•
UC-NRLF
III I Illl
SB 35
IN MEMORIAM
FLOR1AN CAJORI
PHILLIPS-LOOMIS MATHEMATICAL SERIES
ELEMENTS OF TRIGONOMETRY
PLANE AND SPHERICAL
BY
ANDREW W. PHILLIPS, PH.D.
AM>
WENDELL M. STRONG, PH.D.
VALK UN1VKRSITV
NEW YORK AND LONDON
HARPER & BROTHERS PUBLISHERS
1899
THE PHILLIPS-LOOMIS MATHEMATICAL SERIES.
ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By
ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale
University. Crown 8vo, Half Leather.
ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D.,
and IRVING FISHER, Ph.D., Professors in Yale University. Crown
8vo, Half Leather, $1 75. [By mail, $1 92.]
ABRIDGED GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [By
mail, $1 40.]
PLANE GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING
FISHEK, Ph.D. Crown 8vo, Cloth, 80 cents. [By mail, 90 cents.]
LOGARITHMIC AND TRIGONOMETRIC TABLES. Five- Place
and Four-Place. By ANDREW W. PHILLIPS, Ph.D., and WKNDKI.I.
M. STRONG, Ph.D., Yale University. Crown 8vo.
LOGARITHMS OF NUMBERS. Five- Figure Table to Accompany
the "Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and
IKVING FISHER, Ph.D., Professors in Yale University. Crown 8vo,
Cloth, 30 cents. [By mail, 35 cents.]
NEW YORK AND LONDON :
HARPER & BROTHERS, PUBLISHERS.
Copyright, 1898, by HARPER & BROTHERS.
All rights reserved.
PREFACE
IN this work the trigonometric functions are defined as
ratios, but their representation by lines is also introduced at
the beginning, because certain parts of the subject can be
treated more simply by the line method, or by a combination
of the two methods, than by the ratio method alone.
Attention is called to the following features of the book :
The simplicity and directness of the treatment of both
the Plane and Spherical Trigonometry.
The emphasis given to the formulas essential to the solu-
tion of triangles.
The large number of exercises.
The graphical representation of the trigonometric, inverse
trigonometric, and hyperbolic functions.
The use of photo-engravings of models in the Spherical
Trigonometry.
The recognition of the rigorous ideas of modern math-
ematics in dealing with the fundamental series of trigo-
nometry.
The natural treatment of the complex number and the
hyperbolic functions.
The graphical solution of spherical triangles.
Our grateful acknowledgments are due to our colleague,
Professor James Pierpont, for valuable suggestions regard-
ing the construction of Chapter VI.
We are also indebted to Dr. George T. Sellew for making
the collection of miscellaneous exercises.
ANDREW W. PHILLIPS,
WENDELL M. STRONG.
YALE UNIVERSITY, December, 1808.
TABLE OF CONTENTS
PLANE TRIGONOMETRY
CHAPTER I
THE TRIGONOMETRIC FUNCTIONS
PAGE
Angles i
Definitions of the Trigonometric Functions 4
Signs of the Trigonometric Functions .8
Relations of the Functions 10
Functions of an Acute Angle of a Right Triangle 13
Functions of Complementary Angles 14
Functions of o°, 90°, 1 80°, 270°, 360° 15
Functions of the Supplement of an Angle 16
Functions of 45°, 30°, 60° 17
Functions of ( — .r), (180°— .r), (i8o°+.r), (360°— .r) 18
Functions of (90°— y), (90° +j), (270°— y\ (270°+^) 20
CHAPTER II
THE RIGHT TRIANGLE
Solution of Right Triangles 22
Solution of Oblique Triangles by the Aid of Right Triangles . . 28
CHAPTER III
TRIGONOMETRIC ANALYSIS
Proof of Fundamental Formulas (i i)- (14) 32
Tangent of the Sum and Difference of Two Angles 36
Functions of Twice an Angle 36
Functions of Half an Angle 36
Formulas for the Sums and Differences of Functions 37
The Inverse Trigonometric Functions 39
vi TABLE OF CONTENTS
CHAPTER IV
THE OBLIQUE TRIANGLE
PAGE
Derivation of Formulas 41
Formulas for the Area of a Triangle 44
The Ambiguous Case 45
The Solution of a Triangle :
(i.) Given a Side and Two Angles 46
(2.) Given Two Sides and the Angle Opposite One of Them . 46
(3.) Given Two Sides and the Included Angle 48
(4.) Given the Three Sides 49
Exercises 50
CHAPTER V
CIRCULAR MEASURE— GRAPHICAL REPRESENTATION
Circular Measure 55
Periodicity of the Trigonometric Functions 57
Graphical Representation 58
CHAPTER VI
COMPUTATION OF LOGARITHMS AND OF THE TRIGONOMETRIC FUNC-
TIONS—DE MOIVRE'S THEOREM— HYPERBOLIC FUNCTIONS
Fundamental Series 63
Computation of Logarithms 64
Computation of Trigonometric Functions 68
De Moivre's Theorem 70
The Roots of Unity 72
The Hyperbolic Functions 73
CHAPTER VII
MISCELLANEOUS EXERCISES
Relations of Functions . . • 7§
Right Triangles 80
Isosceles Triangles and Regular Polygons 83
Trigonometric Identities and Equations 84
Oblique Triangles 88
TABLE OF CONTENTS vii
SPHERICAL TRIGONOMETRY
CHAPTER VIII
RIGHT AND QUADRANTAL TRIANGLES
PAGE
Derivation of Formulas for Right Triangles 93
Napier's Rules 94
Ambiguous Case 97
Quadrantal Triangles 98
CHAPTER IX
OBLIQUE-ANGLED TRIANGLES
Derivation of Formulas 100
Formulas for Logarithmic Computation 101
The Six Cases and Examples 104
Ambiguous Cases 106
Area of the Spherical Triangle 108
CHAPTER X
APPLICATIONS TO THE CELESTIAL AND TERRESTRIAL SPHERES
Astronomical Problems no
Geographical Problems 113
CHAPTER XI
GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE 115
CHAPTER XII
RECAPITULATION OF FORMULAS 119
APPENDIX
RELATION OF THE PLANE, SPHERICAL, AND PSEUDO-SPHERICAL
TRIGONOMETRIES 125
ANSWERS TO EXERCISES 129
PLANE TRIGONOMETRY
CHAPTER I
THE TRIGONOMETRIC FUNCTIONS
ANGLES
1. In Trigonometry the size of an angle is measured by
the amount one side of the angle has revolved from the
position of the other side to reach its final position.
Thus, if the hand of a clock makes one-fourth of a rev-
olution, the angle through which it turns is one right angle ;
if it makes one-half a revolution, the angle is two right an-
gles; if one revolution, the angle is four right angles; if one
and one-half revolutions, the angle is six right angles, etc.
O'
B
FIG. 2
FIG. 3
The amount the side OB has rotated from OA to reach its final position
may or may not be equal to the inclination of the lines. In Fig. i it is equal
to this inclination ; in Fig. 4 it is not.
Two angles may have the same sides and yet be different. In Fig. 2
I
PLANE TRIGONOME TR 1 '
and Fig. 4 the positions of the sides of the angles are the same ; yet in
Fig. 2 the angle is two right angles, in Fig. 4 it is six right angles. The
addition of any number of complete revolutions to an angle does not change
the position of its sides.
Question. — Through how many right angles does the hour-hand
of a clock revolve in 6^ hours? the minute-hand ?
Question. — If the fly-wheel of an engine makes 100 revolutions per
minute, through how many right angles does it revolve in i second ?
Initial line
RIGHT ANGLES
Initial line
5} RIGHT ANGLES
Def. — The first side of the angle — that is, the side from
which the revolution is measured — is the initial line; the
second side is the terminal line.
Def. — If the direction of the revolution is opposite to that
of the hands of a clock, the angle is positive; if the same
as that of the hands of a clock, the angle is negative.
Initial line
Initial Line
POSITIVE ANGLE
NEGATIVE ANGLE
The angles we have employed as illustrations— those described
by the hands of a clock— are all negative angles.
2. Angles are usually measured in degrees, minutes, and
seconds. A degree is one-ninetieth of a right angle, a min-
ute is one-sixtieth of a degree, a second is one-sixtieth of a
minute.
THE TRIGONOMETRIC FUNCTIONS
The symbols indicating degrees, minutes, and seconds are ° ' ";
thus, twenty-six degrees, forty-three minutes, and ten seconds is
written 26° 43' 10".
3. The plane about the vertex of an angle is divided into
four quadrants, as shown in the figure; the first quadrant
begins at the initial line.
in
IV
THE FOUR QUADRANTS
II
III
ANGLE IN 1ST QUADRANT
ANGLE IN 2D QUADRANT
ANGLE IN 3D QUADRANT
ANGLE IN 4TH QUADRANT
An angle is said to be in a certain quadrant if its terminali
line is in that quadrant.
EXERCISES
4. (i.) Express 2\ right angles in degrees, minutes, and seconds^
In what quadrant is the angle?
(2.) What angle less than 360° has the same initial and terminal
lines as an angle of 745°?
(3.) What positive angles less than 720° have the same sides as am
angle of —73°?
(4.) In what quadrant is an angle of —890°?
4 PLANE TRIGONOMETRY
DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS
5. The trigonometric functions are numbers, and are de-
fined as the ratios of lines.
Let the angle A OP be so placed that the initial line is
horizontal, and from Py any point of the terminal line, draw
PS perpendicular to the initial line.
S A
ANGLE IN THE 1ST QUADRANT
ANGLE IN THE 2D QUADRANT
ANGLH IN THE 30 QUADRANT
Denote the angle A OP by x.
SP
ANGLE IN THE 4TH QUADRANT
- = sine of x (written sin^r).
= cosine of x (written cos^r).
THE TRIGONOMETRIC FUNCTIONS
SP
-y^ — tangent of x (written tan x).
— — )= cotangent of x (written cot x).
OP
— „
OP
= secant of x (written sec^r).
= cosecant of x (written csc^r).
To the above may be added the versed sine (written versin) and coversed
sine (written coversin), which are defined as follows :
versiii x = i - cos a?; coversiii x = i — sin x.
The values of the sine, cosine, etc., do not depend upon
what point of the terminal line is taken as P, but upon the
angle.
S S'
S'S
x
For the triangles OSP and OS'P' being similar, the ratio of any
two sides of OS'P' is equal to the ratio of the corresponding sides
of OSP.
Def. — The sine, cosine, tangent, cotangent, secant, and
cosecant of an angle are the trigonometric functions
of the angle, and depend for their value on the angle
alone.
0. A line may by its length and direction represent a
number; the magnitude of the number is expressed by the
Icngtli of the line ; the number is positive or negative ac-
cording to the direction of the line.
PLANE TRIGONOMETRY
7. In § 5, if the denominators of the several ratios be
taken equal to unity, the trigonometric functions will be rep-
resented by lines.
SP SP
Thus, sin^r= -~p= -— = SP— the number represented by
the line, that is, the ratio of the line to its unit of length.
Hence SP may represent the sine of x.
In a similar manner the other trigonometric functions
may be represented by lines.
In the following figures a circle of unit radius is described
about the vertex O of the angle A OP, this angle being de-
noted by x. Then from § 5 it follows that
FIG. a
Cot
C Cot B
FIG. 3
FIG 4
THE TRIGONOMETRIC FUNCTIONS 7
SP represents the sine of x.
OS represents the co§iiie of x.
A T represents the tangent of x.
BC represents the cotangent of x.
0/^represents the secant of x.
OC represents the co§ecant of x.
For the sake of brevity, the lines SP, OS, etc., of the preceding figures are
often spoken of as the sine, cosine, etc.
Hence, we may also define the trigonometric functions
in general terms as follows :
If a circle of unit radius is described about the vertex of
an angle,
(r.) The iine of the angle is represented by the perpendicular
upon the initial line from the intersection of the terminal line with
the circumference.
(2.) The cosine of the angle is represented by the segment of the
initial line extending from the vertex to the sine.
(3.) The tangent of the angle is represented by a line tangent to
the circle at the beginning of the first quadrant, and extending from
the point of tangency to the terminal line.
(4.) The cotangent of the angle is represented by a line tangent
to the circle at the beginning of the second quadrant, and extending
from the point of tangency to the terminal line.
(5.) The secant of the angle is represented by the segment of the
terminal line extending from the vertex to the tangent.
(6.) The cosecant of the angle is represented by the segment of
the terminal line extending from the vertex to the cotangent.
The definitions in § 5 are called the ratio definitions of the trigonometric-
functions, and those in § 7 the line definitions. The introduction of two
definitions for the same thing should not embarrass the student. We have
shown that they are equivalent. In some cases it is convenient to use the
first definition, and in other cases the second, as the student will observe
in the course of this study. It is therefore important that he should be-
come familiar with the use of both.
8
PLANE TRIGONOMETRY '
SIGNS OF THE TRIGONOMETRIC FUNCTIONS
8. Lines are regarded as positive or negative according
to their directions. Thus, in the figures of § 5, OS is posi-
tive if it extends to the rigJit of O along the initial line,
negative if it extends to the left ; SP \s positive if it extends
upward from OA, negative if it extends downward. OP, the
terminal line, is always positive.
The above determines, from § 5, the signs of the trigono-
metric functions, since it shows the signs of the two terms
of each ratio.
By the line definitions the signs may be determined di-
rectly. The sine and tangent are positive if measured up-
ward from OA, and negative if measured downward.
The cosine and cotangent are positive if measured to the
right from OB, and negative if measured to the left.
B Cot + Cot- B
TIG. 3
FIG. 4
THE TRIGONOMETRIC FUNCTIONS
The secant and cosecant are positive if measured in the
same direction as the terminal line, OP\ negative if measured
in the opposite direction.
The signs of the functions of angles in the different quadrants are as follows :
Quadrant
I
II
Ill
IV
Sine and cosecant
+
+
+
-
-
Cosine and secant
-
-
+
Tangent and cotangent
+
-
+
-
f>. It is evident that the values of the functions of an
angle depend only upon the position of the sides of the
angle. If two angles differ by 360°, or any multiple of 360°,
the position of the sides is the same, hence the values of
the functions are the same.
C cot B
Thus in Fig. i the angle is 120°, in Fig. 2 the angle is 840°, yet
the lines which represent the functions are the same for both angles.
EXERCISE
Determine, by drawing the necessary figures, the sign of tan 1000°;
cos 810°; sin 760°; cot —70°; cos — 550°; tan —560°; sec 300°; cot
1560°; sin 130°; cos 260°; tan 310°.
10
PLANE TRIGONOMETRY
RELATIONS OF THE FUNCTIONS
10. By § 5, whatever may be the length of OP, we have
SP
OS
OP smjr' OP °*x' n^'
SP
—
OS
OP
We have, then, from Figs. 2 and 3,
SP _ _ sinx
OS- -
OS COS X
- — f*Ol JC — -
SP~ "
or
Multiplying (i) by (2),
tana?
i
cot x
Again, from Figs. 2 and 3,
OP
tan x
_
co*x*
OP 1
— = C§C X = — - .
SP sin a?
From Figs. 2 and 3, OS2-f SP*= OP\
or
and
Also, O
or I iKtan2o?=r sec2.x;
OP
B Cot
sin'2jf = i — cos2jr ; cos9 x= i — sin2 jr.
*=OT\ and
FIG. 3
(0
(2)
(3)
(4)
(5)
(6)
(7)
(8)
THE TRIGONOMETRIC FUNCTIONS u
The angle x has been taken in the first quadrant ; the
results are, however, true for any angle. The proof is the
same for angles in other quadrants, except that SP be-
comes negative in the third and fourth quadrants, and OS
in the second and third.
EXERCISES
11. (i.) Prove cos-r sec.i-= i.
(2.) Prove sin^r cscx = i.
(3.) Prove tan x cos x — sin x.
(4.) Prove sin x \f \ — cos'- x = i — cos2.r.
(5.) Prove tan x + cot .r — - - l- --
sin.r cos.r
(6.) Prove sin4.r — Cos4.r = i — 2 cos'.r.
(7.) Prove — = sin.r.
cot-r sec.r
(8.) Prove tan x sin .r -h cosx = sec;tr.
12. The formulas (i)-(8) of § 10 are algebraic equations
connecting the different functions of the same angle. If
the value of one of the functions of an angle is given, we
can substitute this value in one of the equations and solve
to find another of the functions. Repeating the process, we
find a third function, etc.
In solving equation (6), (7), or (8) a square root is extracted;
unless something is given which determines whether to choose the
positive or negative square root, we get two values for some of
the functions. The reason for this is that there are two angles
less than 360° for which a function has a given value.
EXERCISES
13. (i.) Given x less than 90° and sin.r = ^; find all the other
functions of x.
Solution. —
=r rb \/ i — 1—
Since x is less than 90°, we kno\v that cosx is positive.
12 PLANE TRIGONOMETRY
Hence cosx= +
$•3
----
2
(2.) Given tan.r = — £ and .r in quadrant IV; find sin x and cos jr.
Solution, —
_ _ ,
COS.T ~
hence 3 sin .*•=: — COSJF,
sin2.* + cos2.* = i ;
hence 10 sin2jr = i ;
(3.) Given sin( — 30°) = — \\ find the other functions of — 30°.
(4.) Given x in quadrant III and sin^r = — £; find all the other
functions of x.
(5.) Given y in quadrant IV and sin/=z — |, find all the other
functions of y.
(6.) Given cos 6o° = £; find all the other functions of 60°.
(7.) Given sin o°=:o; find coso° and tano°.
(8.) Given tans'rrjand z in quadrant I; find the other functions
of JT.
(9.) Given cot45°= i ; find all the other functions of 45°.
(10.) Given tan/=£\/5 and cos^> negative; find all the other
functions of y.
(ii.) Given cot 30°= \/3 i fi°d tne other functions of 30°.
(12.) Given 2 sin-r=i — cos^r and x in quadrant II; find sin^r
and cos jr.
(13.) Given tan.r4-cot;r = 3 and x in quadrant I ; find sin jr. i
THE TRIGONOMETRIC FUNCTIONS 13
FUNCTIONS OF AN ACUTE ANGLE OF A RIGHT TRIANGLE
14. The functions of an acute angle of a right triangle
can be expressed as ratios of the sides of the triangle.
Remark. — Triangles are usually lettered, as in Fig. 2, the capital
letters denoting the angles, the corresponding small letters the sides
opposite.
In the right triangle ABC, by § 5,
BC a
15. From § 14, for an acute angle of a right triangle, we have
side opposite angle
sme = -- s-s- - ^— ;
hypotenuse
side adjacent to angle
cosine = - — ^ - £—',
hypotenuse
tangent = . sjde opposite angle
side adjacent to angle
side adjacent to angle
cotangent = — —, — - — '— •
side opposite angle
PLANE TRIGONOMETRY
FUNCTIONS OF COMPLEMENTARY ANGLES
16* From § 14, we have
sin ^L = cos^
(o)
tan A = cot B = cot (9O° - A) 5
cot A = tan B = tan (9O° — A).
Because of this relation the sine and cosine are called co-func-
tions of each other, and the tangent and cotangent are called co-
functions of each other.
The results of this article may be stated thus:
A function of an acute angle is equal to the co-function of
its complementary angle.
Ths values of the functions of the different angles are given in " Trigo-
nometric Tables." By the use of the principle just proved, each function
of an angle between 45° and 90° can be found as a function of an angle less
than 45°. Consequently, the tables need to be constructed for angles up to
45° only. The tables are so arranged that a number in them can be read
either as a function of an angle less than 45° or as the co-function of the
complement of this angle.
EXERCISES
17. (i.) Express as functions of an angle less than 45°:
sin 70° ; cos 89° 30' ; tan 63° ;
cos66°; cot47°; sin72°39'.
(2.) cosjr = sin ix ; find x.
(3.) tan x = cot 3-r ; find x.
(4.) sin 2^r = cos3-r ; find x.
(5.) cot(3O° — x) = tan(3o°-J- $x) ; find x.
(6.) A, B, and C are the angles of a triangle; prove that
Hint. — A+B + C=\ So0.
THE TRIGONOMETRIC FUNCTIONS
FUNCTIONS OF O°, 90°, l8o°, 270°, AND 360°
18. As the angle x decreases towards o° (Fig. i), sin* de-
creases and cos* increases. When OP comes into coincidence
with OA, SP becomes o, and OS becomes OA( = i).
Ik-nee sino° = o. coso°=i.
FIG. 3
FIG. 4
As the angle x increases towards 90° (Fig. 2), sin* increases
and cos* decreases. When OP comes into coincidence with OB,
SP becomes OB(—\) and OS becomes o.
Hence sinQO0^!, cosgo^o.
As the angle x decreases towards o° (Fig. 3), tan* decreases
and cot* increases. When OP comes into coincidence with OA,
^/"becomes o and BC has increased without limit.
Hence tano°r=o, coto°=:oo.
As the angle x increases towards 90° (Fig. 4), tan* increases
and cot* decreases. When OP comes into coincidence with OB,
AT has increased without limit, and BC—o.
Hence
Remark. — liy coto°=co we mean that as the angle approaches indefinitely
near to o° its cotangent increases so as to become greater than any finite quan-
tity we may choose. The symbol co does not denote a definite number, but
simply that the number is indefinitely great.
i6
PLANE TRIGONOMETRY
Tn every case where a trigonometric function becomes indefinitely
great it is in a positive sense if the angle approaches the limiting
value from one side, in a negative sense if the angle approaches the
limiting value from the other side. Thus cot o° = -|- oo if the angle
decreases to o°, but cot o°= — oo if the angle increases from a nega-
tive angle to o°. We shall not often need to distinguish between
H-oo and —oo, and shall in general denote either by the symbol oo.
By a similar method the functions of 180°, 270°, and 360° may be
deduced. The results of this article are shown in the following table :
Angle
0°
90°
1 80°
270°
3600
sin
0
I
0
— I
0
cos
i
0
-I
O
I
tan
o
oo
o
00
o
cot
00
O
CO
0
00
19. It may now be stated that, as an angle varies, its sine and cosine
can take on "values from — / to -\- r only, its tangent and cotangent all
values from — oo to -\- oo , its secant and cosecant all values from — oo
to -j- oo , except those between — / and -f- /.
FUNCTIONS OF THE SUPPLEMENT OF AN ANGLE
20. Suppose the triangle OPS (Fig. i) equal to the tri-
angle OP'S ' (Fig. 2), then SP=S'Pf and OS=OS', and the
angle AOP' (Fig. 2) is equal to the supplement of A OP
(Fig. i). Also, in the triangle AOP' (Fig. 3), angle AOP'
= angle AOP' (Fig. 2).
V
8' O
FIG. 2
FIG. 3
THE TRIGONOMETRIC FUNCTIONS
It follows from §§ 5 and 8 that
§in(l§O°— x) = §in
C0§ (1§O°— 05) = — C ,
tan (1§0° - x) = - tan x ; '
cot (1§O°— a?) = — cot x.
The results of this article may be stated thus :
The sine of an angle is equal to the sine of its supplement,
and the cosine, tangent, and cotangent are each equal to minus
the same functions of its supplement.
The principle just proved is of great importance in the solution of tri-
angles which contain an obtuse angle.
FUNCTIONS OF 45°, 30°, AND 60°
21. In the right triangle OSP (Fig. i) angle O = angle P = 4$°*
and OP— i.
Hence OS = SP = i \/2.
Therefore sin 45° = 00545° = £1/2; §§14,16
tan 45° = cot 45°= i.
P
8
FIG. I
c
O
FIG. 2
In equilateral triangle 0/^4 (Fig. 2) the sides are of unit length
PS bisects angle OP A, is perpendicular to OA, and bisects OA.
Hence, in the right triangle OPS, OS = %, SP = ^\/^.
Therefore sin 30° = cos 60° = i; §14
cos 30° = sin 60° = i \/3 I
tan 30° = cot 60° = J y'3 ;
cot 30° = tan 60° = \/3-
2
1 8 PLANE TRIGONOMETRY
22. The following values should be remembered :
Angle
0°
30°
45°
60°
90°
sin
0
i
iv/2
iVi
I
cos
i
4V5
4^2
k
0
EXERCISES
Prove that if x = 30°,
(i.) sin 2x = 2 sin^r COS.T;
(2.) cos 3* = 4 cos3 .r — 3 cos x ;
(3.) cos 2;r =r cos'2 ;r — sin2.r;
(4.) sin 3-r= 3 sin x cos2-r — sin3.r;
2 tan JT
(5.) tan2;r = — — .
i — tan-jir
(6.) Prove that the equations of exercises i and 3 are cor-
rect if .r = 450.
(7) Prove that the equations of exercises (2) and (4) are cor-
rect if .r=i200.
The following three articles, §§ 23-25, are inserted for
completeness. They include the functions of (90— ;r) and
(180— x\ which, on account of their great importance, were
treated separately in §§ 16 and 20.
FUNCTIONS OF (—x), (l8o°— x\ (l8o° + ;r), (360° -*)
23. The line representing any function — as sine, cosine, etc.
— of each of these angles has the same length as the line repre-
senting the same function of x.
Thus in Figs. 2 and 3, triangle OS'P' — triangle OSP, hence SP-^S'P',
THE TRIGONOMETRIC FUNCTIONS
FIG. 3
FIG. 4
In Figs, i and 4, triangle OSP' — triangle OSP, hence SP'=SF.
In Figs, i, 2, and 4, triangle OA T = triangle OA T, hence A T ' = A T.
In Figs, i, 2, and 4, triangle 0/?C' = triangle OBC. hence BC'=BC.
Therefore any function of each of the angles ( — x\ (180°— x),
(i8o°-f •*), (360° —x\ is equal in numerical value to the same function
of x. Its sign, however, depends on the direction of the line repre-
senting it.
Putting in the correct sign, we obtain the following table:
sin (— x] — — sin x
cos (— x) = cos*
tan(— *) = — tan*
cot ( — x) — — cot x
sin (180° + *)= —sin*
cos (180° + *)= — cos*
tanCi8o°-f*) = tan*
cot (180° + *) = cot*
sin ( 1 80° — *) = sin *
cos (180° — *) — — cos*
tan(i8o° — *) — — tan*
cot (180° - *) = - cot*
sin (360° — *) = — sin *
cos (360° — *) — cos*
tan (360° — *) = — tan *
cot (360° — *) = - cot *
20
PLANE TRIGONOMETRY
FUNCTIONS OF (9O° -7), (9O° +j), (270° -y\ (270° +J/)
£4. The line representing the sine of each of these angles is
of the same length as the line representing the' cosine of y; the
cosine, tangent, or cotangent, respectively, are of the same length
as the sine, cotangent, and tangent of y.
FIG. 3
For
Triangle OS'P' = triangle OSP, hence S'P' = OS, and OS' = SP.
Triangle OA T' — triangle OBC, hence A T' = BC.
Triangle OBC — triangle OA T, hence BC — AT.
Therefore any function of. each of the angles (90° —y\ (90° -\-y\
(270°—^), (270°+^), is equal in numerical value to the co-function
THE TRIGONOMETRIC FUNCTIONS 21
of y. Its sign, however, depends on Ihe direction of the line repre-
senting it.
Putting in the correct sign, we .obtain the following table :
sin (90° — y) = cos v sin (90° -f y) = cosy
cos (90° — y) — sin r cos (90° + y) = — sinjv
tan (90° — r) = cot v tan (90° +}')= — cot v
cot (90° —y) —- tan y cot (90° + y) = — tan y
sin (270° —}')= — cosy sin (270° +/) = — cosy
cos (270° — y) — — sin y cos (270° +^) = siny
tan (270° — .r) = cot y tan (270° +y) = — coty
cot (270° - y) = tan r cot (270° + r) = — tan v
25. Either of the two preceding articles enables us directly to-
express the functions of any angle, positive or negative, in terms-
of the functions of a positive angle less than 90°.
Thus, sin 21 2° — sin (180°+ 32°)= —sin 32°;
cos 260° = cos (270°— 10°) = — sin 10°.
•
EXERCISES
(i.) What angles less than 360° have the sine equal to — %\/2? the-
tangent equal to \/3 ?
(2.) For what values of .r less than 720° is sin.r = .Jy^?
(3.) Find the sine and cosine of —30°; 765°; 120°; 210°.
(4.) Find the functions of 405°; 600°; 1125°; —45°; 225°.
(5.) Find the functions of —120°; —225°; —420°; 3270°.
(6.) Express as functions of an angle less than 45° the functions of
233°: -197°: 894°.
(7.) Express as functions of an angle between 45° and 90°, sin 267°;.
tan ( — 254°); cos 950°.
(8.) Given cos 164° = — .96, rind sin 196°.
(9.) Simplify cos (90° + ,r) cos (2 70° — .r) — sin(i8o° — jr)sin(36o° — x)..
(10.) Simplifysin('8o°--'')tan(9o° + .r)+ . . '. .
y sm (270° — x} sin2 (270° — x)
(u.) Express the functions of (.r — 90°) in terms of functions of x.
CHAPTER II
THE RIGHT TRIANGLE
27. To solve a triangle is to find the parts not given.
A triangle can be solved if three parts, at least one of
which is a side, are given. A right triangle has one angle,
the right angle, always given ; hence a right triangle can
be solved if two sides, or one side and an acute angle, are
also given.
The parts of the right triangle not given are found by
the use of the following formulas:
opposite side adjacent side
(i) sine =-~ — ; (2) cosine =—r — ; § 14
hypotenuse hypotenuse
(3) tangent^
opposite side
(4) cotangent =
_ adjacent side ^
16
adjacent side ' opposite side '
To solve, select a formula in which two given parts enter; substituting
in this the given values, a third part is found. Continue this method till
all the parts are found.
In a given problem there are several ways of solving the triangle ; choose
the shortest.
EXAMPLE
The hypotenuse of a right triangle is 47.653, a side is
21.34; find the remaining parts and the area.
THE RIGHT TRIANGLE
SOLUTION WITHOUT LOGARITHMS
The functions of angles are given
in the table of
Natural Functions."
21-34
sin A =-=
f 47.653
47.653)21.3400^4478
190612
227880
190612
372680
333571
391090
381224
9866
sin A = .4478
.4 = 26° 36'
b—c cos A
=47- 653 x -8942
47.653
.8942
95306
190612
428877
381224
42.6113126
* =42.61 f
= (90° -26° 36' 1 = 63° 24
. 34x42.61
21-34
42.61
2134
12804
SOLUTION EMPLOYING LOGARITHMS
It is usually better to solve triangles
by the use of logarithms.
The logarithms of the functions are
given in the tables of " Logarithms of
Functions." *
*
sin A = -
c
log sin A = log a — log c
log 21. 34 =1.32919
log 47. 653 = 1.67809
--- sub.
log sin .4=9.65110— 10
A = 26° 36' 14"
cos A=-
log b = log c + log cos A
log 47- 653 = 1-67809
log cos 26° 36' 14" =9.95140— 10
log £=1.62949
£=42.608
J£=(9o°-26° 36' I4")=63° 23' 46"
area = \ab
1 og area = log 4 + log a + log b
log ^ = 9.69897 -10
Iog2i. 34=1. 32919
log 42 608 = i . 62949
log area=2. 65765
2)909.2974
454.6487
area =4 54- 6
* In this solution the five-place table of the " Logarithms of Functions" is
used.
t No more decimal places are retained, because the figures in them are not
accurate ; this is due to the fact that the table of " Natural Functions" is only
four- place.
PLANE TRIGONOMETRY
CHECK ON THE CORRECTNESS OF THE WORK
= 90.263 x 5.043
90.263
5-Q43
270789
361052
45I3I5Q
a* = 455. 196309
Extracting the square root, a =
21.34, which proves the solution cor-
rect.
a = c- -l>i = (c + b)(c - l>)
= 90.261 x 5.045
log 90.261 = 1.95550
log 5.045 = 0.70286
2)2.65836
log 21. 34 = 1.32918
a = 21.34, which proves the solu-
tion correct.
Remark. — The results obtained in the solution of the preceding
exercise without logarithms are less accurate than those obtained in
the solution by the use of logarithms ; the cause of this is that four-
place tables have been used in the former method, five place in the
latter.
EXERCISES
28. (i.) In a right triangle £ = 96.42, c= 114.81 ; find a and A.
(2.) The hypotenuse of a right triangle is 28.453, a side is 18.197;
find the remaining parts.
(3.) Given the hypotenuse of a right triangle = 747.24, an acute
angle =23° 45' ; find the remaining parts.
(4.) Given a side of a right triangle = 37.234, the angle opposite
= 54° 27'; find the remaining parts and the area.
. — .(5.) Given a side of a right triangle = 1.1293, the angle adjacent
= 74° 13' 27"; find the remaining parts and the area.
(6.) In a right triangle A = 1 5° 22' 1 1 ", c — .01 793 ; find b.
(7.) In a right triangle £ = 71° 34' 53", £ = 896.33; find a.
(8.) In a right triangle c = 3729.4, £ = 2869.1 ; find A.
(9.) In a right triangle a — 1247, b— 1988 ; find c.
(lo.) In a right triangle (7 = 8.6432, £ = 4.7815; find B.
The angle of elevation or depression of an object is the
angle a line from the point of observation to the object
makes with the horizontal.
THE RIGHT TRIANGLE
Thus angle x (Fig. i) is the angle of elevation of P if O is the point of
observation ; angle y (Fig. 2) is the angle of depression of P if O is the
point of observation.
(n.) At a horizontal distance of 253 ft. from the base of a tower the
angle of elevation of the top is 60° 20' ; find the height of the tower.
(12.) From the top of a vertical cliff 85 ft. high the angle of depres-
sion of a buoy is 24° 31' 22"; find the distance of the buoy from the
foot of the cliff.
(13.) A vertical pole 31 f t. h igh casts a horizontal shadow 45 ft. long ;
find the angle of elevation of the sun above the horizon.
(14.) From the top of a tower 115 ft. high the angle of depression
of an object on a level road leading away from the tower is 22° 13' 44";
find the distance of the object from the top of the tower.
(15.) A rope 324 ft. long is attached to the top of a building, and
the inclination of the rope to the horizontal, when taut, is observed
to be 47° 21' 17"; find the height of the building.
(16.) A light- house is 150 ft. high. How far is an object on the
surface of the water visible from the top?
[Take the radius of the earth as 3960 miles.]
(17.) Three buoys are at the vertices of a right triangle; one side
of the triangle is 17,894 ft., the angle adjacent to it is 57° 23' 46".
Find the length of a course around the three buoys.
(i 8.) The angle of elevation of the top of a tower observed from a
point at a horizontal distance of 897.3 ft. from the base is 10° 27' 42" ;
find the height of the tower.
(19.) A ladder 42^ ft. long leans against the side of a building; its
foot is 25! ft. from the building. What angle does it make with the
ground ?
(20.) Two buildings are on opposite sides of a street 120 ft. broad.
S'.Z'b
i
26
PLANE TRIGONOMETRY
The height of the first is 55 ft. ; the angle of elevation of the top of
the second, observed from the edge of the roof of the first, is 26° 37'.
Find the height of the second building.
A -(21.) A mark on a flag-pole is known to be 53 ft. 7 in. above the
j ground. This mark is observed from a certain point, and its angle.
of elevation is found to be 25° 34'. The angle of elevation of the top
of the pole is then measured, and found to be 34° 17'. Find the
height of the pole.
(22.) The equal sides of an isosceles triangle are each 7 in. long ; the
base is 9 in. long. Find the angles of the triangle.
b = 9
Hint. — Draw the perpendicular BD. BD bisects the base, and also the
angle ABC.
In the right triangle ABD, AB—-] in., AD—\\ in., hence ABD can
be solved.
Angle C= angle A, angle ABC— 2 angle ABD.
(23.) Given the equal sides of an isosceles triangle each 13.44 in.,
and the equal angles are each 63° 21' 42"; find the remaining parts
and the area.
(24.) The equal sides of an isosceles triangle are each 377.22 in.,
the angle between them is 19° 55' 32". Find the base and the area
of the triangle.
JL (25.) If a chord of a circle is 1 8 ft. long, and it subtends at the centre
an angle of 45° 31' 10" , find the radius of the circle.
(26.) The base of a wedge is 3.92 in., and its sides are each 13.25 in.
long; find the angle at its vertex.
THE RIGHT TRIANGLE
27
(27.) The angle between the legs of a pair of dividers is 64° 45', the
legs are 5 in. long; find the distance between the points.
(28.) A field is in the form of an isosceles triangle, the base of the
triangle is 1793.2 ft. ; the angles adjacent to the base are each 53° 27'
^49". Find the area of the field.
6 (29.) A house has a gable roof. The width of the house is 30 ft.,
the height to the eaves 25! ft., the height to the ridge-pole 33! ft.
Find the length of the rafters and the area of an end of the house.
^ (30.) The length of one side of a regular pentagon is 29.25 in. ; find
the radius, the apothem, and the area of the pentagon.
b
Hint. — The pentagon is divided into 5 equal isosceles triangles by its radii.
Let AOB be one of these triangles. ^#=29.25 in.; angle AOB=\ of
36o° = 72°. Find, by the methods previously given, OA, OD, and the area
of the triangle A OB.
These are the radius of the pentagon, the. apothem of the pentagon, and
\ the area of the pentagon respectively.
(31.) The apothem of a regular dodecagon is 2 ; find the perimeter.
0(32.) A tower is octagonal ; the perimeter of the octagon is 153.7 ft.
Find the area of the base of the tower.
(33.) A fence extends about a field which is in the form of a regular
polygon of 7 sides; the radius of the polygon is 6283.4 ft. Find the
length of the fence.
(34.) The length of a side of a regular hexagon inscribed in a circle
is 3.27 ft. ; find the perimeter of a regular decagon inscribed in the
same circle.
(35.) The area of a field in the form of a regular polygon of 9 sides
is 483930 sq. ft. ; find the length of the fence about it.
28
PLANE TRIGONOMETRY
SOLUTION OF OBLIQUE TRIANGLES BY THE AID OF
RIGHT TRIANGLES
29. Oblique triangles can always be solved by the aid of
right triangles without the use of special formulas ; the
method is frequently, however, quite awkward ; hence, in a
later chapter, formulas are deduced which render the solu-
tion more simple.
The following exercises illustrate the solution by means
of right triangles :
(i.) In an oblique triangle ^ = 3.72, ^ = 47° 52', £'=109° 10'; find
the remaining parts.
The given parts are a side and two angles.
C
Hint.— A = [i&o°-(B+C)],
Draw the perpendicular CD.
Solve the right triangle BCD.
Having thus found CD, solve the right triangle A CD,
(2.) In an oblique triangle a = 89.7, c— 125.3, B= 39° 8'; find the
remaining parts.
The given parts are two sides and the included angle.
125.3
THE RIGHT TRIANGLE
29
'ii/.— Draw the perpendicular CD.
Solve the right triangle CBD.
Having thus found CD and AD(=c-DB), solve the right triangle ACD.
(3.) In an oblique triangle a = 3.67, b — 5.81, A = 27° 23'; find the
remaining parts.
The given parts are two sides and an anglt opposite one of
them.
C
B'
B
Either of the triangles ACB, ACB' contains the given parts, and
is a solution.
There are two solutions when the side opposite the given angle is
less than the other given side and greater than the perpendicular,
CD, from the extremity of that side to the base.*
Hint.— Solve the right triangle ACD.
Having thus found CD, solve the right triangle CDB (or CDB'\
(4.) The sides of an oblique triangle are 0 = 34.2, £ = 38 A ^- = 55. 12;
find the angles.
The given parts are the three sides.
c =55.12 0
* A discussion of this case is contained in a later chapter on the solution
of oblique triangles.
Hint.—
Hence
PLANE TRIGONOMETRY
a* - ^ = CF? - #• -(c- X}*
In each of the right triangles A CD and BCD the hypotenuse and a side
are now known ; hence these triangles can be solved.
"jsi (5-) Two trees, A and B, are on opposite sides of a pond. The
distance of A from a point C is 297.6 ft., the distance of B from C is
8644 ft., the angle ACS is 87° 43' 12". Find the distance AB.
(6.) To determine the distance of a ship A from a point B on
shore, a line, EC, 800 ft. long, is measured on shore ; the angles, ABC
and ACS, are found to be 67° 43' and 74° 21' 16" respectively. What
is the distance of the ship from the point B?
j^ (7.) A light-house 92 ft. high stands on top of a hill; the distance
from its base to a point at the water's edge is 297.25 ft. ; observed
from this point the angle of elevation of the top is 46° 33' 15". Find
the length of a line from the top of the light-house to the point.
(8.) The sides of a triangular field are 534 ft., 679.47 ft., 474.5 ft.
What are the angles and the area of the field ?
(9.) A certain point is at a horizontal distance of 117^ ft. from a
river, and is u ft. above the river; observed from this point the angle
of depression of the farther bank is i° 12'. What is the width of the river?
(10.) In a quadrilateral ABCD,AB= 1.41, BC— 1.05, CD = 1.76, DA
= 1.93, angle ^=75° 21'; find the other angles of the quadrilateral.
\
n Q . i o
• THE RIGHT TRIANGLE 31
Hint. — Draw the diagonal DB.
In the triangle ABD two sides and an included angle are given, hence the
triangle can be solved.
The solution of triangle ABD gives DB.
I laving found DB, there are three sides of the triangle DBC known, hence
the triangle can be solved.
(ii.) In a quadrilateral ABCD, AB=\2.i, AD = 9.7, angle A —
47° 18', angle 71 = 64° 49'» angle D= 100°; find the remaining sides
Hint.— Solve triangle ABD to find BD.
*h
CHAPTER III
TRIGONOMETRIC ANALYSIS
30. In this chapter we shall prove the following funda-
mental formulas, and shall derive other important formulas
from them :
§in (x + y) = §in x co§ y + cos x §in y,
§in(a?-2/) = §in« co§i/ - cosx siiii/,
cos (x + y] = cos a? cos?/ -sin as §iny,
cos (a? — y) =
(12)
(13)
(14)
PROOF OF FORMULAS (l l)-(l4)
31. Let angle AOQ= AT, angle QOP=y; then angle
The angles * and j are each acute and positive, and in Fig. i
(•'*'+ y) i§ less tn^n 90°, in Fig. 2 (.r-f-j) is greater than 90°.
In both figures the circle is a unit circle, and SP is perpendicular to
OA ; hence SP= sin (x +y), OS= cos (x + y).
TRIGONOMETRIC ANALYSIS 33
Draw DP perpendicular to OQ ;
then DP=siny, OD = cosy,
angle SPD = angle AOQ = x.
(Their sides being perpendicular.)
Draw DE perpendicular to OA, DH perpendicular to SP.
Sin (x +/) = SP= ED + PIP.
cos/.
(For OED being a right triangle, -- = sin.r.)
HP '= (cos x) x DP—CQsx sin/.
HP
(For HPD being a right triangle, - = cos jr.)
Therefore, §in(a? + 2/) = §in.r co§?y 4- cosx siny. (u)
Cos O +/) = (95 = OE - HD. *
= (cos x) x (9Z> = cos x cos/.
(For OED being a right triangle. — ~- = cos jr.)
(For PHD being a right triangle, = sin x.)
Therefore, cos (x + y] = co§x eos?/-§in^ §in//. (13)
5^. The preceding formulas have been proved only for
the case when x and y are each acute and positive. The
proof can, however, readily be extended to include all values
of x and y.
Let/ be acute, and let x be an angle in the second quad-
lant ; then x = (90° 4- xr) where x' is acute.
sin (x -f-/) = sin (90° + x' +y)
= cos(X+/) §24
= cosx' cos/ — sin x' sin/
= sin (90° + x') cos/ + cos (90° 4- x') sin / § 24
•=smx cos/4- cos x sin/.
* If (x +,r) is greater than 90°, OS is negative.
34 PLANE TRIGONOMETRY
Thus the formula has been extended to the case where
one of the angles is obtuse and less than 180°. In a
similar way the formula for cos(x+y) is extended to this
case.
By continuing this method both formulas are proved to
be true for all positive values of x and y.
Any negative angle y is equal to a positive angle y' , minus
some multiple of 360°. The functions of y are equal to
those of y', and the functions of (x-\-y) are equal to those
of
Therefore, the formulas being true for \x -\-y'), are true for
A repetition of this reasoning shows that the formulas are
true when both angles, x and y, are negative.
33. Substituting the angle —y for y in formula (11), it
becomes
s\n(x—y) — s'mx cos(— 7) + cos;r sin (—y).
But cos( — y) = cosy, and sin(— y)— — s\ny. §23.
Therefore, sin (a? — ?/) = §in.x COST/ — cosx *m //. (12)
Substituting (—y) for y in formula (13), it becomes
cos (x—y) — cos x cos ( —y) — sin x sin (—y),
Therefore, cos (a? - y) = cos x cosy + sii»a? siny.* (14)
EXERCISES
34. (i.) Prove geometrically where .r and j are acute and positive :
cos^y — cos^r sinj/,
n,r sinj.
* Formulas (12) and (14) are proved geometrically in § 34. The geometric
proof is complicated by the fact that OD and DP are functions of — y, while
the functions of y are what we use.
TRIGONOMETRIC ANALYSIS
.Q
0,4— -x— ,H
35
Hint.— Angle AOQ-x, angle POQ=y, and angle A OP— (x-y).
Draw /'/) perpendicular to 6>().
Then DP= sin (—;•) = —sin r ; but Z>/* is negative, therefore PD taken
as positive is equal to sin y:
OD=cof,( — ji')=cos y,
Angle HTD — angle AOQ=Jc. their sides being perpendicular.
Draw DI1 perpendicular to SP, DE perpendicular to OA.
sin(x-y)=SP=£D-P//.
From right triangle OED, £D.=(^'mx)x OJr)=sinx cos jr.
From right triangle DHP, P//=(cosx)x PZ)=cosx sin y.
Therefore, . sin (.*•—;')= sin x cos;/ — cos x sin; .
From right triangle 0&D, OE = (cos x)x 0£>=cosx cosjj'.
From right triangle DHP, SJ//=(sin x) x PD = ^\\\x sin;'.
Therefore, cos(.r— _j')=cos x cosj' + sin.r sin;'.
(2.) Find the sine and cosine of (45°+, r), (30°— x\ (6o°-f-,r), in terms-
of sin^r and cos.r.
(3.) Given sin.r=$, sin/ = ^, ,\- and y acute; find sin(,r+j) and
sin(.r— y).
(4.) Find the sine and cosine of 75° from the functions of 30° and 45°.
Hint.— 75°=(45° + 30°).
(5.) Find the sine and cosine of 15° from the functions of 30° and 45°.
(6.) Given x and y, each in the second quadrant, sin x = $, siny = ^ ;
find sin (x-\-y) and cos(.r — y}.
(7.) By means of the above formulas express the sine and cosine of.
(180° — .r), (i8o°-|-,r), (270°— .r), (270°+ •*), in terms of sin^r and cos-r,
(8.) Prove sin (6o°-f 45°) + cos (60° + 45°) = cos 45°.
(9.) Given sin 45° = ^^/I, cos 45°— £ \/2 ; find sin 90° and cos 90°.
(10.) Prove that sin (60° -f- x) — sin (60° — .r) =
36 PLANE TRIGONOMETRY
TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES
_ sin(^r-f-j) sin;r cosj^-f-cos^r sinj/
~~ cos(jr+7)~cos^ cosj — sin x si ny
Dividing each term of both numerator and denominator
of the right-hand side of this equation by COSJT cosj, and
remembering that --- = tan, we have
cos
tan x + tan y
In a similar way, dividing formula (12) by formula (14), we
.obtain
tana? - tan?/
»> = !+ tan*
FUNCTIONS OF TWICE AN ANGLE
36. An important special case of formulas (n), (13), and
(15) is when y~x\ we then obtain the functions of 2x in
terms of the functions of x.
From (n), sin (*?+.*)= sin* cos^-hcos^r sin jr.
Hence §in 2a? = 2 §in x co§ x. (17)
From (13), co§2x = co§2a?-§in2x. (18)
Since cos2^r= i — sin8 JIT* and sinajr= I — cos2.r,
we derive from equation (18),
cos 2^-=: I— 2sin2^r, (19)
and cos2;r = 2 cos2^r— I. (20)
From (15), tanto = >,lf..
FUNCTIONS OF HALF AN ANGLE
57. Equations (19) and (20) are true for any angle; there-
fore for the angle \x.
From(i9), cos;r= I — 2 sin2^;
TRIGONOMETRIC ANALYSIS 37
I — COS.T
or
therefore, sin £* = ±\~ • (22)
From (20), cos.r = 2 cos2 4* — I ;
i -f cos^r
or cos -£.*::= --- ;
therefore, cos-^^rfcy — *j~±-. (23)
Dividing (22) by (23), we obtain
(24)
cos a?
FORMULAS FOR SUMS AND DIFFERENCES OF FUNCTIONS
38. From formulas (u)-(i4), we obtain
sin (x + /)-!- sin (x — j/) = 2sin^r cos}' ;
sin (irH-^)— .sin (^ — j/) = 2cos;tr sinj' ;
cos (x 4-7) + cos (*—}') = 2 COS.T cosj ;
cos (x +y) — cos (x — y) — — 2sin^r sinj/.
Let n - (x +/) and v — (x —y] ;
then x = %(2i + ii), y — %(u — v).
Substituting in the above equations, we obtain
sin ?f + sinr = 2 sin-|(«e +v)cos^(u — v); (25)
siiifr - sin r = 2cos-|(/e + t')sin^(M — r); (26)
cos M + c*o« r= 2 co*lr(u+v) cos|-(?« — f) ; (27)
cos!/-cost'=-28iii-J(w + v) §in-J(i«-f). (28)
Dividing (25) by (26),
§in t « 4- sinv
sin M- sin v
, ,
EXERCISES
,'i,9. Express in terms of functions of x, by means of the formulas
of this chapter,
38 PLANE TRIGONOMETRY
(i.) Tan(i8o° — .r); tan ( 1 80° -f x\
(2.) The functions of (x — 180°).
(3.) Sin (.r — 90°) and cos (^ — 90°).
(4.) Sin (.r — 270°), and cos (x— 270°).
(5.) The sine and cosine of (45°— x); of (45°+.*).
(6.) Given tan 45°= i, tan 30° ^ £ -^3; find tan 75°; tan 15°.
cot./- cot*/-l
(7.) Prove cot (05-1- y) — - - —. (30)
Hint. — Divide formula (13) by formula (n).
COt.X COt 91 + 1
(8.) Prove cot (x-y) = - - . (31)
cot y- cot a?
(9.) Prove cos (30 +y) — cos (30° —y) = — sin y.
(10.) ProVe sin ^x = 3 sin. r — 4 sin3.*-.
/#»/. — Sin 3-r=sin (x+2x).
(n.) Prove cos 3* = 4 cosfc— 3 cos x.
(12.) If x and y are acute and tan-r = £, tanj/ = J, prove that
(i 3.) Prove that tan (.r-|~45°) = —
i — tan x
(14.) Given siny=§ and y acute; find sin \y, cos^y, and tan \y.
(15.) Given cos^r=:— | and x in quadrant II; find sin 2x and
cos 2.r.
(16.) Given cos 45° — i \/2 ; find the functions of 22!°.
(17.) Given tan.r = 2 and .r acute ; find tan \x.
(i 8.) Given cos 30° = | -^3 ; find the functions of 15°.
(19.) Given cos9o° = o; find the functions of 45°.
•*> (20.) Find sin §x in terms of sin x.
(21.) Find COS5-T in terms of COS.T.
(22.) Prove sin(.r-j- y -j-2-) — sin x cosy cos 5- -(-cos ,r sin v coss'-J-cos.r
cosy sin z — sin x sin/ sin^.
Hint. — Sin (x+y + z)— sin (x+y) co$s + cos(.*+j') sin 2.
(23.) Given tan 2^ = 3 tan.r; find x.
^ (24.) Prove sin 32° -|- sin 28° = cos 2°.
(25.) Prove tan x -\- cot x — 2 esc 2x.
(26.) Prove (sin!.r-|-cos£..r)a=: i -)-sin^r.
(27.) Prove (sin \x - cos l.r)- = i - sin x.
TRIGONOMETRIC ANALYSIS 39
^(28.) Prove cos 2.v = cos*.r — sin4.r.
(29.) Prove tan (45° -f x} -f tan (45° — .r) = 2 sec 2x.
2 tan -r
< (30.) Prove sin2.r = —
-f tan2*'
i — tan'.r
(31.) Prove cos 2.1- =
i -h tan-. r
I 4- sin 2x /tan .r-f- i\'
(32^ Prove — - ) •
i —sin 2.x Vtan.r— i/
cin r
(33.) Prove tan|.r =
-|- cos .r
sin JT
, (34.) Prove coti.r=I-_c—.
cos x — cosy
(35.) Express as a product .*
cos x -f- cos_y
COSJT — cos r _— 2 sin •!(«*•+_;') sin^(jr
COS a + COS/ 2 COS \ (x •+• 1') COS ^ (x —
= -tan^(j:+;') tani(jr-.v).
tan ,r -f tan y
/ (36.) Express as a product : .
cot x + cot/
cos (.1- 4- y)
(37.) Prove i — tan x tan y =: - — .
' -
Till: TXVKRSE TRIGONOMETRIC FUNCTIONS
40. Dcf. — The expressions sin— ^, cos-^tan-'tf, etc., de-
note respectively an angle whose sine is a, an angle whose
cosine is a, an angle whose tangent is <7, etc. They are
called the inverse sine of a, the inverse cosine of <?, the
inverse tangent of a, etc., and are the inverse trigono-
metric functions.
Sin-'tf is an angle whose sine is equal to a, and hence de-
notes, not a single definite angle, but each and every angle
whose sine is a.
* Since quantities cannot be added or subtracted by the ordinary operations
with logarithms, an expression must be reduced to a form in which no addition
or subtraction is required, to be convenient for logarithmic computation.
40 PLANE TRIGONOMETRY
Thus, if sin*=£, jr=3o0, 150°, (30° + 360°), etc.,
and sin- 4=30°, 150°, (3O° + 36o°), etc.
Remark. — The sine or cosine of an angle cannot be less than — i
or greater than -|- i; hence sin"1^ and cos~'# have no meaning unless
a is between — i and -f i. In a similar manner we see that sec-'tf
and csc~la have no meaning if a is between — i and -j- *•
EXERCISES
41. (i.) Find the following angles in degrees:
sin~I|-v/2, tan~T(— *)» sin~T(— £).
cos-1!, cos-1!,
(2.) If x — cot-^, find tan x.
(3.) If x = sin-xf , find cos x and tan x.
(4.) Find sin (tan-'i \/3)-
(5.) Find sin(cos-T£).
(6.) Find cot (tan-1 yS).
(7.) Given sin-'fl = 2 cos-1*?, and both angles acute ; find a.
(8.) Given sin"1^ = cos-I^ ; find the values of sin"1^ less than 360°.
(9.) Given tan~xi =-}tan-fo, and both angles less than 360°; find
the angles.
(lo.) Given sin"1^ — cos-V? and sin-^ + cos-1^ = 450°; find sin~V?.
(n.) Prove sin (cos~Vz) = ± \/i—a*.
Hint. — Let jc=cos-1^ ; then a = cos jc,
sin x= ± y I — COS'JJT = ± y I — </2.
(12.) Prove tan^an-^r+tan-1^)^:
r.-b
(13.) Prove tan(tan-rrt — tan-l^)=— — — T-
(14.) Prove cos(2 cos-I^) = 2^2 — i.
