804
200
34
27
03
39
10
194
33
28
05
37
16
188
32
29
07
35
21
182
31
30
0.7009
0.7133
0.9827
1.0176
30
31
11
30
33
170
29
32
13
28
38
164
28
2,2
15
26
44
\m
27
34
17
24
50
152
26
35
0.7019
0.7122
0.9856
1.0147
26
36
22
20
61
14i
24
37
18
67
135
23
38
16
73
129
Z2
39
28
14
79
123
21
40
0.7030
0.7112
0.9884
1.0117
20
41
32
10
90
111
19
42
34
08
9896
105
18
43
36
06
9902
099
17
44
38
04
07
094
16
45
0.7040
0.7102
0.9913
1.0088
16
45
42
7100
19
082
14
47
7098
25
076
13
48
46
96
30
070
12
49
48
94
36
064
11
SO
0.7050
0.7092
0.9942
l.OOSS
10
51
S3
90
48
052
9
52
55
54
047
8
53
57
85
59
041
7
54
59
S3
65
035
6
55
0.7061
0-7081
0.9971
1.0029
5
56
63
79
77
023
57
65
77
83
017
3
5S
67
75
012
2
59
69
73
94
006
1
60
0.7071
0-7071
1.0000
1.0000
T
CO.
Bin
cot
~^
~
46°
45°
102
TABLE
IX
CONVEESION TABLE— DEGEEES TO EADIANS
V* = — — radians 1 radian = — <
180 IT
legrees
0^-45"
o
O'
lO'
20'
30'
40'
50'
O
0.0000
0.0029
0.0058
0.0087
0.0116
0.0145
1
0175
0204
0233
0262
0291
0320
2
0349
0378
0407
0436
0465
0495
3
0524
0553
0582
0611
0640
0669
4
0698
0727
0756
0785
0814
0844
5
0.0873
0.0902
0.0931
0.0960
0.0989
0.1018
6
1047
1076
1105
1134
1164
1193
7
1222
1251
1280
1309
1338
1367
8
13%
1425
1454
1484
1513
1542
9
1571
1600
1629
1658
1687
1716
lO
0.1745
0.1774
0.1804
0.1833
0.1862
0.1891
11
1920
1949
1978
2007
2036
2065
12
2094
2123
2153
2182
2211
2240
13
2269
2298
2327
2356
2385
2414
14
2443
2473
2502
2531
2560
2589
15
0.2618
0.2647
0.2676
0.2705
0.2734
0.2763
16
2793
2822
2851
2880
2909
2938
17
2967
2996
3025
3054
3083
3113
18
3142
3171
3200
3229
3258
3287
19
3316
3345
3374
3403
3432
3462
20
0.3491
0.3520
0.3549
0.3578
0.3607
0.3636
21
3665 ♦
3694
3723
3752
3782
3811
22
3840
3869
3898
3927
3956
3985
23
4014
4043
4072
4102
4131
4160
24
4189
4218
4247
4276
4305
4334
25
0.4363
0.4392
0.4422
0.4451
0.4480
0.4508
26
4538
4567
4596
4625
4654
4683
27
4712
4741
4771
4800
4829
4858
28
4887
4916
4945
4974
5003
5032
29
5061
5091
5120
5149
5178
5207
30
0.5236
0.5265
0.5294
0.5323
0.5352
0.5381
31
5411
5440
5469
5498
5527
5556
32
5585
5614
5643
5672
5701
5730
33
5760
5789
5818
5847
5876
5905
34
5934
5963
5992
6021
6050
6080
35
0.6109
0.6138
0.6167
0.6196
0.6225
0.6254
36
6283
6312
6341
6370
6400
6429
37
6458
6487
6516
6545
6574
6603
38
6632
6661
6690
6720
6749
6778
39
6807
6836
6865
6894
6923
6952
40
0.6981
0.7010
0.7039
0.7069
0.7098
0.7127
41
7156
7185
7214
7243
7272
7301
42
7330
7359
7389
7418
7447
7476
43
7505
7534
7563
7592
7621
7650
44
7679
7709
7738
7767
77%
7825
45
0.7854
0.7883
0.7912
0.7941
0.7970
0.7999
o
O'
10'
20' ,
30'
40'
50'
103
In
using this table, interpolations
may be made
as with other tables.
Thus to find the number of radians corresponding to 49® 15', we
have:
49° 10^ =
: 0.8581 radians
Tabular diff . =
0.0029
T^ of 0.0029 =
Adding, 49° 16' =
0.0016
0.8696 radianR
Afi*-
on*
o
O'
1 nr €^ru
iU\t
±i\f
Ri\l
45
TT - . -- . - ,
0.7854
0.7883 0.7912
0.7941
0.7970
0.7999
46
8029
8058 8087
8116
8145
8i74
47
8203
8232 8261
8290
8319
8348
48.
8378
8407 8436
8465
8494
8523
49
8552
8581 8610
8639
8668
8698
60
0.8727
0.8756 0.8785
0.8814
0.8843
0.8872
51
8901
8930 8959
8988
9018
9047
52
9076
9105 9134
9163
9192
9221
53
9250
9279 9308
9338
9367
9396
54
9425
9454 9483
9512
9541
9570
55
0.9599
0.9628 0.%57
0.9687
0.9716
0.9745
56
9774
9803 9832
9861
9890
9919
57
9948
9977 1.0007
1.0036
1.0065
1.0094
58
1.0123
1.0152 0181
0210
0239
0268
59
0297
0326 0356
0385
0414
0443
60
1.0472
1.0501 1.0530
1.0559
1.0588
1.0617
61
0647
0676 0705
0734
0763
0792
62
0821
0850 0879
0908
0937
0966
63
0996
1025 1054
1083
1112
1141
64
1170
1199 1228
1257
1286
1316
65
1.1345
1.1374 " 1.1403
1.1432
1.1461
1.1490
66
1519
1548 1577
1606
1636.
1665
67
1694
1723 1752
1781
1810^
1839
68
1868
1897 1926
1956
1985
2014
69
2043
2072 2101
2130
2159
2188
70
1.2217
1.2246 1.2275
1.2305
1.2334
1.2363
71
2392
2421 2450
2479
2508
2537
72
2566
2595 2625
2654
2683
2712
73
2741
2770 2799
2828
2857
2886
74
2915
2945 2974
3003
3032
3061
75
1.3090
1.3119 1.3148
1.3177
1.3206
1.3235
76
3265
3294 3323
3352
3381
3410
77
3439
3468 3497
3526
3555
3584
78
3614
3643 3672
3701
3730
3759
79
3788
3817 3846
3875
3904
3934
80
1.3963
1.3992 1.4021
1.4050
1.4079
1.4108
81
4137
4166 4195
4224
4254
4283 .
82
4312
4341 4370
4399
4428
4457
83
4486
4515 4544
4573
4603
4632
84
4661
4690 4719
4748
4777
4806
85
1.4835
1.4864 1.4893
1.4923
1.4952
1.4981
86
5010
5039 5068
5097
5126
5155
87
5184
5213 5243
5272
5301
5330
88
5359
5388 5417
5446
5475
5504
89
5533
5563 5592
5621
5650
5679
90
1 <;7ftR
1 «;7^7 1 1^7/^
1 ';7Qq
1 j;S74
1 JCR*;.^
I
104
TABLE X. CONVERSION OF MINUTES AND SECONDS TO
DECIMALS OF A DEGREE, AND OF DECIMALS OF A DEGREE
TO MINUTES AND SECONDS
r
11
o
' and ff
o
f and ''
O
0.0000
O
0.00000
0.000
0' 0"
0.50
30' 0"
1
0167
1
028
001
0' 4"
51
30' 36"
2
0333
2
056
002
0' 7"
52
31' 12"
3
0500
3
083
003
•O'll"
53
31' 48"
4
0667
4
111
004
0'14"
54
32' 24"
5
0.0833
5
0.00139
0.005
0'18"
0.55
33' 0"
6
1000
6
167
006
0'22"
56
33' 36"
7
1167
7
194
007
0' 25"
57
34' 12"
8
1333
8
222
008
0'29"
58
34' 48"
9
1500
9
250
009
0'32"
59
315' 24"
10
0.1667
lO
0.00278
0.00
0' 0"
0.60
36' 0"
11
1833
11
306
01
0'36"
61
36' 36"
12
2000
12
333
02
1' 12"
62
37' 12"
13
2167
13
361
03
1' 48"
63
37' 48"
14
2333
14
389
04
2' 24"
64
38' 24"
16
0.2500
15
0.00417
0.05
3' 0"
0.65
39' 0"
16
2667
16
444
06
3' 36"
66
39' 36" .