(15.) Prove sin (2 cos—1rt) = ± 2a y/i — a".
2(7
(16.) Prove tan (2 tan-1 a)= --- —•
*
(17.) Prove cos(2tan-I<7)=
(i 8.) Prove sin(sin~Irt + cos-1^) = ab±.y/(\ — a*}(\ —
CHAPTER IV
THE OBLIQUE TRIANGLE
DERIVATION OF FORMULAS
42. The formulas derived in this and the succeeding
articles reduce the solution of the oblique triangle to its
simplest form.
c C C
FIG. 3
Draw the perpendicular CD. Let CD=/i,
Then - — sin ^4;
o
and
(In Fig. 2 -=sin(i8o°-X)=--sin^)
h
- — sin B.
(In Fig. 3 -=
(32)
By division we obtain,
a _ sin A
b ~ silTI* "
Remark. — This formula expresses the fact that the ratio of two sides of an
oblique triangle is equal to the ratio of the sines of the angles opposite, and
does not in any respect depend upon which side has been taken as the base.
Hence if the letters are advanced one step, as shown in the figure, we obtain,
as another form of the same formula,
42 PLANE TRIGONOMETRY
b _ sin/?
Repeating the process, \ve obtain
c sin C
;,=^A' &
The same procedure may be applied to all the formulas for the solution of
oblique triangles. Henceforth only one expression of each formula will lie given.
Formula (32) is used for the solution of triangles in which
a side and two angles, or two sides and an angle, opposite one
of them are given.
43. We obtain from formula (32) by division and compo-
sition, a — b sin^— sin/?
_
a + b ~ sin A + sin B '
By formula (29), denoting the angles by A and B, in-
stead of u and z/,
sin A -sin.#tan-J(^ — B)
Therefore, ~~ - ~-| -^ — ' ^ (33)
This formula is used for the solution of triangles in which
two sides and the included angle are given.
44. Whether A is acute or obtuse, we have ^
C C
FIG.
(If A is acute (Fig. i),AD = bco*A, DH — AB - AD — c - b cos,-/, CD —
bs\nA. UA is obtuse (Fig. 2), AD — ^cos (180°-^) = - ^cos^, DB—AB
, CD— b sin(i8o°— A)- b sin^.)
THE OBLIQUE TRIANGLE 43
— r — 2 be cosA+b* (cos* A +sinM).
Therefore, a2— 62+c2- 2bc co§ A. (34)
This formula is used in deriving formula (37).
// is also used in the solution without logarithms of tri-
angles of which two sides and the included angle or three
sides are given.
45. From formula (34), cos^ = — — 7 —
From formula (22), § 37,
2 sin24/2 = i— cos^4 = I c ~a .
2 be
Hence 2 sin2 ±A =
2 be
(b-_c
~~2bc
2bc
Let J==?±|.±f, then (a-6 + c) = 2(s-d), and (a + b-c)
Substituting, 2 s\n*±A = -
Hence sin^^j = .A^)(S- *\* (35)
V be
From formula (23), § 37,
2 COS' =
bc
* In extracting the root the plus sign is chosen because it is known that
sin A ,/ is positive.
44 PLANE TRIGONOMETRY
Hence cos^A
Dividing (35) by (36), we obtain
tan I A =. \/(s~~ b^~
* \/ \
s {s — a)
(36)
(37)
Let
tan i ^ = - ~
s—a
(38)
Formulas (37) and (38) are used to find the angles of a tri-
angle when tJie three sides are given.
FORMULAS FOR THE AREA OF A TRIANGLE
40. Denote the area by S.
C
D
FIG. I
(In Fig. i, CD=as,\\\B\ in Fig. 2, CD - « sin (180°-^) = asm£.)
In Figs. I and 2, S=$c.CD.
Hence /S'^-JacsinB. (39)
From formula (17),
sinZ? = 2 sin $ cosZ?.
THE OBLIQUE TRIANGLE 45
Substituting for sinj/? and cos^fi the values found in
formulas (35) and (36), we obtain
sin£ = —\ /s(s-a)(s-b}(s-e).
ac*
Therefore, S=*/s(8 — a)(s—b)(s— c). (40)
This formula may also be written,
S=sK. (41)
Formula (39) is used to find the area of a triangle when
two sides and the included angle are knoivn; formula (40) or
formula (41), when the three sides are known.
THE AMBIGUOUS CASE
47» The given parts are two sides, and the angle opposite
one of them.
Let these parts be denoted by a, b, A.
C
If a is less than b and greater than the perpendicular CD
(Fig. i), there are the two triangles ACB and ACS', which
contain the given parts, or, in other words, there are two
solutions.
If a is greater than b (Fig. 2), there is one solution.
If a is equal to the perpendicular CD, there is one solu-
tion, the right triangle A CD.
46 PLANE TRIGONOMETRY
If the given value of a is less than CD, evidently there
can be no triangle containing the given parts.
Since CD=t>sinA, there is no solution when «< bs\\\A ; there is one
solution, the right triangle A CD when a=bs\\\A; there are two solutions
when a <! b and > fisinA.
48. CASE I. — Given a side and two angles.
EXAMPLE
Given a = 36.738, A = 36° 55' 54", B = 72° 5' 56",
C=i8o° — (A + £)= 180°— 109° i' 50" = 70° 58' 10".
To find c.
c sin C
a sin A
= 1.56512
log sin £=9.97559 — 10
colog sin A =0.22 1 23
log r= i. 76194
^•=57.80
To find b.
a sin /4
log rt = i. 56512
log sin .#=9.97845 —
colog sin A =o. 221 23
log 0=1.76480
^ = 58.184
Determine b from r, C, and B by the formula
This check is long, but is quite certain to reveal an error. A check which is
shorter, but less sure, is
b _ sin B
c sin C
Solve the following triangles :
(i.) Given a — 567.25, A — \\° 15', ^ — 47° 12'.
O (2.) Given ^ = 783.29, A = Si° 52', ^ — 42° 27'.
. (3.) Given c= 1125.2, A = 79° 15', ^=55° n'.
(4.) Given ^=15.346, B=it° 51', Cr=580 10'.
(5.) Given a = 5301. 5, ^4 =69° 44', C=4\° 18'.
(6.) Given £=1002.1, ^=48° 59', € = 76° 3'.
t/f>. CASE II. — Given two sides of a triangle and tlie angle
opposite one of them.
THE OBLIQUE TRIANGLE
47
EXAMPLE
Given a = 23.203, b — 35.121, A = 36° 8' 10".
C
To find B and B '.
sin A a
log <$=i. 5.1556
log sin ,4=9.77064—10
colog rz = 8. 6344 5 — 10
log sin #=9.95065 — 10
£=63° 12'
To find C and' C'.
C = i8o°-(/f +£)=So° 39' 50'
') = 2703- 50"
To find c and c .
c sin C
log 11=1.36555
log sin (7=9.99421 — 10
colog sin A =0.22936
log f=i. 58912
^=38.825
log a = i. 36555
log sin C' =9.65800 — 10
colog sin A =0.22936
log c' = i. 25291
^' = 17.902
Check.
Determine b from c, C, and B by the formula
b — a tan4(/?-
tan
This check is long, but is quite certain to reveal an error. A check which is
shorter, but less sure, is
c sin 6"
(i.) How many solutions are there in each of the following?
(i.) A = $o°, a = i$, 6 = 20-,
(2.) A = 30°, a = TO, £ = 20;
(3.) ^ = 30°, a =8, 6 = 20;
(4.) £ = 37° 23', a = 9.1, 6 = 7.$.
v\
48
PLANE TRIGONOMETRY
Solve the following triangles, finding all possible solutions :
7X2.) Given A = 147° 12', a = 0.63735, £ = 0.34312.
(3.) Given A= 24° 31', a = 1.7424, £ = 0.96245.
(4.) Given A= 21° 21', a = 45.693, £ = 56.723.
(5.)Giveny? = 61° 16', # = 9.5124, £ = 12.752.
(6.) Given C= 22° 32', a =0.78727, £ = 0.47311.
£0. CASE III. — Given two sides and the included angle.
Given ^ = 41.003,
parts and the area.
EXAMPLE
' = 48.718, C — 68° 33' 58"; find the remaining
To find A and B.
tan|(Z? — A) _b — a
~~
b-a = 7.715
b + rt = 89.721
log (£-«) = 0.88734
colog (b + a) = 8.04710—10
log tan£(£ + ^) = o. 16639
log tan %(B — A ) = 9. 10083 — 10
-^^ 7° ii' 20"
— ^>2Q 54' 21"
=48° 31' 41"
r _ sin C
a sin A
loga= 1.61281
log sin C= 9. 96888 -10
colog sin A = o. 12535
log<r= 1.70704
c- 50.938
To find the area.
S = \ab sin C
= 9-69897 -10
= 1.61281
= 1.68769
log sin C= 9. 96888 — 10
log S= 2.96835
S= 929.72
Check.
sin C c
siii~5 ~ 7
log sin B = 9.94951 — 10
log c = 1.70704
coiog b — 8.31231 — 10
log sin C — 9.96886 — 10
THE OBLIQUE TRIANGLE 49
Solve the following triangles, and also find their areas :
/(7J Given A= 41° 15', £=0.14726, f =0.10971. Q)
— \^2.) Given C= 58° 47', £=11.726, #=16.147.
(3.) Given £= 49° 50', # = 103 74, ^=99.975.
1 (4.) Given A= 33° 31', £=0.32041, ^=0.9203.
(5.) Given C=i28° 7', £=17.738, #=60.571.
51. CASE IV. — Given the three sides.
EXAMPLE
Given # = 32.456, £ = 41.724, ^ = 53.987 ; find the angles and area.
^ = 64.084
(.»• — </) = 3i.628
(s — ^ = 22.360
log A'= i. 02349
-r)- 10.097
*- / (J - </JU -
A \
- u)(s - <>)(* - 0
log (.f — rt) = 1.50007
lug (-f — ^) =1.34947
log (s — r)=i. 00419
colog j = 8. 19325 — 10
2)2.0461*8
log A'=i.o2349
To find A.
K
s — a
log A'= i. 02 349
log (s-a) = 1.50007
sub.
log tan|^=g. 52342 -10
A^ = i8° 27' 23"
^=36° 54 4""
log (/-/;) = 1.34947
sub.
log tan ^ .#=9.67402 - i o
•§^=25° 16' 16"
^=50° 32' 32"
C*
log A'= i. 02349
log (j-<-)= 1.00419
sub.
log tan£C=o.oi93O
|(7=46° 16' 22"
C=92° 32' 44"
Find the angles and areas of the following triangles:
(i.) Given # = 38.516, £=44.873, ^=14.517.
„ (2.) Given # = 2.1158, £=3.5854, ^=3.5679.
* C could be found from (A +J9)=(i8o°- C), but for the sake of the check it
is worked out independently.
4
Check.
50 PLANE TRIGONOMETRY
(3.) Given ^=82.818. £=99.871, ^=36.363.
(4.) Given ^ = 36.789, £=i 1.698, ^=33.328.
(5.) Given a — i 13.03, £=131.17, c — \ 14.29.
(6.) Given a= .9763, £=1.2489, <• = 1.6543.
EXERCISES
52. (i.) A tree, A, is observed from two points, B and C, 1863 ft.
apart on a straight road. The angle BCA is 36° 43', and the angle
CBA is 57° 21'. Find the distance of the tree from the nearer
point.
(2.) Two houses, A and B, are 3876 yards apart. How far is a third
house, C, from A, if the angles ABC and J5AC are 49° 17' and 58° 18'
respectively ?
(3.) A triangular lot has one side 285.4 ft. long. The angles adja-
cent to this side are 41° 22' and 31° 19'. Find the length of a fence
around it, and its area.
(4.) The two diagonals of a parallelogram are 8 and 10, and the
angle between them is 53° 8' ; find the sides of the parallelogram.
(5.) Two mountains, A and B, are 9 and 13 miles from a town, C;
the angle ACB is 71° 36' 37". Find the distance between the moun-
tains.
(6.) Two buoys are 2789 ft. apart, and a boat is 4325 ft. from the
nearer buoy. The angle between the lines from the buoys to the
boat is 1 6° 13'. How far is the boat from the farther buoy? Are
there two solutions ?
(7.) Given a = 64.256, r= 19.278, C=i6° 19' 11"; find the differ-
ence in the areas of the two triangles which have these parts.
(8.) A prop 13 ft. long is placed 6 ft. from the base of an embank-
ment, and reaches 8 ft. up its face; find the slope of the embank-
ment.
(9.) The bounding lines of a township form a triangle of which the
sides are 8.943 miles, 7.2415 miles, and 10.817 miles; find the area
of the township.
(10.) Prove that the diameter of a circle circumscribed about a
triangle is equal to any side of the triangle divided by the sine of the
angle opposite.
' - i -J / - "' Srf!
'° *~^t b
THE OBLIQUE TRIANGLE
Hint. — By Geometry, angle A OB =2C.
Draw OD perpendicular to AB.
Angle DOB=%AOB=C.
DB=r sin DOB=r sin C.
Hence c=2rs'mC,
c
or 2r=~— ;.
sine
(ii.) The distances AB, BC, and AC, between three cities, A, Bt
and Care i$ miles, 14 miles, and 17 miles respectively. Straight rail-
roads run from A to B and C. What angle do they make ?
(12.) A balloon is directly over a straight road, and between two
points on the road from which it is observed. The points are 15847
ft. apart, and the angles of elevation are found to be 49° 12' and
53° 29' respectively. Find the distance of the balloon from each of
the points.
(13.) To find the distance from a point A to a point B on the op-
posite side of a river, a line, AC, and the angles CAB and ACB were
measured and found to be 315.32 ft., 58° 43', and 57° 13' respectively.
Find the distance AB.
(14.) A building 50 ft. high is situated on the slope of a hill. From
a point 200 ft. away the building subtends an angle of 12° 13'. Find
the distance from this point to the top of the building.
(15.) Prove that the area of a quadrilateral is equal to one-half
the product of the diagonals by the sine of the angle between
them.
(16.) From points A and B, at the bow and stern of a ship respec-
tively, the foremast, C, of another ship is observed. The points A
and B are 300 ft. apart; the angles ABC and BAC are found to be
tftr)
'
52 PLANE TRIGONOMETRY
65° 31' and 1 10° 46' respectively. What is the distance between the
points A and C of the two ships ?
(17.) Two steamers leave the same port at the same time ; one sails,
directly northwest, 12 miles an hour; the other 17 miles an hour, in
a direction 67° south of west. How far apart will they be at the end
of three hours ?
(i 8.) Two stakes, A and />, are on opposite sides of a stream; a
third stake, C, is set 92 ft. from A ; the angles ACB and CAB are
found to be 50° 3' 5" and 61° 18' 20" respectively. How long is a
rope connecting A and Z??
(19.) To find the distance between two inaccessible mountain-tops,
A and B, of practically the same height, two points, C and D, are
taken one mile apart. The angle CDA is found to be 88° 34', the
angle DC A is 63° 8', the angle CDB is 64° 27', the angle DCB is 87° 9'.
What is the distance?
(20.) Two islands, B and C, are distant 5 and V3 miles respectively
from a light-house, A, and the angle BAC is 33°"/>ff; find the dis-
tance between the islands.
(21.) Two points, A and B, are visible from a third point C, but
not from each other; the distances AC, BC, and the angle ACB were
measured, and found to be 1321 ft., 1287 ft., and 61° 22' respectively.
Find the distance AB. \ ^ 1 »^
(22.) Of three mountains, A, B, and C, B is directly north of C 5
miles, A is 8 miles from C and 1 1 from B. How far is A south of B ?
(23.) From a position 215.75 ft. from one end of a building and
198.25 ft. from the other end, the building subtends an angle of
53° 37' 28"; find its length.
(24.) If the sides of a triangle are 372.15, 427.82, and 404.17 ; find
the cosine of the smallest angle.
(25.) From a point 3 miles from one end of an island and 7 miles
from the other end, the island subtends an angle of 33° 55' 15'''; find
the length of the island.
(26.) A point is 13581 in. from one end of a wall 12342 in. long, and
10025 in- from tne other end. What angle does the wall subtend at
this point?
(27.) A straight road ascends a hill a distance of 213.2 ft., and is in-
THE OBLIQUE TRIANGLE 53
clined 12° 2' to the horizontal; a tree at the bottom of the hill
subtends at the top an angle of 10° 5' 16". Find the height of the
tree.
(28.) Two straight roads cross at an angle of 37° 50' at the point A ;
^ miles distant on one road is the town B, and 5 miles distant on the
other is the town C. How far are B and C apart ?
(29.) Two stations, A and B, on opposite sides of a mountain, are
both visible from a third station, C\ AC =11.5 miles, BC = 9.4 miles,
and the angle ACB—^cp $1'. Find the distance from A to B.
(30.) To obtain the distance of a battery, A, from a point, B, of the
enemy's lines, a point, C, 372.7 yards distant from A is taken ; the an-
gles ACB and CAB are measured and found to be j$ 53' and 74° 35'
respectively. What is the distance ABt
(31.) A town, B, is 14 miles due west of another town, A. A third
town, C, is 19 miles from A and 17 miles from B. How far is C west
of A?
(32.) Two towns, A and B, are on opposite sides of a lake. A is
18 miles from a third town, C, and B is 13 miles from C; the angle
ACB is 13° 17'. Find the distance between the towns A and B.
(33.) At a point in a level plane the angle of elevation of the top
of a hill is 39° 51', and at a point in the same direct line from the hill,
but 217.2 feet farther away, the angle of elevation is 2&J 53'. Find
the height of the hill above the plane.
(34.) It is required to find the distance between two inaccessi-
ble points, A and B. Two stations, C and D, 2547 ft. apart, are
chosen and the angles are measured ; they are ACB=2j° 21', BCD
=33° 14', £DA = i8° 17', and ADC=$i° 23'. Find the distance from.
A to B.
(35-) Two trains leave the same station at the same time on straight
tracks inclined to each other 21° 12'. If their average speeds are 40
and -^Smiles an hour, how far apart will they be at the end of the first
fifteen minutes?
(36.) A ship, A, is seen from a light-house, B\ to determine its dis-
tance a point, C, 300 ft. from the light-house is taken and the angles
BCA and CBA measured. If £CA = io8° 34' and CBA=6$Q 27', what-
is the distance of the ship from the light-house?
54
PLANE TRIGONOMETRY
(37.) Prove that the radius of the inscribed circle of a triangle is
equal to a sin-|Z>' sin^Csec^.
Hint. — Draw OB, OC, and the perpendicular OD.
OB and OC bisect the angles B and C respectively, and OD—r.
sm
Hence
sin \ /J sin -^ C sin ^ A' sin \ C
sin ^ v9 sin •
-
cos-i/4
= a sin
sm i c sec
CHAPTER V
CIRCULAR MEASURE— GRAPHICAL REPRESENTATION
CIRCULAR MEASURE
53. The length of the semicircumference of a circle is
irR (77 = 3.14159 + ); the angle the semicircumference sub-
tends at the centre of the circle is 180°. Hence an arc
whose length is equal to the radius will subtend the angle
1 80°
— ; this angle is the unit angle of circular measure,
and is called a radian.
7T R
If the radius of the circle is unity, an arc of unit length
subtends a radian ; hence in the unit circle the length of an
arc represents the circular measure of the angle it subtends.
Thus, if the length of an arc is , it subtends the angle - radians.
Since one radian =
1 80
we have
00° — radians,
2
= 7r radians,
56 PLANE TRIGONOMETRY
270°= - radians,
360° — 2?r radians/etc.
The value of a radian in degrees and of a degree in radians are :
i radian = 57.29578°,
= 57° 17' 45".
1°=:. 0174533 radian.
In the use of the circular measure it is customary to omit the word radian ;
thus we write - , TT, etc., denoting - radians, TT radians, etc. On the other
hand, the symbols ° are always printed if an angle is measured in degrees,
minutes, and seconds ; hence there is no confusion between tlie systems.
EXERCISES
(i.) Express in circular measure 30°, 45°, 60°, 120°, 135°, 720°, 990°.
(Take ^=3.1416.)
(2.) Express in degrees, minutes, and seconds the angles -^, — , - ,-.
8 10 2 4
(3.) What is the circular measure of the angle subtended by an arc
of length 2.7 in., if the radius of the circle is 2 in.? if the radius is
5 in. ?
<T4. The following important relations exist between the
circular measure x of an angle and the sine and tangent of
the angle.
(i .) If x is less than — , sin x < x < tan x.
O S
Draw a circle of unit radius.
By Geometry, SP<arcAP<AT.
Hence sin x <x < tan^r.
CIRCULAR MEASURE 57
sin x tan x
(2.) As x approaches the limit o, — — and — — approach
& Jv
the limit i.
Dividing sin x < x < tan x by sin x, we obtain
,
I <- - <
sinjr cos^r
sin 4: cos JIT
Inverting, i>~ — > •
*-v 1
As x approaches the limit o, COS.T approaches the length
of the radius, that is, i, as a limit.
Therefore, - - approaches the limit I.
sin x
Dividing i > — - > cos^r by cos;r, we obtain
Jv
i tan x
COS X X
As x approaches the limit o, cos;r approaches the limit I ;
hence approaches the limit I.
cos-r
Therefore, - approaches the limit I.
PERIODICITY OF THE TRIGONOMETRIC FUNCTIONS
o&» The sine of an angle x is the same as the sine of
(^+360°), (x + 720°), etc.— that is, of (>+2;/7r), where n is
any integer.
The sine is therefore said to be a periodic* function, hav-
ing the period 360°, or 2?r.
The same is true of the cosine, secant, and cosecant.
* If a function, denoted by /(.*)> of a variable jc, is such that f(x + k}=f(x}
for every value of x, k being a constant, the function f(x) is periodic; if k is
the least constant which possesses this property, k is the period of /(.r).
58 PLANE TRIGONOMETRY
The tangent of an angle x is the same as the tangent of
(x+ 180°), (^4-360°), etc.— that is, of (x + ntr\ where n is any
integer.
The tangent is therefore a periodic function, having the
period 1 80°, or TT.
The same is true of the cotangent.
GRAPHICAL REPRESENTATION
36. On the line OX lay off the distance OA(=x) to rep-
resent the circular measure of the angle x. At the point A
erect a perpendicular equal to sin x. If perpendiculars are
thus erected for each value of x> the curve passing through
their extremities is called the sine curve.
If sin* is negative, the perpendicular is drawn downward.
In a similar manner the cosine, tangent, cotangent, secant,
and cosecant curves can be constructed.
Sine Curve
-1
Cosine Curve
GRAPHICAL REPRESENTA TION
59
TangentCurve
.0
Cotangent Curve
6o
PLANE TRIGONOMETRY
0
3/27T
SECANT CURVE
If the distances on OX are measured from O' instead of
O, we obtain from the secant curve the cosecant curve.
In the construction of the inverse curves the number is
represented by the distance to the right or left from O\
the circular measure of the angle by the length of the per-
pendicular erected.
All of the preceding curves, except the tangent and co-
tangent curves, have a period of 2?r along the line OX; that
is, the curve extended in either direction is of the same
form in each case between 2?r and 477% 4?r and 6?r, — 27r and
o, etc., as between o and 2?r, while the corresponding inverse
curves repeat along the vertical line in the same period.
The period of the tangent and cotangent curves is TT.
GRAPHICAL REPRESENTATION
61
-i o +1
INVERSE SINE CURVE
-1
INVERSE COSINE CURVE
J L
-3 -2 -1 0 +1 + 2 +3
INVERSE TANGENT CURVE
62
PLANE TRIGONOMETRY
87T
-3
+ 3
INVERSE SHCANT
CHAPTER VI
COMPUTATION OF LOGARITHMS AND OF THE TRIG-
ONOMETRIC FUNCTIONS -DE MOIVRE'S THEOREM
—HYPERBOLIC FUNCTIONS
*7f. A convenient method of calculating logarithms and
the trigonometric functions is to use infinite series. In
work? on the Differential Calculus it is shown that
= x-~2+i-j-+ • • • d)
nf*$ OT^ ZM*^
= x -_+__-+...* (2)
/y»2 /y^ /y»6
l-4jj + il-^!+ . (3)
Another development which we shall use later is
or or^ or^ Gt*^
e*=:1 + l! + 2! + 3! + 4!+- (4)
where c— 2. 7 18281 8 ... is the base of the Naperian system
of logarithms.
58. The series (i) converges only for values of x which satisfy the
inequality — I<JT^I. The series (2), (3), and (4) converge for all
finite values of .r.
It is to be noted that the logarithm in (i) is the Naperian, and the
angle x in (2) and (3) is expressed in circular measure.
* 3! denotes 1x2x3; 4] denotes 1x2x3x4, etc.
64 PLANE TRIGONOMETRY
COMPUTATION OF LOGARITHMS
59. We first recall from Algebra the definition and some
of the principal theorems of logarithms.
The logarithm to the base a of the number m is the number .r
\vhich satisfies the equation,
This is written x =
The logarithm of the product of two numbers is equal to the sum
of the logarithms of the numbers.
T h us 1 og^ m 11 = \oga m +'1 oga n.
The logarithm of the quotient of two numbers is equal to the log-
arithm of the dividend minus the logarithm of the divisor.
in
Thus l°£<z — — 1°R«;;/ — l°gtf;z-
n
The logarithm of the power of a number is equal to the logarithm
of the number multiplied by the exponent.
Thus log^ m*=p \oga m.
To obtain the logarithm of a number to any base a from its Na-
perian logarithm, we have
log in
log* m = = Ma log, m,
tog, a
where Mrt= — — ; Ma is called the modulus of the system.
6*0. We proceed now to the computation of logarithms.
The series (i) enables us to compute directly the Naperian
logarithms of positive numbers not greater than 2.
Example.— To compute log*- to five places of decimals.
Substitute - for x in (i):
2/22 2 3 2' 4 2
If the result is to be correct to five places of decimals, we must take enough
terms so that the remainder shall not affect the fifth decimal place. Now we
COMPUTATION OF LOGARITHMS
know by Algebra that in a series of which the terms are each less in numerical
value than the preceding, and are also alternately positive and negative, the re-
mainder is less in numerical value than its first term. Hence we need to take
enough terms to know that the first term neglected would not affect the fifth
place.
Positive terms
2
=0.5000000
I
1
= .0416667
3
2
j
I
= .0062500
5
2
I
1
•=. .OOIIl6l
7
21
i
1
—
= .0002170
9
2
i
I
= .OOOO444
1 1
-
i
I
= .0000094
13
2
.5493036
Negative terms
i
I
2
2* ~~
3.1250000
I
4
I
' 21™
.0156250
I
6
I
.0026042
i
8
I
.0004883
i
T
• — =
.0000977
10
2
I
I
12
*2" =
.OOOO2O3
I
I
14
'2"=
.0000044
.1438399
Subtracting the sum of the negative from the sum of the positive terms, we
obtain
lo&^= -4054637-
Denote the sum of the remaining terms of the series by. R. Then, by Alge-
bra,
< .0000021.
The error caused by retaining no more decimal places in the computation is
less than .0000006. Hence the total error is less than .0000027. Therefore
the result is correct to five decimal places.
61. As remarked, the series (i) does not enable us to
calculate directly the logarithms of numbers greater than 2,
but it can be readily transformed into a series which gives
us the logarithm of any positive number.
Replacing x by — x in (i), we obtain
5
66 PLANE TRIGONOMETRY
x* x3 x*
log, (i -*)=-* --_-_-
This series converges for — i <•*<*•
Subtracting this from (i), we obtain
*v\
.,j-H =2(^f+I+7V"-)' (s)
4 f I
/ which converges for — i < x < i .
Putting y=[- -), we see that j passes from o to oo as x
\i x/
passes from — i to -f-i ; hence, if we make this substitution in
(c), we get a
which converges for all positive values of j, and therefore enables
us to compute the Naperian logarithm of any number.
From (5) we can get another series which is useful : put
i i4-x y+i
x— - ; then, as— -=-=- - , equation (5) gives us
which converges for all positive values of y. Hence,
This series gives us log,(jy-f-i), when log,^ is known. It con-
verges more rapidly than (6), when y is greater than 2, and hence
should be used under these circumstances.
62. To construct a table we need to compute directly
only the logarithms of prime numbers, since the others can
be obtained by the relation
log ,ry = \og ;r + log y.
i-
i
COMPUTATION OF LOGARITHMS
67
Thus, to obtain the logarithms of the integers up to 10,
we need to compute by series only the logarithms of the
numbers 2, 3, 5, and 7.
(For 4=22, 6=2 . 3, 8 = 2*, 9=32, 10=2 . 5, and log 1=0.)
In this case we are computing the logarithms of successive integers, and
should therefore use (7).
6Yf. Example. — Compute the Naperian logarithms of 2, 3, 4, and 5.
.j
3 3 33 5
^.<..l+i.l9+. \
3s 7 37 9 39 /
-=-3333333
.=. 0008230
l-. ^=. 0000653
1.1^.0000056
.3465729
2
Denote the sum of the remaining
terms of this series by A'.
Then, by Algebra,
or A" < .000000573.
The error caused by not retaining
more places of decimals in the pre-
ceding column is less than .0000005.
Hence, the total error is less than
.00000165.
log, 2 = .693 1458
Remark. — We should get the same series if we were to use (6).
- = .2000000
\ ' ^ = .0026667
6
- • —=.0000018
7 5'
.2027325
2
.4054650
Add log, 2= .6931458
log, 3 = 1.0986108
AX- •
or A* < .00000006.
Noting the errors in the pre-
ceding column and in log, 2, we
see that the total error is less than
;OOOOO2I7.
68 PLANE TRIGONOMETRY
Remark. — If we were to use (6) to compute log, 3, we should have
This series converges much more slowly than the above, since its
terms are multiples of powers of \, while the terms of the above are
the same multiples of powers of \. Thus, we should be obliged to
use eight instead of four terms to have the result correct to five
places.
log, 4 = 2 log, 2 = 1.3862916.
or log, 5 = 1.60944.
64. Proceeding in like manner, we may calculate any number of
logarithms.
The following table gives the Naperian logarithms of the first ten
integers :
log* J — -ooooo
log, 2= .69315
log, 3 = 1.09861
log* 4 =1.38629
= 1.60944
log, 6= 1.79176
log, 7 = I-94591
log, 8 = 2.07944
log, 9 = 2. 19722
log, 10 = 2.30259
The common logarithm of any number may be found by multiply-
ing its Naperian logarithm by M10=. 43429448. § 59
Thus loglo 5 = log, 5 X 43429448 = .69897.
65. Remark. — If a table of logarithms were to be computed, the
theory of interpolation and other special devices would be employed.
COMPUTATION OF TRIGONOMETRIC FUNCTIONS
c. sin^r cos^r
f>6. Since tan;tr= , cot;r= — , etc., the computa-
COS.T sin x
tion of all the trigonometric functions depends upon that of
the sine and cosine ; thus the developments (2) and (3) suf-
fice for all the trigonometric functions. Further, since the
^
COMPUTATION OF SINES AND COSINES 69
sine or cosine of any angle is a sine or cosine of an angle
p-, it is never necessary to take x greater than - in the
^4 4
series (2) and (3). § 16
Since - =0.785398 .,.<—, these series converge rapidly; in fact,
4 10
— = .000003 does not affect the fifth decimal place, and — the
9! 11!
seventh.
67. Remark. — In the systematic computation of tables we should
not calculate the functions of each angle from the series independent-
ly. We should rather make use of the formulas (25) and (27) of § 38,
thus obtaining
sinw.r = 2 cos.r sin (n — i)^r — sin (n — 2) x,
cos nx = 2 cos x cos (n — i ) x — cos (;/ — 2) x.
If our tables are to be at intervals of i', we should calculate the
sine and cosine of i' by the series. The above expressions then en-
able us to find successively the sine and cosine of 2', 3', 4', etc., till we
have the sine and cosine of all angles up to 30° at intervals of i'.
To obtain the sine and cosine of angles from 30° to 45° we should
make use of these results by means of the formulas
sin (30°+ y) =cosy — sin (30°— y),
cos (3o°-f-_y) = cos (30°—^) — sin y.
68. To employ series (2) and (3) in computing the sine
and cosine we must first convert the angle into circular
measure.
To do this we recall that
i° = . 017453293, i ' = .0002908882, i " = . 000004848 1 37.
Example. — To compute the sine and cosine of 12° 15' 39".
1 2°= .209439516
15' =.004363323
39" = .000189076
12° 15' 39" = .213991915 in circular measure.
PLANE TRIGONOMETRY
x-. 2139919
(
= . 0000037
.2139956
X*
subtract — = .0016332
Correct to five decimal places.
cos*=i H —
2! 4!
1 = 1.0000000
— = .0000874
4 !
1.0000874
subtract — -= .0228963
cos.r= .9771911
Correct to five decimal places.
DE MOIVRE'S THEOREM
00. In Algebra we learn that the complex number
(8)
may be represented graphically thus :
Y
Take two lines, OX and OY, at right angles to each other.
To the number a will correspond the point A, whose dis-
tances from the two lines of reference are ft and a re-
spectively.
This geometrical representation shows at once that we
can also write a in the form
a=r (cos $+4 sin 3). (9)
7O. From Algebra we recall the definition of the sum of the
complex numbers a — a + //3 and b=y + il-, namely
Subtraction is defined as the inverse of addition, so that
a — b^a — y-M'()3 — c).
DE MOIVRE'S THEOREM 71
Multiplication is most conveniently defined when a and b are
written in form (9). If
a — r (cos-$-M sin £) and d~s (cos^-f / sin^),
their product is defined by the equation
ab — rs [cos (3 + 0)-|-/ sin (£-r-0)]- • (IO)
Division is defined as the inverse of multiplication, so that
Finally, we recall that in an equation between complex numbers,
•+173=7+13,
we have «=y, /3 = S. (n)
*7 71. Consider the different powers of the complex number
# = cos $+*' sin $.
By (10) we have
*a = (cos $ + / sin S) (cos $+/ sin $),
= cos 2-S + / sin 2-$.
jc3=A:2 . ^ = (cos 2^+1 sin 2$) (cos $+*' sin 3),
=cos3^-|-/ sin 3^.
And, in general, for any integer n,
*«=:(cos 3+t sin ^)"=cos n$+i sin n§.
From this equation we have De Moivre's Theorem, which
is expressed by the formula
(12)
72. An interesting application of De Moivre's Theorem
is the expansion of sin nx and cos nx in terms of sin x and
COS.T. Expanding the left-hand side of (12) by the bino-
mial theorem, and substituting x for 3-, we have
cosnx-\-i sin nx=cosnx+n cos*"1 x (i sin x) -f - j — cosw~2^
(• sin*)' + — f*=!l£=!> co8—«*(isin*)' + . . .
72 PLANE TRIGONOMETRY
or
cosnx + i sinnx=(cosnx — j — - cosw~2^ sin2*-}- . . .)
[// (n— i) (n — 2)
n cos""1 x smx — cosw 3 x sin x+ ....
Equating real and imaginary parts, as in (11), we have
cosnx=cosnx cosn~2x sin'2#-f- ; . . (13)
sin//*— ncosn lxs'mx— - cos*"3* sin3 x+. . . (14)
Example. — n = 5.
cos 5-r = cos6 x — 10 cos3.*- sin2^r-f-5 cos^r sin*,r.
sin 5_r= 5 cos*,r sin or— 10 cos2 x sin3 x + sin* x.
THE ROOTS OF UNITY
73. We find another application of De Moivre's Theorem
in obtaining the roots of unity. The #th roots of unity are
by definition the roots of the equation
xn—\.
Every equation has n roots and no more ; hence, if we
can find n distinct numbers which satisfy this equation we
shall have all the #th. roots of unity.
Consider the // numbers
2?rr 2irr
xr = cos \-t sin ,
n n
r=o, i, 2, ... n— i.
Geometrically these numbers are represented by the n
vertices of a regular polygon. They are, therefore, all dif-
ferent. We shall see now that they are precisely the ;/th
roots of unity.
In fact, we have by (12),
* — ( cos — • — \-i sin ) ,
\ ;/ n )
THE ROOTS OF UNITY
(2Ttr\ . . . / 2:rr\
n . - )-f-j sin [«.— -),
« / V « /
73
sin 27T/-,
= 1+*'. 0 = 1.
Therefore xr is one of the roots of unity.
Thus the cube roots of unity are represented by the points A, P,
and Q of the following figure. In the figure OA = i, angle AOP =
— = i2o°, angle AOQ = — = 240°; that is, the circumference is di-
vided into three equal parts by the points A, P, and Q. Then OD =4,
and DP — DQ = ^^/^. Hence we see from the method of represent-
ing a complex number given above that A represents -\-i,P represents
;, Q represents —£ — /'•
P.
EXERCISES
74. (i.) Express sin 4* and cos 4* in terms of sin x and cos*.
(2.) Express sin 6.r and cos 6* in terms of sin x and cos*.
(3.) Find the six 6th roots of unity.
(4.) Find the five 5th roots of unity.
THE HYPERBOLIC FUNCTIONS
75. The hyperbolic functions are defined by the equations
§inha?=— ~ — , (15)
(16)
cosh x = —
in which sinh* and cosh* denote the hyperbolic sine and
74 PLANE TRIGONOMETRY
hyperbolic cosine of x respectively. These functions are
called the hyperbolic sine and cosine on account of their
relation to the hyperbola analogous to the relation of the
sine and cosine to the circle. A natural and convenient
way to arrive at the hyperbolic functions and to study their
properties is by using complex numbers in the following
manner. The series (2), (3), and (4) give the value of sin x,
cos^r, and e* for every real value of x. These series also
serve to define sin^r, cos^r, and ex for complex values of x.
In the more advanced parts of Algebra it is shown that
the following fundamental formulas which we have proved
only for a real variable,
sin (x+y) = s\nx cosjy-}- cos x sin y, (17)
cos (x+y) — CQSx cos^— sin* sinj, (18)
e*+*=e*e", (19)
hold unchanged when the variable is complex.
This fact enables us to calculate with ease sin^r, COS.T, and
ex for any complex value of the variable.
In so doing we are led directly to the hyperbolic func-
tions. At the same time a relation between the trigono-
metric and hyperbolic functions is established by means of
which the formulas of Chapter III. can be converted into
corresponding formulas for the hyperbolic functions.
Taking x and y real and replacing y in (17), (18), and (19) by
*>, we get
sin (aF+/y) = sina: cos/y-f cos* sin iy,
cos (x+iy)=cos x cos iy— sin x sin iy,
Thus the calculation of these functions when the variable
is complex is made to depend upon the case where the vari-
able is a pure imaginary.
HYPERBOLIC FUNCTIONS 75
If we replace x by ix in series (4) we obtain
V
3! 5! 7!
A comparison with series (2) and (3) shows that these two
series are cos^r and sin^r respectively; hence the important
formula due to Euler —
This enables us to calculate ?* from sin^r and cos;r when
ix is a pure imaginary ; that is, when x is real.
To find sin ix and cosix replace x in (20) by ix\ we obtain
e—*=cosix+t sin ix. (21)
Again replacing x by —ix in (20), we obtain
e* = cos ix — i sin ix. (22)
The sum and difference of (21) and (22) give
cos ix = == cosh a?, (23 )
(24)
If we compute the value of e* by the aid of series (4) for
a succession of values of x, we find that sinh^r and cosher
are represented by the curves on page 76.
The system of formulas belonging to the hyperbolic func-
tions is obtained from those of the trigonometric functions
by using (23) and (24). This shows that for every formula
in analytic trigonometry there exists a corresponding for-
mula in hyperbolic trigonometry which we get by this sub-
76
PLANE TRIGONOMETRY
stitution. In the examples which follow, this method is
used to obtain important formulas in hyperbolic trigonome-
try.
Replacing x by — ix in (23) and (24), we get
-• (25)
-» (26)
which are formulas frequently used.
Example.— sinh (jr -}- j) = — / sin / (
= — i [sin ix cos /p + cos ix sin /y],
= — / [/' sinh .r cosh^/ + / cosh x sinhj],
= sinh x cosh_y + cosh x sinh y.
Example. — sinh x -\- sinh_y = — /(sin ix -\- sin iy),
— — i 2 sin £ i(x -\-y} cos ^ i (x— y\
= 2 sinh £ (jr +7) cosh ^ (x—y).
sinh
HYPERBOLIC FUNCTIONS 77
EXERCISES
70. (i.) Prove sinho=o, cosho=i.
(2.) Prove sinh £TT/ = /, cosh^7r/=o.
(3.) Prove sinh TT/ = O, cosh7i7 = — i.
Prove that
(4.) si n (*— /,r) = — sin ix.
( 5 .) cos (— ix) = cos /Jr.
(6.) sinh(— x) = — sinh x.
(7.) cosh( — x} = cosher.
Remark.— The hyperbolic tangent, cotangent, secant, and cosecant
are defined by
sinhjr cosher
tanh-r =
cosher smh^r
sech-r = — \ — , eschar = •
t \^*3\-,LM. ** ^— . .
cosher sinner
Prove that
(8.) tan (tx) — i tanh x.
(9.) coth ( — x} = — coth x.
(10.) sech (— .r) = sech x.
(n.) coshajr — sinh3jr=:i.
(12.) sech a.r-f- tan ha.r= i.
(13.) coth'.r — csch'jr = i .
(14.) sinh(.i- — y) = sinh^r cosh y — cosher sinh^.
(15.) cosh(.i- — _y) = cosher coshj — sinhjr
(,6.) coshi.r=v/l
(17.) sinhw — sinhr/ = 2 cosh \(u + v) sinh \ (u — v).
(i 8.) cosh u -\- cosh v = 2 cosh %(u-\-v) cosh \(u^v).
(19.) cosh u — cosh v = 2 sin h^(#-f-?/) sinh %(u — z/).
CHAPTER VII
MISCELLANEOUS EXERCISES
RELATION OF FUNCTIONS
77. Prove the following :
(i.) cos^r = sin^r cot-r.
(2.) CSC.T tan x = sec x.
(3.) (tan x -|- cot x) s\nx cos^r=:i.
(4.) (sec/ — tan /) (sec y -f tan /) = i .
(5.) (CSC 2 — COt 2) (CSC 2 -f- COt z) = I .
(6.) cos2/ + (tan / — cot/) sin/ cos y = sin2/.
(7.) cos4.r — sin4* -{-1=2 cos2,r.
.) (sinj/ — cos/)2 = 1 — 2 sin j cos/.
— sin^r COSJT).
. cot x-\- tan y
(10.) - — -• - — = cot. r tan y.
tan ^- + cot/
\-<n.) cos2/ — sin2/ = 2 cos2/— i.
(12.) i — tan4^r = 2 sec2^r — sec*;r.
(i 3-) - - 5— = tan jr.
sm^r cot2^:
(14.) sec2/ esc2/ = tan2/ -f- cot2/ +2.
.) cot/ — esc/ sec/ (i — 2 sin2/) = tan/.
, / I \2 I — COS.?
(16.) H — — cot 2^) =
Vsin« / i 4- cos z
I +cos/ sin3/
(i 8.) i+
(19.) — - -- sin3.r = (cos-r — sin x) (i-|-sin.r cos,r).
(20.) (sin,r cos/-f-cos.r sin/)2-j-(cos.r cos/ — sin .r si
MISCELLANEOUS EXERCISES 79
(21.) (a cos.r — b sin .r)2 -(-(,* sin x + b cos.r)2 = &*+&*.
-> ' ^ 4tan2j
— sin2jf ~ (i— tan2;/)2'
Find an angle not greater than 90° which satisfies each of the fol-
lowing equations:
(23.) 4 cos x = 3 sec .r.
(24.) sin^^cscj — f.
(25.) \/2 sin.r — tan.r = o.
(26.) 2 cos.r— \/3 cot.r = o.
(27.) tan j + cot j — 2 = 0.
(28.) 2 sin'-/ — 2 = — \/2 cosj.
(29.) 3 tan2.r — i =4 sin2.r.
(30.) cos2.r-|-2 sina.r — -| sin.r = o.
(31.) csc.r = §tan.r.
(32.) sec .r -(- tan .r — ± \/3-
(33.) tan .r + 2 v/3 cos.r = o.
(34.) 3 sin.r — 2 cos2.r=o.
Express the following in terms of the functions of angles less
than 45°:
(35.) sin 92°.
(36.) cos 1 27°.
(37.) tan 320°.
(38.) cot 350°.
(39.) sin 2f>5 .
(40.) tan 171°.
(41.) Given sin .r = $ and x in quadrant II; find all the other
functions of x.
(42.) Given cos.n= — f and x in quadrant III; find all the other
functions of x.
(43.) Given tan.r = | and x in quadrant III; find all the other
functions of x.
.'44.) Given cot,r = — £ and x in quadrant IV; find all the other
functions of x.
8o PLANE TRIGONOMETRY
In what quadrants must the angles lie which satisfy each of the
following equations :
(45.) sin^r cos.*' — i\/3.
(46.) sec.r tan .*• = 2-^/3.
(47.) tanjj/ + -y/20 cos}> = o. -
(48.) cos x cot x = £.
Find all the values of y less than 360° which will satisfy the fol-
lowing equations :
(49.) tanj-f-2 sinj = o.
(50.) (i -f- tan x) (i — 2 sin x] = o.
(51.) sin^r COS.T (i + 2 cos-r)=o.
Prove the following:
(52.) cos 780° = £.
(53.) sin 1485° = i\/2.
(54.) cos 2550° = ^ -y/3-
(55.) sin (— 3000°) = — cos 30°.
(56.) cos 1 300° = — cos 40°**
(57.) Find the value of a sin 90° + b tano°-\-a cos 180°.
(58.) Find the value of a sin 30° + ^ tan 45° -\-a cos 60° -\-b tan 135°.
(59.) Find the value of (a — b) tan 225° + £ cos 180° — a sin 270°.
(60.) Find the value of (a sin 45°+^ cos 45°) (a sin 135° + ^ sin 225°).
RIGHT TRIANGLES
7£. In the following problems the planes on which distances are measured
are understood to be horizontal unless otherwise stated.
(i.) The angle of elevation of the top of the tower from a point
1 121 ft. from its base is observed to be 15° 17'; find the height of
the tower.
(2.) A tree, 77 ft. high, stands on the bank of a river ; at a point on
the other bank just opposite the tree the angle of elevation of the
top of the tree is found to be 5° if 37". Find the breadth of the
MISCELLANEOUS EXERCISES 81
(3.) What angle will a ladder 42 ft. long make with the ground if its
foot is 25 ft. from the base of the building against which it is placed ?
(4.) When the altitude of the sun is 33° 22', what is the height of a
tree which casts a shadow 75 ft. ?
(5.) Two towns are 3 miles apart. The angle of depression of one,
from a balloon directly above the other, is observed to be 8° 15'.
How high is the balloon ?
(6.) From a point 197 ft. from the base of a tower the angle of ele-
vation was found to be 46° 45' 54" ; find the height of the tower.
(7.) A man 5 ft. 10 in. high stands at a distance of 4 ft. 7 in. from
a lamp-post, and casts a shadow 18 ft. long; find the height of the
lamp-post.
(8.) The shadow of a building 101.3 ft- high is found to be 131.5
ft. long; find the elevation of the sun at that time.
(9.) A rope 112 ft. long is attached to the top of a building and
reaches the ground, making an angle of 77° 20' with the ground ;
find the height of the building.
(10.) A house is 130 ft. above the water, on the banks of a river;
from a point just opposite on the other «eank the angle of elevation
of the house is 14° 30' 21". Find the width of the river.
(u.) From the top of a headland, 1217.8 ft. above the level of the
sea, the angle of depression of a dock was observed to be 10° 9' 13" ;
find the distance from the foot of the headland to the dock.
(12.) 1121.5 ft. from the base of a tower its angle of elevation is
found to be n° 3' 5 "; find the height of the tower.
(13.) One bank of a river is 94.73 ft. vertically above the water, and
subtends an angle of 10° 54' 13" from a point directly opposite at the
water's edge; find the width of the river.
(14.) The shadow of a vertical cliff 113 ft. high just reaches a boat
on the sea 93 ft. from its base ; find the altitude of the sun.
(15.) A rope, 38 ft. long, just reached the ground when fastened to
the top of a tree 29 ft. high. What angle does it make with the
ground ?
(16:) A tree is broken by the wind. Its top strikes the ground 15
ft. from the foot of the tree, and makes an angle of 42° 28' with the
ground. Find the height of the tree before it was broken.
6
82 PLANE TRIGONOMETRY
(17.) The pole of a circular tent is 18 ft. high, and the ropes reach-
ing from its top to stakes in the ground are 37 ft. long; find the
distance from the foot of the pole to one of the stakes, and the angle
between the ground and the ropes.
(i 8.) A ship is sailing southwest at the rate of 8 miles an hour.
At what rate is it moving south ?
(19.) A building is 121 ft. high. From a point directly across the
street its angle of elevation is 65° 3'. Find the width of the street.
(20.) From the top of a building 52 ft. high the angle of elevation
of another building 112 ft. high is 30° 12'. How far are the buildings
apart ?
(21.) A window in a house is 24 ft. from the ground. What is the
inclination of a ladder placed 8 ft. from the side of the building and
reaching the window ?
(22.) Given that the sun's distance from the earth is 92,000,000
miles, and its apparent semidiameter is 16' 2" ; find its diameter.
(23.) Given that the radius of the earth is 3963 miles, and that it
subtends an angle of 57' 2" at the moon; find the distance of the
•moon from the earth.
(24.) Given that when the moon's distance from the earth is 238885
miles, its apparent semidiameter is 15' 34"; find its diameter in miles.
(25.) Given that the radius of the earth is 3963 miles, and that it
subtends an angle of 9" at the sun ; find the distance of the sun
from the earth.
(26.) A light-house is 57 ft. high ; the angles of elevation of the top
and bottom of it, as seen from a ship, are 5° 3' 20" and 4° 28' 8". Find
the distance of its base above the sea-level.
(27.) At a certain point the angle of elevation of a tower was ob-
served to be 53° 51' 1 6", and at a point 302 ft. farther away in the
same straight line it was 9° 52' 10"; find the height of the tower.
(28.) A tree stands at a distance from a straight road and between
two mile-stones. At one mile-stone the line to the tree is observed
to make an angle of 25° 15' with the road, and at the other an angle
of 45° 17'. Find the distance of the tree from the road.
(29.) From the top of a light-house, 225 ft. above the level of the
sea, the angle of depression of two ships are 17° 21' 50" and 13° 50' 22",
MISCELLANEOUS EXERCISES 83
and the line joining the ships passes directly beneath the light-house ;
find the distance between the two ships.
ISOSCELES TRIANGLES AND REGULAR POLYGONS
S^-
79. (i.) The area of a regular dodecagon is 37.52 ft.; find its
apothem.
(2.) The perimeter of a regular polygon of 1 1 sides is 23.47 ft. ; find-
the radius of the circumscribing circle.
(3.) A regular decagon is circumscribed about a circle whose radius
is 3.147 ft. ; find its perimeter.
(4.) The side of a regular decagon is 23.41 ft. ; find the radius of
the inscribed circle.
(5.) The perimeter of an equilateral triangle is 17.2 ft.; find the
area of the inscribed circle.
(6.) The area of a regular octagon is 2478 sq. in. ; find its pe-
rimeter.
(7.) The area of a regular pentagon is 32.57 sq. ft. ; find the radius
of the inscribed circle.
(8.) The angle between the legs of a pair of dividers is 43°, and the
legs are 7 in. long ; find the distance between the points.
(9.) A building is 37.54 ft. wide, and the slope of the roof is 43° 36' ;
find the length of the rafters.
(10.) The radius of a circle is 12732, and the length of a chord is
18321 ; find the angle the chord subtends at the centre.
(u.) If the radius of a circle is taken as unity, what is the length
of a chord which subtends an angle of 77° 17' 40"?
(12.) What angle at the centre of a circle does a chord which is ^
of the radius subtend ?
(13.) What is the radius of a circle if a chord 11223 ft. subtends an
angle of 59° 50' 52"?
(14.) Two light-houses at the mouth of a harbor are each 2 miles
from the wharf. A person on the wharf finds the angle between the
lines to the light-houses to be 17° 32'. Find the distance between the
two light-houses.
(15.) The side of a regular pentagon is 2; find the radius of the
inscribed circle.
84 PLANE TRIGONOMETRY
(i 6.) The perimeter of a regular heptagon inscribed in a circle is
12 ; find the radius of the circle.
(17.) The radius of a circle inscribed in an octagon is 3; find the
perimeter of the octagon.
(18.) A regular polygon of 9 sides is inscribed in a circle of unit
radius; find the radius of the inscribed circle.
(19.) Find the perimeter of a regular decagon circumscribed about
a unit circle.
(20.) Find the area of a regular hexagon circumscribed about a
unit circle.
r (21.) Find the perimeter of a polygon of 11 sides inscribed in a
\ unit circle.
(22.) The perimeter of a dodecagon is 30; find its area.
(23.) The area of a regular polygon of 11 sides is 18; find its pe-
rimeter.
TRIGONOMETRIC IDENTITIES AND EQUATIONS
8O. Prove the following :
(i.) sin
sin 2x -f- si
(4.) cos2/ tan2/ -f sin2/ cot8/ = I.
(5.) -. — = cot x cot y cot z — cot x — cot y — cot z.
sin* sin/ sin z
(6.) cos2 (x —y) — sin2 (x +/) = cos 2 x cos 2/.
. sin jr-4-siny
(7.) - — — = — cot i (x — y).
cos x — cos y
. cosx — sec^r
sin 2x
(o.) cot;r = —
i — cos 2x
I — COS 2/
(10.) tan y = — — — •
i + cos 2/
(11.) cot x — tan .r = 2 cot 2x. v,
/
MISCELLANEOUS EXERCISES 85
(12.) tan^.r + 2 sin2 ^,r cot.r
tan x ± tan /
(n.) - — — = dbsm^r sec.r tan/.
cot .r± cot/
(14.) sin .r — 2 sin3 ,r = sin x cos 2.r.
(15.) 4 sin/ sin (60° — /) sin (60° -(-/) = sin 3/.
suy(,-tan^)/ _1_ _ - ' = si
sec2/ Vcos/ — sin/ cos/+sin/
(17.) i + tan/ tan £/=
(1 8.) sin 4^r = 4 sin.r cos3 ^r — 4 cos^r si
(20.) tan 50° 4- cot 50° — 2 sec 10°.
(21.) cos (x + 45°) 4- sin (x — 45°) = o.
tan^r
(22.)
i — cot 2x tan x
(23.) ( i — tan2 x) sin x cos .r = cos 2.r yj
— cos 2x
-f COS 2X '
-= ec
' cos/ — sin/
(25.) sin (*+/) cos,r— cos(.r+/) sin^r = si
(26.) cos (.r — /) sin/ + sin (x — /) cos/ = sin JJT.
. sin(jr— /) sin(/— g) , sin (g — JT) _
(27 ) - _ _i_ - -- --4- — — — u.
COSX COS/ COS/ COS.? COS 2 COSX
sin£+sirL2£ = co
cos .r — cos 2.r
(29.) 2 sin2 ,r sina/-h 2 cos2 x cos2/ = i + cos 2x cos 2/.
(30.) sin 60° 4- sin 30° = 2 sin 45° cos 15°.
tan (,r-/) + tan/
u '; i— tan (.r—,/) tan/
sin/ tan i/
(33.) sin^r+sin 2.r = 2 sin
sin x 4- sin/ _
cos .*• — cos/ sn/ — sn x
(36.) 2 tan 2/ = tan(45°+/) — tan (45°— /).
86 PLANE TRIGONOMETRY
tan2.r4-tan.r _sin3.r
tan 2 x — tan x ~ sin^r
(38.)
i — 3 tan2/
(39.) sin 60° 4- sin 20° = 2 sin 40° cos 20°.
(40.) sin 40° — sin 10° = 2 cos 2 5° sin 15°.
(41.) cos ix — cos4-r = 2 sin yc sin x.
(42.) tan 15° = 2— v/3-
(43.) (\/i 4sin.r — \/i — sin.r)2=4 sin2
(44.) "V/i H-sin^r--'i — sin^)2 = 4 cos
, ... N siii4.r
(46.) . * = 2 COS 2.T.
(47.) sin 50° — sin7o°-f-sinio° = o.
f.Q\ I? IT ^TT . IT
(40-) cos -- cos- = 2 sin —sin — •
3 J2 12 12
sin 750 - sin 1 5° /f
y cos75°4-cosi5° V '
(51.) tan8£.r(i+cota£.r)3= --—
(52.) tan 7 5° = 2
(53.) sin 3^- -f- sin 5.1- = 2 sin 4^- cos^r.
(54.) cos 5-r -f- cos 9-r = 2 cos 7-r cos 2^r.
~ 1
(55.) sin 1 5° =
\/2
(56.) - - =tan;r.
cos 3-r + cos ^*
(57.) sin 5j = 5 sinj — 20 sin3/-}- 16 sin5/.
(58.) cos5/ = 5 cosj — 20 cos3/ + 16 cos5/.
(60.)
(6 1 .) cos 3^- -f- cos 5a- -}- cos 7* + cos 1 5^- = 4 cos 4^ cos 5 ^r cos £>x
(62.) sin2 \x (cot \. i- — i)2= i — sin.r.
— sin \r
MISCELLANEOUS EXERCISES 87
sin2 \x (c
(63.)
(64.)
4
(65.)
cos .r — sin.r 2
(66.) cosj + cos I1 2O —y} + cos ( 1 2
x si 113.1-
(67.) . =2 cos2.r+ 1.
sin.r
(68)
.
cos 3-r 4- 3 cos x
(64.) sin.r(i-|-tan.r)4-cos.r(i 4-cot.r) = esc ^ + sec ^r.
cos3.r — sin3.r
(sin y — sinj)(cos4_y — cos6y)
i — cos.r
4cos.r
, ,
(70.) —- — - 2- = 2.
sin.r cos.r
s i H- sin a- 4- cos. r
(71.) - — =cotfr.
i +sm.r — cos.r
cos (4-r — 27) 4- cos (4.T —
Sin.r + sin3.r + sin5.r4-sin7.r = ^
cos .r -f- cos 3.r -|- cos $.r 4- cos 7.1-
If A, B, and C are the angles of a triangle, prove the following
(74.) sin2^4-|-sin2/>'4-sin2C = 4 sin A s\n£ sinC
(75.) sin 2^f + sin 27)' — sin 2(7 = 4 cos^4 cos# sinC.
(76.) sinM4-sin2/»'-f-sin2C=2 + 2 cos A cos B cos C
(77.) tan ^ -f tan B 4- tan C = tan A tan ^ tan C.
Solve the following equations for values of x less than 360°.
(78.) cos 2.r 4- cos x = — i .
(79.) sin.r4-sin7.r
(80.) cos^" — sin2.r —
(8 1.) cos.r — sin3^r — cos2.i- = o.
(82.) sin^a- — 2 sin2,r r=o.
(83.) sin 2.r — cos 2.r — sin x + cos x = o.
(84.) sin (60° — .r) - sin (60° + x) = + i y
(85.) sin (30° 4- -r) — cos (60° + x) = — | ^
88 PLANE TRIGONOMETRY
(86.) esc x = i + cot x.
(87.) cos T.X = cos 2.r.
(88.) 2 sin_y=sin 2y.
(89.) sin $y -\- sin 2y -\- s'my = o.
(90.) sina.r + 5 cos'.r = 3.
(91.) tan(45° —
OBLIQUE TRIANGLES
81. (i.) It is required to find the distance between two points, A
and JB.on opposite sides of a river. A line, AC, and the angles BAG
and ACB are measured and found to be 2483 ft., 61° 25', and 52° 17'
respectively.
°(2.) A straight road leads from a town A to a town B, 12 miles
distant ; another road, making an angle of 77° with the first, goes from
A to a town C, 7 miles distant. How far are the towns B and C apart ?
In order to determine the distance of a fort, A, from a battery,
B, a line, BC, one-half mile long, is measured, and the angles ABC
and ACB are observed to be 75° 18' and 78° 21' respectively. Find
the distance AB.
(4.) Two houses, A and B, are 1728 ft. apart. Find the distance of
a third house, C, from A if BAC=tf° 51 'and ABC= 57° 23'.
(5.) In order to determine the distance of a bluff, A, from 'a house,
B, in a plane, a line, BC, was measured and found to be 1281 yards,
also the angles ABC and BCA 65° 31' and 70° 2' respectively. Find
the distance AB.
(6.) Two towns, 3 miles apart, are on opposite sides of a balloon.
The angles of elevation of the balloon are found to be 13° 19' and
20° 3'. Find the distance of the balloon from the nearer town.
(7.) It is required to find the distance between two posts, A and B,
which are separated by a swamp. A point C is 1272.5 ft. from A, and
2012.4 ft- from B. The angle ACB is 41° 9' 1 i".
(8.) Two stakes, A and B, are on opposite sides of a stream ; a
third point, C, is so situated that the distances AC and BC can be
found, and are 431.27 yards and 601.72 yards respectively. The angle
ACB is 39° 53' 13". Find the distance between the stakes A and B.
MISCELLANEOUS EXERCISES 89
(9.) Two light-houses, A and B, are 11 miles apart. A ship, C, is
observed from them to make the angles BAC =31° 13' 31" and ABC
= 21° 46' 8". Find the distance of the ship from A.
(10.) Two islands, A and B, are 6103 ft. apart. Find the distance
from A to a ship, C, if the angle ABC is 37° 25' and BAC is 40° 32'.
(u.) In ascending a cliff towards a light-house at its summit, the
light-house subtends at one point an angle of 21° 22'. At a point
55 ft. farther up it subtends an angle of 40° 27'. If the light-house
is 58 ft. high, how far is this last point from its foot?
(12.) The distances of two islands from a buoy are 3 and 4 miles
respectively. The islands are 2 miles apart. Find the angle sub-
tended by the islands at the buoy.
^•(13.) The sides of a triangle are 151.45, 191.32, and 250.91. Find
the length of the perpendicular from the largest angle upon the
opposite side.
** (14.) A tree stands on a hill, and the angle between the slope of the
hill and the tree is 110° 23'. At a point 85.6 ft. down the hill the
tree subtends an angle of 22° 22'. Find the height of the tree.
! (15.; A light-house 54 ft. high is built upon a rock. From the top
of the light-house the angle of depression of a boat is 19° 10', and
from its base the angle of depression of the boat is 12° 22'. Find the
height of the rock on which the light-house stands.
(16.) Three towns, A, B, and C, are connected by straight roads.
A£ = 4 miles, BC= 5 miles, and AC= 7 miles. Find the angle made
by the roads AB and BC.
(17.) Two buoys, A and B, are one-half mile apart. Find the dis-
tance from A to a point C on the shore if the angles ABC and BAC
are 77° 7' and 67° 17' respectively.
(i 8.) The top of a tower is 175 ft. above the level of a bay. From
its top the angles of depression of the shores of the bay in a certain
direction are 57° 16' and 15° 2'. Find the distance across the bay.
(19.) The lengths of two sides of a triangle are \/2 and -v/3- The
angle between them is 45°. Find the remaining side.
(20.) The sides of a parallelogram are 172.43 and 101.31, and the
angle included by them is 61° 16'. Find the two diagonals.
(21.) A tree 41 ft. high stands at the top of a hill which slopes
X
X
90 PLANE TRIGONOMETRY
10° 12' to the horizontal. At a certain point down the hill the tree
subtends an angle of 28° 29'. Find the distance from this point to
the toot of the tree.
(22.) A plane is inclined to the horizontal at an angle of 7° 33'. At
a certain point on the plane a flag-pole subtends an angle 20° 3', and at
a point 50 ft. nearer the pole an angle of 40° 35'. Find the height of
the pole.
(23.) The angle of elevation of an inaccessible tower, situated in a
plane, is 53° 19'. At a point 227 ft. farther from the tower the angle
of elevation is 22° 41'. Find the height of the tower.
(24.) A house stands on a hill which slopes 12° 1 8' to the horizontal.
75 ft. from the house down the hill the house subtends an angle of
32° 5'. Find the height of the house.
(25.) From one bank of a river the angle of elevation of a tree on
the opposite bank is 28° 31'. From a point 139.4 ft. farther away in a
direct line its angle of elevation is 19° 10'. Find the width of the river.
(26.) From the foot of a hill in a plane" the angle of elevation of
the top of the hill is 21° 7'. After going directly away 211 ft. farther,
the angle of elevation is 18° 37'. Find the height of the hill.
(27.) A monument at the top of a hill is 153.2 ft. high. At a point
321.4 ft. down the hill the monument subtends an angle of 11° 13'.
Find the distance from this point to the top of the monument.
(28.) A building is situated on the top of a hill which is inclined
10° 12' to the horizontal. At a certain distance up the hill the angle
of elevation of the top of the building is 20° 55', and 115.3 ft- farther
down the hill the angle of elevation is 15° 10'. Find the height of
.the building.
(29.) A cloud, C, is observed from two points, A and J3, 2874 ft.
apart, the line AB being directly beneath the cloud. At A, the angle
of elevation of the cloud is 77° 19', and the angle CAB is 51° 18'.
The angle ABC is found to be 60° 45'. Find the height of the cloud
above A.
(30.) Two observers, A and B, are on a straight road, 675.4 ft. apart,
directly beneath a balloon, C. The angles ABC and BAC are 34° 42'
and 41° 15' respectively. Find the distance of the balloon from the
first observer.
MISCELLANEOUS EXERCISES 91
(31.) A man on the opposite side of a river from two objects, A
and B, wishes to obtain their distance apart. He measures the dis-
tance CD = 357 ft., and the angles ACB=2^> 33', BCD = 38° 52', ADB
= 54° 10', and ADC =34° n'. Find the distance AB.
i (32.) A cliff is 327 ft. above the sea-level. From the top of the
cliff the angles of depression of two ships are 15° 11' and 13° 13'.
From the bottom of the cliff the angle subtended by the ships are
122° 39'. How far are the ships apart ?
(33.) A man standing on an inclined plane 112 ft. from the bottom
observed the angle subtended by a building at the bottom to be 33°
52'. The inclination of the plane to the horizontal is 18° 51'. Find
the height of the building.
(34) Two boats, A and B, are 451.35 ft. apart. The angle of ele-
vation of the top of a light-house, as observed from A, is 33° if.
The base of the light-house, C, is level with the water; the angles
ABC and CAB are 12° 31' and 137° 22' respectively. Find the height
of the light-house.
(35.) From a window directly opposite the bottom of a steeple the
angle of elevation of the top of the steeple is 29° 21'. From another
window, 20 ft. vertically below the first, the angle of elevation is 39° 3'.
Find the height of the steeple.
(36.) A dock is i mile from one end of a breakwater, and i£ miles
from the other end. At the dock the breakwater subtends an angle
of 31° n'. Find the length of the breakwater in feet.
(37.) A straight road ascending a hill is 1022 ft. long. The hill
rises i ft. in every 4. A tower at the top of the hill subtends an
angle of 7° 19' at the bottom. Find the height of the tower.
(38.) A tower, 192 ft. high, rises vertically from one corner of a
triangular yard. From its top the angles of depression of the other
corners are 58° 4' and 17° 49'. The side opposite the tower subtends
from the top of the tower an angle of 75° 15'. Find the length of
this side.
(39.) There are two columns left standing upright in a certain ruins ;
the one is 66 ft. above the plain, and the other 48. In a straight line.,
between them stands an ancient statue, the head of which is 100' ft.
from the summit of the higher, and 84 ft. from the top of the lower
92 PLANE TRIGONOMETRY
column, the base of which measures just 74 ft. to the centre of the
figure's base. Required the distance between the tops of the two
columns.
(40.) Two sides of a triangle are in the ratio of 1 1 to 9, and the
opposite angles have the ratio of 3 to i. What are these angles ?
(41.) The diagonals of a parallelogram are 12432 and 8413, and the
angle between them is 78° 44' ; find its area.
(42.) One side of a triangle is 1012.6 and two angles are 52° 21' and
57° 32' ; find its area.
(43.) Two sides of a triangle are 218.12 and 123.72, and the included
angle is 59° 10' ; find its area.
(44.) Two angles of a triangle are 35° 15' and 47° 18', and one side
is 2104.7 I find its area.
(45.) The three sides of a triangle are 1.2371, 1.4713, and 2.0721;
find the area.
(46.) Two sides of a triangle are 168.12 and 179.21, and the included
angle is 41° 14' ; find its area.
(47.) The three sides of a triangle are 51 ft., 48.12 ft., and 32.2 ft. ;
find the area.
(48.) Two sides of a triangle are 1 1 1 . 1 8 and 121.21, and the included
angle is 27° 50' ; find its area.
(49.) The diagonals of a parallelogram are 37 and 51, and they form
an angle of 65° ; find its area.
(50.) If the diagonals of a quadrilateral are 34 and 56, and if they
intersect at an angle of 67°, what is the area ?
SPHERICAL TRIGONOMETRY
CHAPTER VIII
RIGHT AND QUADRANTAL TRIANGLES
RIGHT TRIANGLES
82. Let O be the centre of a sphere of unit radius, and
ABC a right spherical triangle, right angled at A, formed by
the intersection of the three planes A OC, AOB, and BOC
with the surface of the sphere. Suppose the planes DAC"
and BEC passed through the points A and B respectively,
and perpendicular to the line OC. The plane angles DC" A
and BC'E each measure the angle C of the spherical tri-
angle, and the sides of the spherical triangle a, b, c have the
same numerical measure as BOC, AOC, and AOB respec-
94 SPHERICAL TRIGONOMETRY
tively, then, AD = ta.nc, BE — s\\\c, BC1 = sma,
cos6, OE = cosc, AC" = sin b.
In the two similar triangles OEC' and OAC" ',
OA i . cos b ' '
•- LOS 6* LOS 6.
V)
In the triangle BC' E,
^.n BE Qr ^.n
sin^r v
(2)
^C?"0
In the triangle DAC" ,
sin a
DA or
tan ^
(3)
(4)
Combining formulas (2) and (3) with
., tan $
(i).
rrb^
tan a
Again, if AB were made the base of the right spherical
triangle ABC, we should have
sinj5=£^- (5)
4-i, •> A
(6)
£r (7)
From the foregoing equations we may also obtain by
combinations,
cos/?=sin C cos^. (8)
cosC— sin B cose. (9)
cos # = cot B cot £7. (10)
NAPIER'S RULES OF CIRCULAR PARTS
S3. The above ten formulas are sufficient to solve all
cases of right spherical triangles. They may, however, be
RIGHT AND QUADRANTAL TRIANGLES
95
expressed as two simple rules, called, after their inventor,
Napier's rules.
The two sides adjacent to the right angle, the complement
of the hypotenuse, and the complements of the oblique an-
gles are called the circular parts.
The right angle is not one of the circular parts.
comp B
comp
comp C
Thus there are fire circular parts — namely, />, c, comprt, comp/?, compC
Any one of the five parts may be called the middle part, then the two parts next
to it are called adjacent parts, and the remaining two parts are called the oppo-
site parts.
Thus if c is taken for the middle part, comp/? and b are adjacent parts, and
comptf and comp C are opposite parts.
The ten formulas may be written and grouped as follows :
ist Group.
sin comp C = tan comprt tan b.
sin comp /?= tan compi? tan c.
MII comp a = tan comp/? tan comp C.
sin c = tan comp B tan b.
sin
b =tan comp C tan r.
zd Group.
sin comp rt=r cos l> cos c.
sin b = cos comp a cos comp B.
sin r=cos comprt cos comp C.
sin comp j9=cos comp C cos £.
sin comp f=cos comp/? cose-.
Napier's rules may be stated :
I. The sine of the middle part is equal to the product of
the tangents of- the adjacent parts.
II. The sine of the middle part is equal to the product of
the cosines of the opposite parts.
96
SPHERICAL TRIGONOMETRY
84. In the right spherical triangles considered in this work, each
side is taken less than a semicircumference, and each angle less than
two right angles.
In the solution of the triangles, it is to be observed,
(i.) If the two sides about the right angle are both less or both
greater than 90°, the hypotenuse is less than 90°; if one side is less
and the other greater than 90°, the hypotenuse is greater than 90°.
(2.) An angle and the side opposite are either both less or both
greater than 90°.
EXAMPLE
85. Given # = 63° 56', £ = 40° o', to find c, B, and C.
To find c.
cotnp a is the middle part.
c and b are the opposite parts,
sin comp a=cos b cos c,
cos 0=cos b cos c.
cos a
cos c = -•
cos b
log cos 0=9.64288
colog cos £=0.11575
log cos ^-=9.75863
'=54° 59 47"
To find C.
comp C is the middle part.
comp a, and b are adjacent parts.
sin comp C=tan comp a tan£,
cos C= cot a tan£.
log cot a= 9. 68946
log tan £=9 92381
9-61327
C=65° 45' 58"
To find B.
l> is the middle part.
comp a and comp B are the opposite
parts.
sin £=cos comp a cos comp B,
or sin £=sin a sin B.
• sin b
log sin £=9.80807
colog sin rt=o. 04659
log sin £=9. 85466
^ = 45° 41 '28"
Check.
Use the three parts originally required.
comp C is the middle part.
comp.# and c are opposite parts.
sin comp C=cosc cos comp B,
or cos C=cos c sin B,
log cos £-=9.75863
log sin B=<). 85466
log cos (7=9.61329
C=6$° 45' 54"
RIGHT AND QUADRANTAL TRIANGLES 97
AMBIGUOUS CASE
86. When a side about the right angle and the angle opposite
this side are given, there are two solutions, as illustrated by the fol-
lowing figure. Since the solution gives the values of each part in
terms of the sine, the results are not only the values of a, b, B, but
1 8o°— rt, 180°— b, 1 8o°— #.
Given c = 26° 4'.
To find a, a', b, b' and B, B', using Napier's rules.
To find B and B '.
sin comp C= cos comp B cose,
cos C=sin B cos c,
cos C
~~ cos c
log cos C=g. 90796
colog cos £-=0.04659
log sin ^ = 9.95455
B= 64° 14' 30"
r = i8o°-£=ii5° 45' 30"
To find b and b' .
sin £=tan c tan comp C,
sin £=tan c cot C
log tan ^-=9.68946
log cot 67=0.13874
log sin £=9.82820
b— 42° 19' 17"
£=137° 40' 43"
To find a and a'.
sin £-=cos comp a cos Comp C,
sin c=sin a sin (7,
sin c
or
or
log sin £- = 9.64288
colog sin (7=0.23078
log sin 0=9.87366
a= 48° 22' 55"-
fl' = i8o°-a=i3i°37' 5" +
'Discrepancy due to omitted decimals.)
Check.
sin £=cos comp a cos comp /?,
or sin £=sin a sin ^.
log sin a or a'=g. 87366
log sin # or .#'=9.95455
log sin £=9.82821
£= 42° 19' 21"
39"
98 SPHERICAL TRIGONOMETRY
QUADRANTAL TRIANGLES
87. Def. — A quadrantal triangle is a spherical triangle
one side of which is a quadrant.
A quadrantal triangle may be solved by Napier's rules for
right spherical triangles as follows :
By making use of the polar triangle where
C=i8o° — ^ <r=i8o°— C'
we see that the polar triangle of the quadrantal triangle is
a right triangle which can be solved by Napier's rules.
Whence we may at once derive the required parts of the
quadrantal triangle.
EXAMPLE
Given A = 1 36° 4'. B = 1 40° o'. a — 90° o'.
The corresponding parts of the polar triangle are
a' =^3° 56', V = 40° o', A' = 90°.
By Napier's rules we find
B' = 45° 41 ' 28", C' = 65° 45' 58", c - 54° 59' 47" ;
whence, by applying to these parts the rule of polar triangles, we
obtain
b— 134° 18' 32", c= 114° 14' 2", C=i25°o' 13".
EXERCISES
88. (i.) In the right-angled spherical triangle ABC, the side a=
63° 56', and the side £ = 40°. Required the other side, c, and the
angles B and C.
(2.) In a right-angled triangle ABC, the hypotenuse a = 91° 42', and
the angle ^ = 95° 6'. Required the remaining parts.
(3.) In the right-angled triangle ABC, the side b = 2.6° 4', and the
angle ^ = 36°. Required the remaining parts.
- (4.) In the right-angled spherical triangle ABC, the side c = 54° 30',
and the angle £ = 44° 50'. Required the remaining parts.
Why is not the result ambiguous in this case?
RIGHT AND QUADRANTAL TRIANGLES 99
(5.) In the right-angled spherical triangle ABC, the side £ = 55° 28',
and the side ^ = 63° 15'. Required the remaining parts.
(6.) In the right-angled spherical triangle ABC, the angle B = 69°
20', and the angle C = 58° 16'. Required the remaining parts.
(7.) In the spherical triangle ABC, the side # = 90°, the angle C=
42° 10', and the angle ^ = 115° 20'. Required the remaining parts.
Hint. — The angle A of the polar triangle is a right angle.
(8.) In the spherical triangle ABC, the side £ = 90°, the angle C=
69° 13' 46", and the angle A = 72° 12' 4". Required the remaining
parts.
(9.) In the right-angled spherical triangle ABC, the angle C=23°
27' 42", and the side b— 10° 39' 40". Required the angle B and the
sides a and c.
(10.) In the right spherical triangle ABC, the angle £ = 47° 54' 20",
and the angle C=6i° 50' 29". Required the sides.
CHAPTER IX
OBLIQUE-ANGLED TRIANGLES
89. Let O be the centre of a sphere of unit radius, and
ABC an oblique-angled spherical triangle formed by the
three planes AOB, BOC, and AOC. Suppose the plane
AED passed through the point A perpendicular to AO, in-
tersecting the planes A OB, BOC, and AOC, in AE, ED,
and AD respectively. Then AD=tan b, AE-tan c, OD—
In the triangle EOD,
ED'2 = seca£ + secV — 2 sec b sec c cos a.
In the triangle AED,
ED'* = tan2^ -f tanV — 2 tan b tan c cos A.
Subtracting these two equations and remembering that
sec2^ — tan2£=i, we have
0 = 2 — 2 sec£ seer cos#-|-2 tan£ tanr cos A.
Reducing, we have
co§c+§in&
(i)
OBLIQUE-ANGLED TRIANGLES 101
If we make b and c in turn the base of the triangle, we obtain in a
similar way,
cos £ = cos £• cos#-|-sin<: sin a cos B,
and cos<r = cosd! cos/^ + sin<7 s\nb cosC.
Remark. — In this group of formulas the second may be obtained
from the first, and the third from the second, by advancing one letter
in the cycle as shown in the figure ; thus, writing b for
a, c for b, a for c, B for A, C for B, and A for C. The
same principle will apply in all the formulas of Oblique-
Angled Spherical Triangles, and only the first one of
each group will be given in the text.
90. By making use of the polar triangle where
we may obtain a second group of formulas.
Substituting these values of a, b, c, and A in (i), and remembering
that cos ( 1 80° — A) — — cos A and sin (i So0— A) r= sin A, we have
cos^4' = — cos^'cosC'-fsin^' sin C' cosa'.
Since this is true for any triangle, we may omit the accents and
write,
cos A = - cos B cos C + sin B sin C cos a. (2)
FORMULAS FOR LOGARITHMIC COMPUTATION
. Formula (i), cos a = cos b cose + sin & sin c cos A,
cos a — cos^ cos*:
gives cos A —
sne
By § 36, cos^ = i — 2 sin2-J^
cos a — cos^
Whence i —
or sin2 £ A —
sin b sin c
' sin c—cosa
2 sin b sine
102 SPHERICAL TRIGONOMETRY
— c] — cos a
2 sin b sin c
sin sin
sin b sin c • (38)
Putting
c a + b—c , a — b-\-c
-=s, then - — =s— c, and - — =^— b,
we
/sm(s
have sin-J^f=\/- <—.
V si
. . .
sin b sin c
Since, also, cos A — i+ 2 cos*$A,
we have, similarly,
/sin s s\n(s — a]
= V - — — i A
v -
sin b sine
/Ii
Hence
By a like process, formula (2) reduces to
% , —cosScos(S-A) ,
tania^W- ^- (TI)
. If, in formula I, we advance one letter, we have
/sin (s — c) sin (s—a)
=\/- • f L\ •
v situ sin (s— -b)
And dividing tan^A by tan^^, and reducing, we obtain
tan^A sin(s — b)
tan \B~ sin (j— tf) '
By composition and division,
tan %A-\- tan \B sin (j— ^) + sin(^— a)
tan ^ A— tan ^^ ~~ sin (j— b) — sin (j— «)'
§§ 30» 38, this becomes I?1
§in ^(A— B)~~ tan ^ (a — b)'
OBLIQUE-ANGLED TRIANGLES 103
Multiplying tanf A by tan£#, and reducing, we obtain
tan \A tan -J B sin (s— c)
i sin s
By division and composition, and by §§ 30, 38, this be-
comes
tanjc
co»-^(A — B) tan -J- (a + b) '
Proceeding in a similar way with formula II, we obtain
s!n-J-(a + 6)_ cot-J-C ,_,,
*in%(a — b)~ tan%(A — &)'
cow 4- (« -f 6) cot -J C
And ~- -.- = ^-A — ~. (VI)
9»9. In the spherical triangle yi ^{7, suppose C7? drawn per-
pendicularly to ABt then, by the formulas for right spher-
ical triangles,
In triangle A CD, sin / = sin b sin A.
In triangle BCD, sin p~ sin # sin B.
Whence sin a sin /?=sin b sin ^4,
sin a §in 6
Remark.— If (A + B)>i8o°, then
1 80°, then (a
, and if (A-hB)<
104
SPHERICAL TRIGONOMETRY
94. All cases of oblique-angled triangles may be solved
by applying one or more of the formulas I, II, III, IV, V,
VI, VII, as shown in the following cases.
CASES
(i.) Given three sides, to find the angles.
Apply formula I. Check : apply V or VI.
(2.) Given three angles, to find the sides.
Apply formula II. Check : apply III or I V.
(3.) Given two sides and the included angle.
Apply V and Vl\ and VII. Check : apply III or I V.
(4.) Given two angles and included side.
Apply III and I V, and VI L Check : apply V or VI.
(5.) Given two angles and an opposite side.
Apply VII, V, and III. Check : apply IV.
(6.) Given two sides and an opposite angle.
Apply VII, V, and IV. Check : apply III.
EXAMPLE— CASE (l)
95. Given a = 81° 10' b = 60° 20'
To find A, B, and C.
a— 81° 10'
b — 60° 20'
C =:II20 25'
<r=ii2°25'
j = i26° 57' 30"
s-a=45° 47' 30"
^-^=66° 37' 30"
j-^^i4°32' 30"
log sin .f =9. 90259
log sin(.r — ^0=9.85540
log sin (s — £)=9. 96281
log sin (j— ^=9.39982
To find
sin s sin(s — <i)
log sin (s — ^=9.9628 1
log sin(j-<r)=9.39982
colog sin s=o. 14460
colog sin (s— a)=o.ogi4i
2) I Q .60464
log tan £.4:= 9.80232
^=32° 23' 19"
^ = 64° 46' 38"
ur
OBLIQUE-ANGLED TRIANGLES
105
To find B.
tan A B= * /?
V
— b]
log sin(j— a)=
log sin (s— ^=9.39982
colog sin .r= 0.0974 1
colog sin(j — b) =0.03719
2)19.38982
logtan£#= 9.69491
To find C.
/
V
_ sin (j — 0)
sin s sin(j— c)
log sin (s—a)=g. 85540
log sin (s — ^=9.9628 1
colog sin j-=o. 09741
colog sin (s— r) =0.600 1 8
2)20.51580
log tan£ (7= 10.25790
K= 6l° 5' 32"
(7=122° II' 4"
,
Formula V, cot A C=
sin a
A =64° 46' 38"
.£= 52° 42' 12"
fl=8i° 10'
b =60° 20'
=141° 30' ; £(a+£)=70° 45'
a — b— 20° 50'; \(a — ^)=io° 25'
A-B=i2° 4' 26"
4—B)— 6° 2' 13"
log tan ^(A— ^=9.02430
log sin 4r(rt-f-/>)=9 97501
colog sin $(a — ^=0.74279
cot£ (7=9.74210
£ C- 61° 5' 32"
(7=122° II' 4"
EXAMPLE— CASE (3)
96. Given a = 78° 15' £ = 56° 20' C=I2O°
To find ^4, B, and r.
log sin £(a+ ^=9.96498
log cos £ (a + ^=9.58663
log si n^(« — ^=9.27897
log cos £ (a — ^=9.99201
log cot £(7= 9. 76144
+ £)=67° 17' 30
-3)= 10° 57' 30
To fin
Formula V/may be written
cosA(r7 — ^) cot
COS (7<
log COS^(fl! — ^)= 9.99201
log cot £ C= 9. 76 1 44
colog cos £ (« + b) — 0.41337
log tan $(A + 5) = 10. 16682
±(A-B)= 6° 47' 4"
A =62° 31' 40"
^=48° 57' 32"-
To find$(A-B\
Formula V^ may be written
sjnj>-^
log sin 1(^ — ^=9.27897
log cot£ (7=9.76144
colog sin£(rt +^)=o. 03502
io6
SPHERICAL TRIGONOMETRY
To find c.
From Formula VII, sin c=
sin b sin C
sin B
log sin£ =9.92027
log sin £=9.93753'
colog sin 2? =0.12249
log sin ^=9. 98029
<r=io7°8'
Check.
Formula III may be written
_ sin % (A + B} tan £ (a - b}
log sin %(A + B) = 9.91725
log tan \ (a — b) = 9.28696
colog sin %(A—B) — 0.92762
log tan £ r== 10. 1 3 1 83
\c= 53° 33' 56"-
^=107° 7' 51" —
(Discrepancy due to omitted decimals )
AMBIGUOUS CASES
97. (i.) Two sides and an angle opposite one of them are the
given parts.
If the side opposite the given angle differs from po° more than the
other given side, the given angle and the side opposite being either both
less or both greater than 90°, there are two solutions.
(2.) Two angles anc} a side opposite one of them are the given parts.
If the angle opposite the given side differs from 90° more than the
other given angle, the given side and the angle opposite being either
both less or both greater than 90°, there are two solutions.
Remark. — There is no solution if, in either of the formulas.
sin B=
sin A sin b
sin b sin A
sin a sin B
the numerator of the fraction is greater than the denominator.
OBLIQUE-ANGLED TRIANGLES
107
cos /-.
Formula V may be written
cot A C- s
EXAMPLE — CASE (6)
98. Given #=40° 16' £=47°44' ^=52° 30'
To find B, B', C, C, and ct c'.
To find B and B'.
Formula VII may be written
„ sin^4 sin/'
sm B= : .
sin a
log sin ^=9. 89947
log sin ^=9. 86924
colog sin fl=o. 1 8953
log sin jB=g. 95824
B= 65° 16' 30"
B' = 114° 43' 30"
To find c.
Formula IV may be written
tan£r=
log CO!
log tan £ (« + />) =9. 98484
colog cos $(A—B)= 0.002 70
log tan £ i'=g. 70080
£<r=26° 39' 42"
'=53° 19' 24"
To find c.
log tan |(^ + /')=9. 98484
colog cos £ (A— B') =0.06745
log tan ££•' =9.09860
*<•'= 7° 9' 9"
sin£(rt — b)
^)= 9.84177
log tan£(.4— -5)= 9.04901 n
colog sin £(rt — <£)= 1.1863311
log cot $C= 10.0771 1
^C=39o56'24"
C=79° 52' 48"
To find C,
logsin£(rt + £)= 9.84177
log tai4 (.4 -.#')= 9.7815311
colog sin ^ (a — ti\— 1. 18633 n
log cot \ C = 10. 80963
lc= 8° 48' 41"
Check.
Formula III may be written
sin B sin c
sin b= - . _ •
smC
log sin £=9. 95824
log sine =9. 904 1 8
colog sin C*=o. 00682
log sin ^=9.86924
^=47° 44'
<r' = i4° 18' 18"
EXERCISES
99. (i.) In the spherical triangle ABC, the side ^ = 124° 53', the
side b = 31° 19', and the angle A = 16° 26'. Find the other parts.
(2.) In the oblique-angled spherical triangle ABC, angle A = 128°
45', angie C= 30° 35', and the angle ,5 = 68° 50'. Find the other parts.
* The letter " n" indicates that these quantities are negative.
lo8 SPHERICAL TRIGONOMETRY
(3.) In the spherical triangle ABC, the side ^ = 78° 15', £=56° 20',
and A = 120°. Required the other parts.
(4.) In the spherical triangle ABC, the angle ,4 = 125° 20', the an-
gle (7 = 48° 30', and the side ^ = 83° 13'. Required the remaining
parts.
(5.) In the spherical triangle ABC, the side ^ = 40° 35', £ = 39° 10',
and a = 71° 15'. Required the angles.
(6.) In the spherical triangle ABC, the angle A = 109° 55', B— \ 16°
38', and C= 120° 43'. Required the sides.
(7.) In the spherical triangle ABC, the angle ^ = 130° 5' 22", the
angle C= 36° 45' 28", and the side £ = 44° 13' 45". Required the re-
maining parts.
(8.) In the spherical triangle ABC, the angle ^ = 33° 15' 7", B =
3l0 34' 38", and C= 161° 25' 17". Required the sides.
(9.) In the spherical triangle ABC, the side <r=ii2° 22' 58", £ =
52° 39' 4", and a = 89° 16' 53". Required the angles.
(10.) In the spherical triangle ABC, the side ^ = 76° 35' 36", b =
50° 10' 30", and the angle ^ = 34° 15' 3". Required the remaining
parts.
AREA OF THE SPHERICAL TRIANGLE
100. It is proved in geometry that the area of a spherical
triangle is equal to its spherical excess, that is,
area = (A + B + C— 2 rt. angles) X area of the tri-rectangular triangle,
where A, B, and C are the angles of the spherical triangle.
Hence
area _A+£-\-C— 180°
surface of sphere "~ 720°
The surface of the sphere is 477^, therefore
A + B+ C-180°\
The following formula, called Lhuilier's theorem, simpli-
fies the derivation of (A +jB+C—i8o°) where the three
OBLIQUE-ANGLED TRIANGLES 109
sides of the spherical triangle are given ; in it a, b, and c
denote the sides of the triangle, and 2s=
tan /- _ y'tan i s tan i (s-a) tan i(s-6) tan i (s-cj.
EXERCISES
(i.) The angles of a spherical triangle are, ^=63°, £=84° 21',
C=79°; the radius of the sphere is 10 in. What is the area of the
triangle ?
(2.) The sides of a spherical triangle are, a = 6.47 in., £ = 8.39 in.,
^ = 9.43 in.; the radius of the sphere is 25 in. What is the area of
the triangle ?
(3.) In a spherical triangle, ^ = 75° 16', ^ = 39° 20', c = 26 in.; the
radius of the sphere is 14 in. Find the area of the triangle.
(4.) In a spherical triangle, a = 441 miles, ^ = 287 miles, C = 38° 21';
the radius of the sphere is 3960 miles. Find the area of the triangle.
CHAPTER X
APPLICATIONS TO THE CELESTIAL AND TERRES-
TRIAL SPHERES
ASTRONOMICAL PROBLEMS
101. An observer at any place on the earth's surface
finds himself seemingly at the centre of a sphere, one-half
of which is the sky above him. This sphere is called the
celestial sphere, and upon its surface appear all the heavenly
bodies. The entire sphere seems to turn completely around
once in 23 hours and 56 minutes, as on an axis. The im-
aginary axis is the axis of the earth indefinitely produced.
The points in which it pierces the celestial sphere appear
stationary, and are called the north and south poles of the
heavens. The North Star (Polaris) marks very nearly (with-
in i° 16') the position of the north pole. As the observer
travels towards the north he finds that the north pole of the
heavens appears higher and higher up in the sky, and that
its height above the horizon, measured in degrees, corre-
sponds to the latitude of the place of observation.
The fixed stars and nebulae preserve the same relative
positions to each other. The sun, moon, planets, and com-
ets change their positions with respect to the fixed stars
continually, the sun appearing to move eastward among
the stars about a degree a day, and the moon about thir-
teen times as far.
AP PLICA TIONS 1 1 1
The zenith is the point on the celestial sphere directly
overhead.
The horizon is the great circle everywhere 90° from the
zenith.
The celestial equator is the great circle in which the
plane of the earth's equator if extended would cut the ce-
lestial sphere.
The ecliptic is the path on the celestial sphere described
by the sun in its apparent eastward motion among the stars.
The ecliptic is a great circle inclined to the plane of the
equator at an angle of approximately 23^°.
The poles of the equator are the points where the axis
of the earth if produced would pierce the celestial sphere,
and are each 90° from the equator.
The poles of the ecliptic are each 90° from the ecliptic.
The equinoxes are the points where the celestial equa-
tor and ecliptic intersect ; that which the sun crosses when
coming north being called the vernal equinox, and that
which it crosses when going south the autumnal equinox.
The declination of a heavenly body is its distance, meas-
ured in degrees, north or south of the celestial equator.
The right ascension of a heavenly body is the distance,
measured in degrees eastward on the celestial equator, from
the vernal equinox to the great circle passing through the
poles of the equator and this body.
The celestial latitude of a heavenly body is the dis-
tance from the ecliptic measured in degrees on the great
circle passing through the pole of the ecliptic and the
body.
The celestial longitude of a heavenly body is the dis-
tance, measured in degrees eastward on the ecliptic, from
112 SPHERICAL TRIGONOMETRY
the vernal equinox to the great circle passing through the
pole of the ecliptic and the body.
EXERCISES
(i.) The right ascension of a given star is 25° 35', and its decima-
tion is -f-(north) 63° 26'. Assuming the angle between the celestial
equator and the ecliptic to be 23° 27', find the celestial latitude and
celestial longitude.
In this figure AB is the celestial equator, AC the ecliptic, P the pole of
the equator, P' the pole of the ecliptic. ' S is the position of the star, and
the lines SB and SC are drawn through P and P' perpendicular to AB and
AC. AB is the right ascension and BS the declination of the star, while
AC is the longitude and SC the latitude of the star.
In the spherical triangle P'PS, it will be seen that P'S is the comple-
ment of the celestial latitude, PS the complement of the declination, and
P'PS is 90° plus the right ascension. It is to be noted that A is the ver-
nal equinox.
(2.) The declination of the sun on December 2ist is — (south)
23° 27'. At what time will the sun rise as seen from a place whose
latitude is 41° 18' north ?
The arc ZS which is the distance from the zenith to the centre of the sun
when the sun's upper rim is on the horizon is 90° 50'. The 50' is made up
of the sun's semi-diameter of 16', plus the correction for refraction of 34'.
AP PLICA TIONS 1 1 3
(3.) The declination of the sun on December 2ist is — (south)
23° 27'. At what time would the sun set as seen from a place in lati-
tude 50° 35' north ?
SUNRISE SUNSET
In these figures P is the pole of the equator, Z the zenith, EQ the celes-
tial equator. ASh the declination of the sun, ZS=qcP 50', PS—goP + dec-
lination, PZ= 90° -latitude. The problem is to find the angle SPZ. An
angle of 15° at the pole corresponds to I hour of time.
GEOGRAPHICAL PROBLEMS
102. The meridian of a place is the great circle passing
through the place and the poles of the earth.
The latitude of a place is the arc of the meridian of the
place extending from the equator to the place.
Latitude is measured north and south of the equator from o° to 90°.
The longitude of a place is the arc of the equator extend-
ing from the zero meridian to the meridian of the place.
The meridian of the Greenwich Observatory is usually taken
as the zero meridian.
Longitude is measured east or west from o° to 180°.
The longitude of a place is also the angle between the zero meridian and
the meridian of the place.
ii4 SPHERICAL TRIGONOMETRY
In the following problems one minute is taken equal to one geo-
graphical mile.
(i.) Required the distance in geographical miles between two
places, D and E, on the earth's surface. The longitude of D is 60°
15' E., and the latitude 20° 10' N. The longitude of E is 115° 20' E.,
and the latitude 37° 20' N.
In this figure A C represents the equator of the earth, P the north pole,
and A the intersection of the meridian of Greenwich with the equator. PB
and PC represent meridians drawn through D and E respectively. Then
AB is the longitude and BD the latitude of D ; AC the longitude and CE
the latitude of E.
(2.) Required the distance from New York, latitude 40° 43' N.,
longitude 74° o' W., to San Francisco, latitude 37° 48' N., longitude
122° 28' W., on the shortest route.
(3.) Required the distance from Sandy Hook, latitude 40° 28' N.,
longitude 74° i' W., to Madeira, in latitude 32° 28' N., longitude 16° 55,
W., on the shortest route.
(4.) Required the distance from San Francisco, latitude 37° 48'
N., longitude 122° 28' W., to Batavia in Java, latitude 6° 9' S., longi-
tude 1 06° 53' E., on the shortest route.
(5.) Required the distance from San Francisco, latitude 37° 48'
N., longitude 122° 28' W., to Valparaiso, latitude 33° 2' S., longitude
71° 41' W., on the shortest route.
CHAPTER XI
GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE
J.03. The given parts of a spherical triangle may be laid
off, and then the required parts may be measured, by making
use of a globe fitted to a hemispherical cup.
The sides of the spherical triangle are arcs of great circles,
and may be drawn on the globe with a pencil, using the
rim of the cup, which is a great circle, as a ruler. The rim
of the cup is graduated from o° to 1 80° in both directions.
The angle of a spherical triangle may be measured on a
great circle drawn on the sphere at a distance of 90° from
the vertex of the angle.*
CASE I. Given the sides a, b, and c of a spherical triangle,
to determine the angles A , B, and C.
Place the globe in the cup, and draw upon it a line equal
to the number of degrees in the side c, using the rim of the
cup as a ruler. Mark the extremities of this line A and B.
With A and B as centres, and b and a respectively as radii,
draw with the dividers two arcs intersecting at C (Fig. i).
Then, placing the globe in the cup so that the points A and
C shall rest on the rim, draw the line AC=b, and in the
same way draw BC=a.
To measure the angle A place the arc AB in coincidence
* Slated globes, three inches in diameter, made of papier-mache, and held
in metal hemispherical cups, are manufactured for the use of students of
spherical trigonometry at a small cost.
Ii6 SPHERICAL TRIGONOMETRY
with the rim of the cup, and make AE equal to 90°. Also
make AF in AC produced equal to 90°. Then place the
globe in the cup so that E and F shall be in the rim, and
note the measure of the arc EF. This is the measure of the
angle A. In the same way the angles B and C can be de-
termined.
CASE II. Given the angles A, B, and C, to find the sides
a, b, and c.
Subtract A, B, and C each from 180°, to obtain the sides
a1 ', b' , and c' of the polar triangle. Construct this polar tri-
angle according to the method employed in Case I. Mark
its vertices A', B' , and C '. With each of these vertices as
a centre, and a radius equal to 90°, describe arcs with the di-
viders. The points of intersection of these arcs will be the
vertices A, £, and C of the given triangle. The sides of
this triangle a, b, and c can then be measured on the rim
of the cup.
GRAPHICAL SOLUTION
117
CASE III. Given two sides, b and c, and the included angle
A, to find B, C, and a.
Lay off (Fig. 3) the line AB equal to c, and mark the
point D in AB produced, so that AD equals 90°. With the
dividers mark another point, F3 at a distance of 90° from A.
Turn the globe in the cup till D and Fare both in the rim,
and make DE equal to the number of degrees in the angle A.
With A and E in the rim of the cup, draw the line AC equal
to' the number of degrees in the side b. Join C and B. The
required parts of the triangle can then be measured.
FIG. 3
FIG. 4
CASE IV. Given the angles A and B and the included side
c, to find a, b, and C.
Lay off the line AB equal to c. Then construct the given
angles at A and B, as in Case III., and extend their sides to
intersect at C.
CASE V. Given the sides b, a, and the angle A opposite one
of these sides, to find c, B, and C. (Ambiguous case.)
Ii8 SPHERICAL TRIGONOMETRY
Lay off (Fig. 4) AC equal to b, and construct the angle A
as in Case III. Take c in the dividers as a radius, and with
C as a centre describe arcs cutting the other side of the tri-
angle in B and B' , and measure the remaining parts of the
two triangles.
If the arc described with C as a centre does not cut the other side of the
triangle, there is no solution. If tangent, there is one solution.
CASE VI. Given the angles A, B, and the side a opposite
one of the angles.
Construct the polar triangle of the given triangle by
Case V. ; then construct the original triangle as in Case II.,
and measure the parts required.
The constructions given above include all cases of right and quadrantal
triangles.
CHAPTER XII
RECAPITULATION OF FORMULAS
ELEMENTARY RELATIONS (§ IO)
sin x * COSJT
tan x = - , cot x = — - ,
COSJT sinx
i i
sec x = -- , esc x =
cos.r
tan x cot x = i ,
sin3 x -\- cos2 x=. i,
i + cot2 x = csca x.
RIGHT TRIANGLES (§§ 14 AND 2 7)
cos A = - , cos B = - ,
|, tan # = -,
cot A = - , cot B = -, ,
a b
where c=. hypotenuse, a and b sides about the right angle; A and B
the acute angles opposite a and £.
FUNCTIONS OF TWO ANGLES (§§ 30-34)
sin (x-\-y)=.s\nx cos^-j-cos^r sinj,
sin (x— y) = sinx cos y — cos x siny,
cos (x -\-y) = cos x cosy — sin x siny,
cos (.r— j) = cos.r cosj-j-sin^r si
120 RECAPITULATION OF FORMULAS
tan (.r-h)/)=-
i — tan^r
tan x — tan y
tan (.r— y) = — — —
i-f tan^r tanj/
cot^r cot/— i
cot^r cot y+ i
cot (.r — /)= — — .
cot/ — cot x
FUNCTIONS OF TWICE AN ANGLE (§ 36)
sin 2x = 2 sin^r cos,r,
= 2 cos2.r — i,
2 tan .r
tan 2x =
cot 2;r =
i — tan x
cot'.r— i
2 cot x
FUNCTIONS OF HALF AN ANGLE (§ 37)
cos
i — cos x
SUMS AND DIFFERENCES OF FUNCTIONS (§ 38)
sin // -f~ sin ^ = 2 sin ^ (« -+- v) cos ^ (« — T/),
sin u — sin7/ = 2 cos^(« + z/) sin \(u — v\
COS U -f- COS W = 2 COS £ (« + 7/) COS £ (« — 2/),
cos u — cos ?/ = — 2 sin $ (w + v) sin £ (w — v).
sin u -f sin y _ tan \ (u -f ^)
sin u — sin v ~~ tan \ (u — v] '
RECAPITULATION OF FORMULAS 121
OBLIQUE TRIANGLES (§§ 42-45)
a_sin A a__s'mA b sin B
6~sinL" £~s'mC' <r~sinC'
a — ^tan|(^~ £)
tan£C' =
'.'—*)
a+fi+c '
where s=
where AT=
AREA OF A TRIANGLE (§ 46)
=|rtr sin ^. 5"=^ sin C 5=^ sin ^4.
LOGARITHMIC, COSINE, SINE, AND EXPONENTIAL SERIES
(§58)
=*- +~+> etc>
122 RECAPITULATION OF FORMULAS
,.
r2 r3 x*
^=i+.r+- + _+-+,etc.
DE MOIVRE'S THEOREM (§ 71)
- ~ * - cosw~3 ^ sin3^+, etc.
(« — i )
cos «.r rr >z cos .r -- - — ? — - CO8* jr sin2^-f, etc.
HYPERBOLIC FUNCTIONS (§ 75)
ex-e~x
.
cos /.r = — =cosh jr.
2
SPHERICAL TRIANGLES
RIGHT AND QUADRANTAL TRIANGLES (§§ 83, 87)
Use Napier's rules.
OBLIQUE TRIANGLES (§§ 89-93)
cos a =. cos b cos c -j- sin b sin c cos ^4.
cos A = — cos # cos C-\- sin ^ sin C cos #.
sin s sin (.$• — a)
RECAPITULATION OF FORMULAS 123
•v;
tan | a -cos cos -
cos (S—£) cos(S-C)
tan c
sn ^ —
cos (y — ) tan
sin $(a-\-&) cot ^
sini(« — b) tani(A —
cos$(a-\-t>)_ cotj C
cos £ (# — ^)~"tan £ (A-\-£)'
s\na __ sin If
sin A ! ~sin B'
AREA OF SPHERICAL TRIANGLES (§ lOl)
tan (*' —
\ 4
APPENDIX
RELATIONS OF THE PLANE, SPHERICAL, AND PSEUDO-
SPHERICAL TRIGONOMETRIES
We have up to the present considered the trigonometries
which deal with figures on a plane or spherical surface. A
characteristic feature of these two surfaces is that the curv-
ature of the plane is zero, while that of the sphere is a posi-
tive constant p. If the radius of the sphere is increased in-
definitely, its surface approaches the plane as a limit while
its curvature p approaches o.
-In works on absolute geometry it is shown that there ex-
ists a surface which has a constant negative curvature : it is
called a pseudo-sphere, and the trigonometry upon it pseudo-
spherical trigonometry.
We observe that as p passes continuously from positive
to negative values, we pass from the sphere through the
plane to the pseudo-sphere. Thus the formulas of plane
trigonometry are the limiting cases of those of either of the
two other trigonometries.
In the treatment of spherical trigonometry the radius of
the sphere has been taken as unity. If, however, the radius
of the sphere is r, and a, b, and c denote the lengths of the
sides of the spherical triangle, the formulas are changed, in
that a is replaced by -, b by -, and c by - ; thus,
126
APPENDIX
becomes
. „ sin^r
sin C= —
sin a
. c
sin-
r
. a
sm-
r
The formulas for pseudo-spherical trigonometry are the
same as the formulas of spherical trigonometry, except that
the hyperbolic functions of -, -, and - are substituted for
the trigonometric.
Thus, corresponding to the above formula of spherical
trigonometry, is the formula
sin C =
of pseudo-spherical trigonometry.
The pseudo-sphere is generated by revolving the curve whose equation is
r-\- vr't — x1
y=r log
*A>
about its y axis. The radius of the base of the pseudo-sphere is r.
APPENDIX 127
Hence the formulas of plane trigonometry can be derived
from the formulas of either spherical or pseudo- spherical
trigonometry by expressing the functions in series and al-
lowing r to increase without limit.
Example.— Show that if r is increased indefinitely the following
corresponding formulas for the spherical and pseudo-spherical right
triangle
a be
cos = cos - cos - » (i)
, a i b 1 c
cosh - = cosh- cosh-, (2)
r r r
reduce to the corresponding formula for a plane right triangle; that
is, to
a>=F+c\ (3)
Substituting the series cos -, etc., in equation (i), we obtain
I rt! , 1 a4 , I If I S i A4
'-i! 7< +4-, S + ' ' • = '- T\ ?" 7 + r, ?+ ' ' ' ^
Substituting in equation (2) the series for cosh - , etc. , which we obtain from
cosh x = - , we have
Cancelling i in equations (4) and (5), multiplying by r2, and, finally, allowing
r to increase without limit, we get from either equation
EXERCISES
Derive each of the following formulas of plane trigonometry from
the corresponding formula of spherical trigonometry, and also from
the corresponding formula of pseudo-spherical trigonometry.
123 APPENDIX
Right triangles ; A = right angle.
(i.) Plane, sin C=^
sin c
Spherical, sin C = -
sin a
Pseudo-spherical, sin C= . h '
Oblique Triangles.
(2.) Plane, # 2 =£2 + <r2 — 2 be cos A
Spherical, cos a = cos £ cos c-\- sin £ sin <r cos A
Pseudo-spherical, cosh # = cosh b cosh <: -J- sinh b sinh ^ cos A.
(3.) Plane, S=Vs (s-a) (j- J) (j-f).
Spherical,
(^ + 3+ C- .800) = i ^ £ tan (J-.^ ^ ^a-g ^ ^^-^
4 * r r y y
Pseudo-spherical,
(s-a) , (s-6)
i_2 tanh ^ 1— -i tanh | L-
§ 4 (page 3).
(i.) 192° 51' 25f .
Quadrant III.
(2.) 25°.
(3-) 287°, 647^.
(4.) Quadrant III.
§ 9 (page 9).
tan 1000° is negative,
cos 8 10° is o.
sin 760° is positive,
cot — 70° is negative,
cos — 550° is negative,
tan — 560° is negative,
sec 300° is positive,
cot 1560° is negative,
sin 130° is positive,
cos 260° is negative,
tan 310° is negative.
§ 13 (page n).
(3.) cos — 30° = ^ v/3-
tan -3o° = -iy'3.
COt — 30° = — v/3,
sec-3op=§yg;
CSC —30°= — 2.
(4.) cos.r= -§ v/2,
tan x = i v/2,
COt X = 2 V/2,
sec.r = — I v/2,
esc .r = — 3.
9
ANSWERS TO EXERCISES
(5-)
cot y = — §, sec y = J,
(6.) sin 60° = £ v/3.
tan 60° = v/3,
cot 60° = i ^3.
sec 60° = 2,
esc 60° = f v/3.
(7-) cos o° = i , tan o° = o.
(8.) sin 2 = |, cos ,3- = !,
esc ^ = f .
(9.) sin 45° = cos 45° = | -/2,
tan 45°= i,
sec 45° = esc 45° = v/2.
(10.) sin^ = — £ v/5, cosj/ = — f,
cot_>/ = f v/5, sec j = — |,
(11.) sin3o° = i co
tan 30° = i- v/3,
sec 30° = | v/3,
CSC 30° = 2.
(12.)
=— f.
§ 17 (page 14).
(i .) sin 70° = cos 20°,
cos 60° = sin 30°,
cos 89° 31'= sin 29',
cot 47°= tan 43°,
'3°
ANSWERS TO EXERCISES
tan 63°= cot 27°,
sin 72° 39'= cos 17° 21'.
(2.) .r = 300.
(3.) * = 22°30'.
(4.).r=i8°.
(5.) jr=is°.
§ 25 (page 21).
(i.) 225° and 315°,
60° and 240°.
(2.) 60°, 120°, 420°, 480°.
(3.) sin-3o°=-i
cos — 30°=^ -v/3,
sin 765°= cos 765 = £ -v/2,
sin 1 20°= £ -v/3,
cos 1 20° = — |,
sin 210°= — ^,
cos 2io°=— £ -y/3-
(4.) The functions of 405° are
equal to the functions of 45°.
sin 6oo°= — |- \/3»
cos6oo°=— i
tan 600° = -Y/3.
cot 6oo°=£ -v/3,
sec 6oo°= —2,
esc 6oo°= — f -v/3.