17
2833
17
472
07
4' 12"
67
40' 12"
18
3000
18
500
08
4' 48"
68
40' 48"
19
3167
19
528
09
5' 24"
69
41' 24"
20
0.3333
20
0.00556
O.IO
6' 0"
0.70
42' 0"
21
3500
21
583
11
6' 36"
71
42' 36"
22
3667
22
611
12
7' 12"
72
43' 12"
23
3833
23
639
13
7' 48"
73
43' 48"
24
4000
24
667
14
8' 24"
74
44' 24"
25
0.4167
25
0.00694
0.15
9' 0"
0.75
45' 0"
26
4333
26
722
16
9' 36"
76
45' 36"
27
4500
27
750
17
10' 12"
77
46' 12"
28
4667
28
778
18
10' 48"
78
46' 48"
29
4833
29
806
19
11' 24"
79
47' 24"
30
0.5000
30
0.00833
0.20
12' 0"
0.80
48' 0"
31
5167
31
861
. 21
12' 36"
81
48' 36"
32
5333
32
889
22
13' 12"
82
49' 12"
33
5500
33
917
23
13' 48"
83
49' 48"
34
5667
34
944
24
14' 24"
84
50' 24"
35
0.5833
35
0.00972
0.25
15' 0"
0.85
51' 0"
36
6000
36
01000
26
15' 36"
86
51' 36"
37
6167
37
028
27
16' 12"
87
52' 12"
38
6333
38
056
28
16' 48"
88
52' 48"
39
6500
39
083
29
17' 24"
89
53' 24"
40
0.6667
40
0.01111
0.30
18' 0"
0.90
54' 0"
41
6833
41
139
31
IS' 36"
91
54' 36"
42
7000
42
167
32
19' 12"
92
55' 12"
43
7167
43
194
33
19' 48"
93
55' 48"
44
7333
44
222
34
20' 24"
94
56' 24"
45
0.7500
45
0.01250
0.35
21' 0"
0.95
57' 0"
46
?567
46
278
36
21' 36"
96
57' 36"
47
7833
47
306
37
22' 12"
97
58' 12"
48
8000
48
333
38
22' 48"
98
58' 48"
49
8167
49
361
39
23' 24"
99
59' 24"
50
0.8333
50
0.01389
0.40
24' 0"
l.OO
60' 0"
51
8500
51
417
41
24' 36"
10
66' 0"
52
8667
52
444
42
'■$5' 12"
20
72' 0"
53
8833
53
472
43
25' 48"
30
78' 0"
54
9000
54
500
44
26' 24"
40
84' 0"
55
0.9167
55
0.01528
0.45
27' 0"
1.50
90' 0"
56
9333
56
556
46
27' 36"
60
96' 0"
57
9500
57
583
47
28' 12"
70
102' 0"
58
9667
58
611
48
, 28' 48"
SO
108' 0"
59
9833
59
639
49
\ 29' 24"
90
114' 0"
60
1.0000
60
0.01667
0.50
be 0"
2.00
120' 0"
i
_d
o
tt
^and "
o
' and ''
\
i
f^/i
\
H
^/
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