The functions of 1125° are
equal to the functions of 45°
sin — 45° = — ItV*.
cos- 45°= i -v/2,
tan — 45°= cot — 45°= — i ,
sec — 45°=-v/2,
csc — 45°= — -v/2.
sin 225°= cos 225°= — £ V2
tan 225°= cot. 225°= i,
sec 225°= esc 225°= — v/2.
(5.) The functions of — 120° are
the same as those of 600°
given in (4).
sin — 225° = £ 1/2,
cos — 225° = — £ -v/2,
tan — 225°= cot — 225°= — i,
sec — 225°= — "v/2,
esc — 225°= -v/2,
sin — 420° = — £ \/3,
cos — 420° =\,
tan — 420° = — y^
cot— 420°=— iVi
sec — 420° = 2,
csc-42o° = -t-v/3^
The functions of 3270° are
equal to the functions of 30°.
(6.) sin 233° = — cos 37°,
cos 233° — — sin 37°,
tan 233° — cot 37°,
cot 233° = tan 37°,
sec 233° = — esc 37°,
esc 233° = — sec 37°.
sin — 1 97° = sin 17°,
cos — 1 97° = — cos 1 7°,
tan — 197° = — tan 17°,
cot— 197° = — cot 17°,
sec — 1 97° = — sec 17°,
esc— 1 97° = esc 17°.
sin 894° = sin 6°,
cos 894° = — cos 6°,
tan 894° = — tan 6°,
cot 894° = — cot 6°,
sec 894° = — sec 6°,
esc 894° = esc 6°.
(7.) sin 267°=:— sin 87°,
tan — 254° = — tan 74°,
cos 950° = — cos 50°.
(8.) —0.28.
ANSWERS TO EXERCISES
(9.) 2 sin2 x.
(10.) i -f-seca x.
(ii.) sin (*— 90°)= — cos. r,
cos(.r — ooc) = sin-r,
tan (JT — 90°) = — cot x,
cot (JT — 90°) = — tan x,
sec (x — 90°) = esc x,
esc (.r — 90°) = — sec x.
$ 28 (page 24).
(I.) a =62.324,
^ = 32° 52' 40".
(2.) £ = 21.874,
^ = 39° 45' 28",
#=50° 14' 32".
(3.) <* = 300.95.
£ = 683.96,
£ = 66° 15'.
(4.) £ = 26.608,
*• = 45-763.
^ = 35° 33'-
area = 495. 34.
(5.) £ = 3-9973-
? = 4.1537,
^ = 1 5° 46' 33".
area = 2. 257.
(6.) £ = 0.01729.
(7.) <* = 298.5.
(8.) ^ = 39° 42' 24".
(9.) ^- = 2346.7.
(10.) # = 28° 57' 8".
(if.) 444.16 ft.
(12.) 186.32 ft.
(i 3.) 34° 33' 44".
114.) 303.99 ft.
(15.) 238.33 ft.
(16.) 15 miles (about).
(17.) 79,079 ft.
(18.) 165.68 ft.
(I9-) 53° 33'-
(20.) 115.136 ft.
(21.) 76.355 ft.
(22.) £ = 80° 32",
A = C = 49° 59' 44".
(23.) #=53°i6'36",
£= 12.0518 in.,
area = 72. 392 sq. in.
(24.) £= 130.52 in.,
area = 24246 sq. in.
(25.) 23.263 ft.
(26.) 1 7° 48".
(27.) 5.3546 in.
(28.) 1084950 sq. ft.
(29.) 17 ft., 885 sq. ft.
(30.) radius = 24.882 in.,
apothem = 20. 1 3 in.,
area= 1472 sq. in.
(31.) 12.861.
(32.) 1782.3 sq. ft.
(33.) 38168 ft.
(34.) 20.21 ft.
(35.) 2518.2 ft.
§ 29 (page 28).
(I.) ^ = 22° 58',
£ = 7.07,
c = 9.0046.
(2.) £ = 79-435.
A = 45° 27' 14",
C = 95° 24' 46".
(3.) ^^ = 7.6745,
^#' = 2.6435,
^ = 46° 43' 50",
2? ' = 133° 16' 10",
ACB^ 105° 53' 10",
ACB' = 19° 20' 50".
(4.) ^ = 37° 53'-
# = 43° 52' 25",
132
ANSWERS TO EXERCISES
C = 98-14' 35"-
(5.) 902.94.
(6.) 1253.2 ft.
(7.) 357-224 ft.
(8.) ^ = 44° 2' 9".
^ = 51° 28' ii",
C = 84° 29' 40",
area = 126100 sq. ft.
(9.) 407.89 ft.
(io.) B=i2\° 7' 16",
C = 92° 20' 38",
D = J\° n' 6".
(11.) #^ = 6.6885,
v 3V/5 + :
cos (* —y) — 2JL5 —
§ 39 (Page 37).
(5.) sin (45°-*) =
£ 1/2 (cos * — sin *),
cos (45°—*)=:
1 1/2 (cos * + sin*),
sin (45°+*) =
1/2 (cos* + sin*),
§ 34 (page 34).
(2.) sin (45°+ x) =
£l/2 (cos * + sin*),
cos (45°+ f) =
4 I/2 (cos .r — sin*),
sin (30°—*) =
£ (cos*— 1/3 sin*),
cos (30°— *)_ =
i (VX3 cos * + sin*), |
sin (6o°+*)_=
$ (y'3 cos* -f- sin*),
cos (60°+*) =
£ (cos* — -y/3 sin*).
(3.) sin(*+y)=ff,
sin *— ) = -
(4.) sin 75° =
cos 7 5° =
(.5-) sin 15°=
4
1/6+ 1/2"
(15.) sin 2* = — ff,
cos 2* = — T/5.
(i 6.) sin 22|° = 2 -1/2.
(170
cos i5 =
ANSWERS TO EXERCISES
133
tan i5° = 2 — v/3,
cot 15° = 2-f- V^.
sec 1 5° = 2^/2 — \/3,
CSC I5° = 2*/2 + -v/3.
(20.) sin 5.r =
5 sin .r — 20 sin3 .r
+ 1 6 sin5.r.
(21.) cos 5_r =
5 cos x — 20 cos3 x
4- 1 6 cos5 ,r.
(23.) The values of .r <3oo° are
o°, 30°, 1 50°, 1 80°, 210°, 330°.
(36.) tan,v tan^/.
§ 41 (page 40).
(i.) sin-«i ^2=45°. i
45°+ 360°, etc.,
cos- i £ = 60°, 300°, etc.,
tan-' (—0= 1 35°. 3 1 5°. etc.,
cos — 1 i =0°, 360°, etc.,
sin -i (— |) = 210°, 330°, etc.
(2.) tan,r = 3-
(3.) cos.r = d
(5.) sin (cos-1 $) = ±jj.
(6.) cot (tan- 1 3V) =17.
(7.) « = i\/3.
(8.) 45°, 225°.
(9.) ,r = 45°,_y =180°.
(10.) sin — Irt = 225°.
§ 48 (page 46).
(i.) C=i2i°33',
^ = 2133.5,
c = 2477.8.
(2.) C=55°4i',
^ = 534.05,
^- = 653.52.
(3.) C=45°34/,
a= 1548.1,
(4.) ^ = 105° 59',
a =54.018,
^• = 47.738.
(5.) ^ = 68° 58',
^ = 5274.9,
£• = 3730.
(6.) ^ = 54° 58'.
a = 923.4,
c= 1 187.7-
§ 49 (page 47).
(i.) (I.) Two solutions.
(2.) One solution, a right tri-
angle.
(3.) One solution..
(4.) Two solutions.
(2.) Z?=i6°57'2i",
C= 1 5° 50' 39"'.
£•=1:0.32122.
(3.) £- = 2.5719,
B=IT° 15' i",
C=i42° 1 3' 59".
(4.) £- = 93-59. c' = 54-069,
B = 26° 52' 7", B'= 1 33° 7' 53",
C = i3i°46/53",C/=25°3i/7'/.
(5.) No solution.
(6.) £= 1.0916, ^'=0.36276,
B = 1 1 7° 50' 44", B'= 1 7° 5' 1 6".
§ 50 (page 48).
(i.) a = 0.0971,
#=90° 35' 36",
5 = 0.0053261.
134
ANSIVERS TO EXERCISES
(2.) C — 14.211,
^ = 48° 44' 32",
A = 76° 20' 5",
C= 95° 1 5' 56",
£ = 44° 52' 55"
5 = 0.60709.
5 = 80.962.
(3.) £ = 85.892,
§ 52 (page 50).
A = 67° 2l'42",
(i.) 1116.6 ft.
C = 62°48' 18",
5=3962.8.
(4.) « = 0.6767,
(2.) 308 1. 8 yards.
(3-) 638.34 ft.,
14653 sq. ft.
5= 15° 9' 2l",
(4.) 4.1 and 8.1.
C= 131° 19' 39",
(5.) 13.27 miles.
5=0.08141.
(6.) 6667 ft. One solution.
(5.) <: = 72.87,
(7.) 121.97.
^ = 40° 50' 32",
(8.) 44° 2' 56".
^=11° 2' 28",
5 = 422.65.
(9.) 32.151 sq. miles.
(11.) 54° 29' 12".
(12.) a = 12296 ft.,
§ 51 (page 49).
r= 13055 ft.
(i.) ,4 = 55° 20' 42",
^=106° 35' 36",
C=i8°3'42",
5=267.92.
(2.) A = 34° 24' 26",
B = 73° H' 56",
C=72°20' 36",
5=3.6143.
(13.) 294.77 ft.
(14.) 222.1 ft.
(16.) 42Q2J^ftr 4- '
(17.) 72.613 miles,
(i 8.) 50.977 ft.
(19.) 0.85872 miles.
(20.) 2.98 miles.
(21.) 1393.9 ^.
(22.) 8.2 miles.
(3.) /* = 52° 20' 24",
B= 107° 19' 14",
(23.) 187.39 ft.
(24.) 0.6011.
C = 20° 20' 24",
(25.) 4.8112 miles.
k
5= 1437.5.
(4.) A =97° 48',
(26.) 60° 51' 8".
#=l8°2I 48",
(27.) 37.365 ft.
C=63° 50' 12",
(28.) 3.2103 miles.
5=193.13.
(29.) 10.532 miles.
(5.) A = 54° 20' 16",
(30.) 851.22 yards.
B = 70° 27' 46",
(31.) 9.5722 miles.
C=54°72',
(32.) 6.1271 miles.
5 = 6090.
(33.) 280.47 ft.
(6.) A = 35° 59' 30",
(34-) 123.33 ft.
ANSWERS TO EXERCISES
135
(35-) 4-8ii2 miles.
(36.) 2666.1 ft.
* 53 (page 56).
(i.) 30° = 0.5236,
45° = 0.7854,
60° = 1.0472,
I 20 = 2.0944,
135°= 2.3562,
720°= 12.5664,
990° =17.2788.
<2.) I = 22° 30',
1O
| = 28^ 38' 53",
} = 100° 16' 4".
(3.) 1.35,0.54.
§ 74 (page 73).
(i.) sin 4,1- = 4 cos3. i~ sin x
— 4 cos.r sin3^-,
cos 4,1- = cos4 x
— 6 cos2 x sin2 x -j- sin4 x.
(2.) sin 6x = 6 cos5 .r sin .r
— 20 cos3 a- sin3.r
-|-6 cos.r sin5^-,
cos 6x = cos6 .r
— 15 cos*.r sin2^-
-|- 1 5 cos1' x sin4 x — sin6 x.
1 1 \ i- v 1 I „• V 3
(3-)-ro— '• -ri = \ -r z — >
(4.) .r0=i, ^ = o. 3090+ / 0.951 1,
-fa = — 0.8090 -|- /' 0.5878,
.r3 = — 0.8090 — / 0.5878.
.r4 = o. 3090 — / 0.951 1.
§ 77 (page 78).
(23.) .r = 3o°.
(24.) ^ = 30°.
(25.) x = o° or 45°.
(26.) A- = 6o°.
(27.)_y = 45°.
(28.) j = 45°-
(29.) -r = 45°.
(30.) .r = 3o°.
(31.) ^- = 60°.
(32.) ^- = 30°.
(33.) No angle < 90°.
(34.) jr = 3o-.
(35.) sin 92° = cos 2°.
(36.) cos 127° = — sin 37°.
(37.) tan 320° = — tan 40°.
(38.) cot 350° = — cot ioj.
(39.) sin 265° = — cos 5°.
(40.) tan 171°= -tan 9°.
(41.) cos.r= -
tan. r ^ —
esc -r —
(42.)
(43.) sin.r = —
cos x-~
cot .1- = f , sec .r = —
136
ANSWERS TO EXERCISES
(44.) sin,r = — 7^- -v/74.
= — f, sec jt- =
(45.) Quadrant II or IV.
(46.) Quadrant I or II.
(47.) Quadrant III or IV.
(48.) Quadrant I or II.
(49.) .r = o°, 120°, 1 80°, 240°.
(50.) .r = 3o°, 135°, 150°, 315°.
(51.) ,r = o°, 90°, 120°, 1 80°, 240
270°.
(57-) o.
(58.) a.
(59.) 2 (<*—£).
(60.) i(fl'-J9).
§ 78 (page 80).
(i.) 306.32 ft.
(2.) 831.06 ft.
(3.) 53° 28' 14".
(4.) 49.39 ft.
(5.) 0.43498 mile.
(6.) 209.53 ft.
(7.) 7-3188 ft-
(8.) 37° 36' 30".
(9.) 109,28 ft.
(10.) 502.46 ft.
(u.) 6799.8ft.
(12.) 219.05 ft.
(13.) 49i.76ft.
(14.) 50° 32' 44".
(15.) 49° 44' 38".
(i 6.) 34-063 ft.
(17.) 32.326 ft., 29° 6' 35".
(18.) 5.6569 miles an hour.
(19.) 56.295 ft.
(20.) 103.09 ft.
(21.) 71° 33' 54".
(22.) 858,160 miles.
(23.) 238,850 miles.
(24.) 2163.4 miles.
(25.) 90,824,000 miles.
(26.) 432.08 ft.
(27.) 60.191 ft.
(28.) 0.32149 mile.
(29.) 193.77 ft.
§ 79 (page 83).
(i.) 3416 ft.
(2.) 3.7865 ft.
(3.) 20.45 ft-
(4.) 36.024^.
(5.) 8.6058 sq. ft.
(6.) 181.23 in.
(70 2-9943 ft.
(8.) 5.1311 in.
(9.) 25.92 ft.
(io.) 92° i' 24",
(11.) 1.2491.
(12.) 33° 12' 4".
(13.) 11248 ft.
(14.) 0.60965 miles.
(15.) 1.3764.
(16.) 1.9755-
(17.) 19.882.
(i 8.) 0.9397.
(19.) 6.4984.
(20.) 3.4641.
(21.) 6.1981.
(22.) 6.9978.
(23.) 15.25.
§ 80 (page 84).
(78.) X — OO°, I 20°, 2400, 270°.
(79.) ,r = o°, 20°, 45°, 90°, 100°,
I35C, 140°, 1 80°, 220°,
225°, 260°, 270°, 315°,
340°.
ANSWERS TO EXERCISES
137
(8o.) ^- = 0°, 30°, 90°, 150°, 1 80°,
(24.) 55.74 ft.
270°.
(25.) 247.52 ft.
(8 1.) _r = o°, 45°, 120°, 240°, 225°,
(26.) 556.34 ft.
270°.
(27.) 465.72 ft.
(82.) ;r = o0, 90°, 1 80°, 270°.
(28.) 109.22 ft.
(83.) .r = 0°, 90°, 210°, 330°.
(29.) 2639.4 ft.
(84.) ^- = 240°, 300°.
(30.) 396- 54 ft.
(85.) x = 2ioP, 330°-
(31.) 287.75 ft.
(86.) x = o°, 90°.
(32.) 2280.6 ft.
.(87.) ;r = o°, 1 80°.
(33.) 64.62 ft.
(88.) * = o°, 1 80°.
(34.) 127.98 ft.
(89.) ^ = 0°, 90°, 120°, 1 80°, 240°, (35.) 45-l83 ft-
270°.
(36.) 4365-2 ft.
(90.) ^- = 450,1350,2250,3150.
(37.) 140.17 ft.
(91.) ^ = 30°, 150°, 210°, 330°.
(38.) 610.45 ft-
(39.) 1 56.66 ft.
§ 81 (page 88).
(40.) 41° 48' 39" and 125° 25
(i.) 2145.1 ft.
(41.) 51,288,000.
(2.) 12.458 miles.
(42.) 366680.
(3.) 1.1033 miles.
(43-) U586.
(4.) 1508.4 ft.
(44.) 947460.
(5-) I7I9-3 yards.
(45.) 0.89782.
(6.) 1.2564 miles.
(46.) 9929-3-
(7.) 1346.3^.
(47.) 7 5 1. 62 sq.ft.
(8.) 387.1 yards.
(48.) 3H5-9-
(9.) 5.1083 miles.
(49.) 855.1.
(10.) 3791-8 ft.
(50.) 876.34.
(u.) 4.4152 ft-
(12.) 28° 57' 20".
§ 88 (page 98).
(13.) 115.27.
(i.) ^=54° 59' 47",
(14.) 44.358 ft.
^ = 45° 41 '28",
(15.) 92.258 ft.
s~* /* o p> ' rR''
(16.) 101° 32' 16".
(2.) C=7i° 36' 47".
(17.) 0.83732 mile.
^ = 95° 22';
(18.) 539.1 ft.
c = 7i° 32' 14",
(19.) 1.239.
(3.) C=64° 1 4' 30",
(20.) 152.31 and 238.3.
C'— 115° 45' 3° .
(21.) 68.673 ft-
.00 ^^' rr"
a m 4^ 22 55 >
(22.) 32.071 ft.
*' = i3i°37' 5".
(23.) 13778ft.
^=42° 19' 17".
57'
1 38
ANSWERS TO EXERCfSES
c = 137° 40 43"
(4.) C- 65° 49' 54"
a = 63° 10' 6",
£ = 38° 59' 12".
(5.) a = 7S° 13' i",
^=58° 25' 46",
(6.) a = 76° 30' 37",
£ = 65° 28' 58,"
<r = 55° 47' 44".
3'
(8.)
(9-)
, = 64° 36' 39",
= 47° 57' 45"
'-96° 1 3' 23",
= 73° 1 7' 29",
= 70° 8' 38".
= 66° 58',
(10.) tf = 6i°4' 55",
b = 40° 30' 22",
<r=5o° 30' 32".
§ 99 (page 107).
(2.) a
b
(3-) a
(4.) B
131° 36' 36",
116° 36' 38",
29° 1 1' 42".
107° 7' 45",
48° 57' 29",
62° 31 '40".
62° 54' 43",
114° 30' 26",
= 56° 39' 10".
a=
(5.) ^ = 130° 35' 56",
^ = 30° 25' 34",
C = 3i° 26' 32",
(6.) ^=98° 21' 22",
b-=. 109° 50' 8",
*•= 115° 13' 4".
(7.) B = y.° 26' 9'',
a= 84° 14' 32",
^ = 51° 6' 12".
(8.) tf-8o° 5' 8",
£ — 70° 10' 36",
r = i45°5'2".
(9.) .4=70° 39' 4",
# = 48° 36' 2",
C=ii9° 15' 2".
(10.) a = 40° o' 12",
^ = 42° 15' ii",
C=i2i° 36' 19".
§ 100 (page 109).
(i.) 80.895 sq- in-
(2.) 26.869 sq. in.
(3,) 158.41 sq. in.
(4.) 39990 sq. miles.
§ 101 (page 112).
(i.) 5C = 48° 2' 43",
^=52° 53' 9".
(2.) 7 : 24 A.M.
(3.) 4 P.M.
§ 102 (page 114).
(I.) 3029^ miles.
(2.) 2229.8 miles.
(3.) 2748.5 miles.
(4.) 7516.3 miles.
(5.) 5108.9 miles.
THE END
LOGARITHMIC
AND
TRIGONOMETRIC TABLES
FIVE-PLACE AND FOUR-PLACE
PHILLIPS-LOOMIS MATHEMATICAL SERIES
LOGARITHMIC
AND
TRIGONOMETRIC TABLES
FIVE-PLACE AND FOUR-PLACE
BY
ANDREW W. PHILLIPS, PH.D.
AM)
WENDELL M. STRONG, PH.D.
YALE UNIVERSITY
NEW YORK AND LONDON
HARPER & BROTHERS PUBLISHERS
1899
THE PHILLIPS-LOOMIS MATHEMATICAL SERIES.
ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By
ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale
University. Crown 8vo, Cloth.
ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D.,
and IRVING FISHER, Ph.D., Professors in Yale University. Crown
8vo, Half Leather, $1 75. [By mail, $1 92.]
ABRIDGED GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [By
mail, $1 40.]
PLANE GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING
FISHER, Ph.D. Crown 8vo, Cloth, 80 cents. {By mail, 90 cents.]
LOGARITHMIC AND TRIGONOMETRIC TABLES. Five Place
and Four- Place. By ANDREW W. PHILLIPS, Ph.D., and WENDELL
M. STRONG, Ph.D., Yale University. Crown 8vo.
LOGARITHMS OF NUMBERS. Five-Figure Table to Accompany
the "Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D., Professors in Yale University. Crown 8vo,
Cloth, 30 cents. [By mail, 35 cents.]
NEW YORK AND LONDON :
HARPER & BROTHERS, PUBLISHERS.
Copyright, 1898, by HARPER & BROTHERS.
All rights reserved.
CONTENTS
TABLE pAGE
INTRODUCTION TO THE TABLES v
I. FIVE-PLACE LOGARITHMS OF NUMBERS i
II. FIVE -PLACE LOGARITHMS OF THE TRIGONOMETRIC
FUNCTIONS TO EVERY MINUTE 29
III. FIVE-PLACE LOGARITHMS OF THE SINE AND TANGENT
OF SMALL ANGLES 121
IV. FOUR-PLACE NAPERIAN LOGARITHMS 131
V. FOUR-PLACE LOGARITHMS OF NUMBERS 135
VI. FOUR -PLACE LOGARITHMS OF THE TRIGONOMETRIC
FUNCTIONS TO EVERY TEN MINUTES 139
VII. FOUR -PLACE NATURAL TRIGONOMETRIC FUNCTIONS
TO EVERY TEN MINUTES 149
VIII. SQUARES AND SQUARE ROOTS OF NUMBERS 159
IX. THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF
NUMBERS FROM o TO 2.5 AT INTERVALS OF .1 . . 160
X. CONSTANTS— MEASURES AND WEIGHTS AND OTHER
CONSTANTS . 161
INTRODUCTION TO THE TABLES
COMMON LOGARITHMS.
1. The common logarithm of a number is the index of
the power to which 10 must be raised to give the number.
Thus, log IOG = 2, because 100 = io2
log i =o, " i =10°
log .1 = — i, .1 = io-I
log 3 =47712, " 3 =io-47m
In general, log m — x if ;;/ = io*.
2. To multiply two numbers, add their logarithms. The
result is the logarithm of the product.
Proof. — Ifaw = io* so that log m = x,
and n = io> " " log n =y,
then mn = io*+*" " log mn = x+y.
Hence log mn = \ogrn -f log n.
3. To divide one number by another, subtract the loga-
rithm of the divisor from the logarithm of the dividend.
The result is the logarithm of the quotient.
Proof.— — = •
Hence log^~ =x~?
4. To raise a number to a power, multiply the logarithm
of the number by the index of the power. The result is the
logarithm of the power.
vi INTRODUCTION TO THE TABLES.
Proof. — ma = ( i ox)a = i o-ax ;
Hence \ogma = ax = a logm.
5. To extract a root of a number, divide the logarithm of
the number by the index of the root. The result is the loga-
ritJini of the root.
Proof.— "Im =-. * Ao* = 10*.
*v^>s A. / "" ~ 7 »
b b
6*. Restatement of laws :
log nin = log in + log n ;
log— = logm — logn ;
log ma = a log m ;
7. Most numbers are not integral powers of 10; hence
most logarithms are of decimal form.
Thus, log 2. 2 — .34242, Iog4 =1.60206.
S. If a logarithm is negative, it is expressed for conven-
ience as a negative integer plus a positive decimal.
The logarithm of a number less than I is negative.
The negative integer is usually expressed in the form
9—10, 8— 10, etc.
Thus, Iog.2i544 = — i -f .33333, written 9-33333 - 10 :
Iog.o2i544 = — 2 -h. 33333, " 8.33333—10;
log .0021 544 = — 3 + .33333, " 7-33333 ~ 10.
Remark. — In some books the negative integer is written i, 2, etc.,
instead of 9— 10, 8— 10, etc.
The integral part of a logarithm is the characteristic;
the decimal part is the mantissa.
Thus, log 2 1 5.44 = 2. 33333 ; the characteristic is -f- 2 ; the mantissa
COMMON LOGARITHMS. vii
is +-33333: log .021544=8.33333— 10 ; the characteristic is 8— 10
= — 2 ; the mantissa is -f .33333.
9. It is evident that the larger a number the larger its logarithm.
Hence the logarithm of any number
between i and 10 is o -f- a mantissa,
10 " 100 " i+" "
.1 " i "-i+"
.01 " .1 " — 2-f" " etc.
We have, then, the following rule for obtaining the characteristic :
10. Count the number of places the first left-hand digit of
the number is removed from the unit's place.
If this digit is' to the left of the unit's place, the result is the
required characteristic.
If this digit is to the rig/it of the unit's place, the result
taken with a minus sign is the required characteristic.
If this digit is in the unit' s place, t/ic characteristic is zero.
Thus the characteristic of the logarithm of 21550 is 4
" " " ' " 21.55 •• i
2.155 " o
' -2155 "—i
" " " " " » " .02155 " -2
11. The logarithms of numbers which differ only in the
position of the decimal point have the same mantissa.
For to change the position of the decimal point is to multiply or
divide by an integral power of 10; that is, an integer is added to or
subtracted from the logarithm, and consequently only the character-
istic is changed.
Thus, log 2 1 544 =3-33333
log 2.1544 =0.33333
log .21544 =9.33333-10
log .021544 = 8.33333-10
Therefore, in finding the mantissa of the logarithm of a
number the decimal point may be disregarded. The man-
tissa is found from the tables of logarithms.
viii INTRODUCTION TO THE TABLES.
USE OF THE TABLE OF LOGARITHMS OF NUMBERS.
(TABLE i.)
12. To find the logarithm of a number.
Look in the column at the head of which is " N " for the
first three figures of the number, and in the line with "N" for
the fourth figure. In the line opposite the first three figures
and in the column under the fourth is the desired mantissa.
Only the last three figures of the mantissa are found thus; the
first two must be taken from the first column ; they are found either
in the same line or in the first line above which gives the whole man-
tissa, except when a * occurs. If a * precedes the last three figures of
the mantissa the first two are found in the following line :
The characteristic is obtained by § 10.
Example. — To find the logarithm of 105400.
The characteristic = 5. § 10
The mantissa = .02284 (opposite 103 and under 4 in the tables) ;
Hence log 105400 = 5.02284.
13. If there are five or more figures in a number the
figures beyond the fourth are treated as a decimal. The
corresponding mantissa is between two successive mantissas
of the tables.
Example. — To find the logarithm of 10543.
The characteristic = 4. § 10
The mantissa is not in the tables, but is between the mantissa of
1055 = .02325
and the mantissa of 1054 = .02284
Their difference = 41
Hence an increase of one in the fourth figure of the number pro-
duces an increase of 41 in the mantissa. Then an increase of .3 must
produce an increase of 41 X .3 in the mantissa.
41 X. 3 = 12.3 = 12 nearly.
Hence the mantissa of 10543=1.02284-}- 12 = .02296.
Therefore log 10543= 4.02296.
LOGARITHMS OF NUMBERS. ix
An easy method of multiplying 41 by .3 is to use the table of pro-
portional parts at the bottom of the page in the tables.
Under 41 and opposite 3 is 12. 3 (=41 X-3).
14. Figures beyond the fifth are usually omitted in the
use of a five -place table, as their retention does not add
much to the accuracy of the result. For the fifth figure,
however, we choose the one which gives most nearly the
true value of the number.
Thus, if the number is 157.032, we use 157.03;
" 157.036, " " 157.04;
" " " " 157.035. " " 157-04.
13. To find a number from its logarithm.
The process is the reverse of finding the logarithm from
the number; it is illustrated by the following examples:
Find the number of which 9.12872 — 10 is the logarithm.
Since the characteristic = — i, the decimal point will be before the
first figure of the number.
.12872 is opposite 134 and under 5 in the tables.
Hence .12872 = the mantissa of 1345,
and 9.12872— 10 = log. 1 345.
Find the number of which 9.12895 — 10 is the logarithm.
The mantissa .12895 is not i" the tables, but is
between .12905 = mantissa of 1346
and .12872= " " 1345.
.00033 = tne difference.
.12895 — mantissa given,
.12872 = mantissa of 1345, the smaller number,
23 = the difference.
Change §§ into a decimal. The first figure of this decimal will be
the figure in the fifth place of the number.
§f = .7 nearly.
Hence 9.12895 — 10 — log. 13457.
x INTRODUCTION TO THE TABLES.
An easy method of changing §§ into a decimal is to use the table
of proportional parts.
Under 33 is found 23.1 (= 23 nearly), which is opposite 7.
Hence H = -7 nearly.
The process we have employed in finding the logarithm
of a number of more than four figures, or the number corre-
sponding to a mantissa not given in the table, is called in-
terpolation.
EXAMPLES FOR THE USE OF LOGARITHMS.
16. Multiply 5789.2 by .018315.
tog 5789.2 = 3.76262
log .018315 =8.26281 — 10
2.02543 = log 106.03
Multiply 9.8764 by .10013.
log 9.8764 = 0.99460
log. 10013 = 9.00056— 10
9.99516 — 10 = log .98892
Find the value of 3.1416 X 7638.6 x .017829.
log 3.1416=0.49715
log 7638.6 = 3.88302
log .017829 = 8.251 13— 10
2.631 30 = log 427. 86
Divide 81.321 by 3.1416.
Iog8i.3i2 = 1.91021
log 3. 141 6 = 0.497 1 5
1.41306 =log 25.886
Find the value of (2.1345)'.
log 2. 1 345 =0.32930
5
1.64650 = log 44.310
Find the value ofy/.oio2i.
log .01021 = 8.00903 — 10
= 28.00903 — 30
28.00003 — 30
^ — ^-=9.33634 -10 = log. 21694
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS, xi
17. The logarithm of — is called the cologarithm of m,
and is obtained by subtracting logm from zero.
Thus, if log m = 9.76423— 10, cologw =0.23577.
It is frequently shorter to add cologm than to subtract
logm when we wish to divide by a number m.
The following example illustrates this :
Find the value of 57fX42'24.
644.32
log 57. 98= 1.76328
log 42. 24= 1.62572
colog 644.32 = 7.19090— 10
0.57990 = log 3.801
USE OF THE TABLE OF LOGARITHMS OF TRIGONOMETRIC
FUNCTIONS. (TABLE n.)
18. For an angle less than 45°, the degrees are at the
head of the page, the minutes in the column at the left, and
"L. Sin.," "L. Tang.," etc., at the head of the correspond-
ing columns. For angles between 45° and 90°, the degrees
are at the foot of the page, the minutes in the column at
the right, and " L. Sin.," " L. Tang.," etc., at the foot of the
corresponding columns.
The characteristic is printed 10 too large where it would
otherwise be negative. Hence, in using this table, — 10 is
to be supplied, except for the cotangent of angles less than
45° and the tangent of angles from 45° to 90°.
EXAMPLES.
log sin 15° 25' = 9.42461 — 10.
log tan 28° 1 7' = 9.73084— 10.
log cos 62° 14' = 9.66827 — 10.
log cot 25° 34' = 0.3 2020.
xii INTRODUCTION TO THE TABLES.
19. If the given angle contains seconds, we may reduce
the seconds to a decimal of a minute and proceed as in
finding the logarithms of numbers. It must be remem-
bered, however, that log cos and log cot decrease as the
angle increases.
In practice we remember that 6" is one-tenth of a minute, and di-
vide the number of seconds by 6", then use the table of proportional
parts at the bottom of the page.
EXAMPLES.
Find log sin 28° 14' 36" (=log sin 28° 14.6').
log sin 28° 1 5' — log sin 28° 14' = 23 (found in column "d.")
log sin 28° 14' = 9.67492 — 10
23 X. 6 = 13.8= 14 nearly
log sin 28° 14' 36" = 9.67506— 10
Find log cos 39° 17' 22" (=log cos 39° 17.3!').
log cos 39° 1 7' = 9.8887 5— 10
log cos 39° 17' 22" = 9.88871 — 10
Find log tan 51° 27' 44" (=log tan 51° 27.7^').
log tan 51° 27' = .09862
log tan 51° 27' 44" = .0988 1
Find log cot 67° i8'46".
log cot 67° 1 8' =9.62150 — 10
6X.= 28
Hence log cot 67° 18' 46" = 9.62122 — 10
20. The 'process of finding an angle, if its logarithmic
sine or tangent, etc., is given, is the reverse of the pre-
ceding.
EXPLANATION OF THE TABLES. xiii
EXAMPLES.
Given log sin x = 9.67433 — 10 ; find x,
log sin 28° ii' = 9.6742 1 — 10
log sin ,r — log sin 28° ii' = 12
and log sin 28° 12' — log sin 28° H' = 24
Hence .r = 28° 1 1' 30" (££ of if being 30").
Find the angle whose log 005 = 9.88231 — 10.
log cos 40° 1 8' = 9.88234— 10.
60" x ft= 16".
Hence log cos 40° 18' 1 6" = 9.88231 — 10.
Find the angle whose log tan =0.17844.
log tan 56° 27 =0.17839.
6o"x&=u".
Hence ' log tan 56° 27' ii" = 0.17844.
Find the angle whose log cot = 9.87432 — 10.
log cot 53° 10' = 9.87448 — 10.
6o"xi£=37"-
Hence log cot 53° 10' 37" = 9.87432— 10.
EXPLANATION OF THE TABLES.
21. A dash above the terminal 5 of a mantissa, as 5, de-
notes that the true value is less than 5.
Thus, log 389 = 2.5899496 to seven places, but to five places
log 389 = 2.58995.
Tables I and II have already been explained.
TABLE III.
22. The logarithmic sine and tangent cannot be obtained
very accurately from Table II if the angle contains seconds
and is less than 2°.
Table III is to be used when greater accuracy in the sine
or tangent of a small angle is desired than can be obtained
xivr INTRODUCTION TO THE TABLES.
by the use of Table II. It is to be noted that the first page
of Table III gives the sine and tangent to every second for
angles less than 8'.
TABLE IV.
23. Naperian or "natural" logarithms are logarithms to
the base e ( = 2.71828 + ). The whole logarithm is given,
since the integral part cannot be supplied by inspection, as
with common logarithms.
TABLES V AND VI.
24. Four-place logarithms and logarithmic functions are
used instead of five-place if the results are sufficiently ac-
curate for the purpose in view.
In Table VI both the degrees and minutes are in the col-
umns at the sides of the page, otherwise this table does not
differ in form from Table II.
TABLE VII.
23. This table is identical with Table VI in form, but
gives the trigonometric functions themselves, instead of
their logarithms.
TABLES VIII, IX, X.
26. These tables require no explanation.
TABLE I
FIVE-PLACE LOGARITHMS
OF NUMBERS
100-130
N
O
1
2
3
4
5
O
7
8
9
100
oo ooo
o43
o87
i3o
i73
2I7
260
3o3
346
389
IOI
432
475
5i8
56i
6o4
647
689
732
775
817
102
860
9o3
945
988
*o3o
*072
*ii5
*i57
io3
01
284
326
368
4io
452
494
536
578
620
662
104
7o3
745
787
828
87o
912
953
995-
*o36
*o78
io5
02
119
1 60
202
243
284
325
366
407.
449
490
106
53i
572
612
653
694
735
776
816
857
898
107
938
979
'9
*o6o
*IOO
*i4i
*i8i
*222
*262
*302
108
o3
342
383
423
463
5o3
543
583
623
663
7o3
109
743
782
822
862
902
94 1
98i
*O2I
*o6o
*IOO
110
o4 1 3g
i79
218
258
297
336
376
415
454
493
1 1 1
532
57i
610
650
689
727
766
8o5
844
883
I 12
922
961
999
*o38,
*°77
*n5
*i54
*I92
*23l
*269
n3
o5
3o8
346
385
423
46 1
500
538
576
6i4
652
ii
4
690
729
767
805
843
881
9i8
956
994
*032
ii
5
06 070
1 08
i45
i83
221
258
296
333
37i
4o8
116
446
483
521
558
595
633
67o
7°7
744
78i
117
819
856
893
93o
967
*oo4
*o4i
*o78
*ii5
*i5i
1x8
07
188
225
262
298
335
372
4o8
445
482
5i8
119
555
59i
628
664
7oo
737
773
809
846
882
120
918
954
99°
*027
*o63
*°99
*i35
*i7i
*2O7
*a43
121
08
279
3i4
35o
386
422
458
493
529
565
600
122
636
672
707
743
778
8i4
849
884
92O
^955
123
991
*O26
*o6i
"096
*l32
*i67.
*2O2
*237
*272
124
09
342
377
4l2
447
482
617
552
587
621
656
125
691
726
760
796
83o
864
899
934
968
*oo3
126
10
o37
O72
1 06
i4o
175
209
243
278
3l2
346
I27
38o
4i5
449
483
5i7
55i
585
619
653
687
128
•721
755
789
823
857
890
924
958
992
*025
I29
ii
o5g
093
126
1 60
i93
227
261
294
327
36i
130
394
428
46 1
494
528
56i
594
628
661
694
N
0
1
2
3
4
5
6
7
8
9
PP
44
43 42
41 40 39
38 37 36
i
4.4
4.3 4.2
i
4.i
4.o
3.9
i
3.8
3.7
3.6
2
8.8
8.6 8.4
2
8.2
8.0
7-8
2
7.6
7-4
7-2
3
13.2
12.9 12.6
3
12.3
I2.O
n.7
3
n.4
ii. i
10.8
4
17.6
1-7.2 16.8
4
16.4
16.0
i5.6
4
15.2
1 4.8
i4.4
5
22. 0
21.5 21. 0
5
20. 5
20. o
i9.5
5
19.0
i8.5
18.0
6
26.4
25.8 25.2
6
24.6
24.0
23.4
6
22.8
22.2
21.6
7
3o.8
3o.i 29.4
7
28.7
28.0
27.3
7
26.6
25.9
25.2
8
35.2
34.4 33.6
8
32.8
32.0
3l.2
8
3o.4
29.6
28.8
9
39.6
9
36. 9
36.o
35.i
9
34.2
32.4
13O— 16O
N
0
1
2
3
4
5
6
7
8
9
130
1 1
394
428
46i
494
528
56i
594
628
661
694
i3
i
727
76o
793
826
860
893
926
959
992
*O24
I 32
12 057
090
123
i56
189
222
254
287
320
352
i33
385
4i8
45o
483
5i6
548
58 1
6i3
646
678
1 34
710
743
775
808
840
872
9°5
937
969
*OOI
i35
i3
o33
066
098
i3o
162
i94
226
258
290
322
1 36
354
386
4i8
45o
48 1
5i3
545
577
6o9
64o
i37
672
704
735
767
799
83o
862
893
925
956
1 38
988
*c
'9
*o5i
*082
*n4
*i4g
*i76
*208
*270
139
i4
3oi
333
364
395
426
457
489
52O
55i
582
140
6i3
644
675
7o6
737
768
799
829
860
89i
U
i
922
953
983
*oi4
+045
*o76
*io6
*i37
*i68
*i98
142
i5
229
259
290
320
35i
38i
412
442
473
5o3
M
3
534
564
594
625
655
685
7i5
746
776
806
i44
836
866
897
927
957
987
*OI7
*o47
*077
*IO7
i45
16
i37
167
i97
227
256
286
3i6
346
376
4o6
i46
435
465
495
524
554
584
6i3
643
673
702
i47
732
761
7<
pi
820
850
879
909
938
967
997
i48
i7
026
o56
085
n4
i43
i73
202
23l
260
289
149
3i9
348
377
4o6
435
464
493
522
•55i
58o
150
609
638
667
696
725
754
782
811
84o
869
i5
i
898
926
955
984
*oi3
*o4i
"070
*o99
*I27
*i56
i52
18
1 84
2l3
241
270
298
327
355
384
4l2
44 1
i53
469
498
526
554
583
611
639
6t>7
696
724
1 54
752
780
808
837
865
893
921
949
977
*oo5
i55
19 o33
06 1
o89
117
i45
i73
201
229
257
285
i56
312
34o
368
396
424
45i
479
5o7
535
562
i57
590
618
645
673
700
728
756
783
811
838
1 58
866
893
921
948
9-76
*oo3
*o3o
*o58
*o85
*II2
i59
20
i4o
167
i94
222
249
276
3o3
33o
358
385
160
412
439
466
493
520
548
575
602
629
656
N
O
1
2
3
4
5
6
7
8
9
PP
35
34
33
32 31 30 29 28 27
i
3.5
3.4
3.3
i
3.c
3.1
3.o i 2.9
2.8
2.7
2
7.0
6.8
6.6
2
6.4
6^2
6.0 2 5.8
5.6
5.4
3
io.5
IO.2
9-9
3
9.6
9-3
9.0 3 8.7
8.4
8.1
4
i4.o
i3.6
13.2
4
12.8
12.4
12. o 4 ii. 6
I 1. 2
10.8
5
i7.5
I7.0
i6.5
5
16.0
i5.5
i5.o $ i4.5
i4.o
i3.5
6
21. 0
20.4
19.8
6
19.2
18.6
18.0 6 17.4
16.8
16.2
7
24.5
.23.8
23.1
7
22.4
21,7
21. o 7 20. 3
19.6
18.9
8
28.0
27.2
26.4
8
25.6
24.8
24.0 8 23.2
22.4
21.6
°
3i.5
29.7
9
s8.8
57.9
27.0 9 26.1
25.2
24.3
16O-190
N
0
1
2
3
4
5
6
7
8
9
160
20 4l2
439
466
493
520
548
575
602
629
656
161
683
710
737
763
79°
817
844
871
898
925
162
952
978
*oo5
*032
*o59
*o85
*II2
*:39
*i65
*I92
i63
21
219
245
272
299
325
352
378
405
43i
458
1 64
484
5u
537
564
59o
617
643
669
696
722
i65
748
775
801
827
854
880
906
932
958
985
166
22
on
037
o63
089
u5
i4i
167
i $4
220
246
167
272
298
324
350
376
4oi
427
453
479
5o5
168
53i
557
583
608
634
660
686
712
737
763
169
7%
8i4
84o
866
891
917
943
968
994
*oi9
170
23 045
070
096
121
i47
172
198
223
249
274
171
3oo
325
35o
376
4oi
426
452
477
502
5a8
172
553
578
6o3
629
654
679
704
729
754
779
i73
805
83o
855
880
9°5
93o
955
980
*oo5
*o3o
1 74
24
055
080
105
i3o
155
180
204
229
254
379
i75
3o4
329
353
378
4o3
428
452
477
502
537
176
55i
576
601
625
650
674
699
724,
,748
773
177
797
822
846
871
895
920
944
969
993"
*oi8
178
25
042
066
091
n5
i39
1 64
188
212
237
261
179
285
3io
334
358
382
4o6
43i
455
479
5o3
180
527
55i
575
600
624
648
672
696
720
744
181
768
792
816
84o
864
888
912
935
959
983
182
26
007
o3i
055
079
1 02
126
i5o
1 74
198
221
i83
245
269
293
3i6
34o
364
387
4n
435
458
1 84
482
5o5
529
553
576
600
€23
647
670
694
i85
717
74i
764
788
811
834
858
881
9°5
928
186
95i
975
998
*02I
*o45
*o68
*09i
*u4
*i38
*i6i
187
27
1 84
207
23l
254
277
3oo
323
346
37o
393
1 88
4i6
439
462
485
5o8
53i
554
577
600
623
189
646
669
692
7i5
738
761
784
807
83o
85a
190
875
898
921
944
967
989
*OI2
*o35
*o58
*o8i
N
O
1
2
3
4
5
6
7
8 9
PP 27
26
25
24 23 22
21 20 19
i
2-7
2.6
2.5
:
2.4
2.3
2.2
i
2.1
2.O
1.9
2
5.4
5.2
5.o
2
4.8
4.6
4.4
2
4.2
4.o
3.8
3
8.1
7.8
7-5
3
7.2
6.9
6.6
3
6.3
6.0
5-7
4
10.8
10.4
IO.O
4
9.6
9.2
8.8
4
8.4
8.0
7.6
5
i3.5
i3.o
12.5
5
12.0
ii.5
n.o
5
io.5
IO.O
9-5
6
16.2
i5.6
i5.o
6
i44
i3.8
13.2
6
12.6
I2*.0
n.4
7
18.9
18.2
i7.5
7
16.8
16.1
i5.4
7
i4.7
i4.o
i3.3
8
21.6
20.8
20. o
8
19.2
18.4
17.6
8
16.8
16.0
15.2
9
24.3
23.4
22.5
9
21.6
20.7
19.8
9
18.9
18.0 17.1
190-230
N
0
1
2
3
4
5
6
7
8
9
190
27876
898
921
944
967
989
*OI2
*o35
*o58
*o8i
191
28 io3
126
149
171
194
2I7
240
262
285
3o7
192
33o
353
375
398
421
443
466
488
5n
533
i93
556
578
601
623
646
668
691
7i3
735
758
194
780
8o3
825
847
87o
892
914
937
959
98i
i95
29 oo3
026
o48
O7O
092
115
i37
i59
181
203
196
226
248
270
292
3i4
336
358
38o
4o3
425
197
447
469
491
5i3
53s
557
579
601
623
645
198
667
688
710
732
754
776
798
820
842
863
199
885
9°7
929
95i
973
994
*oi6
*o38
*o6o
*o8i
200
3o io3
125
i46
1 68
190
211
233
255
276
298
201
32O
34 1
363
384
4o6
428
449
47i
492
5i4
202
535
557
578
600
621
643
664
685
707
728
203
750
771
792
8i4
835
856
878
899
920
942
2O4
963
984
*oo6
*027
*o48
*o69
*O9i
*II2
*i33
*i54
205
3i i75
197
218
239
260
281
302
323
345
366
206
387
4o8
429
450
47i
492
5i3
534
555
576
207
597
618
639
660
681
702
723
744
765
785
208
806
827
848
869
890
911
93i
962
973
994
209
32 015
o35
o56
077
098
118
i39
160
181
2OI
210
222
243
263
284
305
325
346
366
387
4o8
211
428
449
469
49o
5io
53i
552
572
593
6i3
212
634
654
675
695
7i5
736
756
777
797
818
2l3
838
858
879
899
919
94o
96o
980
*OOI
*O2I
2l4
33o4i
062
082
102
122
i43
i63
i83
203
224
2l5
244
264
284
3o4
325
345
36s
385
4o5
425
216
445
465
486
5o6
52.6
546
566
586
606
626
217
646
666
686
706
726
746
766
786
806
826
218
846
866
885
906
925
945
965
985
*oo5
*O25
219
34o44
o64
o84
io4
124
i43
i63
i83
203
223
220
242
262
282
3oi
321
34i
36i
38o
4oo
420
221
439
459
4?9
498
5i8
537
557
577
596
616
222
635
655
674
694
7i3
733
753
772
792
811
223
83o
850
869
889
908
928
947
967
986
*oo5
224
35 025
o44
o64
o83
102
122
i4i
1 60
1 80
I99
225
218
238
257
276
295
315
334
353
372
392
226
4n
43o
449
468
488
5o7
526
545
564
583
227
6o3
622
64i
660
679
698
717
736
755
774
228
793
8i3
832
85i
87o
889
908
927
946
965
229
984
*oo3
*O2I
*o4o
*o59
*o78
*097
*n6
*i35
*i54
230
36 i73
192
21 I
229
248
26-7
286
305
324
342
X
O
1 2
3
4
5
6
7
8 9
23O— 26O
N
0
1
2
3
4
5
0
7
8
0
230
36 173
192
211
229
248
267
286
3o5
324
342
s3i
36i
38o
399
4i8
436
455
474
493
5n
53o
232
549
568
586
605
624
642
661
680
698
717
233
736
754
773
79I
810
829
847
866
884
9o3
234
922
94o
959
977
996
*oi4
*o33
*o5i
"070
*o88
235
37 io7
125
1 44
162
181
199
218
236
254
273
236
291
3io
328
346
365
383
4oi
420
438
457
237
475
493
5n
53o
548
566
585
6o3
621
639
238
658
676
694
7I2
73i
749
767
785
8o3
822
239
84o
858
876
894
912
93i
949
967
985
*oo3
240
38 021
039
057
o75
093
112
i3o
1 48
166
1 84
241
202
220
238
256
274
292
3io
328
346
364
242
382
399
4i7
435
453
47i
489
507
525
543
243
56i
578
596
6i4
632
650
668
686
7o3
72I
244
739
757
775
792
810
828
846
863
881
899
245
9i7
934
952
970
987
*oo5
*023
*o4i
*o58
*o76
246
39o94
in
I29
i46
1 64
182
199
217
235
252
247
270
287
3o5
322
34o
358
375
393
4io
428
248
445
463
48o
498
5i5
533
55o
568
585
602
249
620
637
655
672
690
707
•724
742
759
777
250
794
811
820.
846
863
881
898
9i5
933
95o
•Si
967
985
*002
*oi9
*o37
*o54
*o
7i
*o88
*io6
*I23
252
4o i^
0
i57
175
I92
209
226
243
261
278
295
253
3l2
329
346
364
38i
398
4i5
432
449
466
254
483
5oo
5i8
535
552
569
586
6o3
620
637
255
654
67i
688
705
722
739
756
773
79°
807
256
824
84i
858
875
892
909
926
943
960
976
257
993
*OIO
*027
*o44
*o6i
*078
*°95
*ii
i
*I28
*i45
258
4i 162
179
i96
212
229
246
263
280
296
3i3
269
33o
347
363
38o
397
4i4
43o
447
464
48 1
260
497
5i4
53i
547
564
58i
597
6i4
63i
647
N
0
1
2
3
4
5
6
7
8
9
PP
19
18 17 16
15
14
i
T^
1.8
1.7 i 1.6
7s"
i.4
2
3'.8
3.6
3.4 2 3.2
3'.o
2.8
3
5-7
5.4
5.i 3 4.8
4.5
4.2
4
7.6
7.2
6.8 4 6.4
6.0
5.6
5
9.5
9.0
8.5 5 8.0
7-5
7.0
6
1 1. 4
10.8
10.2 6 9.6
9.0
8.4
7
i3.3
12.6
11.9 7 ii. 2
io.5
9.8
8
15.2
i4.4
i3.6 8 12.8
12.0
I 1. 2
9
17.1
16.2 i5.3 9 r4.4
i3.5
12.6
260-300
N
0
1
2
3
4
5
6
7
8
9
260
4i 497
5i4
53i
547
564
58i
597
6i4
63i
647
261
664
681
697
714
73i
747
764
780
797
8i4
262
83o
847
863
880
896
9i3
929
946
963
979
263
996
*OI2
*029
*o45
*o62
*078
*°95
*in
*I27
^i44
a64
42 1 60
177
i93
210
226
243
259
275
292
3o8
265
325
34 1
357
374
39o
4o6
423
439
455
472
266
488
5o4
521
537
553
-570
586
602
6i9
635
267
65i
667
684
700
716
732
?49
765
781
797
268
8i3
83o
846
862
878
894
9u
927
943
959
269
975
991
*oo8
*024
*o4o
*o56
*O72
*o88
*io4
*I20
270
43 i36
152
169
185
201
217
233
249
265
28l
271
297
3i3
329
345
36i
377
393
4o9
425
44 1
272
457
473
489
505
521
537
553
569
584
600
273
616
632
648
664
680
696
712
727
743
759
274
775
791
807
823
838
854
870
886
9O2
9i7
275
933
949
965
981
996
*OI2
*028
*o44
*o59
*o75
276
44091
107
122
i38
1 54
I70
i85
2OI
217
232
277
248
264
279
295
3n
326
342
358
373
389
278
4o4
420
436
45i
467
483
498
5i4
529
545
279
56o
576
592
607
623
638
654
669
685
7oo
280
716
73i
747
762
778
793
8o9
824
84o
855
281
871
886
902
9i7
932
948
963
979
994
*OIO
282
46025
o4o
o56
071
086
IO2
117
i33
i48
i63
283
179
I94
209
225
240
255
271
286
3oi
3i7
284
332
347
362
378
393
4o8
423
439
454
469
285
484
500
5'5
53o
545
56i
576
59i
606
621
286
637
652
667
682
697
712
728
743
758
773
287
788
8o3
818
834
849
864
879
894
9°9
924
288
939
954
969
984
*ooo
*oi5
*o3o
*o45
*o6o
*075
289
46 090
105
I2O
i3s
150
165
1 80
195
2IO
225
290
240
255
270
285
3oo
315
33o
345
359
374
291
389
4o4
419
434
449
464
479
494
5o9
523
292
538
553
568
583
598
6i3
627
642
657
672
293
687
702
716
73i
746
761
776
79°
8o5
820
294
83,5
850
864
879
894
9°9
923
938
953
967
296
982
997
*OI2
*O26
*o4i
*o56
*O70
*o85
*IOO
*n4
296
47 129
1 44
1*9
I73
188
202
217
232
246
261
297
276
290
305
3i9
334
349
363
378
392
4o7
298
422
436
45i
465
48o
494
5o9
524
538
553
299
567
582
596
611
625
64o
654
669
683
698
300
712
727
74i,
^756
770
784
799
8i3
828
842
N
0
1
a
3
4
5
6
7
8
9
3OO— 33O
N
O
1
2
3
4
5
6
7
8
9
800
47712
727
74i
756
77°
784
799
8i3
828
842
3oi
857
871
885
9oo
914
929
943
958
972
986
302
48 ooi
oi5
029
o44
o58
073
087
101
116
i3o
3o3
i44
i59
I73
187
202
216
230
244
259
273
3o4
287
302
3i6
33o
344
359
373
387
4oi
4i6
3o5
43o
444
458
473
487
5oi
5i5
53o
544
558
3o6
572
586
601
615
629
643
657
671
686
7oo
3o7
7i4
728
742
756
770
785
799
8i3
827
84i
3o8
855
869
883
897
911
926
954
968
982
309
996
*OIO
*024
*o38
*052
*o66
*o8o
*o94
*io8
*I22
310
49 1 36
i5o
1 64
178
192
206
220
234
248
262
3u
276
290
3o4
3i8
332
346
36o
374
388
4O2
3l2
4i5
429
443
457
471
485
499
5i3
527
54i
3i3
554
568
582
596
610
624
638
65i
665
679
3i4
693
707
721
734
748
762
776
79°
8o3
817
3i5
83i
845
859
872
886
900
914
927
941
^955
3i6
969
982
996
*OIO
*024
*o37
*o5i
*c
,g?
*o79
3i7
5o 106
I2O
i33
i47
161
1 74
188
202
2l5
229
3i8
243
256
270
284
297
3n
325
338
352
365
3i9
379
393
4o6
420
433
447
46i
474
488
5oi
320
5^5
520
542
556
569
583
596
610
623
637
321
65i
664
678
691
705
718
732
745
759
772
322
786
799
8i3
826
84o
853
866
880
893
9°7
323
920
934
947
961
974
987
*OOI
*oi4
*028
*o4i
324
5i 055
068
08 1
095
1 08
121
138
i48
162
'75
325
1 88
202
2l5
228
242
255
268
282
295
3o8
326
322
335
348
362
375
388
402
4i5
428
44 1
327
455
468
48 1
495
5o8
521
534
548
56i
574
328
587
601
6i4
627
64o
654
667
680
693
7o6
329
720
733
746
759
772
786
799.
812
825
838
330
85i
865
878
891
904
917
93o
943
957
97°
N
0
1
2
3
4
5
0
7
8 t>
PP 15 14 13 12
11
!
i.5
i.4
i.3
I 1.2
i.i
2
S'.Q
2!8
•*.6
2 2.4
2.2
3
4.5
4.2
3.9
3 3.6
3.3
4
6.0
5.6
5.2
4 4.8
4.4
5
7-5
7.0
6.5
5 6.0
5.5
6
9.0
8.4
7.8
6 7.2
6.6
7
io.5
9-8
9.1
7 8.4
7-7
8
12. 0
II. 2
10.4
8 9.6
8.8
9
i3.5
12.6
11.7
9 10.8
9-9
330-370
Jg
0
1
2
3
4
5
6
7
8
9
330
5i 85i
865
878
891
904
917
93o
943
957
97°
33i
983
996
*oo9
*022
*o35
*o48
*o6i
*°75
*o88
*IOI
332
52 u4
127
i4o
i53
166
179
192
2O5
218
23l
333
244
257
270
284
297
3io
323
336
349
362
334
375
388
4oi
4i4
427
44o
453
466
479
492
335
5o4
5i7
53o
543
556
569
582
595
608
621
336
634
64?
660
673
686
699
711
•724
737
75o
337
763
776
789
802
815
827
84o
853
866
879
338
892
9°5
917
930
943
956
969
982
994
*OO7
339
53 020
o33
o46
o58
071
o84
097
I IO
122
i35
340
i48
161
i73
186
199
212
224
237
250
263
34i
275
288
3oi
3i4
326
339
352
364
377
39o
342
4o3
4i5
428
44 1
453
466
479
49i
5o4
5i7
343
529
542
555
567
58o
593
6o5
618
63i
643
344
656
668
681
694
706
719
732
744
757
769
345
782
794
807
820
832
845
857
870
882
895
346
908
920
933
945
958
97°
983
995
*oo8
*O2O
347
54o33
o45
o58
070
o83
o95
1 08
120
i33
i45
348
1 58
170
i83
i95
208
220
233
245
258
270
349
283
295
3o7
320
332
345
357
370
382
394
350
407
419
432
444
456
469
48 1
494
5o6
5i8
35i
53i
543
555
568
58o
593
605
617
63o
642
352
654
667
679
691
7o4
716
728
74 1
753
765
353
111
79°
802
8i4
827
839
85i
864
876
888
354
900
9i3
925
937
949
962
974
986
998
*OII
355
55 023
o35
047
060
O72
o84
096
108
121
i33
356
i45
i57
169
182
194
206
218
230
242
255
357
267
279
291
3o3
3i5
328
34o
352
364
376
358
388
4oo
4i3
425
437
449
46i
473
485
497
359
5og
522
534
546
558
57o
582
594
606
618
360
63o
642
654
666
678
691
7o3
7i5
727
739
36i
75i
763
775
787
799
8n
823
835
847
859
362
871
883
895
907
919
93i
943
955
967
979
363
991
*oo3
*oi5
*027
*o38
*o5o
*062
*o74
*o86
*o98
364
56 no
122
1 34
i46
i58
170
182
194
2O5
217
365
229
24 1
253
265
277
289
3oi
3l2
324
336
366
348
36o
372
384
396
407
419
43i
443
455
367
467
478
490
502
5i4
526
538
549
56i
573
368
585
597
608
620
632
644
656
667
679
691
369
7o3
7i4
726
738
750
761
773
785
797
808
370
820
832
844
855
867
879
891
902
9i4
926
N
0
1
2
3
4
5
6
7
8
9
37O-4OO
N
0
1
2
3
4
5
0
7
8
9
370
56 820
832
844
855
867
879
891
902
914
926
37i
937
949
961
972
984
996
*oo8
*oi9
*o3i
*o43
372
57o54
066
078
089
101
u3
124
1 36
1 48
i59
373
171
i83
1 94
206
217
229
241
252
264
276
374
287
299
3io
322
334
345
357
368
38o
392
375
4o3
426 438
449
46 1
473
484
496
5o7
376
5i9
53o
542
553
565
576
588
600
611
623
377
634
646
657 669
680
692
7o3
715
•726
738
378
379
749
864
76i
875
887
784
898
795
910
807
921
818
933
83o
944
84i
955
852
967
380
978
99°
*OOI
*oi3
*024
*o35
*o47
*o58
*O7O
*o8i
38i
58 o92
io4
n5
127
1 38
149
161
172
1 84
J95
382
206
218
229
240
252
263
274
286
297
309
383
320
33i
343
354
365
377
388
399
4io
422
384
433
444
456
467
478
490
5oi
5l2
524
535
385
546
557
569
58o
59i
602
6i4
625
636
647
386
659
67o
681
692
704
715
726
737
749
760
387
77i
782
794
805
816
827
838
850
861
872
388
883
894
906
917
928
939
95o
961
973
984
389
995
*oo6
*oi7
*028
*o4o
*o5i
*062
*°73
*o84
*095
390
59 1 06
118
129
i4o
i5i
162
i73
1 84
i95
207
39i
218
229
240
25l
262
273
284
295
3o6
3i8
392
329
34o
35i
362
373
384
395
4o6
4i7
428
393
439
45o
46i
472
483
494
5o6
5i7
528
539
3g4
550
56r
572
583
594
605
616
627
638
64g
395
660
671
682
693
764
7i|
>
•726
737
748
759
396
77°
780
791
802,
8i3
824
835
846
857
868
397
879
890
901
912
923
934
945
956
966
977
398
988
999
*OIO
*O2I
*032
*o43
*o54
"065
*o76
*o86
399
60 o97
1 08
119
i3o
i4i
l52
i63
I73
1 84
i95
400
206
217
228
239
249
260
271
282
293
3o4
N
0
1
2
3
4
5
0
7
8
9
PP 12 11
10 9
I
1.2 1,1 I
i.o 0.9
2
2.4 2.2 2
2,0 1.8
3
3.6 3.3 3
3.o 2.7
4
4.8 4.4 4
4.o 3.6
5
6.0 5.5 5
5.o 4.5
6
7.2 6.6 6
6.0 5.4
7
8.4 7-7 7
7.0 6.3
8
9.6 8.8 8
8.0 7.2
9
10.8 9.9 9
9.0 8.1
10
400-440
N
0
1
2
3
4
5
6
7
8
9
400
60 206
217
228
239
249
260
271
282
293
3o4
4oi
3i4
325
336
347
358
369
379
390
4oi
412
402
423
433
444
455
466
477
487
498
509
520
4o3
53i
54i
552
563
574
584
595
606
617
627
4o4
638
649
660
670
681
692
7o3
7i3
724
735
4o5
746
756
767
778
788
799
810
821
83i
842
4o6
853
863
874
885
895
906
917
927
938
949
4oy
959
970
981
991
*OO2
*oi3
*023
*o34
*o45
*o55
4o8
6 1 066
077
087
098
I09
119
i3o
i4o
i5i
162
409
172
i83
194
204
215
225
236
247
257
268
410
278
289
3oo
3io
321
33i
342
352
363
374
4n
384
395
4o5
4i6
426
437
448
458
469
479
412
4go
5oo
5n
621
532
542
553
563
574
584
4i3
5,5
606
616
627
637
648
658
669
679
690
4i4
7OO
711
721
73i
742
752
763
773
784
794
4i5
805
8i5
826
836
847
857
868
878
888
899
4i6
909
920
93o
94i
95i
962
972
982
993
*oo3
4i7
62 oi4
024
o34
045
o5'5
066
076
086
097
I07
4i8
118
128
i38
149
i59
170
1 80
190
2OI
211
419
221
232
242
252
263
273
284
294
3o4
315
420
325
335
346
356
366
377
387
397
4o8
4i8
421
428
439
449
459
469
48o
490
5oo
5n
521
422
53i
542
552
562
572
583
593
6o3
6i3
624
423
634
644
655
665
675
685
696
7o6
7i6
726
424
?3?
747
757
767
778
788
798
808
818
829
426
839
849
859
870
880
890
900
910
921
93i
426
941
95i
961
972
982
992
*OO2
*OI2
*O22
*o33
427
63 o43
o53
o63
073
o83
094
io4
u4
124
1 34
428
i44
155
165
175
185
i95
205
2l5
225
236
429
246
256
266
276
286
296
3o6
3i7
327
337
430
347
357
367
377
387
397
407
4i7
428
438
43i
448
458
468
478
488
498
5o8
5i8
528
538
432
548
558
568
579
589
599
669
619
629
639
433
649
659
669
679
689
699
709
7i9
729
739
434
749
759
769
779
789
799
809
819
829
839
435
849
859
869
879
889
899
909
919
929
939
436
949
959
969
979
988
998
*oo8
*oi8
*028
*o38
437
64o48
o58
068
078
088
098
108
118
128
i37
438
i47
•57
167
177
187
197
207
217
227
237
439
246
256
266
276
286
296
3o6
3x6
326
335
440
345
355
365
375
385
395
4o4
4i4
424
434
N
0
1
2
3
4
5
6
7
8
9
440-470
N
0
1
2
3
4
5
6
7
8
9
440
64345
355
365
375
385
395
4o4
4i4
424
434
44 1
444
454
464
473
483
493
5o3
5i3
523
532
442
542
552
562
572
582
5oi
601
611
621
63i
443
64o
65o
660
670
680
689
699
7°9
719
729
444
738
748
758
768
777
787
797
8o7
816
826
445
836
846
856
865
875
885
895
904
914
924
446
933
943
953
963
972
982
992
*002
*OII
*02I
44?
65o3i
o4o
o5o
060
070
079
089
099
108
118
448
128
i37
i47
i57
167
176
186
196
2O5
215
449
225
234
244
254
263
273
283
292
302
3l2
450
321
33i
34 1
35o
36o
369
379
389
398
4o8
45i
4i8
427
437
447
456
466
475
485
495
5o4
452
5i4
523
533
543
552
562
57i
58i
59i
600
453
610
619
629"
639
648
658
667
677
686
696
454
.706
7i5
725
734
744
753
763
772
•782
792
455
801
811
820
83o
839
849
858
868
877
887
456
896
906
916
925
935
944
954
963
973
982
457
992
*OOI
*OII
*O2O
*o3o
*o39
*o49
*o58
*o68
*°77
458
66087
096
106
"5
124
1 34
i43
i53
162
172
459
181
191
200
2IO
219
229
238
247
257
266
460
276
285
295
3o4
3i4
323
332
342
35i
36i
46 1
37o
38o
389
398
4o8
4i7
4a7
436
445
455
462
464
474
483
492
5O2
5u
621
53o
539
549
463
558
567
577
586
596
605
6i4
624
633
642
464
652
661
671
680
689
699
708
717
727
736
465
745
755
764
773
783
792
801
811
820
829
466
839
848
857
867
876
885
894
904
9i3
922
467
932
94i
95o
960
969
978
987
997
*oo6
*oi5
468
67025
o34
o43
o52
062
071
080
089
099
1 08
469
117
127
i36
i45
1 54
1 64
I73
182
191
201
470
2IO
219
228
237
247
256
265
274
284
293
N
O
1
2
3
4
5
6
7
8
9
PP 10 9 8
i
I.O
i
0.9
i 0.8
2
2.0
2
1.8
2 1.6
3
3.o
3
2.7
3 2.4
4
4.o
4
3.6
4 3.2
5
5.o
5
4.5
5 4.0
6
6.0
6
5.4
6 4.8
7
7.0
7
6.3
7 5.6
8
8.0
8
7.2
8 6.4
9
9.0
9
8.1
9 7-2
47O-51O
N
0
1
2
3
4
5
6
7
8
9
470
67 2IO
219
228
"^
*47
256
265
274
284
293
4?i
302
3ii
321
33o
339
348
357
367
376
385
472
394
4o3
4i3
422
43i
44o
449
459
468
477
473
486
495
5o4
fri4
523
532
54i
55o
56o
569
4?4
578
587
596
6o5
6i4
624
633
642
65i
660
475
4?6
669
761
679
77°
688
779
788
706
797
7i5
806
•724
8i5
733
825
742
834
752
843
477
85a
861
87o
879
888
897
906
916
925
934
478
943
952
961
97o
979
988
997
*oo6
*oi5
*024
479
68o34
o43
052
06 1
070
o79
088
o97
1 06
n5
480
124
i33
142
i5i
1 60
169
178
187
196
205
48 1
215
224
233
242
25l
260
269
278
287
296
482
3o5
3i4
323
332
34i
350
359
368
377
386
483
395
4o4
4i3
422
43i
44o
449
458
467
476
484
485
494
502
5u
520
529
538
547
556
565
485
574
583
592
601
610
619
628
637
646
655
486
664
673
68 1
690
699
•708
7i7
•726
735
744
487
753
762
77i
78o
789
797
806
8i5
824
833
488
842
85i
860
869
878
886
895
904
9i3
922
489
93i
940
949
958
966
975
984
993
*002
*on
490
69 020
028
o37
o46
o55
o64
o73
082
090
099
491
1 08
117
126
i3fi
1 44
l52
161
I70
I79
188
492
197
2O5
2l4
223
232
241
249
258
267
276
493
285
294
302
3u
320
329
338
346
355
364
494
373
38i
390
399
4o8
417
425
434
443
452
4g5
46 1
469
478
487
496
5o4
5i3
522
53i
539
496
548
557
566
574
583
592
601
609
618
627
497
636
644
653
662
671
679
688
697
7o5
7i4
498
723
732
740
749
758
767
775
784
793
801
499
810
819
827
836
845
854
862
87I
880
888
500
897
906
9i4
923
932
94o
949
958
966
975
5oi
984
992
*OOI
*OIO
*oi8
*O27
*o36
*o44
*o53
*062
5O2
70 070
079
088
096
105
n4
122
i3i
i4o
i48
5o3
i57
i65
174
i83
191
200
209
217
226
234
5o4
243
252
260
269
278
286
295
3o3
3l2
321
5o5
329
338
346
355
364
372
38i
389
398
4o6
5o6
4i5
424
432
44 1
449
458
467
475
484
492
607
5oi
Sog
5i8
526
535
544
552
56i
569
578
5o8
586
595
6o3
612
621
629
638
646
655
663
609
672
680
689
697
706
7i4
723
73i
74o
749
510
757
766
774
783
791
800
808
817
825
834
N
O
1
2
3
4
5
6
7
8
9
i3
510-540
N
O
1
2
3
4
5
6
7
8
9
510
70767
766
774
783
791
800
8o«
817
825
»34
5n
842
85i
859
868
876
885
893
9O2
9io
9i9
5l2
927
935
944
952
961
969
978
986
995
*oo3
5x3
71 012
020
029
037
o46
o54
o63
071
079
088
5x4
o96
105
n3
122
i3o
i39
i47
i55
1 64
172
5x5
181
189
198
2O6
2l4
223
23l
240
248
257
5i6
265
273.
282
29O
299
3o7
3i5
324
332
34i
5x7
349
357
366
374
383
39i
399
4o8
4i6
425
5x8
433
44 1
450
458
466
475
483
492
500
5o8
5i9
5x7
525
533
542
55o
559
567
575
584
592
520
600
609
617
625
634
642
65o
659
667
675
521
684
692
700
7<>9
717
725
734
742
75o
759
522
767
775
784
792
800
8o9
817
825
834
842
523
85o
858
867
875
883
892
9oo
9o8
917
925
524
933
941
950
958
966
975
983
99i
999
*oo8
525
72 016
024
032
o4i
049
067
066
o74
082
o9o
526
099
107
ix5
123
132
i4o
1 48
i56
165
i73
527
181
189
198
206
214
222
230
239
247
255
528
263
272
280
288
296
3o4
3i3
321
329
337
529
346
354
362
370
378
387
395
4o3
4u
4i9
580
428
436
444
452
46o
469
477
485
493
5oi
53i
5o9
5i8
526
534
542
55o
558
567
575
583
532
59i
599
607
616
624
632
64o
648
656
665
533
673
681
689
697
7o5
7i3
722
73o
738
746
534
754
762
770
779
787
795
8o3
811
819
827
535
835
843
852
860
868
876
884
892
900
9o8
536
916
925
933
94 1
949
957
965
973
981
989
537
997
*oo6
*oi4
*O22
*o3o
*o38
*o46
*o54
*062
*070
538
73o78
086
o94
IO2
in
u9
127
135
i43
i5i
539
i59
167
IT!
r
>
i83
191
i99
207
2l5
223
23l
540
239
247
255
263
272
280
288
296
3o4
3l2
N
O
1
2
3
4
5
O
7
8
9
PP 9
8 7
i
0.9
j
0.8
i
0.7
2
1.8
2
1.6
2
i .4
3
2-7
3
2.4
3
2.1
4
3.6
4
3.2
4
2.8
5
4.5
5
4.o
5
3.5
6
5.4
6
4.8
6
4.2
7
6.3
7
5.6
7
4.9
8
7-2
8
6.4
8
5.6
9
8.1
9
7.2
9
6.3
i4
54O— 58O
N
1)
1
15
3
4
5
6
7
8
9
540
73239
247
255
263
272
280
288
296
3o4
3l2
54i
32O
328
336
344
352
36o
368
376
384
392
542
4oo
4o8
4i6
424
432
44o
448
456
464
472
543
48o
488
496
5o4
5l2
520
528
536
544
552
544
56o
568
576
584
592
600
608
616
624
632
545
64o
648
656
664
672
679
687
695
7o3
711
546
•719
727
735
743
75i
759
767
775
783
79i
547
799
807
815
823
83o
838
846
854
862
870
548
878
886
894
902
910
918
926
933
94 1
949
549
957
965
973
981
989
997
*oo5
*oi3
*O2O
*028
550
74o36
o44
o52
060
068
o76
o84
092
099
107
55i
n5
123
i3i
i39
i47
155
162
170
I78
186
55a
194
202
2IO
218
225
233
24 1
249
257
265
553
273
280
288
296
3o4
3l2
320
327
335
343
554
35i
359
367
374
382
390
398
4o6
4i4
421
555
429
437
445
453
46i
468
476
484
492
500
556
5o7
5i5
523
53i
539
547
554
562
57o
578
557
586
593
601
609
617
624
632
64o
648
656
558
663
671
679
687
695
702
710
7i8>
•726
733
559
74i
749
757
764
772
780
788
796
8o3
811
560
819
827
834
842
850
85.8
865
873
881
889
56i
896
904
912
920
927
935
943
95o
958
966
562
974
981
989
997
*oc>5
*OI2
*O2O
*028
*o35
*o43
563
75 o5i
oSg
066
o74
082
089
097
105
u3
120
564
128
1 36
i43
i5i
i59
166
I74
182
i89
I97
565
205
2l3
220
228
236
243
25l
269
266
274
566
282
289
297
3o5
3l2
320
328
335
343
35i
567
358
366
374
38i
389
397
4o4
412
420
427
568
435
442
45d
458
465
473
48 1
488
496
5o4
569
5n
5i9
526.
-*
534
542
549
557
565
572
58o
570
587
595
6o3
610
618
626
633
64 1
648
656
57i
664
671
679
686
694
702
709
717
724
732
572
74o
747
755
•762
77<>
778
785
793
800
808
573
8i5
823
83i
838
846
853
861
868
876
884
574
891
899
906
914
921
929
937
944
952
959
575
96-7
974
982
989
997
*oo5
*OI2
*O2O
*O27
*o35
576
76 042
050
057
065
072
080
087
095
io3
no
577
118
125
i33
i4o
i48
i55
i63
170
178
i85
578
193
200
208
2l5
223
230
238
245
253
260
579
268
275
283
290
298
3o5
3i3
320
328
335
580
343
35o
358
365
373
38o
388
395
4o3
4io
N
0
1
2
3
4
5
6
7
8
9
i5
580-61O
N
0
1
a
3
4
5
6
7
8
9
580
76343
35o
358
365
373
38o
388
395
4o3
4io
58i
4i8
425
433
44o
448
455
462
47o
477
485
582
492
500
5o7
5^5
522
53o
537
545
552
559
583
567
574
582
589
597
6o4
612
6i9
626
634
584
64 1
649
656
664
671
678
686
693
7OI
7o8
585
7i6
723
73(
>
738
745
753
76o
768
775
782
586
79°
797
805
812
8i9
827
834
842
849
856
587
864
871
879
886
893
9oi
9o8
916
923
93o
588
938
945
953
96o
967
975
982
989
997
*oo4
589
77OI2
019
026
o34
o4i
o48
o56
o63
O7O
078
590
o85
o93
IOO
107
"5
122
I29
i37
i44
i5i
59i
i59
166
I73
181
188
i95
203
2IO
2I7
225
602
232
240
247
254
262
269
276
283
29I
298
593
3o5
3i3
320
327
335
342
349
357
364
37i
594
379
386
393
4oi
4o8
4i5
422
43o
437
444
595
452
459
466
474
48i
488
495
5o3
5io
5i7
596
525
532
539
546
554
56i
568
576
583
59o
597
597
605
612
6i9
627
634
64 1
648
656
663
598
670
677
685
692
699
7o6
7i4
721
728
735
599
743
750
757
764
772
779
786
793
801
808
600
8i5
822
83o
837
844
85i
859
866
873
880
601
887
895
902
9°9
9i6
924
93i
938
945
952
602
96o
967
974
98 1
988
996
*oo3
*OIO
*OI7
*025
6o3
78 o32
o39
o46
o53
06 1
068
075
082
089
°97
6o4
io4
in
118
125
132
i4o
i47
1 54
161
168
6o5
176
i83
190
i97
204
211
219
226
233
240
606
247
254
262
269
276
283
290
297
305
3l2
6o7
3i9
326
333
34o
347
355
362
369
376
383
608
39o
398
4os
412
4u;
426
433
44o
447
455
6o9
462
469
47e
i
483
49o
497
5o4
5l2
5i9
526
610
533
54o
547
554
56i
569
576
583
59o
597
N
0
1
2
3
4
5
6
7
8
9
PP 8
7 6
i
0.8
i
0.7
i
0.6
2
1.6
2
i.4
2
1.2
3
2.4
3
2.1
3
1.8
4
3.2
4
2.8
4
2.4
5
4.o
5
3.5
5
3.o
6
4.8
6
4.2
6
3.6
7
5.6
7
4-9
7
4.2
8
6.4
8
5.6
8
4.8
9
7-2
9
6.3
9
5.4
16
010-650
N
0
1
2
3
4
5
6
7
8
9
610
78 533
54o
54?
554
56i
569
576
583
59o
597
611
6o4
611
618
625
633
64o
647
654
661
668
612
675
682
689
696
704
711
718
725
732
739
6i3
746
753
760
767
774
781
789
796
8o3
810
6i4
817
824
83i
838
845
852
859
866
873
880
6i5
888
895
902
909
916
923
93o
937
944
95i
616
958
965
972
979
986
993
*ooo
*oo7
*oi4
*O2I
617
79029
o36
o43
050
057
o64
071
078
085
O92
618
099
1 06
n3
I2O
127
1 34
i4i
1 48
i55
l62
619
169
176
i83
190
197
204
211
218
225
232
620
239
246
253
260
267
274
28l
288
295
302
621
309
3i6
323
33o
337
344
35i
358
365
372
622
379
386
393
4oo
407
4i4
421
428
43s
442
623
449
456
463
470
477
484
491
498
505
5u
624
5i8
525
532
539
546
553
56o
567
574
58i
626
588
595
602
609
616
623
63o
637
644
65o
626
657
664
671
678
685
692
699
706
7i3
720
627
727
734
74i
748
754
761
768
775
782
789
628
796
8o3
810
817
824
83i
837
844
85i
858
629
865
872
879
886
893
900
906
9i3
920
927
630
934
941
948
955
962
969
975
982
989
996
63i
8ooo3
OIO
017
024
o3o
o37
o44
o5i
o58
065
632
072
079
o85
092
099
1 06
n3
120
127
1 34
633
i4o
i47
1 54
161
168
175
182
1 88
i95
202
634
209
216
223
229
236
243
25o
257
264
271
635
277
284
291
298
3o5
312
3i8
325
332
339
636
346
353
359
366
373
38o
387
393
4oo
407
637
4i4
421
428
434
44 1
448
455
462
468
475
638
482
489
496
5O2
509
5i6
523
53o
536
543
639
55o
557
564
57o
577
584
591
598
6o4
611
640
618
625
632
638
645
652
659
665
672
679
64 1
686
693
699
706
7i3
720
726
733
740
747
642
754
760
767
774
781
787
794
801
808
8i4
643
821
828
835
84i
848
855
862
868
875
882
644
889
895
902
909
916
922
929
936
943
949
645
956
g63
969
976
983
990
996
*oo3
*OIO
*OI7
646
81 023
o3o
o37
o43
o5o
057
o64
070
077
o84
647
090
097
io4
1 1 1
117
124
i3i
i37
1 44
i5i
648
i58
1 64
171
178
i84
191
198
204
211
218
649
224
23l
238
245
25l
258
265
271
278
285
650
291
298
3o5
3u
3i8
325
33i
338
345
35i
N
O
1
2
3
4
5
6
7
8
9
650-680
N
0
1
ii
3
4
5
a
7
8 !>
650
81 291
298
305
3u
3i8
325
33i
338
345
35i
65i
358
365
37i
378
385
39i
398
405
4n
4i8
662
425
43i
438
445
45i
458
465
47i
478
485
653
491
498
505
5n
5i8
525
53i
538
544
55i
654
558
564
57i
578
584
59i
598
6o4
6n
6i7
655
624
63i
637
644
65i
657
664
671
677
684
656
690
697
7o4
7IO
7i7
723
73o
737
743
750
657
757
763
77°
776
783
79°
796
8o3
809
816
658
823
829
836
842
849
856
862
869
875
882
659
889
895
902
908
9i5
921
928
935
94 1
948
660
954
961
968
974
981
987
994
*ooo
*oo7
*oi4
661
82 020
027
o33
o4o
o46
o53
060
066
o73
°79
662
086
092
099
io5
112
119
125
132
i38
i45
663
i5i
1 58
1 64
171
I78
1 84
191
197
2O4
210
664
217
223
23o
236
243
249
256
263
269
276
665
282
289
295
302
3o8
3i5
321
328
334
34i
666
347
354
36o
367
373
38o
387
393
4oo
4o6
667
4i3
419
426
432
439
445
452
458
465
47i
668
478
484
491
497
5o4
5io
5i7
523
53o
536
669
543
549
556
562
569
575
582
588
595
601
670
607
6i4
620
627
633
64o
646
653
659
666
671
672
679
685
692
698
7°5
•711
718
724
73o
672
737
743
750
756
763
769
776
782
789
795
673
802
808
8i4
821
827
834
84o
847
853
860
674
866
872
879
885
892
898
9°5
911
918
924
675
93o
937
943
950
956
963
969
975
982
988
676
995
*OOI
*oo8
*oi4
*O2O
*O27
*o33
*o4o
*o46
*052
677
83o59
o65
O72
078
085
091
097
io4
no
II7
678
123
129
1 36
i4a
1 49
155
161
1 68
i74
181
679
187
198
200
206
2l3
219
225
232
238
245
680
a5i
257
264
270
276
283
289
296
302
3o8
N
O
1
2
3
4
5
6
7
8
9
PP 7
6
i 0.7 i
0.6
2 1.4 2
1.2
3 2.1 3
1.8
4 2.8 4
2.4
5 3.5 5
3.o
6 4.2 6
3.6
7 4.9 7
4.2
8 5.6 8
4.8
Q 6.3 9
5.4
18
68O— 72O
N
0
1
2
3
4
5
6
7 8
9
680
83 261
257
264
270
276
283
289
296
302
3o8
68 1
315
321
327
334
34o
347
353
359
366
372
682
378
385
391
398
4o4
4io
4i7
423
429
436
683
442
448
455
46i
467
474
48o
487
493
499
684
5o6
5l2
5i8
525
53i
537
544
55o
556
563
685
569
575
582
588
594
601
6o7
6i3
620
626
686
632
639
645
65i
658
664
67o
677
683
689
687
696
702
708
7i5
721
727
734
74o
746
753
688
759
765
77i
778
784
79°
797
8o3
809
816
689
822
828
835
84 1
847
853
860
866
872
879
690
885
891
897
904
910
916
923
929
935
942
691
948
9^4
960
967
973
979
985
992
998
*oo4
692
84 on
017
023
029
o36
042
o48
055
061
o67
693
o73
080
086
092
098
105
in
n7
123
i3o
694
1 36
1 4a
1 48
155
161
i67
I73
180
1 86
I92
695
198
205
211
2I7
223
230
236
242
248
255
696
261
267
273
280
286 292
298
3os
3u
3i7
697
323
33o
336
342
348 354
36i
367
373
379
698
386
392
398
4o4
4io
4i7
423
429
435
442
699
448
454
46o
466
473
479
485
491
497
5o4
700
5io
5i6
522
528
535
54i
547
553
559
566
701
572
578
584
590
597
6o3
609
6i5
621
628
702
634
64o
646
652
658
665
67i
677
683
689
7o3
696
•702
•708
7i4
720
•726
733
739
745
75i
704
757
763
•770
776
782
788
794
800
807
8i3
7o5
819
825
83i
837
844
850
856
862
868
874
706
880
887
893
899
9o5
911
9i7
924
93o
936
707
942
948
954
960
967
973
979
985
991
997
708
85oo3
009
016
022
028
o34
o4o
o40
052
o58
709
065
07I
077
o83
089
o95
IOI
I07
n4
1 20
710
126
I 32
i38
i44
i5o
1 56
i63
169
175
181
711
i87
io3
199
205
211
2I7
224
23o
236
242
712
248
254
260
266
272
278
285
291
297
3o3
7i3
309
3i5
321
327
333
339
345
352
358
364
7i4
37o
376
382
388
394
4oo
4o6
412
4i8
425
7i5
43i
437
443
449
455
46 1
467
473
479
485
716
491
497
5o3
509
5i6
522
528
534
54o
546
717
552
558
564
57o
576
582
588
594
600
606
718
612
618
625
63i
637
643
649
655
661
667
719
673
679
685
691
69-7
7o3
7°9
7i5
72I
727
720
733
739
745
75i
757
763
•769
775
781
788
N
0
1
2
3
4
5
6
7
8
9
72O-750
N
0
1
2
3
4
5
6
7
8
9
720
85 733
739
745
75i
757
763
-769
775
78i
788
72I
794
800
806
812
818
824
83o
836
842
848
722
854
860
866
872
878
884
890
896
902
908
723
9i4
920
926
932
938
944
950
956
962
968
724
974
980
986
992
998
*oo4
*OIO
*oi6
*O22
*028
725
86o34
o4o
o46
Of)2
o58
064
070
o76
082
088
726
094
IOO
1 06
112
118
124
i3o
i?6
lAi
i47
727
i53
i59
i65
171
177
i83
189
i95
201
207
728
2l3
219
225
23l
237
243
2
49
255
26l
729
273
279
285
291
297
3o3
3o8
3i4
320
326
730
332
338
344
35o
356
362
368
374
38o
386
73i
392
398
4o4
4io
4i5
421
427
433
439
445
732
45i
457
463
469
475
48 1
487
493
499
5o4
733
5io
5i6
522
528
534
54o
546
552
558
564
734
57o
576
58i
587
593
599
6o5
611
617
623
735
629
635
64 1
646
652
658
664
67o
676
682
736
688
694
7oo
7o5
711
7i7
723
729
735
74 1
737
747
753
759
764
770
776
782
788
794
800
738
806
812
817
823
829
835
84i
847
853
859
739
864
87o
876
882
888
894
900
906
911
917
740
923
929
935
94 1
947
953
958
964
97°
976
74i
982
988
994
999
*oo5
*OII
*o
17
*023
*029
*o35
742
87 o4o
o46
052
o58
o64
O7O
o75
08 1
087
093
743
099
105
III
116
122
128
1 34
i4o
i46
i5i
744
i57
i63
1 69,
175
181
186
192
198
2O4
210
745
216
221
227
233
239
245
25l
256
262
268
746
274
280
286
291
297
3o3
3o9
315
320
326
747
332
338
344
349
355
36i
367
373
379
384
748
390
396
402
4o8
4i3
419
425
43i
437
442
749
448
454
46o
466
47i
477
483
489
495
5oo
750
5o6
5l2
5i8
523
529
535
54 1
547
552
558
N
0
1
2
3
4
5
6
7
8
9
PP
6
5
i
0.6 . i
o.5
2
1.2 2
I.O
3
1.8 3
i.5
4
2.4 4
2.O
5
3.o 5
2.5
6
3.6 6
3.o
7
4-2 7
3.5
8
4.8 8
4-0
9
5.4 9
4.5
750-79O
N
O
1
2
3
4
5
6
7 8
9
750
87 5o6
5l2
5x8
523
529
535
54i
547
552
558
761
564
57o
576
58 1
587
593
599
6o4
610
616
752
622
628
633
639
645
65i
656
662
668
674
753
679
685
691
697
7o3
7o8
7i4
720
726
73i
754
737
743
749
754
76o
766
772
111
783
789
755
795
800
806
812
818
823
829
835
84 1
846
756
852
858
864
869
875
881
887
892
898
9o4
767
910
9i5
921
927
933
938
944
950
955
96i
758
967
973
978
984
99°
996
*OOI
*oo7
*oi3
*oi8
759
88 024
o3o
o36
o4i
o47
o53
o58
o64
O7O
076
760
08 1
087
o93
o98
io4
I IO
116
121
I27
i33
761
1 38
1 44
150
i56
161
167
i73
I78
1 84
I9O,
•762
i95
2OI
207
2l3
218
224
230
235
241
247
763
262
258
264
270
275
281*
287
292
298
3o4
764
309
3i5
321
326
332
338
343
349
355
36o
765
366
372
377
383
389
395
4oo
4o6
412
4i7
766
423
429
434
44o
446
45i
457
463
468
474
767
48o
485
491
497
502
5o8
5i3
5io
525
53o
768
536
542
547
553
559
564
57o
576
58i
587
769
593
598
6o4
610
6i5
621
627
632
638
643
770
649
655
660
666
672
677
683
689
694
7oo
771
7o5
7n
717
722
728
734
739
745
75o
756
772
762
767
773
779
784
79°
795
801
8o7
812
773
818
824
829
835
84o
846
852
857
863
868
774
874
880
885
891
897
902
9o8
9i3
9i9
925
775
93o
936
94i
947
953
958
964
969
975
981
776
986
992
997
*oo3
*oo9
*oi4
*020
*025
*o3i
*o37
777
89 042
o48
o53
oSg
o64
070
o76
081
o87
092
778
098
io4
109
"5
120
126
i3r
i37
i43
i48
779
yi54
i59
165
170
176
182
i87
i93
i98
2O4
780
209
2l5
221
226
232
237
243
248
254
260
781
265
271
276
282
287
293
298
3o4
3io
3i5
782
321
326
332
337
343
348
354
36o
365
37i
783
376
382
387
393
398
4o4
4o9
4i5
421
426
784
432
437
443
448
454
459
465
47o
476
48 1
785
48 7
492
498
5o4
5o9
5i5
520
526
53i
537
•786
542
548
553
559
564
57o
575
58i
586
592
787
597
6o3
609
6i4
620
625
63i
636
642
647
788
653
658
664
669
675
680
686
69i
697
7O2
789
708
7i3
7i9
724
73o
735
74i
746
752
757
790
763
768
774
779
785
79°
796
801
807
812
N
0
1
2
3
4
5
6
7
8
9
79O— 82O
N
0
1
2
3
4
5
G
7
8
9
790
89 763
768
774
779
785
79°
796
801
807
812
791
818
823
829
834
84o
845
85i
856
862
867
792
873
878
883
889
894
900
9
o5
911
9i6
922
793
927
933
938
944
949
955
960
966
97i
977
794
982
988
1
993
998
*oo4
*oo9
*o
jr
*O2O
*O26
*o3i
795
90 o37
o4s
t
o48
o53
o59
064
069
075
080
086
796
091
097
102
1 08
n3
119
124
129
135
i4o
797
1 46
i5i
i57
162
1 68
i73
1-79
1 84
i89
^95
798
200
206
211
2I7
222
227
233
238
244
249
799
255
260
266
27I
276
282
287
293
298
3o4
800
309
3i4
320
325
33i
336
342
347
352
358
801
363
369
374
38o
385
390
396
4oi
407
4l2
802
4i7
423
428
434
439
445
450
455
46 1
466
8o3
472
477
482
488
493
499
5o4
509
5I5
620
8o4
526
53i
536
542
547
553
558
563
569
574
8o5
58o
585
590
596
601
6o7
612
617
623
628
806
634
639
644
650
655
660
666
671
677
682
8o7
687
693
698
7o3
7°9
7i4
720
725
73o
736
808
74 1
747
752
757
763
768
773
779
784
789
809
795
800
806
811
816
822
827
832
838
843
810
849
854
859
865
87o
875
881
886
891
897
811
902
9°7
9i3
9i8
924
929
934
940
945
95o
812
956
961
966
972
977
982
988
993
998
*oo4
8i3
91 009
oi4
020
O25
o3o
o36
o4i
o46
OD2
o57
8i4
062
068
o73
o78
o84
089
094
IOO
io5
I IO
8i5
116
121
126
I 32
i37
142
i48
i53
i58
1 64
816
169
i74
180
185
190
196
201
206
212
2I7
817
222
228
233
238
243
249
254
259
265
270
818
275
281
286
291
297
302
307
312
3i8
323
819
328
334
339
344
350
355
36o
365
37i
376
820
38i
387
392
397
4o3
4o8
4i3
4i8
424
429
N
O
1
2
3
4
5
G
7
8
9
PP
6
5
i
0.6 i
o.5
2
1.2 2
I.O
3
1.8 3
i.5
4
2.4 4
2.O
5
3.o 5
2.5
6
3.6 6
3.o
7
4.2 7
3.5
8
4.8 8
4.o
9
5.4 9
4.5
82O-860
N
0
1
a
3
4
5
0
7
8
9
820
91 38 1
387
392
397
4o3
4o8
4i3
4i8
424
429
821
434
44o
445
45o
455
46i
466
471
477
482
822
48 7
492
498
5o3
5o8
5i4
5i9
524
529
535
828
54o
545
55i
556
56i
566
572
577
582
587
824
593
598
6o3
609
6i4
6i9
624
63o
63s
64o
825
645
65i
656
661
666
672
677
682
687
693
826
698
7o3
7°9
7i4
719
724
73o
735
74o
745
827
75i
756
761
766
772
777
782
787
793
798
828
8o3
808
8i4
819
824
829
834
84o
845
85o
829
855
861
866
871
876
882
887
892
897
9o3
830
908
9i3
918
924
929
934
939
944
950
955
83i
960
965
97i
9-76
98i
986
991
997
*002
*oo7
83a
92 OI2
018
023
028
o33
o38
o44
049
o54
o59
833
065
070
075
080
o85
091
096
101
1 06
in
834
117
122
1*7
132
i37
i43
1 48
i53
i58
i63
835
169
1 74
i79
i84
i89
195
200
205
2IO
2l5
836
221
226
23l
236
241
247
252
25?
262
267
837
273
278
283
288
293
298
3o4
309
3i4
3i9
838
324
33o
335
34o
345
35o
355
36i
366
37i
839
376
38i
387
392
397
402
407
4l2
4i8
423
840
428
433
438
443
449
454
459
464
469
4?4
84i
48o
485
490
495
5oo
5o5
5n
5i6
521
526
842
53i
536
542
547
552
557
562
567
572
578
843
583
588
593
598
6o3
609
6i4
619
624
629
844
634
639
645
650
655
660
•665
670
675
681
845
686
691
696
7OI
7o6
•711
716
722
727
732
846
737
742
747
752
758
763
768
773
778
783
84?
788
793
799
8o4
8o9
8i4
819
824
829
834
848
84o
845
850
855
860
865
870
875
881
886
849
891
896
901
9o6
9n
916
921
927
932
937
850
942
94?
952
957
962
96-7
973
978
983
988
85i
993
998
*oo3
*oo8
*oi3
*oi8
*024
*029
*o34
*o39
852
93o44
o49
o54
o59
o64
069
°75
080
085
o9o
853
095
IOO
io5
no
u5
120
125
i3i
i36
i4i
854
i46
i5i
i56
161
166
I7I
176
181
186
192
855
197
202
20-7
212
2I7
222
227
282
237
242
856
247
252
258
263
268
273
278
283
288
293
857
298
3o3
3o8
3i3
3i8
323
328
334
339
344
858
349
354
359
364
369
374
379
384
389
394
859
399
4o4
409
4i4
420
425
43o
435
44o
445
860
450
455
46o
465
47o
475
48o
485
490
495
N
o
1
2
3
4
5
6
7
8
9
23
860-890
N
0
1
2
3
4
5
6
7
8 9
860
93450
455
46o
465
470
475
48o
485
490
495
861
5oo
5o5
5xo
5x5
52O
526
53i
536
54 1
546
862
55i
556
56i
566
57i
576
58i
586
59i
596
863
60 1
606
611
616
621
626
63i
636
64i
646
864
65i
656
661
666
671
676
682
687
692
697
865
702
707
712
717
722
727
732
737
742
747
866
752
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
9I2
917
922
927
932
937
942
947
870
952
957
962
967
972
977
982
987
992
997
871
94 002
007
OI2
017
022
027
032
o37
042
047
872
052
o57
062
067
O72
°77
082
086
091
096
873
IOI
1 06
III
116
121
126
I
3i
1 36
i4i
1 46
874
i5i
1 56
161
1 66
I7I
176
181
186
191
196
875
2OI
206
21 I
216
221
226
23l
236
240
245
876
25o
255
260
265
27O
275
280
285
290
295
877
3oo
305 i 3io
3x5
320
325
33o
335
34o
345
878
349
354
359
364
369
374
379
384
389
394
879
399
4o4
409
4x4
419
424
429
433
438
443
880
448
453
458
463
468
473
478
483
488
493
881
498
5o3 5o7
5l2
5i7
522
527
532
537
542
882
547
552 557
562
567
57i
576
58i 586
691
883
596
601
606
6x1
616
621
626
63o
635
•64o
884
645
65o
655
660
665
67o
675
680
685
689
885
694
699
7o4 7°9
7i4
•719
724
729
734
738
886
743
748
753
758
763
768
773
778
783
787
887
792
797
802
807
812
817
822
827
832
836
888
84i
846
85i
856
861
866
871
876
880
885
889
890
895
900
9°5
910
9*5
919
924
929
934
890
939
9U
949
954
959
963
968
973
978
983
N
O
1
2
3
4
5
a
7
8
9
PP 5
4
i o.5 i
o.4
2 I.O 2
0.8
3 i.5 3
I .2
4 2.0 4
1.6
5 2.5 5
2.O
6 3.o 6
2.4
7 3.5 7
2.8
8 4.o 8
3.2
9 4.5 9
3.6
24
89O— 93O
N
O
1
a
3
4
5
6
7
8
9
890
94939
944
949
954
959
963
968
973
978
983
89i
988
993
998
*OO2
*oo7
*OI2
*OI7
*022
*027
*032
892
95o36
o4i
o46
o5i
o56
061
066
07I
o75
080
893
o85
o9o
095
100
105
109
n4
124
129
894
1 34
i39
i43
1 48
i53
i58
i63
168
i73
177
895
182
187
192
197
202
207
211
216
221
226
896
a3i
236
240
245
25o
255
260
265
270
274
897
279
284
289
294
299
3o3
3o8
3i3
3i8
323
898
328
332
337
342
347
352
357
36i
366
37i
899
376
38i
386
390
395
4oo
405
4io
4i5
419
900
424
429
434
439
444
448
453
458
463
468
901
472
477
48a
487
492
497
5oi
5o6
5u
5i6
9O2
521
525
53o
535
54o
545
55°
554
559
564
9o3
569
574
578
583
588
593
598
602
607
612
904
617
622
626
63i
636
64 1
646
65o
655
660
9©5
665
670
674
679
684
689
694
698
7o3
7o8
906
7i3
718
722
727
732
737
742
746
761
756
907
761
766
770
775
78o
785
789
794
799
8o4
908
809
8i3
8l8
823
828
832
837
842
847
852
909
856
861
866
871
875
880
885
89o
895
899
910
9o4
9°9
914
918
923
928
933
938
942
947
911
952
957
961
966
*9?I
976
98o
985
99°
*995
912
999
*oo4
*oo9
*oi4
*023
*028
*o33
*o38
9i3
96 047
052
o57
06 1
066
07I
o76
080
o85
090
914
°95
°99
io4
109
n4
118
123
128
i33
i37
9i5
142
i47
I 52
i56
161
166
171
i75
1 80
185
916
190
i94
199
204
209
2l3
218
223
227
232
917
237
242
246
a5i
256
261
265
27O
275
280
918
284
289
294
298
3o3
3o8
3i3
3i7
322
327
919
332
336
34i
346
35o
355
36o
365
369
374
920
379
384
388
393
398
402
4o7
4l2
4i7
421
921
426
43i
435
44o
445
450
454
459
'464
468
922
473
478
483
487
492
497
5oi
5o6
5n
5x5
923
52O
525
53o
534
539
544
548
553
558
562
924
567
572
577
58i
586
59i
595
600
605
609
925
6i4
6i9
624
628
633
638
642
647
652
656
926
661
666
67o
675
680
685
689
694
699
7o3
927
7o8
7i3
717
722
727
73i
736
74i
745
75o
928
755
759
764
769
774
778
783
788
792
797
929
802
806
811
816
820
825
83o
834
839
844
930
848
853
858
862
867
872
876
881
886
890
N
O
1 2
3
4
5
6
7
8
9
93O-96O
N
O
1
2
3
4
5
6
7
8
9
980
96848
853
858
862
867
872
876
881
886
890
93i
895
900
9o4
9°9
9i4
918
923
928
932
937
932
942
946
95i
956
960
965
97°
974
979
984
933
988
993
997
*OO2
*oo7
*OI I
*oi6
*O2I
*025
*o3o
934
97035
o39
o44
o49
o53
o58
o63
067
O72
077
935
08 1
086
o9o
o95
IOO
104
109
n4
118
123
936
128
132
i37
142
i46
i5i
i55
160
165
i69
937
1 74
*79
i83
188
192
19-7
202
206
211
216
938
220
225
230
234
239
243
248
253
257
262
939
267
271
276
280
285
290
294
299
3o4
3o8
940
3i3
3i7
322
327
33i
336
34o
345
350
354
94i
359
364
368
373
377
382
387
391
396
4oo
942
4o5
4io
4i4
4i9
424
428
433
437
442
447
943
45 1
456
46o
465
470
474
479
483
488
493
944
945
497
543
5O2
548
5o6
552
557
5i6
562
520
566
5^5
57i
529
575
534
58o
539
585
946
589
594
598
6o3
607
612
617
621
626
63o
947
635
64o
644
649
653
658
663
667
672
676
948
681
685
69o
695
699
7o4
708
7i3
717
722
949
727
73i
736
74o
745
749
754
759
763
768
950
772
777
782
786
791
795
800
8o4
809
8i3
95i
818
823
827
832
836
84 1
845
85o
855
859
952
864
868
873
877
882
886
891
896
9oo
9°5
953
909
9i4
9i8
923
928
932
937
94i
946
95o
954
955
959
964
968
973
978
982
987
99i
996
955
98 ooo
005
oo9
oi4
019
023
028
o3a
o37
o4i
956
o46
o5o
055
o59
o64
068
073
o78
082
o87
957
091
o96
IOO
105
109
n4
118
123
127
I 32
958
i37
i4i
i46
i5o
155
i59
1 64
168
i73
177
959
182
186
I9i
i95
200
2O4
209
2l4
218
223
960
227
232
236
241
245
250
254
259
263
268
N
O
1
2
3
4
5
O
7
8
9
PP
5
4
i
o.5 i
o.4
2
I.O 2
0.8
3
i.5 3
1.2
4
2.0 4
1.6
5
2.5 5
2.O
6
3.o 6
2.4
7
3.5 7
2.8
8
4.o 8
3.2
9
4.5 9
3.6
26
960-1OOO
N
0
1
ti
3
4
5
6
7
8 9
960
98227
232
236
24 1
245
250
254
259
263
268
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
272
3i8
363
4o8
453
498
543
588
632
277
322
367
412
457
5O2
547
592
637
281
327
372
4i7
462
507
55b
597
64 1
286
33i
376
421
466
5n
556
601
646
29O
336
38i
426
47i
5i6
56i
6o5
65o
295
34o
385
43o
475
520
565
610
655
299
345
390
435
48o
525
57o
6i4
659
3o4
349
394
439
484
529
574
619
664
3o8
354
399
444
489
534
5-79
623
668
3i3
358
4o3
448
493
538
583
628
673
677
682
686
691
695
700
704
709
7i3
717
722
76.7
in
856
900
945
989
99 o34
078
726
771
816
860
9°5
949
994
o38
o83
73i
776
820
865
909
954
998
o43
087
735
780
825
869
914
958
*oo3
047
092
74o
784
829
874
918
963
*oo7
052
096
744
789
834
878
923
967
*OI2
o56
IOO
749
793
838
883
927
972
*oi6
06 1
105
753
798
843
887
932
976
*02I
065
109
758
802
847
892
936
981
*025
069
n4
762
807
85 1
896
94i
985
*029
074
118
123
127
i3i
i36
i4<>
i45
1 49
1 54
i58
162
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
167
21 I
255
3oo
344
388
432
476
52O
171
216
260
3o4
348
392
436
48o
524
176
220
264
3o8
352
396
44 1
484
528
180
224
269
3i3
357
4oi
44s
489
533
185
229
273
3i7
36i
4o5
449
493
537
189
233
277
322
366
4io
454
498
542
I93
238
282
326
37o
4i4
458
502
546
198
242
286
33o
374
419
463
5o6
55o
202
247
291
335
379
423
467
5n
555
207
25l
295
339
383
427
47i
5i5
559
564
568
572
577
58i
585
590
594
599
6o3
607
65i
695
739
782
826
870
9i3
957
612
656
699
743
787
83o
874
917
961
616
660
704
747
791
835
878
922
965
621
664
708
752
795
839
883
926
97°
625
669
712
756
800
843
887
93o
974
629
673
717
760
8o4
848
891
935
978
634
677
721
765
808
852
896
939
983
638
682
726
769
8:3
856
9oo
944
987
642
686
73o
774
817
861
9o4
948
99i
647
691
734
778
822
865
909
952
996
00 OOO
oo4
009
oi3
017
022
026
o3o
o3§
o39
X
O
1
2
3
4
5
6
7
8
9
27
TABLE II
FIVE -PL ACE LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
TO EVERY MINUTE
0°.
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
0
—
—
—
0 . OO OOO
60
I
6.46 373
30103
6.46 373
3.53627
O.OO OOO
59
2
3
6.94085
17609
6.76476
6.94085
30103
17609
3.23 524
3.o59i5
O.OO OOO
0.00 000
58
57
4
7.06679
12494
9691
7.06 579
12494
nfinr
2.9
3421
O.OO OOO
56
5
7.16 270
7018
7.16 270
909
2.8
373o
O.OO OOO
55
6
7.24 188
7910
7.24 188
791?
2.75 812
O.OO OOO
54
6694
6694
7
7.30882
7.30882
c8oo
2.69 1 1 8
O.OO OOO
53
8
7.36682
7.36682
2.633:8
O.OO OOO
52
9
7-4i 797
5"5
7-4i 797
5"5
2.58 2o3
O.OO OOO
5i
10
7.46373
7.46 373
457°
2.53 627
O.OO OOO
50
1 1
7.5o 5i2
4J39
7.5o 5i2
4139
2.49488
O.OO OOO
49
12
i3
7.54 291
7.57 767
3476
7.54 291
7.57767
3476
2.45 709
2.42 233
O.OO OOO
O.OO OOO
48
47
i4
i5
7.60985
7.63 982
3218
2997
7.60 986
7.63 982
3219
2996
2. Sg Ol4
2.36 018
0.00 000
O.OO OOO
46
45
16
7.66784
7.66785
2803
2.332:5
O.OO OOO
44
2633
263
1
17
7-69417
2483
7.69418
2482
2.3o 582
9-99999
43
18
7.71 900
7-74248
2348
7.71 900
7.74248
2348
2.28 :oo
2.25 752
9-99999
9-99999
42
4i
20
7.76475
2227
7.76476
2.23 524
9-99999
40
21
7-7*
J 594
7.78595
2.2: 405
9-99999
39
22
7.80615
7.80 6i5
2.:9385
9-99999
38
23
7.82545
1930
7.82546
*93
2.:7 454
9-99999
37
1848
184
1
24
7.84393
7-84394
2.: 5 606
9 -.99 999
36
25
7.8(
5 166
7.86167
2.:3833
9-99999
35
26
7.87870
1704
7.87871
1704
2.:2 :29
9-99999
34
1639
i63(
1
27
28
7«89 509
7.91 088
1579
7.89 5io
7.91 089
1579
2. :o 49o
2.08 9: i
9-99999
9-99999
33
32
29
7.92 612
1524
7.92 6i3
152-
\
2.07 387
9-99 99s
3:
30
7.94 o84
7.94 086
1
2.o5 9:4
9-99998
30
L. Cos.
d.
L. Cotg. d.
L. Tang.
L. Sin.
'
89° 3O .
PP
9691
4576
2997
2483
2119
I848
1704
1579
1472
.1
969
458
300
.x
248
212
185
.!
170
158
*47
.2
1938
.2
497"
424
37°
.2
341
316
294
•3
2907
1372
899
•3
745
636
554
•3
5"
474
442
•4
3876
1830
"99
•4
993
848
739
•4
682
632
589
5
4846
2288
1498
•5
1242
1060
924
•5
8.S2
789
736
.6
2646
1798
.6
i49p
I27I
1109
.6
IO22
947
883
•7
6784
3203
2098
• 7
1738
M83
1294
.7
"93
1105
1030
.8
7753
3661
2398
.8
1986
l695
1478
.8
1263
1178
4118 2697
1663
1421
1325
O° 3O .
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
30
7 . 94 o84
7.94 086
2.o5 914
9.99998
30
3i
32
7.95 5o8
7.96 887
1424
1379
7.96 5io
7.96 889
1424
1379
2.04 490
2.O3 III
9.99998
9.99998
29
28
33
7.98 223
133°
7-9*
1 225
2.01 775
9.99998
27
1297
129;
34
7.99 520
7.99 522
1259
2.OO 478
9.99998
26
35
36
8.00 779
8. 02 002
1223
1190
8.00 781
8.02 004
1223
1190
I.992I9
1.97996
9.99998
9.99998
25
24
37
8.o3 192
1158
8.o3 194
1159
I .96 806
9-99 997
23
38
8.o4 35o
1128
8.04353
i .96 647
9-99997
22
39
8.o5
478
8.o548i
i . 94 5 1 9
9-99997
21
40
8.06 578
8.06 58i
1.93419
9-99997
20
4i
8.07 650
1046
8.07 653
1047
1.92 347
9-99997
I9
42
8.08 696
8.08 700
i .91 3oo
9-99997
18
43
8.09 718
8.09 722
1022
i .90 278
9-99997
17
999
998
44
8. 10 717
976
8. 10 720
i .89 280
9.99996
16
45
8. ii
693
8. ii
696
1.88 3o4
9.99996
i5
46
8.12
647
954
8.12 65 1
955
1.87 349
9.99996
i4
47
8.i3 58i
934
8.i3 585
934
1.86415
9.99996
i3
48
8.i4
495
8. 1 4 500
i.85 5oo
9.99996
12
49
8.16391
896
8.i5395
895
0_0
i.84 605
9.99996
II
50
8.16268
877
8.16273
i. 83 727
9.99995
10
5i
8.17 128
840
8.17 i33
843
i .82 867
9.99996
9
52
8.17 971
8.17976
828
i .82 024
9-99 995
8
53
8.18 798
27
812
8.18 8o4
812
i .81 196
9-99995
7
54
55
' 8.19 610
8. 20 407
797
8.19 616
8.2o4i3
797
_0_
i. 80 384
1.79687
9-99995
9.99994
6
5
56
8.21
189
782
8.21 196
702
1.78 805
9.99994
4
57
8.21
968
769
8.21 964
769
7rA
1.78036
9.99994
3
58
8.22 713
755
8.22 720
1.77 280
9.99994
2
59
8.23456
743
8.23462
742
1.76 538
9-99 994
I
60
8.24 186
730
8.24 192
73°
1.75808
9.99993
0
L. Cos. d.
L. Cotg.
d.
L, Tang.
L.
Sin.
'
89°.
PP 1379
1223
IIOO
999
914
860
812
769
730
r
138
122
no
.1
IOO
9*
86
i
81
77
73
.2
276
245
220
.2
200
183
172
2
162
146
•3
414
367
330
•3
300
274
258
3
244
231
219
• 4
552
489
440
•4
400
366
344
4
325
308
292
690
612
500
457
430
5
406
tf-5
365
.6
827
734
660
.6
599
548
6
487
461
438
•7
96s
856
770
7
699
640
602
7
568
538
5"
.8
1103
978
880
.8
799
73i
688
8
650
615
584
uoi 990
9
899 823
774
9 731
692
657
3 1
1°.
,
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
0
8.24 186
8.24 192
i.75 808
9.99993
60
I
8.24903
717
706
8.24 910
710
706
i .75 090
9.99 993
59
2
8.25 609
8.25 616
1.74384
9.99 993
58
3
8.26 3o4
095
684
8.26 3i2
2J
1.73688
9.99993
5?
4
8.26 988
673
8.26 996
671
1.73 oo4
9.99992
56
5
8.27 661
8.27 669
i .72 33i
9.99 992
55
6
8.28 324
663
8.28 332
663
1.71 668
9.99992
54
7
8.28977
653
8.28 986
64?
1.71 oi4
9.99992
53
8
8 .29 621
8 .29 629
1.70 37i
9.99992
52
9
8.3o 255
°34
8.3o 263
634
i.69737
9-99991
5i
10
8,3o 879
024
616
8.3o888
625
i .69 112
9-99 991
50
n
8.3i
495
608
8.3i 5o5
17
607
i.68495
9-99 991
49
12
8.32 io3
8.32 112
1.67888
9.99990
48
:3
8.32 702
599
8.32 711
599
1.67 289
9-9999°
47
14
8.33 292
59°
8.33 3o2
^
1.66698
9.99990
46
i5
8.33875
8.33 886
5 4
i .66 1 14
9.99990
45
16
8.3445o
568
8.3446r
575
568
i.65539
9.99989
44
17
8.35oi8
560
8.35 029
i .64 971
9.99989
43
18
8.35 578
8.35 590
5
i .64 4io
9.99989
42
19
8.36 i3i
553
8.36 i43
553
i.63857
9-99 989
4i
20
8. 36 678
8. 36 689
M°
i.63 3n
9.99 988
40
21
8.37 217
533
8.37 229
54°
i .62 771
9-99988
39
22
8.37750
526
8.37 762
1.62 238
9.99988
38
23
8.38 276
520
8.38 289
527
520
i .61 711
9.99987
3?
24
8.38 796
514
8.38 809
i .61 191
9.99987
36
25
8.39 3io
8.39 323
i .60 677
9.99987
35
26
8. 398i8
SOB
8.39832
s°9
i. 60 168
9.99986
34
27
8.4o 320
502
406
8.4o 334
502
406
1.59666
9.99986
33
28
8.4o 816
8.4o83o
i .5
9 170
9.99 986
32
29
8.4i 307
491
8.4i 32i
491
i.5
8679
9.99 985
3i
30
8.4i 792
485
8.4i 807
486
i.58 i93
9.99985
30
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
'
88° 30 .
PP
706
663
634
599
575
553
533
SM
496
.!
70.6
66.3
63.4
.1
59-9
57-5
55-3
.!
53-3
51-4
49-6
.2
141.2
132.6
126.8
.2
119.8
115.0
1 10. 6
.2
106.6
102.8
99.2
•3
211. 8
198.9
190.2
•3
179.7
172-5
165.9
•3
159-9
I54-2
148.8
•4
282.4
265.2
253-6
•4
239.6
230.0
221.2
•4
213.2
205.6
198.4
• 5
353-°
33I-5
317-°
t e
299-5
287.5
276.5
• 5
266.5
257.0
248.0
.6
423.6
397-8
380.4
6
359-4
345-0
33^.8
.6
319.8
308.4
297.6
• 7
494.2
464.1
443-8
•7
4r9-3
402.5
387.1
.7
373-1
359-8
347-2
.8
564.8
53°- 4
507.2
.8
479-2
460.0
442-4
.8
426.4
411.2
396.8
.9 635.4
596.7 ' 570.6
539.1 5T7-5
497 -J
•9 479-7
446.4
32
>
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
30
8.4i 792
8.4i 807
i .5s i93
9.99985
30
3i
32
8.42 272
8.42 746
480
474
8.42 287
8.42 762
475
i.577i3
1.57 238
9.99985
9.99 984
29
28
33
8.43 216
470
464
8.43 232
470
464
i.56 768
9.99984
27
34
35
8.43 680
8.44 i39
459
8. 43 696
8.44 i56
460
i.56 3o4
i.55 844
9.99984
9.99 983
26
25
36
8 .44 594
455
8.446u
455
i.55389
9.99983
24
37
38
8.45 o44
8.45489
45<>
445
8.45o6i
8.45 507
450
446
1.54939
1.54493
9.99983
9.99982
23
22
39
8.45 930
441
8.45 948
44'
i.54o52
9.99982
21
40
8.46 366
430
8.46385
437
i.536i5
9.99982
20
4i
8.46 799
433
8.46817
432
428
i.53 i83
9.99 981
19
42
8.47 226
8.47 245
1.52 755
9.99981
18
43
8.47650
424
8.47 669
424
i.52 33i
9.99981
'7
419
4*
44
45
8.48 069
8.48485
416
8.48 089
8.485o5
416
i . 5i 911
i.5i 495
9.99980
9.99980
16
i5
46
8.48896
411
408
8.4*
J 917
412
408
j.5i o83
9.99979
i4
47
8.49 3o4
8.49 325
i 5o 675
9.99979
i3
48
8.49 708
404
8.49 729
i .5o 271
9.99979
12
49
8.5o 1 08
400
8.5o i3o
401
i .49 870
9.99 978
I I
50
8.5o 5o4
39°
8.5o 527
397
i .49 473
9.99978
10
5i
8.5o 897
393
8.5o 920
393
i .49 080
9.99977
9
52
8.5i
287
-Of:
8.5i 3io
-Of.
i .48 690
9.99977
8
53
8.5i
673
330
382
8.5i 696
300
383
i.48 3o4
9 99977
7
54
8.52 o55
8.52 079
180
i .47 921
9.99976
6
55
8.52434
8.52459
i.4754i
9.99976
5
56
8.52 810
376
8.52 835
376
1.47 i65
9.99975
4
373
37:
57
8.53 i83
060
8.53 208
i .46 792
9-99975
3
58
8.53 552
8.53 578
i .46 422
9.99974
2
59
8.53 919
367
8. 53 945
367
,6,
i.46o55
9.99974
I
60
8.54 282
8.54 3o8
i .45 692
9-99 974
0
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin.
!
88°.
PP
470
455
441
424
4o8
396
386
376
367
.!
47.0
45-5
44.1
.1
42.4
40.8
39-6 -i
38.6
376
36.7
. 2
94.0
91 o
88.2
.2
84.8
81.6
79.2 .2
77-2
752
73-4
•3
141.0
136-5
132-3
•3
127.2
122.4
118.8 .3
1128
1 10. 1
•4
188.0
182.0
176.4
•4
169.6
163.2
158.4 .4
154-4
1504
146.8
• 5
235-0
227.5
220.5
• 5
2I2.O
204.0
198.0 .5
193.0
1880
183-5
6
282.0
273.0
264.6
.6
254-4
244.8
237-6 -6
231.6
225.6
220.2
.7
329.0
318.5
308.7
• 7
296.8
285.6
277.2 .7
270.2
263.2
256.9
.8
376.0
364.0
352.8
.8
339-2
326.4
316.8 .8
308.8
300.8
293-6
•9 423-
409. 5 396. 9
381.6 367.2
356-4 -9 347-4
338.4
33°-3
33
2°.
/
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
0
8.54282
760
8.54 3o8
i .45 692
9.99974
60
I
8.54642
357
8.54669
358
i.45 33i
9.99 973
59
2
8.54 999
8.55 027
i.44 973
9.99973
58
3
8.55 354
355
8.55 382
355
i.446i8
9.99 972
57
4
8.557o5
8.55734
352
1.44266
9.99 972
56
5
8.56o54
8.56o83
349
i .43 917
9.99 971
55
6
8.56 4oo
346
8.56429
346
i.43 57i
9.99971
54
7
8.56 743
343
8.56 773
344
i.43 227
9.99 97o
53
8
8.67 o84
8.57 n4
1.42 886
9.99 9-70
52
9
8.57 421
337
8.57452
338
1.42 548
9-99 969
5i
10
8.57757
33°
8.57788
336
I .42 212
9-99 969
50
1 1
8.58 089
8.58 121
333
33°
I .4l 879
9.99968
49
12
i3
8.584i9
8.58747
328
8.5845i
8.58779
328
i .41 549
I .4l 221
9.99968
9.99 967
48
47
i4
i5
8.59 072
8.59395
325
323
8.59 io5
8.69428
326
323
i.4o 895
1 .40 672
9.9996-7
9.9996-7
46
45
16
8.69 716
318
8.59 749
321
i .4o 261
9.99966
44
17
18
8.6oo33
8.60 349
3i6
8.60068
8.60 384
316
1.39 932
1.39 616
9.99966
9.99965
43
42
'9
8.60 662
3*3
8.60 698
3i
|
1.39 3o2
9.99 964
4i
20
8.60 973
3°9
8. 6 r 009
3"
i.38 991
9.99 964
40
21
8.61 282
307
8.61 3i9
310
i. 3868i
9.99963
39
22
8.61 589
8.61 626
i.38 374
9.99963
38
23
8.61 894
8.61 931
3°5
i.38 069
9.99962
37
302
3°
1
24
8.62 196
301
8.62234
i. 37766
9.99962
36
25
8.62 497
8.62 535
1.37465
9.99961
35
26
8.62795
298
8.62 834
299
1.37 166
9.99961
34
27
28
8.63 091
8.63 385
296
294
8.63 i3i
8.63426
297
295
i.36 869
i.36574
9.99960
9.99960
33
32
29
8. 63 678
293
8.63 718
292
1.36282
9.99 959
3i
30
8. 63 968
290
8.64 009
291
i.35 991
9.99959
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
87° 3O'.
PP
360
350
34°
33°
320
310
300
290
285
.1
36
35
34
.1
33
32
31 .1
30
29
28.5
.2
72
7°
68
.2
66
64
62 .2
60
58
57-°
•3
1 08
105
1 02
•3
99
96
93 -3
90
87
85-5
•4
144
140
136
•4
132
128
124 .4
120
116
114.0
180
170
•65
160
ISO
145
M2.5
.6
216
2IO
204
.6
198
192
1 86 .6
1 80
171.0
•7
252
245
238
• 7
231
224
217 .7
2IO
203
199-5
.8
288
280
272
.8
264
256
248
240
232
228.0
.0
324
315
306
2Q7
288
279 .9 270
161
256.5
2° 3O .
/
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
30
8.63 968
8 .64 009
i .35 991
9.99 959
30
3i
32
8.64256
8.64543
287
8.64 298
8.64 585
289
287
i .35 702
i.35 415
9.99958
29
28
33
8.6
4827
284
283
8.64 870
285
284
i.35 i3o
9.99957
27
34
8.65 no
8.65 i54
1.34846
9.99 956
26
35
8.6539i
8.65435
1.34565
9.99956
25
36
8. 65 670
279
8. 65 715
280
1.34285
9.99955
24
37
38
8.65 947
8.66223
277
276
8.65 993
8.66 269
278
276
i .34 007
i.33 73i
9.99955
9.99954
23
22
39
8.66 497
274
8.66543
274
1.33457
9-99954
21
40
8.66 769
272
8.66816
273
i.33 1 84
9-99 953
20
4i
42
8.67 039
8.673o8
270
269
8.67 087
8. 67 356
271
269
i.32 913
i.32 644
9.99952
9.99952
«9
18
43
8.67575
267
266
8.67624
266
i.32376
9.99961
«7
44
45
8.67841
8.68 io4
263
8.67 890
8.68 1 54
264
i .32 no
i.3i 846
9.99961
9.99950
16
i5
46
8.68 367
263
260
8.684i7
263
261
i.3i 583
9.99949
i4
47
8.68627
8.68678
260
i.:
»I 322
9.99949
i3
48
8.68886
259
8.68 938
1.2
(I 062
9.99948
12
49
8.69 1 44
258
8.69 196
258
i.3o 8o4
9.99948
I I
50
8.69 4oo
256
8.69453
257
i.3o547
9.99 947
10
5i
8. 69 654
254
8.69 708
255
i .3o 292
9.99946
9
52
8.69 907
8.69 962
i.3oo38
Q. 99946
8
63
8.70 i59
252
8.70 214
252
1.29 786
9.99945
7
250
25
54
8.70 409
8.70465
i. 29 535
9.99944
6
55
8.70658
8.70 714
.29 286
9.99944
5
56
8.70 905
247
246
8.70 962
240
246
.29 o38
9.99943
4
57
8.71 i5i
8.71 208
.28 792
9.99942
3
58
8.71 395
8.7i453
.28547
9.99942
2
59
8.71 638
243
8.7i 697
244
.28 3o3
9.99941
I
60
8.71 880
8.71 940
i .28 060
9-9994o
0
L.
Cos.
d.
L. Cotg.
d. L. Tang.
L.
Sin.
f
87°.
PP
280
275
270
265
260
255
250
245
240
.1
28.0
27-5
27.0
.1
26.5
26.0
25-5
.1
25.0
24-5
24.0
.2
56.0
55-0
54-o
.2
53-o
52.0
51.0
.2
50.0
49-°
48.0
•3
84.0
82.5
81.0
•3
79-5
78.o
76.5
•3
75-0
73-5
72.0
• 4
112. 0
1 10.0
108.0
•4
106.0
104.0
102.0
•4
100.0
98.0
96.0
t e
140.0
137-5
i35-o
•5
132-5
130.0
I27-5
•5
125.0
122.5
12O.O
.6
168.0
165.0
162.0
.6
159.0
156.0
i53-o
.6
150.0
147.0
144.0
.7
196.0
192.5
189.0
•7
185-5
182.0
178-5
•7
175-0
i7i-5
168.0
.8
224.0
22O.O
216.0
.8
2I2.O 2O8.O
204.0
.8
200.0
196.0
192.0
•9
247-5
243.0
•9
238.5 234.0
.9 225.0
216.0
35
(
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
0
8.71 880
340
239
238
237
235
234
232
232
230
229
228
226
226
224
223
222
2 2O
2 2O
2I9
2I7
216
216
2I4
213
212
211
210
209
208
208
8.71 940
241
239
239
237
236
234
234
232
231
229
229
227
226
225
224
222
222
220
2I9
2I9
217
216
215
214
213
211
211
210
209
208
i .28 060
9 .99 940
60
I
2
3
4
5
6
8
9
8.72 I2O
8. 72359
8.72 597
8.72 834
8.73069
8.73 3o3
8.73 535
8.73 767
8.73 997
8.72 181
8.72 420
8.72 659
8.72 896
8.73i32
8.73366
8.73 600
8.73832
8 . 74 06-3
i .27 819
1.27 58o
1.27 34 i
1.27 io4
1.26868
1.26634
i .26 4oo
1.26 168
i .25 937
9.99 940
9.99939
9.99938
9.99938
9.99937
9.99936
9.99936
9.99935
9.99934
59
58
57
56
55
54
53
52
5i
10
8.74 226
8.74 292
i.25 708
9.99934
50
ii
12
i3
i4
i5
16
17
18
'9
8.74454
8.74 680
8.74 906
8.75 i3o
8. 75353
8.75575
8.75795
8.76oi5
8.76234
8.74 52i
8. 74748
8.74974
8.75 199
8.75 423
8.75 645
8.75867
8.76087
8.76 3o6
i.25 479
I .25 252
I .25 026
i .24 801
1.24 577
1.24355
1.24 i33
i.23 913
i.23 694
9.99933
9.99932
9.99932
9.99931
9.99930
9.99929
9.99929
9.99928
9.99 927
49
48
47
46
45
44
43
42
4r
20
8.7645i
8.76 525
I .23 475
9.99926
40
21
22
23
24
25
26
27
28
29
8.76667
8. 76 883
8.77097
8.77 3io
8.77 522
8.77943
8.78 i52
8.78 36o
8.76 742
8.76958
8.77173
8.77387
8.77 600
8.77 811
8.78 022
8.78232
8.7844i
i.23 258
I .23 O42
I .22 827
1.22 6l3
I .22 400
I .22 189
I .21 978
I. 21 768
I .21 559
9.99926
9.99925
9.99924
9.99923
9.99923
9.99922
9.99921
9.99920
9.99 920
39
38
37
36
35
34
33
32
3i
30
8.78 568
8.78 649
i. 2 1 35 1
9.99 919
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
80
°3O
.
PP
238
234
229
.2
•3
•4
:78
•9
225
220
216
2X2
208
204
.2
•3
•4
• 9
23.8
47.6
71.4
95-2
119.0
142.8
166.6
190.4
23-4
46.8
70.2
93-6
117.0
140.4
163.8
187-2
22.9
a;
91.6
"4-5
137-4
160.3
183.2
206. i
22.5
45-0
67-5
90.0
112.5
135-0
157-5
180.0
52-5
22. 0
44.0
66.0
88.0
IIO.O
132.0
176.0
21.6 .1
43-2 ' .2
64.8 .3
86.4 1 .4
108.0 .5
129.6 ! .6
151-2 .7
172.8 .8
194.4 .9
21.2
42.4
63.6
84.8
ro6.o
127.2
148.4
[69.6
90.8
20.8
41.6
62.4
83.2
104.0
124.8
145.6
166.4
187.2
20.4
40.8
61.2
81.6
IO2.O
122-4
142.8
163.2
183.6
36
3° 3O'.
/
L. Sin.
d.
L. Tang. d.
L. Cotg.
L.
Cos.
30
8.78 568
8.78 649
i .21 35i
9-99 9'9
30
3i
8.78 774
8.78 855
206
206
1.21 145
9.99918
29
32
8.78979
8.70. 061
i .20 939
9.99917
28
33
8.79 i83
204
8.79266
205
I .20 734
9.99917
27
203
20<
34
35
8.79 386
8.79 588
202
8.79 4?o
8.79673
203
i .20 53o
i .20 327
9.99916
9.99915
26
25
36
8.79789
8.79875
I .20 125
9.99 9i4
24
20J
37
8.79990
8.80076
1.19 924
9.99913
23
38
8.80 189
199
8.80 277
1 .19 723
9.99913
22
39
8. 80 388
199
8.80476
199
i . 19 524
9.99912
21
40
8. 80 585
197
8.8o674
198
1.19 326
9.99911
20
4i
8.80782
197
1 06
8.80 872
198
1 06
1.19 128
9.99910
i9
42
43
8.80 978
8.81 173
'95
8.81 068
8.81 264
I96
1. 18 932
i.i8736
9.99909
9.99909
1 8
17
194
X9S
44
8.81 367
8.81 459
1. 18 54i
9.99908
16
45
8.81 56o
Z93
8.81 653
X94
i.i8347
9.99907
i5
46
8.81 752
192
8.81 846
X9-.
1. 18 i54
9.99906
i4
47
8.81 944
192
8.82o38
192
i . 17 962
9.99905
i3
48
8.82 1 34
8.82 23o
1.17 770
9.99904
12
49
8.82 324
190
8.82 420
190
1.17 58o
9.99904
I I
50
8.82 5i3
8.82 610
i.i739o
9.999o3
10
5i
8.82 701
8.82 799
1 88
I.I7 201
9.99902
9
r
8.82888
8.820,87
18
5
1.17 oi3
9.99901
8
53
8.83075
lOJ
186
8.83i75
1 81
i
1.16825
9.99900
7
54
8.83 261
18
8.8336i
i8C
(
1.16 639
9.99899
6
55
8.83446
8.83547
1. 16453
9.99898
5
56
8.8363o
104
183
8.83732
105
184
1.16268
9.99898
4
57
8.83 8i3
181
8.839i6
1 8 A
1. 1 6 o84
9-99 897
3
58
8.83 996
8.84 ioo
0
i . 1 5 900
9.99896
2
59
8.84 177
181
8.8
4282
182
i .
[5 718
9-99 895
I
60
8.84 358
8.84464
i.i5536
9.99 894
0
L.
Cos. d.
L. Cotg. d
. L. Tang.
L.
Sin.
'
86°.
PP
201
198
195
192
I89
187
185
183
181
.1
20.1
19.8
19-5
.1
19.2
,8.q
i8.7
.x
18.5
18.3
18.1
.2
40.2
39.0
.2
38.4
37-8
37-4
.2
37-°
36.6
36-2
•3
60.3
59-4
58-5
•3
57-6
56-7
56.1
•3
55-5
54-9
54-3
•4
80.4
79-2
78.0
•4
76.8
75-6
74-8
•4
74.0
73-2
72.4
100-5
So
97-5
.5
96.0
Q4-5
93-5
•5
92-5
91.5
90-5
.6
1 20. 6
8
117.0
.6
115-2
"3-4
II2.2
.6
III.O
109.8
co8.6
140.7
138.6
136-5
•7
134-4
132.3
130.9
•7
129.5
128.1
126.7
160.8
158.4
156.0
.8
151.2
149.6
.8
148.0
146.4
r44.8
•9
180.9
178.2 175.5
168.3
164.7 '62.9
4°.
/
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
0
8.84358
_0
8.84464
1. 15536
9.99 894
60
I
8. 84539
179
8.84646
1 80
.16 354
9.99893
r
2
8.8
4718
8.84 826
.i5 174
9.99 892
Do
3
8.8
4897
179
8.85 006
1 80
.14994
9.99891
5?
4
8.85 075
178
8.85 185
179
T,o
.i48i5
9.99 891
56
5
8.85 252
8.85363
.i4637
9.99890
55
6
8.85429
177
8.85 54o
177
. i4 46o
9.99889
54
<7
8.85605
176
8.85 717
177
176
i.i4283
9-^
>9888
53
8
8.85 780
8.85893
i . i4 107
9.99887
52
9
8.85 955
175
8.86069
176
i.i393i
9.99 886
5i
10
8.86 128
173
8.86 243
174
i.!3757
9.99 885
50
1 1
8.86 3oi
8.86417
1. 13583
9.99884
49
12
8.86474
8.8659i
1. 1 3 409
9.99 883
48
i3
8.86645
171
8.86763
172
i.i3237
9.99 882
4?
i4
8.86816
171
8.86935
172
i.i3 065
9.99 881
46
i5
8.86987
1 60
8.87 106.
I .12 894
9.99 880
45
16
8.87 i56
169
8.87277
171
170
I . 12 723
9.998-79
44
18
8. 87 325
8.87494
169
8. 87 447
8.87 616
169
1. 12 553
1. 12 384
9-99 879
9-99878
43
42
'9
8.87661
107
168
8.87785
169
I .12 2l5
9.99 877
4i
20
8.87 829
1 66
8.87953
.£.-
1. 12 047
9.99 876
40
21
8.87995
1 66
8.88 120
1. 1 1 880
9.99 875
39
22
8.88 161
8.88 287
1 7
i. ii 7i3
9.998-74
38
23
8.8
8326
164
8.8
8453
i. ii 547
9.99873
3?
24
8.88 490
164
8.88 618
.fie
i. ii 382
9.99872
36
25
8.8
8654
if>i
8.8
8783
i. ii 217
9.99871
35
26
8.88817
103
163
8.8
8948
163
I. 1 1 052
9.99870
34
27
8.88 980
162
8.89 in
163
1.10889
9.99869
33
28
8.8
9 142
8.89 274
i . 10 726
9.99868
32
29
8.89 3o4
162
8.89437
163
i.io563
9.99867
3i
30
8.89464
8.89 598
I.I0402
9.99 866
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
9
85° 30 .
PP
181
179
177
175
173
171
1 68
166 164
.1
18.1
17.9
17.7
.!
17-5
»7-3
17.1 ; i
16.8
16.6 16.4
.2
36.2
SS-8
35-4
.2
35-o
34-6
34-2 2
33.6
33-2 32.8
•3
54-3
53-7
•3
52-5
51.3 3
50.4
49.8 49.2
•4
72.4
71.6
70.8
•4
70.0
6q.2
68.4 4
67.2
66.4 65.6
9°-5
88.5
87- S
86. S
85-5 ' 5
84.0
83.0 82.0
.6
108.6
107.4
106.3
.6
105.0
103.8
102.6 | 6
100.8
99.6 98.4
•7
126.7
125-3
123.9
•7
122.5
121. 1
119.7 • .7
117.6
116.2 114.8
.8
144-8
143.2
141.6
.8
140.0
138.4
136.8 : .8
134-4
132.8 131.2
•9,,
162.9
161.1
*59-3
•9
'57-5
155-7
153.9 -9 iS1-2
149.4 r47-6
38
4° 30 .
;
L. Sin.
d.
L. Tang.
d. L.
Cotg.
L
Cos.
30
8.89464
8.89 598
I .
10 402
9.99 866
30
3i
32
8.C
8.C
*9 625
19784
159
8.89 760
8.89 92o
162
160
I .
I .
IO 240
10 080
9.99 865
9.99864
29
28
33
8.89943
159
159
8.9o 080
160
160
I .
09 920
9.99 863
27
34
8.9o 1 02
I eg
8.9o 240
I .09 760
9.99 862
26
35
8.9o 260
8.9o 399
I .
09 601
9.99 861
25
36
8.9o 417
157
8.9o 557
158
1.09 443
9.99 860
24
J57
15
0
37
38
8.9o 674
8.9o 730
156
8.90715
8.90872
157
1.09 285
i .09 128
9.99859
9.99 858
23
22
39
8.9o885
155
8.91 029
157
i .08 971
9-99857
21
40
8.9; o4o
155
8.91 185
150
1.08 8 1 5
9.99 856
20
4i
42
8.9i i95
8.9i 349
155
154
8.91 34o
8.91 495
155
i .08 660
i. 08 505
9.99855
9.99 854
i a
43
8.9i 5o2
I53
8.91 650
X55
i. 08 35o
9.99853
17
44
45
8.9i 655
8.9i 807
153
152
8.91 8o3
8.91 957
153
154
i .08 197
i. 08 o43
9.99 852
9.99 85i
16
i5
46
8.9i 959
152
8.92 110
J53
i .07 890
9.99850
i4
47
8.92 IIO
151
8.92 262
152
1.07 738
9.99848
i3
48
8.92 261
151
8.924i4
i. 07 586
9.99847
12
49
8.92 4i i
150
8.92 565
*5
1.07435
9.99 846
II
50
8.92 56 1
15°
8.92 716
15
i .07 284
9.99845
10
5i
8.92 710
149
8.92866
•S"
1.07 i34
9.99 844
9
52
8.92859
8.93 016
i .06 984
9.99843
8
53
8.93 007
148
M7
8.93 165
149
148
i.o6835
9.99 842
7
54
55
8.93 i54
8.93 3oi
*47
8.933i3
8.93462
149
i. 06 687
i.o6538
9.99 84i
9.99 84o
6
5
56
8.93448
X47
146
8.93 609
J4
*4
/
7
i .06 391
9-99 839
4
57
8.93 594
1 46
8.93 756
i .06 244
9.99 838
3
58
8.93 74o
8.93 903
i .06 097
9.(
?983-
1
2
59
8.93885
145
8.94 049
140
.,«
i.o5 951
9.99 836
I
60
8.94 o3o
8.94 195
i.o5 805
9.99 834
0
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L
Sin.
'
85°.
PP
162
160
159
157
155
153
151
149
147
.1
16.2
16.0
15-9
.!
J5-7
i-vS
15-3
I
iS-i
14.7
.2
32-4
32.0
31-8
.2
31.0
30.6
2
30.2
2J.8
29.4
•3
48.6
48.0
47-7
•3
47.1
4b-5
45-9
3
45-3
44-7
44.1
•4
64.8
64.0
63.6
•4
62.8
62.0
61.2
4
60.4
59-6
58.8
• 5
81.0
80.0
79-5
•5
78.5
77-5
76.5
5
75-5
74-5
73- S
.6
97.2
96.0
95-4
.6
94.2
93.0
91.8
6
90.6
89.4
88.2
• 7
"3-4
112. 0
111.3
• 7
109.9
08.5
107.1
7
i°5-7
104.3
102.9
.8
129.6
128.0
127.2
.8
125.6
124 o
122.4
8
120.8
119.2
117.6
• 9 145-8
144.0
143- 1
I4J-3 139-5
137-7
9 -135-9
J34-i
39
5°.
f
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L
Cos.
0
8 .94 o3o
8.94 195
i .o5 805
9.99 834
60
I
8 ,g4 174
144
4340
MS
i .o5 660
9.99 833
59
2
8.94 3i7
8.94485
i.o5 515
9.99 832
58
3
8.94461
8.94 63o
M
b
i .o5 370
9.99 83i
57
142
i^
^
I
4
8.94 6o3
8.5
4 773
i .o5 227
9.99 83o
56
5
8.94746
8.5
4 917
i.o5 o83
9.99 829
55
6
8.94 887
141
8.95 060
X4
j
i ,o4 940
9.9982
9
54
142
i/
7
7
8.95 029
8.95 202
i .04 798
9.99827
53
8
8.95 170
8.95 344
i.o4656
9.99 825
52
9
8.95 3io
140
8.95486
142
i .04 5i4
9.99 824
5i
10
8.95 450
8.95 627
141
i.o4 373
9.99 823
50
ii
8.95 589
I39
8. 95767
140
i.o4233
9.99 822
49
12
8.95 728
8.95 908
i .o4 092
9.99 821
48
i3
8.95 867
Z39
138
8.96 047
140
i.o3 953
9.99 820
47
i4
i5
8.96 oo5
8.96 i43
138
8.96 187
8.96 325
138
i.o3 8r3
i.o3 675
9.99819
9.99 817
46
45
16
8.96 280
J37
8.96464
138
i.o3 536
9.99 816
44
17
8.96417
i 6
8.96 602
i.o3 398
9.99 8i5
43
18
8. 96 553
13
8.96 739
*3
i ,o3 261
9.99 8i4
42
'9
8.96 689
136
8.96 877
i.o3 123
9.99 8i3
4i
20
8.96 825
I*
8.97 oi3
130
i .02 987
9.99 812
40
21
8 . 96 960
135
8.97150
135
i .02 85o
9.99 810
39
22
8.97 095
8.97 285
1-36
i .02 715
9.99809
38
23
8.97 229
8.97 421
i .02 579
9.99 808
37
24
8. 97 363
133
8.97 556
1.02 444
9.99807
36
25
8.97496
8.97 691
i .02 3og
9.99 806
35
26
8.97 629
8.97825
X34
i .02 175
9.99 8o4
34
X3
4
27
8.97762
8.97 959
I .02 O4l
9.99 8o3
33
28
8.97 894
8.98 092
I .OI 908
9.99 802
32
29
8.c
)8 026
132
8.98 225
133
i. oi 775
9.99 801
3i
30
8.1
>8i57
8.98 358
i .01 642
9.99 800
30
L.
Cos.
d.
L. Cotg.
d. L. Tang.
L.
Sin.
'
84° 3O .
PP
145
143
141
139
138
136
135
133
131
.2
14-5
2Q.O
28.6
14.1
28.2
.1
.2
13-9
27.8
13.8
27.6
13-6 -I
27.2 .2
13-5
27.0
11:1
2&2
•3
43-5
42.9
42.3
•3
41-7
41.4
40-8 .3
40-5
39-9
39-3
•4
S8.o
57-2
56.4
•4
55-6
55-2
54-4 -4
54- °
53.2
52.4
.5
72.5
7z-5
70.5
. e
60 5
69.0
68.0 .5
07- s
66.5
6s* 5
.6
87.0
85.8
84.6
.6
83-4
82.8
81.6 .6
81.0
79-8
78.6
• 7
101.5
100. 1
98.7
•7
97-3
96.6
95-2 .7
94-5
93-1
91.7
•9
1 30. s
114.4
128.7
126.9
25-1
124.2
122.4 -9 ]
119.7
117.9
5° 30 .
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
30
8.98 157
8.98 358
I .01 642
9.99 8ot
30
3 1
8.98 288
8.9
8 49o
132
i .01 5io
9-99 798
29
32
8.98419
8.9
8 622
i. 01 378
9.99 797
28
33
8.98 549
130
8.9
8 753
13
i .01 247
9.99 796
27
34
8.98 679
8.9
8884
i .01 116
9-99 795
26
35
36
8.98 808
8.98 937
129
8.99 015
8.99 i45
130
i .00 985
i. oo 855
9v-99 79 3
9.99 792
25
24
37
8.99 066
129
128
8.99275
130
i .00 725
9.99791
23
38
8.99 i94
8.99405
i .00 595
9.99790
22
39
8.99 322
128
8.99 534
129
T_o
i .00 466
9.99 788
21
40
8.99 450
8.99 662
i. oo 338
9.99 787
20
4i
8.99577
127
127
8.99791
129
128
I .OO 2O9
9.99 786
'9
42
8.99 704
126
8.99919
I .00 08 I
9-99 785
18
43
8.99 83o
126
9 . oo o46
I2/
128
o.99 954
9.99783
17
44
8.99 g56
126
9.00 174
0.99 826
9.99 782
16
45
9.00 082
9.00 3oi
0.99699
9.99 781
i5
46
9.00 207
125
125
9.00 427
126
0.99 573
9.99 780
i4
47
48
9.00 332
9.00 456
124
9.00 553
9.00 679
126
0.99 447
0.99 32i
9.99778
9.99 777
i3
12
49
9.00 58 1
125
9.00 805
126
o. 99 195
9.99 776
I I
50
9.00 704
123
9.00 gSo
0.99 070
9-99775
10
5i
9.00 828
124
123
9.01 o55
124
0.98 945
9-99 773
9
52
9.00 95 1
9.01 179
0.98 821
9.99 772
8
53
9.01 074
123
9.01 3o3
O » '
?8697
9.99771
7
4
54
9.01 196
9.01 427
0.98 573
9-99 769
6
55
9.01 3i8
9.01 55o
O . '
?845o
9-99 768
5
56
9.01 44o
9.01 673
123
0.98 327
9.99 767
4
121
1
57
9.01 56i
9.01 796
o.98 204
9.99 765
3
58
9.01 682
9.01 918
0.
58 082
9.99 764
2
59
9.01 8o3
9.02 o4o
o.97 960
9-99 763
I
60
9.01 923
9.02 162
0.97838
9-99 76i
0
L.
Cos.
d.
L. Cotg.
d
.
L.
Tang.
L.
Sin.
'
84°.
PP
13°
129
128
126
»5
123
122
121
I2O
.1
13-0
12.9
12.8
.1
12.6
12.5
12.3
.!
12.2
12. I
12.0
.2
20. o
25.8
25.6
.2
25.2
25.0
24.6
.2
24.4
24.2
24.0
•3
39-o
38.7
38.4
•3
37-8
37-5
36-9
•3
36.6
36.3
36.0
•4
52.0
51.6
51.2
•4
5o.4
50.0
49.2
•4
48.8
48.4
48.0
• 5
65.0
64- S
64.0
63.0
62. S
61.5
61.0
60.5
6o.O
.6
78.0
77-4
76.8
.6
75-6
75-o
73-8
.6
73-2
72.6
72.0
•7
91.0
9°-3
89.6
.7
88.2
87-5
86.1
• 7
8.S-4
84.7
84.0
.8
104.0
117.0
103.2
16. i
102.4
IIS- 2
100.8
"3-4
loo.o
112.5
98-4
110.7
.8
9
97.6
109.8
108.9
96.0
I08.0
6°.
/
L.
Sin.
d.
L. Tang.
d,
L.
Cotg.
L.
Cos.
0
9.01 923
9.02 162
0.97 838
9-99 7bl
60
I
9.02 o43
120
9.02 283
121
0.97 717
9.99760
59
2
9.02 i63
9.02 4o4
0.97 596
9.99 759
58
3
9.02 283
9.02 525
°-97475
9.99 757
57
119
12
>
4
9.O2 4O2
118
9.02 645
0.97 355
9.99 756
56
5
9.O2 52O
9.02 766
121
0.97 234
9.99 755
55
6
9.02 63g
119
9.02 885
II9
0.97 115
9.99 753
54
118
12
j
7
9.02757
9.o3 005
0.96 995
9.99752
53
8
9.02 874
9.o3 124
0.96 876
9.99 751
52
9
9.02 992
9.0
3 242
118
0.96 758
9-99 749
5i
10
9«o3 109
117
9-o3 36i
II
)
3
0.96 63g
9-99 748
50
ii
g.oS 226
116
9.o3 479
a
0.96 52i
9.99 747
49
12
9.o3 342
9.o3 597
0.96 4o3
9.99 745
48
1 3
9. o3458
9.0
3 7i4
117
0.96 286
9.99 744
4?
116
II
*
i4
9.o3 574
116
9.03 832
0.96 168
9.99 742
46
i5
9.o3 690
9.o3 948
0.96 o52
9.99 74 1
45
16
9.o3 805
"5
9.04 o65
117
116
0.95 935
9.99 740
44
'7
9.o3 920
114
9.04 181
116
o.<
?58i9
9.99738
43
18
9.04 o34
9.04 297
0.
P57o3
9.99737
42
'9
9.04 1 49
9.04 4i3
0.95 587
9.99 736
4i
20
9.04 262
114
9.04 528
"5
0.95 472
9.99 734
40
21
9.04 376
114
9.04 643
"5
0.95 357
9.99 733
39
22
9.04 490
9.04 ?58
0.95 242
9.99 73i
38
23
9.04 6o3
112
9.04873
"5
114
0.96 127
9.99 730
37
24
9 . o4 7 1 5
"3
9.04 987
0.95 oi3
9.99728
36
25
9.04828
9.o5 101
0.94 899
9.99 727
36
26
9.04 940
9.0
5 214
"3
0.94 786
9.99726
34
27
9-o5 o52
112
9.o5 328
114
0.94 672
9.99 724
33
28
g.oS 1 64
9.o5 44i
"
3
0.94 559
9.99 723
32
29
9.o5 275
9. o5553
112
0.94 447
9.99 721
3i
30
9-o5 386
9-o5 666
"3
0.94334
9.99 720
30
L.
Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
83?° 30 .
PP
121
120
"9
118
117
116
"5
114
"3
.1
12. 1
I2.O
11.9
.1
ii. 8
ii.7
ii. 6
.1
"•5
11.4
"•3
.2
24.2
24.0
23-8
2
23.6
23-4
23.2
.2
23.0
22.8
22.6
•3
36.3
36.0
35-7
•3
35-4
35-i
34-8
•3
34-5
34-2
33-9
•4
48.4
48.0
47-6
•4
47-2
46.8
46.4
•4
46.0
4S.6
45-2
• 5
60-5
60.0
59-5
•5
59-o
S8.S
58.0
•5
57-5
56-5
.6
72.6
72.0
71.4
.6
70.8
70.2
69.6
.6
69.0
68.4
67.8
•7
84.7
84.0
83-3
•7
82.6
81.9
81.2
•7
80. S
79.8
79.1
8
96.8
96.0
95-2
.8
94-4
93-6
92.8
.8
92.0
91.2
90.4
.9 108.9
I08.0
107.1 .9
106.2
105.1
104.4
9 I03-5
IO2.6
101.7
6° 3O .
;
L.
Sin.
d.
LvTang.
d.
L.
Cotg.
L.
Cos.
30
9.o5 3«6
9.06 666
0.94 334
9-99 720
30
3i
32
9-o5 497
9«o5 607
IIO
9-o5 778
9.o5 890
112
112
0.94 222
0.94 no
9.99 718
9.99 717
29
28
33
9.o57i7
no
9.06 002
112
0.93 998
9.99 716
27
34
35
g.oS 827
9.o5937
IIO
9.06 1 13
9.06 224
III
o'.
93887
33 776
9.99714
9.99 713
26
25
36
9.06 o46
109
9.06 335
III
o.93 665
9.99 711
24
37
38
9.06 i55
9.06 264
109
109
9.o6445
9.o6556
IIO
III
0.93555
0.93444
9.99 710
9.99 708
23
22
39
9.06 372
108
9.06 666
IIO
o.93 334
9.99707
21
40
9.06 48i
109
9.06 775
109
0.93 225
9.99 705
20
4i
42
9.06 589
9.06 696
107
9.o6 885
9.06 994
109
0.93 n5
0.93 006
9.99 704
9.99 702
18
43
9.06 8o4
107
9.07 io3
109
108
0.92 897
9.99 701
17
44
45
46
9.06 911
9.07 018
9.07 124
107
106
107
9.07 211
9.07 32O
9.07 428
109
108
108
0.92 789
0.92 680
0.92 572
9.99699
9.99698
9.99696
16
i5
i4
47
48
9.07 23i
9.07337
106
9.07 536
9.07 643
107
0.92 464
0.92 357
9.99695
9.99 693
i3
12
49
9.07 442
105
infi
9.07 75i
0.92 249
9.99 692
I I
50
9.07 548
9.07 858
107
0.92 142
9.99690
10
5i
9.07 653
105
9.07 964
0.92 o36
9.99689
9
62
9.07 768
9.08 071
ir/i
0.91 929
9.99687
8
53
9.07863
105
105
9.08 177
106
0.91 823
9.99 686
7
54
9.07 968
9.08 283
106
0.91 717
9.99 684
6
56
9.08 072
9.08 389
0.91 611
9.99 683
5
56
9.08 176
9.08495
0.91 5o5
9.99 681
4
104
10
i
57
9.08 280
9.08 600
0.91 4oo
9.99 680
3
58
9.08 383
9.08 705
3
0.91 295
9.99678
2
59
9.08 486
103
9.08 810
105
0.91 190
9.99677
I
60
9.08 589
9.08 914
0.91 086
9.99 676
0
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L
Sin.
83°.
PP
112
in
IIO
109
108
107
1 06
105
104
.1
II. 2
ii.i
II. 0
.1
10.9
10.8
10-7
:i
10.6
10.5
10.4
.2
22-4
22.2
22.0
.2
21.8
21.6
21-4
.2
21.2
21.0
20.8
3
33-6
33-3
33-o
•3
32.7
32.4
32.1
3
31.8
31-5
31.2
•4
44.8
44-4
44.0
•4
43.6
43-2
42.8
•4
42.4
42.0
41.6
:J
56.0
67.2
55-5
66.6
55-°
66.0
:I
54-5
54-o
64.8
£5
5
.6
53-°
63.6
52-5
63.0
52.0
62.4
•7
78.4
77-7
77.0
•7
76-3
75-6
74-9
•7
74-2
73-5
72.8
8
89.6
88.8
88.0
.8
87.2
86.4
85.6
.8
84.8
84.0
83-2
•
Q
98.1
97.2
96-3
•9 95-4
94-5
93-6
43
1
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
0
9.08 589
103
103
IO2
102
102
101
IO2
1OI
101
1OO
101
IOO
100
•99
IOO
. 99
99
98
99
98
98
98
98
97
97
97
97
96
97
96
9.08 914
i°s
104
104
103
104
103
103
102
103
102
102
101
102
IOI
IOI
IOI
IOI
IOO
IOO
IOO
IOO
99
99
99
99
99
98
98
98
98
0.91 086
9.99675
60
I
2
3
4
5
6
8
9
9.08 692
9.08 795
9.08 897
9.08 999
9.09 101
9.09 202
9.09 3o4
9.09405
9.09 5o6
9.09 019
9.09 123
9.09 227
9.09 33o
9.09434
9.o9537
9.09 64o
9.09 742
9.09 84s
0.90 981
0.90 877
0.90 773
0.90 670
0.90 566
0.90 463
0.90 36o
0.90 258
0.90 i55
9.99 674
9.99 672
9.99670
9.99 669
9.99 667
9.99 666
9.99 664
9.99 663
9.99 661
59
58
57
56
55
54
53
5l
10
9.09 606
9.09947
0.90 o53
9.99659
50
1 1
12
i3
i4
i5
16
«7
18
J9
9.09 707
9,09 807
9.09907
9. TO OO6
9.IO 1 06
9. 10 205
9.10 3o4
9-10 4O2
9.10 5oi
9.10 049
9.10 i5o
9. IO 252
9.10353
9.10 454
9.10555
9.10 656
9. 10 756
9.10 856
0.89 g5i
0.89 850
0.89 748
0.89 647
0.89 546
0.89445
0.89 344
0.89 244
0.89 i44
9.99 658
9.99 656
9.99655
9.99 653
9.99 65i
9.99 650
9.99 648
9.99647
9-99 645
49
48
47
46
45
44
43
42
4i
20
9.10 599
9.10 g56
0.89 o44
9.99 643
40
21
22
23
24
25
26
27
28
29
9.10 697
9. 10 795
9. 10 893
9. 10 990
9.11 087
9.11 i 84
9.11 281
9.11 377
9.11 474
9.11 o56
9. ii i55
9.11 254
9.1 353
9. 452
9. 55i
9. 649
9- ?4?
9. 845
0.88 944
0.88 845
0.88 746
0.88 647
0.88 548
0.88 449
0.88 35i
0.88 253
0.88 165
9.99 642
9.99 64o
9.99 638
9.99 637
9 .99 635
9.99 633
9.99 632
9.99 63o
9.99 629
39
38'
37
36
35
34
33
32
3i
30
9.11 570
9.1
I 943
0.88 057
9.99627
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
82° 3O .
PP
105
104
103
102
IOI
IOO
99
98 97
.1
.2
3
•4
• 5
.6
:i
i&i
10.5
21.0
31-5
42.0
52.5
63.0
73-5
84.0
04. 5
10.4
20.8
31.2
41.6
52.0
62.4
72.8
83.2
10.3
20.6
30-9
41.2
I,':i
72.1
82.4
^2^
.0
.0
.0
.0
.0
.0
.0
.0
.0
. i
.2
•3
•4
:!
:l
9-9
19.8
29.7
39- 6
49-5
59-4
69-3
79.2
SQ.I
9.8 9.7
19.6 19.4
29.4 29.1
39.2 38-8
49.0 48.5
58.8 58.2
68.6 67.9
78.4 77-6
88.2 87.3
.2
•3
4
•5
.6
:1
•9
20.4
30.6
40.8
51.0
6l.2
71.4
8l.6
QT.8
20.2
3°-3
40.4
50-5
60.6
70.7
80.8
2C
3C
4c
5c
6c
70
8c
DC
7° 30 .
:
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L,
Cos.
30
9.1
i 570
9.11 943
0.88 067
9-99 627
30
3i
9.11 666
90
95
9. 12 C
>4o
97
08
0.87 960
9.99 625
29
32
33
9.11 761
9.11 857
96
9.12 i 38
9. 12 235
97
0.87 862
0.87 765
9.99 624
9.99 622
28
27
34
9.11 952
95
95
9.12 332
97
06
0.87 668
9.99 620
26
35
9.12 047
9. 12 428
0.87 572
9.99 618
25
3b
9.12 142
95
94
9. 12 525
97
96
0.87 475
9.99617
24
38
9.12 236
9.12 33i
95
9.12 621
9.12 717
96
o. 87379
0.87 283
9.99 6i5
9.99 6i3
23
22
39
9.12 425
94
9.12 8i3
96
0.87 187
9.99 612
21
40
9.12 519
94
9. 12 909
96
0.87 091
9.99 610
20
4i
9.12 612
93
94
9 . 1 3 oo4
95
qe
0.86 996
9.99 608
'9
42
9.12 706
9. 1 3 099
0.86 901
9.99607
18
43
9.12 799
93
9 . 1 3 1 94
95
0.86 806
9.99605
17
93
95
44
9.12 892
9-i3 289
0.86 711
9.99 6o3
16
45
9. 12 985
9. 1 3 384
0.86 616
9.99 601
i5
46
9-i3 078
93
9.13478
94
0.86 522
9.99 600
i4
93
95
47
48
9.i3 171
9-i3 263
92
9.i3573,
9.13667
94
0.86 427
0.86 333
9.99 598
9.99 596
i3
12
49
9.i3 355
92
9. 1 3 761"
94
0.86 239
9.99595
I I
50
9-i3 447
92
9.i3 854
0.86 i46
9.99 593
10
5i
9.i3 539
92
9.i3 948
94
93
o.86oi
>2
9.9959i
9
52
9. 1 3 63o
9. 1 4 o4i
o.85 959
9.99 589
8
53
9. i3 722
92
9.i4 i
34
93
93
o.85 866
9.99 588
7
54
9.i38i3
9. i4 227
93
o.85773
9.99 586
6
55
9. 1 3 904
9.i43
20
o.85 680
9.99 584
5
56
9-i3 994
9°
9.14^
12
92
o.85 588
9.99 582
4
91
92
57
9.i4 o85
9. i4 5o4
o.85 496
9.99 58i
3
58
9.14 175
9-i4597
o.854o3
9-99 579
2
59
9. i4 266
91
9.i4 688
91
o.85 3i2
9.99577
I
60
9.14 356
9. i4 780
o.85 220
9.99 575
0
L.
Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
V
82°.
PP
97
96
95
94
93
92
91
90
.j
9-7
9-6
9-5
. i
9-4
9-3
9.2
.1
9-i
9.0
.2
19.4
19.2
19.0
.2
18.8
18.6
i8.4
.2
18.2
18.0
•3
29.1
28.8
28.5
•3
28.2
27.9
27.6
•3
27.3
27.0
4
,,8.8
38.4
38.0
•4
37-6
37-2
36.8
-4
36-4
36.0
5
48- s
48.0
47-5
•5
47.0
46-5
46.0
•5
45-5
45-o
6
58.2
57-6
57-o
.6
56.4
55-8
55-2
.6
54-6
54-o
I
67.9
77.6
67.2
76.8
66.5
76.0
'I
65-8
75-2
65.1
74-4
64.4
73-6
i
63.7
72.8
63.0
72.0
9
87.s
86.4
85-5
•9
84.6
83-7
82.8
•9
81.9
81.0
45
,
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos.
0
9.i4356
0_
9.14 780
o.85 220
9
.99575
60
I
9-14445
09
9°
9.14 872
92
o.85 128
9
.99 574
59
2
9.14535
9.14 963
o.85 037
9
.99 572
58
3
9.14624
89
9. 1 5 o54
91
o.84 946
9
.99 57o
57
4
9.14 71
4
90
80
9.i5 i45
9i
o.84 85s
9
.99 568
56
5
9.i48o3
9.i5 236
9*
o.84 764
9
.99 566
55
6
9.14 891
9.i5 327
91
0.84673
9
.99565
54
7
9. i4 980
89
80
9.i5 417
90
0.84583
9
.99 563
53
8
9. 1 5 069
9-i5 5o8
91
o.84 492
9
.99 56i
52
9
9oi5 157
9..i5 598
90
o.84 i
{02
9
.99 559
5i
10
9. 1 5 245
88
9.i5688
90
o.84 3i2
9
•99 557
50
ii
9.i5333
88
9.i5 777
59
0.84 223
9
.99 556
49
12
9. 1 5 421
9.i5 867
o.84 i33
9
.99554
48
i3
9.i5 5o8
87
9. i5 956
89
o . 84 o44
9
.99 552
47
i4
9. i5 596
88
87
9.1
6o46
90
80
o.83«
,54
9
.99 55o
46
i5
9.i5 683
Q_
9.16 135
o.83*
365
9
.99 548
45
16
9.i5 770
87
9. 16 224
89
88
o.83 776
9
.99 546
44
'7
9.i5 857
87
9. 16 3i2
80
0.83688
9
.99545
43
18
9. 1 5 944
86
9.16 4oi
o.83 599
9
.99 543
42
'9
9. 16 o3o
86
9. 16 489
o.83
Sn
9
.99 54i
4i
20
9.16 1 16
87
9.16 577
88
00
o.83 423
9
.99 539
40
21
9. 16 2o3
86
9.16 665
88
o.83 335
9
.99537
39
22
9.16 289
g.
9.16 753
o.83 247
9
.99 535
38
23
9.16 374
86
9.16 84i
87
o.83 159
9
.99533
37
24
9.16 46o
j5
9. 16 928
88
o.83 072
9
.99 532
36
25
9.i6545
86
9. 17 016
0.82 984
9
.99 53o
35
26
9.16 63i
8s
9. 17 io3
87
87
0.82 897
9
.99 528
34
27
9.16 716
9.17 190
87
0.82 810
9
.99 526
33
28
9.16 801
J
9.17 277
0.82 723
9
.99 524
32
29
9.16 886
85
9.17 363
86
0.82 637
9
.99 522
3i
30
9. 16 970
84
9.17450
87
0.82 55o
9
.99 52O
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
L. Sin.
'
81°3O.
PP 92
9»
9°
89
88
87
86
.1 9.2
9.1
9.0
.1
8.9
8.8
.1
8.7
8.6
.2 18.4
1 8. 2
18.0
.2
17.8
17.6
.2
17.4
17.2
•3 27.6
27-3
27.0
•3
26.7
26.4
•3
26.1
25.8
•4 36.8
36.4
36.0
•4
35-6
35-2
•4
34-8
34-4
• 5 46-0
45-5
45-o
.5
44-5
44.0
43-5
43-°
•6 55-2
54- 6
54-o
.6
53-4
52-8
.6
52.2
51.6
•7 64-4
63-7
63.0
•7
62.3
61.6
• 7
60.9
60.2
.8 73.6
72.8
72.0
.8
71.2
70.4
.8
69.6
68.8
.9 82.8
81.9
81.0
^— _
78-3
77-4
46
8° 3D .
;
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
30
9
. 16 970
9.17 450
0.82 55o
9
>99 52O
30
3i
9
.17055
85
8-1
9.17 536
86
86
0.82 464
9
99 5i8
29
32
9
.17 i39
9.17 622
0.82 378
9
99 5i7
28
33
9
. 17 223
84
84
9.17 708
86
86
0.82 292
9
99515
27
34
9
.I73o7
0 .
9.17 794
Rfi
0.82 206
9
995i3
26
35
9
.i739i
9.17 880
O.82 120
9
99 5i i
25
36
9
.17474
83
84
9.17 965
85
86
0.82 c
35
9
995o9
24
37
9
.17 558
9.18 o5i
0.8 1 949
9
99 507
23
38
9.17 64i
°3
9.18 i36
°5
0.81 864
9
.99 5o5
22
39
9
.17 724
9.18 221
85
0.81 779
9
.99 5o3
21
40
9.17807
83
9.18 3o6
°5
0.8 1 694
9
.99 5oi
20
4i
9
.17 890
83
8?
9.18 391
85
0.81 609
9
99499
19
42
9
.17973
9.18475
0.81 525
9
99497
1 8
43
9
.i8o55
82
9.18 56o
85
84
0.8 1 44o
9
.99495
17
44
9
.18 i37
0,
9.18 644
o .
0.81 356
9
99494
16
45
9
. l8 220
°3
9.18 728
0.81 272
9
.99492
i5
46
9.18 3o2
81
9.18 812
84
84
0.81 1 88
9
.99 490
i4
47
9.18 383
9.18 896
o..
0.81 io4
9
.99488
i3
48
9.18465
9.18 979
0.81 021
9
.99486
12
49
9.18 547
82
9. 19 o63
84
0_
0.80 937
9
• 99484
I I
50
9.18 628
9.19 i 46
°3
Ro
0.80 854
9
.99 482
10
5i
9.18 709
81
9.19 229
°3
83
0.80 771
9
.99 48o
9
52
9. 18 790
9. 19 3i2
0.80688
9
.99478
8
53
9.18 871
8r
9.i9395
83
83
0.80 605
9
.99476
7
54
9. 18 952
81
9.19478
83
O.8o 522
9
.99474
6
55
9. 19 o33
9. 19 56i
P_
0.80439
9
.99472
5
56
9.19 1 13
80
9.19 643
82
0.80 357
9
.99470
4
57
9. 19 193
80
9.19 725
82
O.8o 275
9
.99 468
3
58
9.19 273
9.19 807
O.8o ]
93
9.99 466
2
59
9.19353
80
9. 19 889
82
0.8o 1
1 1
9
.99 464
I
60
9.19433
9.19 971
o. 80 029
9
.99 462
0
"
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
81°.
PP
86
85
84
83
82
81
80
8.6
8-5
8.4
.1
8.3
8.2
>z
8.1
8.0
.2
17.2
17.0
16.8
.2
16.6
,6.4
.2
16.2
16.0
•3
25-8
25-5
25.2
•3
24.9
24.6
•3
24-3
24.0
•4
34-4
34-°
33-6
•4
33-2
32.8
•4
32-4
32.0
• 5
43- o
42-5
42.0
• 5
41- s
41.0
5
40.5
40.0
.6
51-6
51-0
50.4
.6
49.8
49.2
.6
48.6
48.0
.7
60.2
59-5
58.8
.7
58.1
57-4
.7
56-7
56.0
.8
68.8
68.0
67.2
.8
66.4
65.6
.8
64.8
64.0
•9
7*-4
• 9 74-7 73-8
9
72.9
72.0
47
9°.
I
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
0
9.19 433
9. 19 971
0.80 029
9
.99 462
60
I
9,19 5i3
9.20 o53
82
81
0.79 947
9
.99 46o
59
2
9.19 592
9.20 1 34
0.79 866
9
.99458
58
3
9. 19 672
79
9.20 216
81
0.79 784
9
.99 456
57
4
9.19 761
9.20 297
81
0.79 703
9
• 99 454
56
b
9.19 83o
9.20 378
0.79 622
9
.99 452
55
6
9.19909
79
9.20 459
0.79 54i
9
.99450
54
79
81
7
9.1998
8
9.20 54o
0.79 46o
9
• 99448
53
8
9.20 067
9.20 621
0.79 379
9
.99 446
52
9
9.20 i45
7°
_Q
9.20 701
0
°-79 299
9
• 99 444
5i
10
9.20 223
9.20 782
0.79 218
9
.99 442
50
1 1
9.20 3o2
79
78
9.20 862
80
0.79 i38
9
.99 44o
49
12
9.20 38o
9.20 942
o. 79 o58
9
.99 438
48
i3
9.20458
7°
77
9.21 022
80
0.78 978
9
.99 436
47
i4
9.20 535
78
9.21 102
80
0.78 898
9
.99434
46
i5
9.20 6i3
9.21 l82
0.78 818
9
.99432
45
16
9.20 691
JO
77
9.21 26l
79
80
0.78739
9
.99 429
44
17
9.20 768
9.21 34i
0.78 659
9
.99427
43
18
9.20 845
9.21 420
0.78 58o
9
.99 425
42
'9
9.20 922
77
9.21 499
79
0.78 5oi
9
.99423
4i
20
9.20999
9.21 578
79
o. 78 422
9
.99 421
40
21
9.21 076
77
9.21 657
79
o.78'343
9
.99419
39
22
9.21 i53
•76
9.21 736
_0
0.78 264
9
.99417
38
23
9.21 229
77
9.21 8i4
7°
79
0.78 186
9
.99415
37
24
9.21 3o6
76
9.21 893
78
0.78 107
9
.99 4i3
36
25
9.21 382
9.21 971
0.78 029
9
.99411
35
26
9.21 458
76
9 . 22 o4g
7°
78
o.7795i
9
.99 409
34
27
9.21 534
76
9-22 127
78
0.77873
9
.99407
33
28
9,21 610
9-22 2O5
_Q
°-77 795
9
.99 4o4
32
29
9.21 685
•76
9.22 283
78
0.77 717
9
.99 402
3i
30
9.21 76
i
9.22 36i
0.77 639
9
. 99 4oo
30
L. Cos.
d.
L. Cotg. d.
L. Tang.
L. Sin.
8O° 3O .
PP 82
81
so
79
78
77
76
.1 8.2
8.1
8.0
.1
7-9
7.8
. i
7-7
7.6
.2 16.4
16.2
16.0
.2
15-8
15.6
.2
15-4
15.2
3 24.6
24-3
24-0
•3
23-7
23-4
•3
23.1
22.8
•4 32-8
32.4
32.0
•4
31-6
31.2
•4
30.8
30-4
• 5 4i-o
4°-5
40.0
•5
39-5
39- °
•5
38.5
38.0
.6 49.2
48.6
48.0
.6
47-4
46.8
.6
46.2
45-6
-7 57-4
56.7
56.0
•7
55-3
54-6
• 7
53-9
S3-2
.8 65.6
64.8
64.0
.8
63-2
62.4
.8
61.6
60.8
•9 73-8
72.9
72.0
•9 ?i-i
70.2
•9
69-3
68.4
48
9° SO.
,
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos.
30
9.21 761
9.22 36i
0.77 639
9
.99 4oo
30
3i
9.21 836
75
76
9.22 438
77
78
0.77 562
9
.99398
29
32
9.21 912
9.22 5i6
o.77484
9
.99 396
28
33
9.21 987
75
9.22 5g3
77
0.77 407
9
• 99 394
27
34
9.22 062
75
9.22 670
77
0.77 33o
9
.99392
26
35
9.22 187
9.22 747
0.77 253
9
.99 390
25
3b
9.22 211
74
9.22 824
77
0.77 176
9
.99 388
24
75
77
3?
9.22 286
9.22 901
0.77099
9
.99 385
23
38
9.22 36i
9.22 977
7
0.77 O23
9
.99 383
22
39
9.22435
74
9.23 o54
77
0.76 946
9
.99 38i
21
40
9.22 5og
9.23 i3o
0.76 870
9
•99 379
20
4i
9.22 583
74
74
9.23 206
70
0.76 794
9
•99 377
19
42
9.22 657
9.23 283
0.76 717
9
.99375
18
43
9.22 731
74
74
9.23 SSg
7°
76
0.76641
9
•99 372
17
44
9.22 805
9.23435
0.76 565
9
.99 370
16
45
9.22 878
9.23 5io
0.76 490
9
.99 368
i5
46
9.22 952
74
9.23 586
7°
0.76^
n4
9
.99 366
i4
47
9.23 O25
73
9.23 661
75
76
0.76 339
9
.99 364
i3
48
9.23 098
9.23 737
0.76 263
9
.99 362
12
49
9.23 171
73
9.23 812
75
0.76 188
9
.99359
I I
50
9.23 244
9.23 887
0.76 1 13
9
.99 357
10
5i
9.33317
73
9.23 962
75
0.76 o38
9
.99 355
9
52
9.23 390
9. 24 037
0.75 963
9
.99353
8
53
9.23462
72
73
9.24 112
74
0.75888
9
.9935i
7
54
9.23535
9.24 186
0.75 8i4
9
.99348
6
55
9.23 607
9.24 261
0.75 739
9
.99 346
5
56
9.23 679
72
9.24335
74
0.75 665
9
.99 344
4
73
75
57
9.23 752
9.24 4io
0.75 590
9
.99 342
3
58
9.23 823
9.24484
0.75 5i6
9
. 99 34x
»
2
59
9.23 895
72
9.24558
74
0.75 442
9
.99337
I
60
9.23 967
9.24 632
o.75368
9
.99335
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
'
80°.
PP 77
76
75
74
73
72
71
.1 7.7
•2 15-4
7.6
15-2
7-5
15-0
.1
.2
a
,5:1
.1
.2
7-2
14.4
14.2
•3 23.1
22.8
22-5
•3
22.2
21.9
•3
21.6
21.3
• 4 30-8
3°-4
30.0
•4
29.6
29.2
•4
28.8
28.4
•5 38-5
38-0
37-5
• 5
37-°
36.5
•5
36.0
35-5
.6 46.2
45-6
45-o
.6
44-4
43-8
.6
43-2
42.6
•7 53-9
.8 61.6
g;
60.0
• 7
.8
51-8
59-2
51-!
58.4
:i
50.4
57-6
8:1
•9 69-3
68.4
67-5
.9 66.6
65-7
•9
64.8
63.9
I
L. Sin.
d.
L.
Tang. d.
L. Cotg.
L. Cos.
d.
0
9.23 967
9-
24 632
74
73
74
73
74
73
73
73
73
73
72
73
72
73
72
72
72
72
72
71
72
71
72
71
7°
0.75 368
9-99
335
60
I
2
3
4
5
6
7
8
9
9.24 no
9.24 181
9.24253
9.24 324
9.24395
9.24466
9.24536
9.24 607
71
71
72
71
71
70
71
9-
9-
9-
9-
9-
9-
9-
9-
9-
24 706
24 779
24853
24 026
25 ooo
25 073
25^219
2*5 292-
0.75 294
O.75 221
0.75 147
0.75 074
0.75 ooo
0.74927
o. 74854
0.74 781
0.74 708
ooo ooo ooo
ooo ooo ooo
ooo ooo ooo
333
33i
328
326
324
322
3i9
3i7
3i5
2
3
2
2
2
3
2
2
2
3
2
2
2
3
2
2
3
2
2
2
3
2
2
3
2
2
3
2
2
59
58
5?
56
55
54
53
52
5i
10
9.24677
7°
9-
25365
o.74635
9.99
3i3
50
II
12
i3
U
i5
16
17
18
9.24 748
9.24818
9.24888
9.24 g58
9.25 028
9.25 098
9.25 168
9.25 237
9.25 307
70
70
70
70
70
70
69
70
000 000 000
25437
25 5io
25 582
25655
25 727
25 799
25 871
25 943
26 015
0.74563
0.74 490
0.74418
0.74345
0.74273
0.74 201
O.74 129
0.74 057
0.73 985
ooo ooo ooo
ooo ooo ooo
ooo ooo ooo
3io
3o8
3o6
3o4
3oi
299
297
294
292
49
48
47
46
45
44
43
42
4i
20
9.25 376
09
9-
26 086
0.73 914
9.99
290
40
21
22
23
24
25
26
27
28
29
9.25445
9.25 5i4
9.25 583
9-25652
9.25 721
9.25 790
9.25 858
9.25927
9.25 995
°9
68
69
68
68
ooo ooo ooo
26 i58
26 229
26 3oi
26 372
26443
265i4
26 585
26655
26 726
0.73 842
o.7377i
0.73 699
0.73 628
o.73557
0.73 486
o!73345
o.73 274
9.99
9.99
9.99
9-99
9.99
9.99
9.99
9.99
9-99
288
285
283
281
278
276
274
271
269
39
38
37
36
35
34
33
32
3i
30
9. 26 o63
9-
26 797
0.73 2o3
9.99
267
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
L. Sin.
d.
79° 30'.
PP
.2
•3
•4
i
.9
74
73 72
.1
.2
•3
•4
71
70
69
.2
•3
• 4
• 5
.6
68 3
22.2
29.6
44-4
7-3 7-2
14.6 14.4
21.9 21.6
29.2 28.8
36.5 36.0
43-8 43-2
Si. i 5°-4
58.4 57.6
65.7 64.8
14.2
21.3
28.4
35-5
42.6
7.0
21. 0
28.0
35-o
42.0
49-o
56.0
6.9
13-8
20.7
27.6
34-5
41.4
48-3
SS-2
62.1
6.8 0.3
13.6 0.6
20.4 0.9
27.2 1.2
34.0 1.5
40.8 1.8
47.6 2.1
54-4 2.4
61.2 2.7
5o
3O.
/
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9.26 o63
9.26
797
o.
73 2o3
9-99
267
30
3i
9.26 i3i
68
9.26
867
70
o.
73 i33
9.99
264
2
29
32
9.26 199
9.26 937
0.
73o63
9.99
262
28
33
9.26 267
9.27
008
71
o.
72 992
9.99
260
27
34
9-26335
68
68
9.27
078
7o
0.72 922
9.99
257
3
2
26
36
9.26 4o3
9.27
i48
7°
0.72 852
9.99
255
25
36
9.26 470
67
9.27
218
70
0.72 782
9.99
252
24
37
9.26 538
68
67
9.27
288
70
60
0.72 712
9.99
25o
2
23
38
9.26 6o5
9.27
357
o.
72643
9 . 99 248
22
39
9.26 672
07
fit
9.27 427
70
o.
72573
9.99
245
21
40
9
.26 739
07
9.27 496
69
0.
72 5o4
9.99
243
20
4i
9
.26 806
6?
9.27
566
7°
60
o.
72434
9.99
241
3
19
42
9
.26873
9.27
635
0.72 365
9.99
238
1 8
43
9.26 940
07
67
9.27
704
69
69
0.
72 296
9-99
236
3
17
44
9
.27007
66
9.27
773
60
o.
72 227
9.99
233
2
16
45
9.27073
9.27
842
0.
72 i58
9.99
23l
i5
46
9.27 i4o
67
66
9.27
911
69
69
o.
72 089
9.99
229
i4
47
9.27 206
67
9.27
980
60
0.
72 020
9.99
226
2
i3
48
9.27 273
9.28
049
o.
71 95i
9.99
224
12
49
9.27 339
9.28
117
o.
71 883
9.99
221
I I
50
9
.27 405
9.28
186
69
0.
71 8i4
9.99
219
10
5i
9.2747i
66
9.28
254
60
0.
71 746
9.99
2I7
3
9
52
9.2-7 537
9.28
323
o.
71 677
9.99
2l4
8
53
9.27 602
65
9.28
39i
o.
71 6o9
9.99
212
7
66
68
3
54
9.27 668
66
9.28459
68
o.
71 54i
9.99209
2
6
55
9.27734
9.28
527
0.
7i473
9.99
207
5
56
9.27799
65
9.28
595
0.
71 4o5
9.99
204
4
65
67
57
9.27 864
66
9.28
662
68
o.
71 338
9.99
202
2
3
58
9 . 27 g3o
9.28
73o
0.
71 2-70
9.99
2OO
2
59
9.27 995
9.28 798
67
o.
7i 202
9.99
197
I
60
9.28 060
°S
9.28 865
o.
71 135
9.99
195
0
L. Cos. | d.
L. Cotg. d.
L.
Tang.
L. Sin. d.
'
79°.
PP
70
69
68
67
66
65 3
.1
7.0
6.9
6.8
.1 6.7
6.6
.1
6.5 0.3
.2
14.0
13-8
13-6
.2 13.4
13.2
.2
13.0 0.6
•3
21.0
20.7
20.4
, 3 20. i
19.8
•3
19.5 0.9
•4
28.0
27.6
27.2
.4 26.8
26.4
•4
26.O 1.2
•5
35-°
34-5
34-o
•5 33-5
33-o
•5
32-5 1-5
.6
42.0
41.4
40.8
.6 40.2
39- 6
.6
39.0 1.8
• 7
49.0
48-3
47.6
•7 46-9
46.2
• 7
45-5 2.1
.8
56.0
55-2
54-4
•8 53-6
52.8
.8
52.0 2.4
•9
63.0
62.1
•9 6o-3
59-4
58-5 2.7
5i
11°.
'
L. Sin.
d.
L.
Tang. d.
L. Cotg.
L. Cos. i d.
0
9.28 060
65
65
64
65
65
64
64
65
64
9
28 865
68
67
67
67
67
67
67
67
66
67
66
67
66
66
66
66
66
66
66
65
66
65
65
66
65
65
65
65
64
0.71 135
9.99 195
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
60
J
2
3
4
5
6
8
9
9.28 125
9 .28 190
9.28 254
9.28 319
9.28 384
9.28448
9.28 5i2
9.28577
9.28 64 1
9
9
9
9
9
9
9-
9-
9-
28 933
29 ooo
29 067
29 1 34
29 201
29 268
29 335
29 4O2
29 468
o. 71 067
0.71 ooo
0.70 933
0.70866
0.70 799
0.70 732
0.70 665
0.70 598
0.70 532
OOO OOO OOO
ooo ooo ooo
192
I9O
I87
185
182
180
177
172
59
58
5?
56
55
54
53
52
Si
10
9.28 705
9.29535
0.70 465
9.99
170
50
1 1
12
i3
i4
i5
16
18
'9
9.28 769
9.28 833
9.28 896
9.28 960
9.29 024
9.29 087
9.29 i5o
9.29 214
9.29 277
04
64
63
64
64
63
63
64
63
9
9-
9-
9-
9-
9-
9-
9-
9
29 601
29 668
29 734
29 800
29 866
29932
29998
3oo64
3o i3o
0.70 399
0.70 332
0.70 266
0.70 200
0.70 1 34
0.70068
0.70 002
0.69 936
0.69 870
9.99
9.99
9.99
9.99
9.99
9-99
9-99
9-99
9.99
167
165
162
1 60
157
162
150
i47
49
48
4?
46
45
44
43
42
4i
20
9.29 34o
03
9-
3o i95
0.69 805
9.99
145
3
2
3
2
3
2
3
3
2
40
21
22
23
24
25
26
27
28
29
9.29 4o3
9.29 466
9.29 529
9.29 591
9 .29 654
9.29 716
9-29779
9.29 84i
9.29 903
63
63
63
62
63
62
63
62
62
9
9-
9-
9-
9-
9-
9-
9-
9-
3o 261
3o326
3o 391
30 522
3o587
3o652
3o7i7
3o 782
0.69 739
0.69 674
0.69 609
o,69 543
0.69 478
0.69 4i3
0.69 348
o.69 283
0.69 218
ooo ooo ooo
ooo ooo ooo
ooo ooo ooo
142
i4o
i37
132
i3o
127
124
122
39
38
37
36
35
34
33
32
3i
30
9.29 966
63
9-
3o846
0.69 1 54
9.99
119
30
L. Cos.
d.
L
,Cotg
.
d.
L. Tang.
L. Sin.
d.
'
78° 30 .
PP
t. .2
3
•4
.u
68
67 66
.2
•3
•4
• 5
.6
.7
.8
65
64
63
62 3
6.8
13-6
20.4
27.2
34-o
40.8
47.6
54-4
6.7 6.6
13.4 13.2
20. i 19.8
26.8 26.4
33-5 33-o
40.2 39.6
46.9 46.2
53-6 52-8
60.3 59.4
6-5
13-0
19-5
26.0
32-5
45-5
52.0
58. s
6.4
12.8
19.2
256
32.0
38.4
44.8
63
12.6
18.9
25.2
37-8
44.1
50-4
.2
•3
•4
mti
6.2 .3
12.4 .6
18.6 .9
24.8 1.2
31.0 1.5
37.2 1.8
43-4 2.1
49.6 .2.4
5S.8 2.7
1 1° 3D .
L. Sin. d.
L.
Tang.
d.
L. Cotg.
L. Cos. d.
30
9.29 966
9
3o 846
0.69 1 54
9.99 119
30
3i
32
9-3o 028
9»3o 090
62
9
9-
3o 91 1
3o 975
64
0.69 089
0.69 025
9.99
9-99
117
u4
3
29
28
33
9.3o
i5i
9-
3 1 o4o
65
0.68 960
9.99
I 12
2
27
34
9.3o 2i3
62
62
9-
3i io4
0.68 896
9.99
109
3
26
35
9.30275
9-
3i 168
0.68 832
9.99
1O6
3
25
36
9.3o
336
9-
3i 233
65
0.68 767
9.99
104
2
24
37
9-3o 398
62
61
9-
3i 297
64
0.68 703
9.99
1OI
3
23
38
9«3o 459
9-
3i 36i
4
0.68 639
9.99
099
22
39
9 . 3o
521
61
9-
3i 425
64
0.68 575
9-99
096
3
21
40
9.3o 582
61
9-
3i 489
04
6?
0.68 5n
9.99
093
3
20
4i
9.3o643
61
9-
3i 552
61
0.68448
9.99
091
I9
42
9.3o 704
61
9-
3i 616
fio
0.68 384
9-99
088
18
43
9. 3o 765
61
9-
3i 679
°3
0.68 32i
9.99
086
3
17
44
9-3o 826
61
9-
3i743
6-j
0.68 257
9.99
o83
16
45
9.30887
9-
3 1 806
0.68 194
9.99
080
i5
46
9.30947
61
9-
3i 870
64
0.68 i3o
9.99
078
3
i4
47
9. 3 1 008
60
9-
3i 933
0.68 067
9.99
076
i3
48
9.3i 068
9-
3 1 996
°3
0-68 oo4
9.99072
12
49
9.3i
129
60
9-
32 o5c
1
°3
0.67 941
9.99070
1 I
50
9. 3i
189
61
9-
32 122
fio
0.67 878
9-99
067
3
10
5i
9.3i 250
60
9-
32 i85
°3
63
0.67815
9.99 o64
9
52
9-3i 3io
60
9-
32 248
0.67 752
9-99
062
8
53
9.3i 370
60
9-
32 3u
63
62
0.67 689
9.99
059
3
7
54
9. 3 1 43o
60
9-
32373
0.67 627
9.99
066
6
55
9. 3 1 490
9-
32436
3
0.67 564
9.99
o54
5
56
9.3i 549
59
60
9-
32498
62
0.67 5o2
9.99
o5i
3
4
57
9. 3 1 609
60
9-
32 56i
62
0.67 439
9.99
o48
2
3
58
9. 3 1 669
9
32 623
o.67377
9.99
o46
2
59
9-3i
728
59
9-
32685
62
o.673i5
9.99
o43
3
I
60
9.3i
788
9
32747
o.67253
9,99
o4o
0
L. Cos. ! d.
L,
Cotg
. d.
L. Tang.
L. Sin.
d.
'
78°.
PP
65
64 63
6a
61
60
59 3
.1
6.5
6.4 6.3
.!
6.2
6.1
6.0
. i
5-9 0.3
.2
13.0
12.8 12.6
.2
12.4
12.2
I2.O
.2
ii. 8 0.6
•3
19.2 18.9
• 3
18.6
l8-3
18.0
•3
17.7 0.9
4
26.0
25.6 25.2
-4
24.8
24.4
24.0
4
23.6 1.2
32.5
32.0 31-5
.5
31.0
3°-5
30.0
.5
29-5 1-5
.6
38.4 37-8
.6
37-2
36.6
36.0
.6
35-4 1-8
.7
45-5
44.8 44.1
•7
43-4
42.7
42.0
•7
41-3 2.1
.8
52.0
51.2 50.4
.8
49-6
48.8
48.0
.8
47.2 2.4
•9 58-5
54-9
54-° -9
S3-1 2.7
53
12C
,
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.3i 788
9-
32 747
61
0.67 253
9.99040
60
I
9.3i 847
60
9-
32 810
62
0.67 190
9.99
o38
3
59
2
9.3i 907
9-
32 872
0.67 128
9.99
o35
58
3
9-3i 966
59
9-
32 933
62
0.67 067
9.99
032
3
2
$7
4
9.32 025
59
9-
32 995
62
0.67 005
9.99
o3o
56
5
9.32 o84
9-
33o57
0.66 943
9.99
027
55
6
9.32 i43
59
9-
33 119
61
0.66 881
9.99
024
3
2
54
7
9.32 202
59
9-
33 180
62
0.6
6 820
9.99
022
53
8
g.32 261
_0
9-
33242
0.66 758
9.99
019
52
9
9.32 319
9-
33 3o3
0.66 697
9.99
016
3
5i
10
9.32 378
59
9-
33 365
0.66635
9-99
oi3
50
ii
9.32437
58
9-
33426
61
0.66 574
9.99
on
49
12
9.32 L
fo5
9-
33487
o.665i3
9.99
008
48
i3
9.32 553
58
9-
33548
61
0.66 452
9.99
oo5
3
47
59
61
3
i4
9.32 612
9-
33609
61
o.6639i
9.99
002
46
i5
9.32 670
9-
33 670
0.66 33o
9.99
ooo
45
16
9.32 728
58
9-
3373i
61
0.66 269
9.98
997
3
44
58
61
3
17
9.32 786
9-
33 792
61
0.66 208
9.98
994
3
43
18
9.32 844
9-
33853
0.66 i47
9.98
991
42
r9
9.32 902
S8
9-
33 9i3
60
0.66 087
9.98
989
4i
20
9.32 960
S8
9-
33974
61
0.66 026
9.98
986
40
21
9.33 018
58
9-
34o34
61
o.65 966
9.98
983
3
39
22
9.33 075
9
34 095
o.65 905
9.98
98o
38
23
9.33 i33
58
9-
34 1 55
o.65 845
9.98
978
37
57
60
3
24
9-33 190
eg
9-
342i5
61
o.65 785
9.98
975
36
25
9.33 248
9-
34276
o.65 724
9.98
972
35
26
9.33 3o5
57
9-
34336
o.65 664
9.98
969
3
34
57
60
2
27
9.33 362
,0
9-
34396
60
o.65 6o4
9.98
967
33
28
9.33 420
9-
34456
0.65544
9.98
964
32
29
9-33477
57
9.345i6
0.65484
9.98
96i
3
3i
30
9.33 534
57
9-
34576
o.65 424
9.98
958
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
L. Sin. d.
77° 3O.
PP
63
62 61
60
59
58
57 3
.,
6.3
6.2 6.1
.!
6.0
5-9
5-8
.!
5-7 0.3
.2
12.6
12-4 12.2
2
12.0
ii. 8
n.6
2
11.4 0.6
•3
18.9
18.6 18.3
• 3
18.0
17.7
17.4
3
17.1 0.9
•4
25.2
24.8 24.4
4
24.0
23.6
23.2
4
22.8 1.2
•5
31.5
31.0 30.5
5
30.0
29-5
29.0
5
28.5 1.5
.6
37-8
37.2 36.6
6
36.0
35-4
34- 8
6
34.2 1.8
.7
44.1
43-4 42-7
7
42.0
4i-3
40.6
•7
39.9 2. I
.8
50.4
49.6 48.8
8
48.0
47.2
46.4
.8
45-6 2.4
9 56-7
55-8 54-9
9
54-°
53- !
52.2
.9
5r-3 2.7
54
12° 30 '.
;
L. Sin.
d.
L. Tang.
d.
L
, Cotg.
L. Cos.
d.
30
9.33 534
57
9.34
576
o.65 424
9.98 958
30
3i
9.33 591
S6
9-34
635
60
o.
65 365
9.98955
3
29
32
9.33647
9-34
695
60
0.
653o5
9.98
953
28
33
9-33 704
57
9-34
755
59
o.
65245
9.98
950
3
27
34
9.33 761
57
9.34
8i4
60
o.
65 186
9.98 947
26
35
9.338i8
-<
9.34
874
0.
65 126
9.98
944
25
36
9.33874
5°
9-34
933
59
o.
65 067
9.98
94 1
3
24
37
38
9.3393i
9.33 987
57
56
efi
9.34
9.35
992
o5i
59
59
60
o.
o.
65 008
64 949
9.98 938
9.98 936
3
2
23
22
39
9.34o43
5°
9.35
in
0.
64 889
9.98
933
3
21-
40
9.34 100
cfi
9.36
170
o.
6483o
9.98 930
20
4i
9.34i56
5°
9.35
229
59
o.
64 771
9.98
927
3
19
42
9.34 212
5
9.35
288
0.
64 712
9.98
924
18
43
9.34268
56
56
9.35
347
59
58
o.
64653
9.98
921
3
2
i?
44
45
9.34324
9.34380
56
9.35
9.35
4o5
464
59
o.64 595
o.64 536
9.98
9.98
919
916
3
16
i5
46
9.34436
56
9-35
523
59
o.
64477
9.98 913
3
i4
55
58
3
47
48
9.34491
9.34547
56
9.35
9.35
58i
64o
59
o.
o.
644i9
64 36o
9.98 910
9.98 907
3
i3
12
49
9.34 602
55
9.35
698
58
o.
64 3o2
9.98
904
3
I I
50
9.34658
5°
9.35
757
59
eg
0.
64243
9.98
901
3
10
5i
9.34 71
1
56
9.35
815
58
o.
64 1 85
9.98
898
2
9
52
9.34 769
9.35
873
o.
64 127
9.98
896
8
53
9.34 824
55
55
9.35
58
58
o.
64 069
9.98
893
3
7
54
9.34879
55
9.35
989
eg
o.
64 01 1
9.98
890
3
6
55
9.34934
9.36 047
o.
63953
9.98
887
5
56
9.34989
55
9.36
io5
58
0.
63895
9.98
884
4
57
9.35 o44
55
9.36
i63
58
eg
0.
63837
9.98
881
3
3
3
58
9-35 099
9.36
221
o.
63779
9.98
878
2
59
9.35 1 54
55
9. 36
279
58
o.
63 721
9.98
875
I
60
9.35 209
55
9. 36
336
57
0.
63 664
9.98
872
0
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
77V
.
PP 60
59
58
57
56
?
55 3
.1 6.0
5-9
5-8
•i 5-7
5-6
j
5-5 0.3
.2 I2.O
n.8
11.6
.2 II-4
II. 2
2
ii. o 0.6
.3 18.0
17.7
17.4
•3 i7-i
16.8
3
16.5 0.9
4 24.0
23.6
23-2
.4 22.8
22-4
4
22.0 1.2
•5 3°-°
29-5
29.0
•5 28.5
28.0
5
27-5 i-5
.6 36.0
35-4
34-8
.6 34.2
33-6
6
33-o 1.8
.7 42-0
4x-3
40.6
•7 39-9
39-2
7
38-5 2.1
.8 48.0
47-2
46.4
.8 45-6
44-8
8
44.0 2.4
•9 54-°
S3- 1 52-2
•9 1 5i-3
50.4 9
49-5 2-7
55
'
L. Sin.
d.
L. Tang-.
d.
L.
Cotg.
L. Cos. d.
0
9. 35 209
9.36
336
0.
63 664
9.98
872
60
I
9.35 263
55
9. 36 394
5°
c8
o.
63 606
9.98 869
59
2
9.35 3i8
9. 36
452
0.
63 548
9.98
867
58
3
9.35373
9. 36
509
57
o.
63491
9.98
864
3
57
54
57
-
4
9.35427
54
9. 36
566
rg
o.
63434
9.98
861
56
5
9-35481
9.36
624
o .
63 376
9.98
858
55
6
9.35536
55
9. 36
681
57
0.
63 3i9
9.98
855
:
54
7
9.35 590
54
9.36
738
57
0.
63 262
9.98
852
3
53
8
9.35 644
9-36
795
0.
63 205
9.98
849
52
9
9.35 698
54
9.36
852
57
0.
63 i48
9.98
846
3
5i
10
9.35 752
9.36
9°9
57
o.
63 091
9.98
843
"
50
ii
9.35 806
54
9.36
966
57
0.
63o34
9.98
84o
3
49
12
9.35 860
9-37
023
o.
62 977
9.98
837
48
i3
9.35 914
54
9.37
080
57
o.
62 920
9.98
834
-
>
47
i4
9. 35 968
54
9.37
i37
57
cfi
0.
62863
9.98
83i
3
46
i5
9-36 022
9.37
I93
5°
0.
62 8o7
9.98828
45
16
9.36 075
53
9.37
250
57
o.
62 75o
9.98825
•
i
44
17
9.36 129
54
9.37
3o6
5t>
o.
62 6g4
9.98
822
3
43
18
9.36 182
9.37
363
57
0.
62 637
9.98
8i9
42
19
9. 36 236
54
9.37
4i9
56
o.
62 58i
9.98816
3
4i
20
9.36 289
53
9.37
476
57
o.
62 524
9.98
8i3
•
>
40
21
9. 36 342
53
9.37
532
56
o.
62468
9.98
810
39
22
9.36395
53
9.37
588
56
o.
62 412
9.98
8o7
38
23
g.36 449
54
9.37
644
56
o.
62 356
9.98
8o4
3
37
24
9-36 5o2
53
9.37
700
56
0.
62 3oo
9-98
801
3
36
25
9.36555
53
9.37
756
56
o.
62 244
9.98
798
35
26
9.36 608
53
9.37
812
50
o.
62 188
9.98
795
•
34
27
28
~9?3\66o
9.36 7i3
52
53
9.37
9.37
868
924
56
56
o.
0.
62 1 32
62 o76
9.98
9.98
792
789
3
3
33
32
29
9. 36 766
53
9.37
980
56
0.
62 020
9.98
•786
3i
30
9.36 819
53
9. 38
o35
55
0.
61 965
9.98
783
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
-
76° 3O.
PP 58
57
56
55
54
53
3
5-8
5-7
5.6
• i 5-5
5-4
.1
5-3
• 3
.2 II. 6
n-4 .
II. 2
.2 II. 0
10.8
.2
10.6
.6
•3 17-4
17.1
1 6. 8
•3 16.5
16.2
•3
159
•9
•4 232
22.8
22.4
.4 22.O
21.6
•4
21.2
1.2
• 5 29.0
28.5
28.0
•5 27.5
27.0
.5
26.5
i-5
.6 34-8
34-2
33-6
• 6 33.0
32-4
.6
31.8
1.8
.7 4°-6
39-9
392
•7 38.5
37-8
•7
37- *
2.1
.8 46.4
45-6
44-8
.8 44.0
43-2
.8
42.4
2-4
•9 52-2
51.3 ! 50.4
•9 49-5
48.6
47-7
2.7
56
13°3O
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.36
819
9
.38 o35
0.61 965
9.98
783
30
3i
32
9-36
9-36
87I
924
52
53
9
9
.38 091
.38 147
56
0.61 909
0.61 853
9.98
9.98
780
777
3
3
29
28
33
9.36
976
52
9
.38 202
55
0.61 798
9.98
774
3
27
34
9.37
028
52
9
.38257
55
eft
0.61 743
9.98
771
3
26
35
9.37
08 1
9
.383i3
5°
0.61 687
9.98
768
3
25
36
9.37
i33
52
9
.38368
55
0.61 632
9.98
765
3
24
37
9.37
i85
52
9
.38423
55
0.61 577
9.98
762
3
23
38
9-37
237
9
.38479
J
0.61 52i
9.98
759
3
22
39
9.37
289
52
9
.38 534
55
0.61 466
9.98
756
3
21
40
9.37
34i
9
.38 589
0.61 4n
9.98
753
3
20
4i
9.37
393
52
9
38 644
55
0.61 356
9.98
75o
3
'9
42
9.37
445
9
38 699
0.61 3oi
9.98
746
18
43
9.37
497
9
38754
0.61 246
9.98
743
3
ij
52
54
3
44
9.37
549
51
9
38 808
0.61 192
9.98
74o
16
45
9-37
600
9
38863
0.61 137
9.98
737
i5
46
9.37 652
9
38 918
55
0.61 082
9.98
734
J
i4
47
9-37
7o3
51
52
9
38 97
2
54
0.61 028
9.98
73i
3
i3
48
9.37
755
9
39 027
0.60 973
9.98
728
12
49
9.37 806
51
9
39 082
55
0.60 9i8
9.98
725
3
II
50
9.37 858
9
39 1 36
54
0.60 864
9.98
722
3
10
5i
9.3-7909
51
9
39 190
0.60 810
9.98
719
9
52
9.3-7 960
9
39 24
5
0.60 755
9.98
7i5
8
53
9-38 on
9
39299
54
0.60 701
9.98
712
3
7
51
54
3
54
9.38 062
9
39 353
0.60 647
9.98
7°9
6
55
9. 38
n3
9
39 4o
7
0.60 593
9.98
7o6
5
56
9.38
1 64
51
9
3946i
54
0.60 539
9.98
7o3
3
4
51
54
3
57
9.38
2l5
9
39 5i5
0.60485
9.98
7oo
3
58
9.38 266
9
39 569
o.6o43i
9.98
697
2
59
9. 38
3i7
51
9
39623
54
0.60 377
9.98
694
3
I
60
9. 38
368
51
9
39677
54
0.60 323
9.98
69o
4
0
L. Cos. d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
76°.
PP
56
55
54
53
52
51
4 3
!
5.6
5-5
5-4
.1
5-3
5-2
5.1
.1
0.4 0.3
2
II. 2
II. 0
10.8
.2
10.6
10.4
IO.2
.2
0.8 0.6
3
16.8
16.5
16.2
•3
15-9
15.6
15-3
•3
1.2 0.9
4
22.4
22. 0
21.6
•4
21.2
20.8
20.4
•4
1.6 1.2
5
28.0
27.5
27.0
• 5
26.5
26.0
25-5
•5
2.O 1.5
6
33-6
33-0
32.4
.6
31-8
31.2
30.6
.6
2.4 1.8
7
39-2
38. s
37-8
• 7
37-1
*4
35-7
• 7
2.8 2.1
8
44-8
44.0
43-2
.8
42.4
41.6
40.8
.8
3.2 2.4
9 5°-4
49.5 48.6
47-7
46.8
45-9 -9
3.6 2.7
14°.
f
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos. d.
0
9. 38 368
9-
39677
54
54
53
54
53
54
53
54
53
54
53
53
53
53
53
53
52
53
53
52
52
53
52
52
S2
53
52
0.60 323
9.98
690
60
2
3
4
5
6
7
8
9
9.384i8
9.38469
9. 38 5i9
9.38 570
9. 38 620
9. 38 670
9.38 721
9. 38 771
9-38 821
51
50
51
5°
5°
Si
50
So
9-
9-
9-
9-
9-
9-
9-
9-
9-
3973i
39785
39838
39 892
39945
39999
4o o52
4o 1 06
4o i59
0.60 269
0.60 2i5
0.60 162
0.60 1 08
0.60 055
0.60 ooi
0.59 948
0.59 84i
9.98 687
9.98 684
9.98 681
9.98 678
9.98675
9.98 671
9.98668
9.98 665
9.98 662
3
3
3
3
4
3
3
3
59
58
56
55
54
53
52
5i
10
9-38 871
9-
4o 212
0.59 788
9.98
659
50
1 1
12
i3
i4
i5
16
«7
18
'9
9-38 921
9.38 971
9.39 02 1
9.39 071
9.39 121
9.39 170
g.Sg 22O
9.39270
9.393i9
So
So
50
50
49
So
5°
49
9-
9-
9-
9-
9-
9-
9-
9-
9-
4o 266
4o 3i9
40372
40425
40478
4o53i
4o584
4o636
40689
o.59 734
o.5968i
0.59 628
o.Sg 575
O.Sg 522
O.Sg 469
0.59 4i6
0.59 364
o.Sg 3i i
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
656
652
649
646
643
64o
636
633
63o
4
3
3
3
3
4
3
3
49
48
4?
46
44
43
42
4i
20
9.39 369
5°
9-
4o 742
0.59 258
9.98
627
40
21
22
23
24
25
26
27
28
29
9.39 4i8
9.39467
9.39 5i7
9.39 566
9.39615
9.39 664
9.397i3
9.39 762
9.39 8n
49
49
5°
49
49
49
49
49
49
49
9-
9-
9-
9-
9-
9-
9-
9-
9-
40795
40847
4o 900
4o 952
4i 005
4i 057
4i 109
4i 161
4i 214
o.Sg 2o5
0.59 i53
0.59 100
0.59048
o.58 995
0.58943
o.58 891
o.58 839
o.58 786
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
623
620
617
6i4
610
607
6o4
601
597
3
3
3
4
3
3
3
4
39
38
3?
36
35
34
33
32
3i
30
9.39 860
9-
4i 266
o.58 734
9.98
594
3
30
L. Cos.
d.
L,
Cotg.
d.
L. Tang.
L. Sin.
d.
'
75° 30 .
PP
.2
•3
•4
:I
•7
-8
• q
54
53 53
.1
.2
•3
•4
:I
:l
~
51
50
49
4 | 3
,11
16.2
21.6
27.0
32-4
37-8
|f
53 5-2
10.6 10.4
15.9 15.6
21.2 20.8
26. 5 26.0
31.8 31.2
37-1 36-4
42.4 41.6
47-7 46.8
IO.2
15-3
20.4
25 I
30.6
35-7
40.8
45 9
5-o
IO.O
15-0
20.0
25.0
30.0
35-o
40.0
4.9
9.8
14.7
19.6
24-5
29-4
34-3
39-2
44.1
.2
•3
•4
1
:i
• 9
:J :i
1.2 .9
1.6 1.2
2.0 I.5
2.4 1.8
2.8 2.1
3.2 2.4
3.6 2.7
58
14°3O
>
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.39 860
49
9.41 266
o.58 734
9.98 594
30
3i
9.39909
49
9.41 3i8
52
o.58 682
9.98 591
29
32
9.39 958
48
9.41 370
o.5863o
9.98 588
28
33
9.40 006
49
9.41 422
52
o.58 578
9.98 584
^
27
34
9.40 c
55
48
9.41 4?4
s2
o.58 526
9.98 58 r
3
26
35
9.40 io3
9-
4i 526
0.58474
9.98
578
25
36
9.40 i
52
9.41 578
0.58422
9.98574
24
48
5*
3
37
9.40 200
49
9-
4i 629
s2
o.58 37i
9.98
57i
3
23
38
9.40 249
9-
4r 681
o.58 319
9.98 568
22
39
9.40 297
4°
9-
4i 733
52
o.58 267
9.98
565
3
21
40
9.40 346
.0
9-
4i 784
o.58 216
9.98
56i
20
4i
9.4o 3
94
48
9-
4i 836
o.58 i64
9.98
558
3
19
42
9.40 442
9-
4i 887
o.58 n3
9.98
55,5
18
43
9.40490
48
9-
4i 939
52
51
o.58 061
9.98
55i
ll
44
9.40 538
9-
4i 990
o.58 oio
9.98
548
3
16
45
9.40 586
9-
42 o4i
0.57 959
9.98
545
i5
46
9.40 634
48
9-
42 093
52
0.57 907
9.98
54i
i4
48
51
3
47
9.40 682
4.8
9-
42 i44
o.57856
9.98
538
3
i3
48
9.40 730
9-
42 195
0.57 805
9.98
535
12
49
9.40778
48
9-
42 246
51
0.57 754
9.98
53i
3
I I
50
9.40 825
47
.0
9-
42 297
51
0.57 703
9.98
528
10
5i
9.40 873
48
9-
42 348
51
0.57 652
9.98
525
4
9
52
9.40 921
9-
42 399
0.57 601
9.98
521
8
53
9.40968
47
48
9-
42 45o
Si
51
0.57 550
9.98
5i8
3
7
54
9.41 016
9?
42 5oi
0.57 499
9.98
515
4
6
55
9.41 o63
9-
42 552
o.57448
9.98
5n
5
56
9.41 in
4°
9-
426o3
51
o.57397
9.98
5o8
4
57
9.41 i58
47
9-
42653
So
o.57347
9.98
505
3
4
3
58
9.41 2o5
9-
42 704
0.57 296
9.98
5oi
2
59
9-4l 252
47
9-
42755
bl
0.57 245
9.98
498
I
60
9.41 3oo
40
9-
42 8o5
5°
o.57 195
9.98
494
0
"
L. Cos.
d.
L.
Cotg.
d. L. Tang.
L. Sin.
d.
75°.
PR
52
51 50
49
48
47
4
3
.!
52
5.1 5.0
.1 4.9
4.8
4-7
.1
0.4
°-3
2
10.4
10.2 10. 0
.2 9.8
9.6
9.4
.2
0.8
0.6
3
15-6
15-3 15°
3 14-7
14.4
14.1
•3
1.2
0.9
4
20.8
20. 4 20. o
.4 19.6
19.2
18.8
•4
1.6
1.2
I
26.0
31 2
25.5 25.0
30.6 30.0
• 5 24.5
.6 29.4
24.0
28.8
23-5
28.2
•5
.6
2.O
2.4
1^8
7
36.4
35-7 35-o
•7 34-3
33-6
32-9
• 7
2.8
2.1
8
41.6
40.8 40.0
.8 39.2
38-4
37-6
.8
3-2
2-4
.9 46.8
459 45'°
.9 44.1 43.2
42.3 .9
59
15°.
•
L. Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
0
9-4i
3oo
9.42 bo5
51
50
51
50
50
51
50
50
50
5°
5°
5°
50
50
50
49
So
5°
49
50
49
5o
49
5°
49
49
49
5°
49
49
o.57 195
9.98 494
60
2
3
4
5
6
7
8
9
9-4i
9>4i
9.41
9,41
9.41
9-4i
9-4i
9.41
9.41
34?
394
44 1
488
535
582
628
670
722
47
47
47
47
47
46
47
47
9.42 856
9.42 906
9.42 957
9.43 007
9.43o57
9.43 108
9.43i58
9.43 208
9.43258
0.57 1 44
0.57 094
0.57 043
0.56993
o.56 943
o.56 892
0.56842
o.56 792
o. 56 742
9.98 491
9.98 488
9.98484
9.98 481
9.98477
9.98 474
9.98 471
9.98 467
9. 98 464
3
4
3
4
3
3
4
3
59
58
5?
56
55
54
53
52
5i
10
9.41
768
47
46
47
46
47
46
46
47
46
9.43 3o8
o.56 692
9
. 98 460
50
1 1
12
i3
i4
i5
16
17
18
19
9.41
9.41
9-4i
9.41
9.42
9.42
9.42
9.42
9.42
815
861
908
954
OOI
o47
ogS
i4o
186
9.43358
9.434o8
9-43458
9.435o8
9.43558
9.43 607
9.43657
9.43 707
9.43756
0.56642
o.56 592
o., 56 542
o.56 492
0.56442
0.56393
0.56343
o.56 293
o.56 244
9
9
9
9
9
9
9
9
9
.98457
.98453
.98450
.98447
.98443
.98440
.98436
.98433
.98 429
4
3
4
3
4
3
4
49
48
4?
46
45
44
43
42
4i
20
9.42
232
46
9.43 806
o.56 194
9
.98426
4
3
4
3
3
4
3
4
3
4
40
21
22
23
24
25
26
27
28
29
9.42
9.42
9.42
9.42
9.42
9.42
9.42
9.42
9.42
278
324
37o
4i6
46 1
507
553
599
644
46
46
46
46
45
46
46
46
45
46
9.43855
9.43 905
9.43954
9.44 oo4
9.44 o53
9.44 102
9.44 i5i
9.44 20 1
9-44 250
o.56 i4§
o.56 095
o.56 o46
o.55 996
o.55 947
o.55 898
0.55849
o.55 799
0.55 75o
9
9
9
9
9
9
9
9
9
.98 422
.98419
.98415
.98412
.98 409
.98 4o5
.98 402
.98 398
.98395
39
38
37
36
35
34
33
32
3i
30
9.42
690
9.44 299
o.55 701
9
.98 391
30
L. Cos.
d.
L. Cotgr. d.
L. Tang.
L. Sin.
d.
74° 3D '.
PP
.1
.2
•3
•4
:l
:i
51
50
49
48 47 46
.1
.2
•3
•4
• 5
.6
:l
•9
45
4 3
5-i
10.2
15-3
20-4
25.5
30.6
35-7
40.8
45 9
5-o
10.0
15-0
20. o
25-0
30.0
35-o
40.0
45.0
49
9.8
14.7
19.6
24-5
29.4
34-3
39-2
44.1
.2
-3
•4
•5
.6
3
4.8 4-7 4-6
9.6 9.4 9.2
14.4 14.1 13.8
19.2 18.8 18.4
24.0 23.5 23.0
28.8 28.2 27.6
33-6 32.9 32.2
38.4 37.6 36.8
43.2 423 41 4
4-5
9.0
i3-5
18.0
22.5
27.0
3'-5
36.0
4°-5
0.4 0.3
0.8 0.6
1.2 0.9
1.6 1.2
2.0 1.5
2.4 1.8
2.8 2.1
3.2 2.4
3.6 2.7
60
15° 3O
L. Sin. d.
L
. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.42
690
9
.44 299
o.55 701
9.98
39i
30
3i
9.42
735
46
9
.44348
49
0.55652
9.98
388
3
29
32
9.42 781
9
.44397
o.556o3
9.98
384
28
33
9.42
826
46
9
.44446
0.55554
9.98
38i
3
4
27
34
9.42 872
45
9
.44495
49
o.555o5
9.98
377
26
35
9.42
917
9
.44544
Q
0.55456
9.98
373
25
36
9.42
962
45
9
.44 592
o. 554o8
9.98
370
3
24
46
49
37
9.43 008
9
,4464i
o.55359
9.98
366
23
38
9-43o53
9
44 690
.0
o.55 3io
9.98
363
22
39
9-43
098
45
9
44738
o.55 262
9.98
359
4
21
40
9.43
i43
45
9
.44 787
o.55 2i3
9.98
356
20
4i
9-43
188
45
9
.44836
o.55 1 64
9.98
352
4
'9
42
9.43
233
9
4488
4
1
o.55 116
9.98
349
18
43
9.43
278
45
9
44933
49
o.55 067
9.98
345
4
17
44
9-43
323
45
9
.44981
48
o.55 019
9.98
342
3
16
45
9.43
367
9
45 029
1
0.54 971
9.98
338
i5
46
9.43412
45
9
45 078
49
o.54 922
9.98
334
4
i4
47
9.43457
45
9
.45 126
4H
0.54874
9.98
33i
3
i3
48
9.43
502
9
45 174
0.54826
9.98
327
12
49
9-43
546
44
9
.45 222
48
o.54 778
9.98
324
3
I I
50
9.43
59i
45
9
.45 271
49
o.54 729
9.98
320
10
Si
9.43635
44
9
.453l9
48
o.5468i
9.98
3i7
3
9
52
9.43 680
45
9
.45367
4
0.54633
9.98
3i3
8
53
9-43
724
44
.4541
s
48
0.54585
9.98
309
4
7
54
9-43
769
45
9
.45463
48
.0
0.54537
9.98
3o6
3
6
55
9.43
8i3
44
9
.455n
0.54489
9.98
302
5
56
9.43
857
44
9
.45559
48
o.5444i
9.98299
i
4
57
9.43
901
44
9
.456o6
47
.Q
o.54394
9.98 295
4
3
58
9-43
946
45
9
.45654
0.54346
9.98
291
2
59
9.43
990
44
9
.45 702
o.54 298
9.98 288
3
I
60
9-44
o34
44
9
.45750
4°
0.54 25o
9.98
284
0
L. Cos.
d.
L
. Cotg.
d.
L. Tang.
L. Sin.
d.
'
74°
.
PP
49
48 47
46
45
44
4
3
.1
4-9
4-8 4-7
,i
4.6
4-5
4-4
.1
0.4
°'3
.2
9.8
9.6 9.4
.2
9-2
9.0
8.8
.2
0.8
0.6
•3
14.7
14.4 14.1
•3
13-8
13-5
13-2
•3
1.2
0.9
•4
19.6
19.2 18.8
•4
18.4
18.0
17.6
•4
1.6
1.2
f c
24-5
24.0 23.5
. tj
23.0
22.5
22.0
• 5
2.0
1.5
.6
29.4
28.8 28.2
.6
27.6
27.0
26.4
.6
2.4
1.8
34-3
33-6 32-9
•7
32.2
3i-5
30.8
•7
2.8
2.1
39 2
38-4 37-6
.8
36.8
36.0
35-2
.8
3-2
2.4
•9 44- x
43 .2 4^
41.4
40-5
39-6 -9
3-6
2-7
61
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.44 o34
9.45 750
o.54 25o
9
.98 284
60
I
9.44078
9.45 797
47
0.54 203
9
.98 281
59
2
9.44 122
9.45845
4
o.54i55
9
.98277
58
3
9.44 1 66
44
9.45 892
47
o.54 108
9
.98273
57
44
48
3
4
9 . 44 2 1 o
9.45 940
o.54 060
9
.98 270
56
5
9.44253
9.45987
o.54oi3
9
.98 266
55
6
9.44 297
44
9.46 035
48
o.53965
9
.98262
54
7
9.4434i
44
9.46 082
47
j.8
o.53 918
9
.98 259
3
53
8
9.44385
9.46 i3o
o.53 870
9
.98 255
52
9
9.44428
43
9-46 177
47
0.53823
9
.98 261
5i
10
9.44 472
9.46 224
47
o.53 776
9
.98 248
50
ii
9.44 5i6
9.46 271
47
.0
o.53 729
9
.98244
49
12
9.44559
9.46 319
0.5368T
9
.98 240
48
i3
9-44 602
43
9.46 366
47
0.53634
9
.98237
3
47
i4
9-44646
44
9-464i3
47
o.53587
9
.98 233
4
46
i5
9.44689
9-46 46o
47
o.53 54o
9
.98 229
45
16
9.44733
44
9-46 507
47
0.53493
9
.98 226
3
44
17
9-44 776
43
9.46554
47
0.53446
9
.98 222
4
4
43
18
9.44 819
9.46 601
47
o.53 399
9
.98 218
42
'9
9.44 862
43
9.46648
47
o.53 352
9
.98215
3
4i
20
9.44 go5
9.46 694
46
o.533o6
9
.98 211
4
40
21
9.44948
9.46 741
47
o.53 259
9
.98 207
3
39
22
9.44 992
9.46 788
47
O.53 212
9
.98 204
38
23
9.45 035
43
9.46835
47
o.53i65
9
.98 2OO
4
37
24
9.45077
42
9.46 881
46
o.53 119
9
.98 196
4
4
36
25
9.45 120
43
9.46 928
47
o.53 072
9.98 192
35
26
9.45 i63
43
9.46975
47
o.53 025
9
.98 189
34
27
9.45 206
43
9-47 021
46
o.52 979
9
.98 185
4
33
28
9.45 249
9.47 068
47
o.52 932
9
.98 181
32
29
9.45 292
43
9-47
i4
46
o.52 886
9
.98 177
3i
30
9.45334
42
9.47 160
46
o.52 84o
9
.98 174
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.Sin. Id.
'
73° 3D .
PP
48
47
46
45
44
43
42
4 3
,!
4.8
4-7,
4.6
.1
4-5
4-4
4-3
.i
4.2
0.4 0.3
.2
q.6
9-4\
9.2
.2
q.o
8.8
8.6
.2
8.4
0.8 0.6
•3
14.4
14.1
13-8
•3
13-5
13.2
12.9
•3
12.6
1.2 0.9
•4
19.2
18.8
18.4
•4
18.0
17.6
17.2
•4
16.8
1.6 1.2
.6
24.0
28.8
S5
23-0
27.6
:!
22.5
27.0
22.0
26.4
21-5
25-8
•5
.6
21. 0
25.2
2.0 J-5
2.4 1.8
•7
336
32.9
32.2
.7
§*.5
30.8
30.1
• 7
29.4
2. 8 2. I
.8
38.4
37-6
36.8
.8
36.0
35-2
•tf-4
.8
33-6
3.2 2.4
•9 43.2
42.3 41-4
40. s 3Q.6 38.7
37.8
3.6 2.7
16° 30 '.
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos.
d.
30
9.45 344
9.47 160
o.52 84o
9
.98 174
30
3i
32
9.45 419
43
42
9.47 207
9.47 253
47
46
o.52 793
o.52 747
9
9
.98 170
.98 1 66
4
4
29
28
33
9-45
462
43
9.47299
46
o.52 701
9
.98 162
4
27
34
35
9-45
9.45
5o4
547
42
43
9.47346
9-47392
47
46
o.52 654
0.52 608
9
9
.98 159
.98 i5§
3
4
26
25
36
9.45
589
9-47438
46
o.52 562
9
.98 i
5i
4
24
37
9-45
632
43
9-47484
46
46
o.52 5i6
9
.98147
4
23
38
9-45 674
9.47 53o
o.52 470
9
.98 1 44
3
22
39
9.45
716
42
9.47 576
46
O.52 424
9
.98 i4o
4
21
40
9.45
758
9.47 622
4°
o.52378
9
.98 1 36
4
20
4i
9-45
801
42
9.47668
46
o.52 332
9
.98 i32
4
19
42
9-45
843
9-47 7*4
0.52 286
9
.98 129
18
43
9-45
885
42
9.47 760
46
0.52 24O
9
.98125
4
17
44
9.45
927
42
9.47 806
.f.
o.52 194
9
.98 121
16
45
9-45
969
9.47852
0.52 li
18
9
.98 117
i5
46
9-46
OI I
9.47897
45
o.52 io3
9
.98 ii3
4
i4
47
9.46o53
42
9-47943
46
.f.
O.52 of
>7
9
.98 no
3
i3
48
9.46 095
9.47989
4°
O.52 Oil
9
.98 106
12
49
9-46
i36
41
9.48 035
4''
o.5i 965
9
.98 102
4
I I
50
9-46
178
9.48 080
45
o.5i 920
9
.98 098
4
10
5i
9-46
220
42
9.48 126
o.5i 874
9
.98 094
4
9
52
9-46
262
9.48 171
o.5i 829
9
.98 090
8
53
9-46
3o3
9.48 217
40
o.5i 783
9
.98 087
3
7
54
9-46
34.5
42
9.48 262
45
o.5i 738
9
.98083
4
6
55
9.46 386
9.48 307
o.5i 693
9
.98 079
5
56
9-46
428
42
9.48 353
46
o.5i 647
9
.98 075
4
4
4>
45
4
57
9.46469
9.48 398
o.D i 602
9
.98 071
3
58
9-46
5ii
9-48443
o.5i 557
9
.98 067
2
59
9.46
552
41
9.48489
46
o.5i 5ii
9
.98 o63
4
I
60
9-46
594
9.48 534
45
o.5i 466
9
.98 060
0
L. Cos.
d.
L. Cotg.
d.
L, Tang.
L. Sin.
d.
'
73°.
PP
47
46
45
44
43
42
41
4 3
.2
4-7
9-4
4-6
9.2
4-5
9.0
.2
a
tl
t:
.1
2
4.1
8.2
0.4 0.3
0.8 0.6
3
14.1
i3-8
13-5
•3
13-2
12.9
12.6
•3
12.3
1.2 0.9
•4
18.8
18.4
18.0
•4
17.6
17.2
16.8
•4
16.4
1.6 1.2
•5
.6
23-5
28.2
23.0
27.6
•22.5
27.0
.6
22.O
26.4
21.5
25.8
21.0
25.2
•5
.6
20.5
24.6
2.0 1.5
2.4 1.8
• 7
32.9
32.2
3i 5
•7
30.8
30.1
29.4
•7
28.7
2.8 2.1
.8
37-6
36.8
36.0
.8
35-2
34-4
33-6
.8
32.8
3.2 2.4
•9 42.3
41.4 40.5
39-6 38.7 37.8
63
17
f
L. Sin.
d.
L.
Tang
.
d.
L. Cotg.
L. Cos.
d.
0
9 .46 594
9-
48 534
45
45
45
45
45
45
45
45
45
45
45
44
45
45
44
45
44
45
44
45
44
45
44
44
45
44
44
44
44
44
o.5i 466
9 . 98 060
60
I
2
3
4
5
6
8
9
9-46635
9.46 676
9.46717
9.46 758
g.46 800
9.46 84i
9.46 882
9.46 923
9.46 964
41
41
42
41
4r
40
41
41
41
40
40
40
41
40
4*
40
4°
41
40
40
40
9.48 579
-£•48 624
9.48 669
9.48 714
9-48 759
9.48 8o4
9.48849
9.48 894
9.48 939
o.5i 421
o.5i 376
o.5i 33i
o.5i 286
o.5i 241
o.5i 196
o.5i i5i
o.5 1 1 06
o.5i 06 1
9.98 o56
9.98o52
9.98 o48
9.98 o44
9.98 o4o
9.98 o36
9.98 o32
9.98 029
9.98 025
4
4
4
4
4
4
3
4
59
58
57
56
55
54
53
52
5i
10
9.47 005
9-
48 984
o.5i 016
9.98
O2I
50
1 1
12
i3
i5
16
17
18
'9
9.47 o45
9.47 086
9.47 127
9.47 168
9.47 209
9.47 249
9.47 290
9.47371
9.49 029
9.49 073
9.49 118
9.49 i63
9.49 207
9.49 252
9.49 296
9.49 34i
9.49 385
o.5o 971
o.5o 927
o.5o882
o.5o 837
o.5o 793
0.50748
o.5o 704
o.5o 65g
o.5o 615
9.98 017
9.98 oi3
9.98 009
9.98 oo5
9.98 ooi
9.97 997
9.97993
9.97989
9.97986
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
49
48
47
46
45
44
43
42
4i
20
9.47 4i i
9-
49 43o
o.5o 570
9-97
982
40
21
22
23
24
25
26
27
28
29
9.47 452
9.47 492
9.47533
9.47 6i3
9.47654
9.47 694
9.47 734
9.47 774
9.49 474
9 .49 5ig
9.49 563
9.49 607
9.49 652
9.49 696
9.49 74o
9.49 784
9.49 828
o.5o 526
o.5o48i
o.5o 437
o.5o 393
o.5o348
o.5o 3o4
o.5o 260
o.5o 216
o.5o 172
9.97978
9.97 974
9.97970
9.97966
9.97962
9.97 958
9.97954
9.97950
9.97 946
39
38
37
36
35
34
33
32
3i
30
9.47 8i4
40
9-
49872
o.5o 128
9.97 942
30
L. Cos.
d.
L.
Cotg. d.
L. Tang.
L. Sin.
d.
'
72° 3O .
PP
.1
.2
-3
•4
1
•9
45
44 43
.2
•3
•4
42
41
40
4 3
4-5
9.0
13-5
18.0
22.5
27.0
3r-5
36.0
40.5
tl *i
13.2 12.9
17.6 17.2
22.0 21-5
26.4 25.8
30.8 30.1
35-2 34-4
8.4
12.6
16.8
21. 0
25-2
29.4
33-6
8.2
12.3
16.4
20.5
24.6
28.7
32.8
8.0
12. 0
16.0
20. o
24.0
28.0
32.0
36.0
.2
• 3
•4
• 5
.6
• 7
.8
0.8 0.6
1.2 0.9
1.6 1.2
2.O 1-5
2.4 1.8
2.8 2.1
3-2 2.4
3-6 2.7
64
17° 30
J
L. Sin.
d.
L
. Tang. d.
L. Cotg.
L. Cos.
d.
30
9.47 8i4
9
.49 872
o.5o 128
9-97
942
30
3i
9.47
854
9
.49 916
44
o.5o o84
9-97
938
29
32
9.47
894
9
.49 960
o.5o o4o
9-97
934
28
33
9-47
934
40
9
. 5o oo4
44
0.49 996
9-97
4
27
34
35
36
9.47
9-48
9-48
974
014
o54
40
40
9
9
9
.5oo48
. 5o 092
.5o 1 36
44
44
44
0.49 952
0.49 908
0.49 864
9.97926
9.97922
9.97918
4
4
4
26
25
24
37
38
39
9-48
9-48
9.48
094
i33
i73
40
39
40
9
9
9
.5o 180
.50 223
. 5o 267
44
43
44
0.49 820
0.49 777
0.49 733
9-97
9-97
9-97
910
906
4
4
4
23
22
21
40
9-48
2l3
9
.5o3n
0.49 689
9-97
902
20
4i
9.48
252
39
9
.5o355
44
0.49 645
9-97
898
19
42
9-48
292
9
.5o398
0.49 602
9-97
8o4
18
43
9.48
332
4°
9
. 5o 442
44
0.49558
9-97
890
'7
44
9-48
37i
39
9
.5o485
43
0.49 5i*j
9-97
886
4
16
45
9-48
4u
9
.5o 529
0.49 471
9-97
882
i5
46
9-48
45o
39
9
.5o 572
43
0.49 428
9-97
878
i4
4°
44
4
47
9.48 490
39
9
.5o6i6
0.49 384
9.97874
4
i3
48
9.48
529
9
.5o659
0.49 34i
9-97
870
12
49
9-48
568
39
9
.5o 703
44
0.49 297
9-97
866
I I
50
9.48
607
39
9
.50746
43
0.49 254
9-97
861
10
5i
9-48
647
9
.5o 789
43
0.49 211
9-97
857
9
52
9-48
686
9
5o833
O.49 167
9-97
853
8
53
9.48
725
39
9
.50876
43
O.49 124
9-97
849
7
54
9-48
764
39
9
.5o 919
43
0.49 08 I
9-97
845
4
6
55
9-48
8o3
9
. 5o 962
0.49 o38
9-97
84 1
5
56
9-48
842
39
9
. 5 1 oo5
43
0.48 995
9-97
837
4
57
9-48
881
39
9
.5i o48
43
0.48952
9-97
833
4
4
3
58
9-48
920
9
.5l 092
o.48 908
9-97
820
2
59
9-48
959
39
9
.5i 135
43
0.48865
9-97
825
I
60
g.48
998
39
9
.5i 178
43
o.48 822
9-97
821
0
L. Cos. d.
L
. Cotg. d. L. Tang.
L. Sin. d.
f
72°
PP
44
43
42
*
40
39
5 4
,
4-4
4-3
4.2
.1
4.1
4.0
3-9
.1
0.5 0.4
.2
8.8
8.6
8.4
.2
8.2
8.0
7.8
.2
i.o 0.8
•3
13.2
12.9
12.6
•3
12.3
I2.O
11.7
• 3
1-5 1-2
•4
17.6
17.2
16.8
•4
,6.4
16.0
15-6
•4
2.0 1.6
•5
22. 0
21-5
21. 0
•5
20.5
20. o
*9* 5
•5
2.5 2.0
.6
26.4
25-8
25-2
.6
24.6
24.0
23-4
.6
3.0 2.4
•7
30.8
30.1
29.4
• 7
28.7
28.0
27-3
• 7
3-5 2.8
.8
35-2
34-4
33-6
.8
32-8
32.0
31.2
.8
4.0 3.2
9 39-6
38-7 37- 8
.9
36.0
35- x
4-5 3-6
65
18C
L. Sin.
! d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.48 998
9-5i 178
40
0.48 822
9
97821
60
I
9.49 037
39
9-51 221
43
0.48 779
9
97817
59
2
9.49 076
9.61 264
o.48 736
9
97812
58
3
9.49 115
39
38
9. 5 1 3o6
43
o.48 694
9
97 808
4
57
4
9.49 i53
9.5i 349
43
o.4865i
9
97 8o4
56
5
9.49 192
9.5i 392
o.48 608
9
97 800
55
6
9.49 23i
39
9.5i4
*5
o.48 565
9
97796
4
54
38
43
4
7
9.49 269
9.51478
42
0.48 522
9
97792
53
8
9.49 3o8
9.5i 5-
20
o.4848o
9
97788
52
9
9.49 347
39
9.5i 563
43
0.48437
9
97784
4
5i
10
9.49 385
JC
9. 5 1 606
o.48 3g4
9
97779
50
1 1
9.49 424
39
9.61 648
0.48352
9
97775
49
12
9.49 462
3
9.5i 691
o.48 309
9
97 771
48
i3
9.49 5oo
38
9.5i 734
43
o.48 266
9
97 767
4
47
i4
9.49 539
39
,0
9.61 776
42
0.48 224
9
97763
4
46
i5
9.49 577
3°
9-5i 819
0.48 181
9
97 759
45
16
9.49 6 1 5
38
9-5i 861
42
o.48 139
9
97754
5
44
17
9.49654
39
-0
9-5i 903
42
0.48 097
9
9775o
4
43
id
9.49692
9. 5 1 946
o.48o54
9
97 746
42
19
9.49 73o
38
9.5i 9!
48
42
O.48 012
9
97 742
4i
20
9.49 768
38
9.52 o3i
43
0.47 969
9
97738
40
21
9.49806
38
-0
9.52 073
42
0.47 927
9
97734
4
39
22
9.49844
3°
9.52 1 15
0.47885
9
97 729
38
23
9.49882
38
9.52 157
42
0.47843
9
97725
4
37
24
9.49920
38
_0
9.52 200
43
42
O.47 8OO
9
97 721
4
36
25
9.49958
3°
9.52 242
0.47758
9
97 7*7
35
26
9.49996
38
9.52 284
42
0.47 716
9
97 7i3
4
34
27
9.5o o3^
38
9.52 326
42
42
0.47 674
9
97708
5
33
28
9.5o 072
3°
9.52 368
0.47 632
9
97 7°4
32
29
9.5o 1 10
38
9.524
10
42
0.47 590
9
97 700
4
3i
30
9.5o i48
38
9.52 452
0.47548
9
97696
4
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
71° 3D.
PP
43
42
39
38
5
4
.1
4-3
4.2
.T
3-9
3-8
.1
o-5
0.4
.2
8.6
8.4
.2
7.8
7.6
.2
I.O
o.tf
•3
12.9
12.6
• 3
11.7
11.4
•3
i-5
1.2
•4
17.2
16.8
•4
15-6
15-2
•4
2.0
1.6
•5
21.5
21.0
• 5
IQ- 5
19.0
. 5
2-5
2.0
.6
25-8
25.2
.6
23-4
22.8
.6
2-4
•7
30-1
29.4
•7
27-3
26.6
•7
3-5
2.8
.8
34-4
33-6
.8
31.2
30.4
.8
4.0
3-2
•9
38.7
37-8
35- l 34-2
•9
4-5 3-6
66
18° 3O
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9. 5o i48
9.52 452
o.
47548
9.97696
30
3i
9.5o i85
38
9.52
494
42
o.
47506
9-97
69I
29
32
9. 5o 223
,g
9.52
536
0.
47 464
9.97 687
28
33
9-5o 261
9.52
578
o.
47422
9-97
683
4
27
37
42
-
34
9«5o 298
38
9.52
620
41
o.
4738o
9-97
679
26
35
9.5o336
18
9.52
661
o.
47339
9.97674
25
36
9-5o 374
9.52
7o3
o.
47297
9-97
670
4
24
37
42
37
9.5o4n
38
9.52
745
42
0.
47255
9-97
666
23
38
9-5o 44
9
9.52
787
0.
9-97
662
22
39
9-5o486
37
9.52
829
42
o.
47 171
9-97
657
b
21
40
9.5o 523
37
9.52
870
0.
47 i3o
9-97
653
20
4i
9.5o 56i
38
9.52
912
0.47 088
9-97
649
19
42
9.5o 598
9.52
953
o.
47 047
9-97
645
18
43
9.5o635
37
9.52 995
42
0.47 005
9-97
64o
5
'7
44
9.5o 673
38
9.53 037
42
o.46 963
9.97636
4
16
45
9-5o 710
9.53078
o 46 922
9-97
632
i5
46
9-5o 747
37
9.53
I2O
42
o.
4688o
9-97
628
i4
47
9.5o 784
37
9.53
161
41
o.
46839
9-97
623
5
i3
48
9«5o 821
37
9.53
2O2
o.
46798
9.97619
1
12
49
9-5o858
37
9.53
244
42
o.
46756
9-97
6,5
4
II
50
9.50896
38
9.53
285
41
o.46 715
9-97
610
10
5i
9.5o933
37
9.53
327
42
o.
46673
9-97
606
1
9
52
9.5o 970
37
9.53
368
o.
46632
9-97
602
8
53
9-5i 007
37
9.53 409
41
0.
465gi
9-97
597
5
7
54
9. 5 1 o43
36
9.5345o
41
0.
46550
9-97
593
4
6
55
9. 5 1 080
37
9.53
492
o.
465o8
9-97
589
5
56
9.5i 117
37
9.53
533
41
o.
46 467
9-97
584
•
4
57
9.5i i54
37
9.53
574
41
o.
46426
9-97
58o
4
3
58
9-5i 191
37
9.53
6i5
o.
46385
9-97
576
2
59
9-5i 227
36
9.53
656
4i
0.
46 344
9-97
671
I
60
9. 5 1 264
37
9.53
697
4i
o.
46 3o3
9-97
567
0
L. Cos.
d.
L. Cotg.
d. j L.
Tang.
L. Sin.
d.
71°.
PP 42
4i
38
37
36
5
4
.1 4.2
4.1
3-8
•i 3-7
3-6
.!
0.5
04
8.4
8. a
7.6
.2 7.4
7-2
.2
l.O
0.8
• 3 I2-6
12.3
11.4
3 "-1
10.8
•3
i-5
1.2
.4 16.8
16.4
15-2
.4 14.8
14.4
•4
2.O
1.6
•5 21.0
20.5
19.0
•5 18.5
18.0
2-5
2.0
.6 25.2
24.6
22.8
.6 22.2
21.6
.6
3-°
2.4
•7 29.4
28.7
26.6
•7 25.9
25.2
•7
3-5
2.8
.8 33-6
32.8
30.4
.8 29.6
28.8
.8
4.0
3-2
36.9 34.2
32-4
4-5
67
19°.
,
L. Sin.
d.
L. Tang.
d.
L
. Cotg.
L. Cos. d.
0
9.5i 264
9.53 697
0
46 3o3
9-97
567
60
I
9.5i 3oi
37
9.53738
o
46 262
9-97
563
5
59
2
9.5i 338
,6
9.53779
O.46 221
9-97
558
58
3
9.5i 374
37
9. 53 820
41
0
46 180
9-97
554
4
4
57
4
9.5i 4i
I
36
9.53 861
o
46 i39
9-97
550
56
5
9. 5 1 447
9.53 902
o
46 098
9-97
545
55
6
9.5i 484
36
9.53943
41
41
o.46 057
9-97
54 1
4
5
54
7
9.5i 52O
37
9.53984
o.
46 016
9-97
536
53
8
9.5i 557
9.54 025
o
45 975
9-97
532
52
9
9.5i 593
Jj
9.54
o65
40
0.
45 935
9-97
528
4
5i
10
9.5i 629
9.54 1 06
0.
45894
9-97
523
50
ii
9.5i 666
•*6
9.54 147
41
o.
45853
9-97
5i9
4
49
12
9.5i 702
9-54
187
o.
458i3
9-97
515
48
i3
9.5i 738
30
36
9-54
228
41
0.
45 772
9-97
Sio
b
4
47
i4
9.5i 774
37
9-54
269
0.
4573i
9-97
5o6
46
i5
9.5i 81
I
9-54
309
0.
45 691
9-97
5oi
45
16
9.5i 847
36
9-54
350
41
0.
45 65o
9-97
497
4
44
'7
9.5i 883
,6
9-54
390
40
o.
456io
9-97
492
b
43
18
9.5i 91
q
9-54
43i
0.
45 569
9-97
488
42
J9
9.5i95
5
36
9-54
471
4°
0.
45 529
9-97
484
4
4i
20
9.5i99i
3°
9-54
5l2
41
0.
45488
9-97479
40
21
9.52 027
36
9-54
552
4°
o.
45448
9-97475
4
39
22
9.52 o63
9-54
593
0.
45407
9-97
47o
38
23
9.52 099
3°
9.54633
40
0.
45 367
9-97
466
4
37
36
4°
5
24
9.52 i3
5
16
9-54
673
0.
45 327
9-97
46i
36
25
9.52 171
9-54
7i4
0.
45 286
9-97
457
35
26
9.52 207
36
9-54
4°
o.
45 246
9-97
453
4
34
27
9.52 242
35
16
9-54
794
40
0.
45 206
9-97
448
5
33
28
9.52 278
9-54
83,5
0.
45 i65
9-97
444
32
29
9.52 3i4
36
9.54
875
40
o.
45 125
9-97
439
b
3i
30
9.52 350
36
9.54
915
4°
0.
45 o85
9-97
435
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
7O° 3D .
PF
> 41
40
37
36
35
5 4
.1
4-1
4.0
3-7
3-6
3-5
.1
0.5 0.4
.2
8.2
8.0
7-4
.2 7.2
7.0
.2
i.o 0.8
•3
12.3
12.0
n. i
.3 10.8
10.5
•3
1-5 !-2
•4
16.4
16.0
14.8
•4 14-4
14.0
•4
2.0 1.6
•5
20.5
20.0
18.5
.5 18.0
17-5
•5
2.5 2.0
.6
24.6
24.0
22.2
.6 21.6
21. 0
- .6
3.0 2.4
•7
28.7
28.0
25.9
•7 25.2
24-5
•7
3-5 2.8
.8
32.8
32.0
20.6
.8 28.8
28.0
.8
4-0 3-2
•9 36-9
36.0 33.3
•9 32-4
31-5
•9
4-5 3-6
68
19° 3D .
t
L. Sin.
d.
L. Tang.
d.
L
, Cotg.
L. Cos.
d.
30
9.52 350
35
36
35
36
35
36
35
36
35
36
35
35
36
35
35
35
35
35
35
9-54
9*5
40
4o
40
40
4o
40
40
40
4o
40
40
40
39
40
40
40
39
40
40
39
40
39
40
39
40
39
40
39
39
40
o.
45o85
9.97435
5
4
5
4
5
4
5
4
5
30
3i
32
33
34
35
36
37
38
39
9.52 385
9.52 421
9.52456
9.52 492
9.52 527
9.52 563
9.52 598
9.52634
9.52 669
9.54905
9.54 995
9.55 o35
9-55 075
9.55 n5
9.55 i55
9.55 i95
9. 55 235
9.55 275
0.
o.
o.
0.
o.
o.
0.
o.
o.
45 045
45 005
44 965
44925
44885
44845
44805
44765
44 725
9.97430
9.97 426
9.97421
9.97417
9.97 412
9.97 4o8
9.97 4o3
9.97 399
9.97 394
29
28
27
26
25
24
23
22
21
40
9.52 705
9.55
315
o.
44685
9-97
390
20
4i
42
43
44
45
46
47
48
49
9.52 740
9.52 775
9.52 811
9.52 846
9.52 881
9.52 916
9.52 gSi
9.52 986
9.53 021
9-55355
9.55395
9.55434
9.55474
9.555i4
9.55554
9.55593
9.55633
9.55673
0.44645
o.44 6o5
0.44566
o.44 526
0.44486
0.44446
o.44 407
o.44 367
o.44 327
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
385
38i
376
372
367
363
358
353
349
4
5
4
5
4
5
5
4
«9
1 8
17
16
i5
i4
i3
12
I I
50
9-53 o56
35
-e.
9.55
712
o.
44288
9-97
344
10
Si
52
53
54
55
56
57
58
59
9.53 092
9.53 126
9.53 161
9-53 196
9.53 23i
9-53 266
9.53 3oi
9.53 336
9.53 37o
3°
34
35
35
35
35
35
35
34
9.55 752
9-55 791
9. 5583i
9-55 870
9. 55 910
9-55 949
9.55 989
9.56 028
9.56 067
o.44 248
o.44 209
0.44 169
o.44 i3o
o.44 090
o.44 o5 1
0.44 on
o.43 972
o.43 933
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
34o
335
33i
326
322
3i7
3l2
3o8
3o3
5
4
5
4
5
5
4
5
9
8
7
6
5
4
3
2
60
9.534o5
35
9.56
107
o.43893
9-97
299
0
L.Cos. d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
/
70°.
PP 40
39
36
35
34
.1
.2
•3
•4
:I
:l
•9
5 4
.1 4.0
.2 8.0
• 3 12.0
.4 16.0
• 5 20.0
.6 24.0
.7 28.0
.8 32.0
.g 36.0
1:1
11.7
15-6
19-5
23-4
27-3
31.2
35. i
3-6
7.2
10.8
14.4
18.0
21.6
25.2
28.8
32.4
•i 3-5
.2 7.0
•3 i°-5
.4 14.0
•5 17-5
.6 21.0
•7 24.5
.8 28.0
•9 3T-5
u
10.2
13.6
I7.0
20.4
23-8
27.2
30.6
0.5 0.4
i.o 0.8
1.5 i-2 *
2.0 1.6
2-5 2.0
3-0 2.4
3-5 2.8
4-0 3-2
4-5 3-6
69
2O°.
L. Sin.
d.
L. Tang. d.
L.
Cotg.
L. Cos. d.
0
9.534o5
9.56
I07
0.
43 893
9-97
299
60
I
9.53440
35
9.56
i46
39
o.
43854
9.97 294
5
59
2
9-53475
9.56
i8b
0.
438i5
9-97
289
58
3
9.53 509
34
9.56
224
39
o.
43776
9-97
285
4
57
35
4°
4
9.53544
9.56
264
39
o.
43736
9-97
280
56
5
9.53578
9.56
3o3
o.
43697
9-97
276
55
6
9.536i3
35
9.56
342
39
o.
43658
9-97
271
b
54
7
9.53 647
34
9.56
38i
39
39
0.43 619
9-97
266
5
53
8
9.53682
9.56
420
o.
4358o
9-97
262
52
9
9.53 716
34
9.56
459
39
o.
4354i
9-97
25?
5
5i
10
9.5375i
35
9.56
498
39
0.
43 5o2
9-97
252
50
1 1
9.53 785
9.56
537
39
o.
43463
9-97
248
4
49
12
9.53819
9.56
576
o.
43424
9-97
243
'
48
i3
9.53854
35
9.56
6i5
o.
43385
9-97
238.
b
47
34
39
i4
9.53888
34
9.56
654
39
o.
43 346
9-97
234
46
i5
9-53 922
9.56
693
o.
433o7
9-97
229
'
45
16
9.53957
35
9.56
732
0.
43268
9-97
224
-
44
34
39
»7
9.53991
34
9.56771
39
o.
43 229
9-97
220
43
18
9.54 025
9.56
810
o.
43 190
9-97
2l5
42
19
9.54 059
34
9.56
849
39
,a
o.
43 i5i
9-97
210
-
4i
20
9.54093
34
9.56
887
o .
43 n3
9-97
2O6
4
40
21
9.54 127
34
9.56
926
39
o.
43o74
9-97
2OI
5
39
22
9.54 161
9.56
965
o.
43o35
9-97
196
38
23
9.54 195
34
34
9.57
oo4
18
0.
42996
9-97
192
4
5
37
24
9.54 229
9.57
042
39
o.
42958
9-97
l87
36
25
9.54263
9.57
08 1
o.
42 919
9-97
182
35
26
9.54297
34
9.57
I2O
39
o.
42 880
9-97
178
4
34
27
9.5433i
34
9.57
i58
3°
39
o.
42 842
9-97
I73
)
33
38
9.54365
9.57
197
_Q
o.
42803
9-97
1 68
32
29
9.54399
34
9.57
235
3°
o.
42765
9-97
i63
b
3i
30
9.54433
34
9.57
274
o.
42 726
9-97
i59
30
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.Sin.
d.
'
69° 3O .
PP 40
39
38
35
34
5
4
I 4.0
3-9
3-8
•i 3-5
3-4
.,
0.5
0.4
2 8.0
7.8
7.6
.2 7.0
6.8
.2
I.O
0.8
3 12.0
11.7
11.4
•3 JQ-S
IO. 2
•3
i-5
1.2
4 16.0
15-6
1-5-2
.4 14.0
I3.6
•4
2.0
1.6
5 20. o
19-5
19.0
•5 17-5
I7.0
•5
2-5
2.0
6 24.0
23-4
22.8
.6 21. 0
20-4
.6
3-o
2-4
7 28.0
27-3
26.6
•7 24.5
238
•7
3-5
2.8
8 32.0
31.2
3°-4
.8 28.0
27.2
.8
4.0
3-2
Q 36-0
34.2
•9 V-S
4-5
3-6
70
2O° 3O
'
L. Sin. d.
L. Tang. d.
L.
Cotg.
L. Cos.
d.
30
9.54433
9.57
274 38
0.
42 726
9-97
i59
30
3i
9.54466
34
9.57
3i2 M
o.
42688
9-97
1 54
5
29
32
9.54 5oo
9.57
35i
o.
42 649
9-97
i4g
28
33
9.54534
34
9.57
389
38
0.42 611
8.97
145
4
27
34
9.54567
33
34
9.57428
39
38
o.
42 572
9-97
i4o
5
26
35
9.54601
9.57466
0.
42 534
9-97
i35
25
36
9.54635
34
9.57
5o4
38
o.
42496
9-97
i3o
b
24
3?
9-54668
33
9.57 543
39
-,%
0.42457
9-97
126
4
23
38
9. 54 702
9.57
58i
o.
42 4i9
9-97
121
22
39
9.54735
33
9.57
619
38
0.
42 38 1
9-97
116
s
21
40
9-54 769
34
9.57
658
08
0.42 342
9-97
in
20
4i
9.54802
33
9.57
696
3°
og
o.
42 3o4
9-97
107
4
i9
42
9-54836
9.57
734
0.
42266
9-97
102
1 8-
43
9.54869
33
9.57
772
38
0.
42 228
9-97
°97
b
17
44
45
9.54903
9.54936
34
33
9.57 810
9-57 849
38
39
0.
o.
42 I90
42 i5i
9-97092
9-97 °87
5
5
16
i5
46
9.54969
33
9.57
887
38
o.
42 n3
9.97o83
4
i4
4?
9-55 oo3
34
9.57925
38
18
o.
42 075
9-97
078
5
i3
48
9-55o36
9.57963
o.
42o37
9-97
o73
12
49
9.55 069
33
9-58 ooi
38
o.
4i 999
9-97
068
b
I I
50
9-55 102
33
9.58 039
38
o.
4i 96i
9-97
o63
5
10
5i
9.55 i36
34
9.58077
38
18
0.
4i 923
9-97
o59
9
52
9.55 169
9.58
"J
o.4i 885
9-97
o54
8
53
9-55 202
33
9.58
i53
38
0.
41847
9.97o49
S
7
54
9-55235
33
9. 58
191
38
18
o.
4 1 809
9-97
o44
5
6
55
9.55 268
9.58
229
0.
4i 771
9-97
o39
5
56
9.55 3oi
33
9-58
267
38
o.
4i 733
9-97035
4
4
5?
9.55334
33
9-58
3o4
37
0.
4 1 696
9-97
o3o
5
3
58
9.55 367
33
9.58
342
3°
o.
4i 658
9-97
°25
2
59
9.55 4oo
33
9.58
38o
38
0.
4i 620
9-97
020
5
I
60
9.55433
33
9.584i8
S8
o.
4i 582
9-97
oi5
0
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L.Sin.
d.
'
69°.
PP 39
38
37
34
33
5 4
•i 3-9
3-8
3-7
.! 3.4
3-3
.1
0.5 0.4 .
7-8
7.6
7-4
.2 6.8
6.6
.2
i.o 0.8
•3 "-7
11.4
u. i
• 3 IO. 2
9-9
•3
1-5 1-2
•4 »5-6
15-2
14.8
•4 13-6
13.2
•4
2.0 1.6
•5 19-5
10. 0
18.5
•5 i7-°
16.5
• 5
2.5 2.O
.6 23.4
22.8
22.2
.6 20.4
19.8
6
3-0 2.4
•7 27-3
26.6
25-9
•7 23-8
23.1
7
3-5 2.8
.8 31.2
3°-4
29. 6
.8 27.2
26.4
.8
4.0 3.2
•9 35- *
34-2 33-3
• 9 30-6
29.7
•9
4-5 3-6
71
21°.
1
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.55433
33
33
33
32
33
33
33
32
33
9.584i8
37
38
38
38
37
38
37
38
38
37
38
37
38
37
37
38
37
38
37
37
37
38
37
37
37
37
38
37
37
37
o.4i 582
9.97 oi5
5
5
4
5
5
5
5
5
5
5
4
5
5
5
5
5
5
5
5
5
5
5
4
5
5
5
5
5
5
5
60
I
2
3
4
5
6
7
8
9
9.55466
9.55499
9.55532
9. 55 564
9.55597
9.55 63o
9.55663
9.55695
9.55 728
9.58 455
9.58493
9. 5853i
9.58 569
9.58 606
9.58 644
9.58 681
9 58 719
9.58757
o.4i 545
o.4i 507
o.4i 469
o.4i 43i
o.4i 394
o.4r 356
o.4i 319
o.4i 281
o.4i 243
9.97 oio
9.97 oo5
9.97 ooi
9.96996
9.96991
9.96 986
9.96 981
9.96 976
9.96971
59
58
56
55
54
53
52
5i
10
9-55 761
9-58 794
0.4 1 206
9.96 966
50
ii
12
i3
i4
i5
16
17
18
'9
9-55 793
9.55826
9.55858
9-55 891
9.55 923
9.55956
9.55988
9.56 02 1
9.56o53
33
32
33
32
33
32
33
32
32
33
32
32
33
32
32
32
32
32
33
9.58832
9.58869
9-58 907
9.58944
9.58 981
9.59 019
9.59 o56
9.59 094
9.59 i3i
o.4i 168
o.4i i3i
o.4i 093
o.4i o56
o.4i 019
o.4o 981
o.4o 944
o.4o 906
o.4o 869
9.96 962
9.96957
9.96 952
9.96947
9.96 942
9.96937
9.96 932
9.96927
9 . 96 922
49
48
47
46
45
44
43
42
4i
20
9-56o85
9.59 168
o.4o 832
9.96917
40
21
22
23
24
25
26
27
28
29
9,56 118
9.56 iScf
9-56 182
9.56 215
9.56 247
9.56 279
9. 563u
9. 56 343
9.56 375
9.59 2o5
9.59 243
9.59 280
9-59 317
9.59 354
9.59 391
9.59 429
9.59 466
9.59 5o3
o.4o 795
o.4o 757
o.4o 720
o.4o683
o.4o646
o.4o 609
o.4o 571
o.4o534
o.4o 497
9.96 912
9.96907
9.96 903
9.96 898
9.96 893
9.96 888
9.96 883
9.96 878
9.96 873
39
38
37
36
35
34
33
32
3i
30
9.56 4o8
9.59^540
o.4o 46o
9.96 868
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
68° 30'.
PP
.1
.2
•3
•4
• 7
.8
—
38
37
33
32
5
4
11.4
15-2
19 o
22.8
26.6
3°-4
34-2
3-7
7-4
n. i
14.8
18.5
22.2
25-9
29.6
. i
.2
• 3
•4
• 5
.6
'
U
9.9
13.2
16.5
19.8
23.1
26.4
3-2
6.4
9.6
12.8
16.0
19.2
22.4
25.6
.1 0.5
.2 1.0
•3 i-5
.4 2.0
•5 2.5
•7 3-5
.8 4.0
•9 4-S
0.4
0.8
1.2
1.6
2.0
2.4
2.8
72
21° 3O
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos.
d.
30
9.564o8
32
32
32
32
32
31
32
32
32
9. 59 54o
37
37
37
37
37
37
37
36
37
37
37
37
36
37
37
37
36
37
37
36
37
36
37
36
37
36
37
36
37
36
o.4o 46o
9
96 868
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
5
5
5
5
5
5
5
5
5
5
5
30
3i
32
33
34
35
36
38
39
9-56 44o
9.56 472
9.565o4
9.56536
9.56 568
9. 56 599
9. 5663i
9.56663
9-56 695
9.59 577
9.59 6i4
9.59 65i
9.59 688
9-59 725
9.59 762
9.59799
9.59835
9.59 872
0.40 423
o.4o 386
o.4o 349
o.4o 3i2
o.4o 275
o.4o238
O.4o 2OI
o.4o 165
o.4o 128
9.96 863
9.96 858
9.96853
9.96848
9.96 843
9.96 838
9.96833
9.96 828
9.96 823
29
28
27
26
25
24
23
22
21
40
9.56 727
9.59909
o.4o 091
9
96 818
20
4i
42
43
44
45
46
47
48
49
9.56 759
9-56 790
9.56822
9.56854
9.56886
9.56917
9-j>6 949
9.56 980
9.57 012
S2
32
32
32
32
9.59 946
9.59 983
9 .60 019
9-6oo56
9.60 i3o
9.60 166
9.60 2o3
9.60 240
o.4o o54
o.4o 017
0.39 981
0.39 944
0.39 907
0.39 870
0.39834
o. 39797
0.39 760
9.96 8i3
9.96 808
9.96 8o3
9.96 798
9.96793
9.96 788
9.96 783
9.96 778
9.96772
'9
18
17
16
i5
i4
i3
12
I I
50
9.57 o44
9.60 276
0.39 724
9-
96 767
10
5i
52
53
54
55
56
57
58
59
9.57075
9.57 107
9.57 i38
9.57 169
9.57 2OI
9.57 232
9,57 264
9.57 326
32
32
3'
32
3'
9.60 3 1 3
9.60 349
9.60 386
9.60 422
9.60 459
9.60 495
9.60 532
9.60 568
9.60 605
0.39 687
0.39 65i
0.39 6i4
0.39 578
0.39 54 1
0.39 505
0.39468
0.39 432
0.39 395
000 OOO 000
96 762
96 757
96 752
96747
96 742
96737
96 732
96 727
96 722
9
8
7
6
5
4
3
2
I
60
9.57358
9 .60 64 1
0.39 359
9-
96 717
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.Sin. d.
68°.
PP
.1
.2
•3
•4
•9
37
36
32
31
2
3
4
I
• 9
6
5
3-7
7-4
n. i
14.8
18.5
22.2
25-9
29.6
33-3
3-6
,o'.8
21 6
25 2
288
32.4
.1
.2
• 3
•4
:78
12.8
16.0
19.2
22.4
25.6
K
9-3
12.4
15-5
18.6
21.7
24.8
27.9
06
I 2
i 8
24
t!
5-4
0.5
I.O
2.0
2-5
3-5
4.0
4-5
/
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos. d.
0
9.57
358
9. 60 64 1
o. 3g
359
9.96
60
I
9.57389
31
9.6o 677
37
0.39
323
9.96
711
59
2
9.57 420
9.60 714
,6
0.39
286
9.96
706
58
3
9.5745i
31
9.60 750
36
o.Sg
25o
9-96
7oi
5
S
57
4
9.57 482
32
9.60 786
37
0.39
214
9.96
696
56
5
9-57
5i4
9-
60823
,5
o. 39
177
9.96
691
55
6
9 57
545
31
31
9.60 85g
36
o. 39
i4i
9.96686
5
5
54
7
9.57
576
9-
60 895
36
o. 3g
105
9.96
681
53
8
9 . 57 607
9-
60931
16
0.39
069
9.96
676
52
9
9.57638
31
9-
60 967
3°
0.39
o33
9.96
670
5i
10
9.57 669
3r
9-
6 1 oo4
36
o.38
996
9.96
665
50
1 1
9-57
7oo
31
9-
6 1 o4o
36
o.38
960
9.96
660
49
12
9-57
73i
9-
61 076
o.38
924
9.96
655
48
i3
9.57
762
31
9-
61112
36
o,38
888
9.96
650
5
5
47
i4
9.57
793
31
9-
61 1 48
36
o.38
852
9.96
645
46
i5
9.57
824
9-
61 1 84
,6
o.38
816
9.96
64o
45
16
9 57
855
31
3°
9-
6l 220
36
o.38
780
9.96
634
S
44
17
9. 57 885
31
9-
6r 256
36
o.38
744
9.96
629
43
18
9.57
916
9-
61 292
16
o.38
708
9.96
624
42
r9
9.57947
9-
61 328
06
o.38
672
9.96
619
4i
20
9.57
978
9-
61 364
16
0.38
636
9.96
6i4
40
21
9.58 008
3°
9-
6 1 4oo
36
o.38
600
9.96 608
39
22
9.58
o39
9-
61 436
o.38
564
9.96
6o3
38
23
9.58
070
31
9.61 472
3°
36
o.38
528
9.96
598
b
5
37
24
9.58
IOI
9.61 5o8
16
o.38
492
9.96
593
36
25
9.58
i3i
9.61 544
o.38
456
9.96
588
35
26
9.58
162
31
9,61 579
35
o.38
421
9.96
582
34
27
9.58
192
30
9.61 6i5
36
o.38
385
9.96
577
5
33
28
9.58
223
9.61 65i
o.38
349
9.96
572
32
29
9.58
253
30
9.61 687
36
o.38
3i3
9,96
567
b
3r
30
9.58 284
31
9.61 722
35
o.38
278
9.96
562
5
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
L. Sin.
d.
67° 3O .
PP
37
36 35
32
31
3°
6 5
.1
3-7
3-6 3 5
,
3-2
I'1
3°
I
o. 6 o. 5
7-4
7-2 70
.2
6.4
6.2
6.0
2
12 1.0
•3
u. i
10 8 10,5
•3
9.6
9-3
9.0
3
1.8 1.5
•4
14.8
14-4 14-0
4
12.8
12.4
12. 0
4
2 4 2.0
:i
18.5
22.2
18.0 17.5
21.6 21.0
• 5
6
16.0
19.2
III
15.0
18.0
.1
3.0 2.5
36 3-0
• 7
25-9
25.2 24 5
,7
22 4
21 7
21. O
• 7
42 3-5
.8
29.6
28.8 280
,8
25-6
248
24.0
.8
4.8 4.0
9 33 3
32.4 31 5
•9
28.8 27.9
9
5-4 4-5
74
22° 3O
,
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9-
58 284
9.61 722
o.38 278
9.96 562
6
30
3i
9.583i4
3°
9.61 758
3°
76
o.38 242
9. 96 556
s
29
32
9
58 345
9.61 '
r94
o.38 206
9.96 55i
28
33
9
58 375
3°
9.61 83o
36
o.38 170
9.96 546
27
34
9
58 4o6
3'
9.61 865
35
_<
o.38 135
9.96 54 1
5
6
26
35
9
58436
9 61 901
3
o. 38 099
9.96535
25
36
9
58467
3'
9.61 936
35
o.38 o64
9>96 53o
24
37
9
58497
30
9,61 972
36
3*
o.38 028
9.96525
23
38
9
58 527
9.62 008
0.37992
9.96 52O
6
22
39
9
58557
3°
9.62 o43
35
0.37 957
9.96 5i4
5
21
40
9
58588
31
9.62 079
3°
0.37 921
9.96 5o9
5
20
4i
9
.586i8
3°
9.62 u4
35
06
o.37 886
9.96 5o4
6
19
42
9
58648
9,62
50
0.37 85o
9.96 498
18
43
9
.58 678
3°
9.62 i85
35
0.37815
9.96 493
17
44
9
.58 709
3'
9.62 221
36
0.37 779
9.96488
5
5
16
45
9
.58739
30
9.62 256
0.37 744
9.96483
6
i5
46
9
.58 769
30
9.62 292
36
0.37 708
9.96477
i4
47
9
.58799
30
9.62 327
35
o.37673
9.96472
i3
48
9
.58 829
9.62 362
o.37 638
9.96 467
fi
12
49
9
.58859
3°
9.62 398
36
0.37 602
9.96 46 1
5
I I
50
9
.58 889
3°
9.62 433
o.37567
9.96 456
10
5i
9
. 58 9i9
3°
9.62 468
35
16
0.37 532
9.96
45i
6
9
52
9
.58949
3°
9. 62 5o4
0.37 496
9.96 445
8
53
9
.58979
30
9.62
539
35
0.37461
9.96 44o
7
54
g
. 59 oo9
30
9.62
574
35
0.37426
9.96
435
6
6
55
g
.59 o3^
I
3°
9.62 609
0.37 391
9.96
429
5
56
9
.59 o69
3°
9.62
645
36
o.37 355
9-96
424
4
57
58
9
9
.59 o98
.5o 128
29
30
9.62
9.62
680
35
35
0.37 320
o.37285
9.96 4i9
9.96 4i3
6.
3
2
59
9.59 i 58
3°
9.62
75o
35
0.37 250
9.96 4o8
5
I
60
9
.59 188
3°
9.62
785
35
o.
37215
9.96 4o3
0
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
67°.
PP
36
35
31
30
29
6
5
.2
3.6
7.2
3-5
7.0
K
.1
2
3-°
6.0
!;§
.1
.2
0.6
1.2
o-S
I.O
•3
10.8
10.5
9-3
•3
9.0
8.7
•3
1.8
15
•4
14.4
14.0
12.4
4
12. 0
it 6
•4
2.4
2.0
18.0
21.6
21 0
SI
1
15-0
18.0
M-5
17.4
:1
3-°
3-6
25
.7
25.2
24-5
21.7
•7
21.0
20.3
•7
4-2
3-5
.8
28.8
28.0
24.8
.8
g
24.0
27.0
£
.8
•9
4.8
40
23°.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
0
9.69 i»8
3°
29
3°
3°
29
30
3°
29
30
29
3<>
29
3°
29
30
29
29
3°
29
29
30
29
29
29
29
3°
29
29
29
29
9.62
785
35
35
35
36
35
35
35
35
35
34
35
35
35
35
35
35
34
35
35
35
35
34
35
35
34
35
34
35
35
34
0.
37215
9.96 4o3
6
5
5
6
5
6
5
5
6
5
6
5
5
6
5
6
5
6
5
6
5
5
6
5
6
5
6
5
6
5
60
I
2
3
4
5
6
7
8
9
9.69 218
9.69 247
9.69277
9.59 807
9.59 336
9.59 366
9.59 396
9.59 425
9.59455
9.62 820
9.62 855
9.62 890
9.62 926
9.62 961
9.62 996
9.63 o3i
9.63 066
9-63 101
o.
o.
0.
0.
o.
0.
o.
o.
0.
37 180
37 145.
37 1 10
37074
37o39
37 oo4
36 969
36934
36899
9.96
9.96
9.96
9.96
9.96
9.96
9.96
9.96
9.96
397
392
387
38i
376
37o
365
36o
354
59
58
57
56
55
54
53
52
5i
10
9.59 484
9.63
i35
0.
36865
9.96
349
50
1 1
12
i3
i4
i5
16
'7
18
'9
9.59 5i4
9.59543
9.59573
9.59 602
9.59 632
9.59661
9.59 690
9.59 720
9.59749
9-63 170
9.63 2o5
9.63 24o
9.63 275
9«63 3io
9. 63 345
9.63379
9.634i4
9.63 449
0.
o.
o.
o.
o.
0.
o.
o.
0.
3683o
36795
36 760
36725
36 69o
36655
36621
36586
36 55i
9.96
9.96
9.96
9.96
9.96
9.96
9.96
9.96
9.96
343
338
333
327
322
3i6
3ii
3o5
3oo
49
48
47
46
45
44
43
42
4i
20
9.59778
9.63
484
0.
365i6
9.96
294
40
21
22
23
24
25
26
27
28
29
9.59808
9.59837
9.59866
9.59895
9.59924
9.59 954
9.59 983
9.60 012
9.60 o4i
9.63
9.63
9.63
9.63
9.63
9.63
9.63
9.63
9.63
5i9
553
588
623
657
692
726-
761
796
o.
o.
o.
o.
o.
o.
o.
•o.
0.
3648i
36447
364i2
36377
36 343
363o8
36274
36239
362o4
9.96 289
9.96 284
9.96 278
9.96 273
9. 96 267
9.96 262
9.96 256
9.96 25i
9.96 245
39
38
37
36
35
34
33
32
3i
30
9. 60 070
9.63
83o
0.
36 170
9.96
240
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin. d.
'
66° 30 .
PP 36
35
34
3°
29
.1
.2
•3
•4
!
:l
—
6 5
•i 3-6
.2 7.2
.3 10.8
•4 M-4
.5 18.0
.6 21.6
•7 25-2
.8 28.8
.q ^2.4
3-5
7.0
10.5
14.0
I7-S
2I.O
24-5
28.0
31.5
3-4
6.8
IO.2
13-6
17.0
20.4
23.8
27.2
'
3.0
2 6.0
3 9-°
.4 12.0
• s 15-0
.6 18.0
.7 21.0
,8 24.0
•9 27-°
2.9
5-8
8.7
n.6
M-5
17.4
20.3
a:
0.6 0.5
1.2 1.0
1.8 1.5
2.4 2.O
3.0 2-5
3.6 3.0
4-2 3-5
4.8 4.0
5- 4 4-5
76
23° 3O
'
L. Sin.
d.
L. Tang. d.
L. Cotg.
L.
Cos.
d.
30
9
.60 070
9. 6383o
35
34
35
34
35
34
35
34
34
35
34
34
35
34
34
35
34
34
34
34
35
34
34
34
34
34
34
34
34
34
o.36 170
9.96 240
30
3i
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
.60 o99
.60 128
.60 157
.60 186
.60 215
.60 244
.60273
.60 3o2
.6o33i
29
29
29
29
29
29
29
29
9. 63 865
9.63899
9.63 934
9. 63 968
9.64oo3
9.64 037
9.64 072
9.64 1 06
9.64 i4o
o.36i35
o.36 101
o.36 066
o.36o32
o.35997
o.35 963
o.35 928
o.35894
o.35 860
9.96 234
9.96 229
9.96 223
9.96 2 1 8
9.96 212
9.96 207
9.96 2O1
9.96 196
9.96 190
5
6
5
6
5
6
5
6
5
29
28
27
26
25
24
23
22
21
40
9.6o359
9. 64 175
0.35825
9-96 185
20
4i
42
43
44
45
46
47
48
49
9
9
9
9
9
9
9
9
9
.60 388
.60 417
.60 446
.60474
.6o5o3
.60 532
,6o56i
.6o589
60618
29
29
29
28
29
29
29
28
29
9.64 209
9.64243
9.64 278
9.64 3i2
9.64346
9.6438i
9.64415
9-64449
9.64483
o.35 791
0.35757
o.35 722
0.35688
0.35654
o.35 619
0.35585
o.3555i
o.355i7
9.96179
9.96 174
9.96 168
9.96 162
9.96 i57
9.96i5i
9.96 1 46
9.96 i4o
9.96135
5
6
6
5
6
5
6
5
g
19
17
16
i5
i4
i3
12
I I
50
9
60 646
29
29
28
29
28
29
. 28
. 29
28
9.64 5 1 7
0.35483
9.96 i29
10
5r
52
54
55
56
57'
58
59
9
9
9
9
9
9
9
9
9
6o675
60 704
.60 732
60 761
60 789
60818
.6o846
.60875
9.64 552
9.64586
9.64 620
9.64654
9.64688
9.64 722
9.64756
9.64 790
9.64 824
0.35448
o.354i4
o.3538o
0.35346
o.35 3i2
o.35 278
0.35244
o.35 210
o.35 176
9.96 123
9.96 118
9.96 112
9.96 107
9.96 101
9.96 095
9.96 090
9.96 o84
9.96079
5
6
5
6
6
5
6
5
1
7
6
5
4
3
2
I
60
9
.60 93i
28
9. 64 858
o.35 142
9.96073
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
66°.
PP
.2
•3
•4
•J
>9
35
34
n
28
.2
3
•4
•9
6
5
3-5
7.0
10.5
14.0
J7 5
21.0
24-5
28.0
31.5
U
IO. 2
13-6
17.0
20.4
23-8
27.2
3O.O
,1
.2
3
•4
•Jt
•9
tl
n.6
17.4
20.3
23.2
26.1
2.8
5-6
8.4
11.2
14.0
16.8
19.6
22.4
0.6
1.2
1.8
2.4
tJ
5-4
0-5
I.O
2.O
2-5
3-5
4.0
77
24C
'
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
0
9.60 931
9.64
858
0.
35 142
9.96
o73
60
I
9.60 960
28
9-64
892
34
o.
35 108
9.96
067
59
2
9.60 988
28
9-64
926
o.
35o74
9.96
062
58
3
9.61 016
9-64
960
34
o.
35 o4o
9.96
o56
57
29
34
6
4
9.61 045
28
9-64
994
0.
35 006
9.96
o5o
56
5
9.61 o73
_0
9. 65
028
o.
34972
9.96
045
55
6
9.61 10
i
9.65
062
34
o.
34938
9.96
o39
54
7
9.61 129
29
9. 65
096
34
o.
34 904
9.96
o34
5
6
53
8
9.61 i58
9.65
i3o
o.
3487o
9.96
028
52
9
9.61 186
28
9.65
1 64
34
0.
34836
9.96
022
5i
10
9.61 214
9. 65
197
33
o.
348o3
9.96
OI7
50
ii
9.61 242
28
9.65
23l
34
0.
34769
9.96
Ol I
6
49
12
9.61 270
9.65
265
o.
34 73<)
9.96
oo5
48
i3
9.61 298
28
9.65
299
34
o.
34 701
9.96
ooo
5
47
i4
9.61 326
28
28
9.65
333
34
0.
34667
9.95
994
fi
46
i5
9.61 354
9. 65
366
33
o.
34634
9-95
988
45
1 6
9.61 382
28
9. 65
4oo
34
o.
34 600
9.95
982
44
I?
9.61 4i
i
29
9. 65
434
34
0.
34 566
9.95
977
6
43
18
9.61 438
9. 65
467
33
o.
34533
9.95
971
42
'9
9.61 466
9.65
5oi
34
o.
34499
9.95
965
4i
20
9.61 494
9.65
535
34
o.
34465
9.95
960
5
g
40
21
9.61 522
28
9.65
568
33
0.
34432
9.96
954
6
39
22
9.61 55o
9-65
602
34
o.
34398
948
38
23
9.61 578
28
9. 65
636
34
o.
34364
9.95
942
37
24
9.61 606
28
28
9.65
669
33
o.
3433i
9.95
937
b
6
36
25
9.61 634
9.65
7o3
34
o.
34 297
9.95
93i
5
35
26
9.61 662
28
9.65
736
33
o.
34 264
9.95
925
34
27
9.61 689
27
28
9.65
770
34
o.
3423o
9.95
920
b
6
33
28
9.61 717
9-65
8o3
33
0.
34 197
9.95
914
g
32
29
9.61 745
28
9.65
837
34
o.
34 i63
9.95
908
6
3i
30
9.61 773
9.65
87o
33
o .
34 i3o
).95
902
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin. |d.
65° 30 .
PP 34
33
29
28
27
6 5
•i 3-4
3-3
2.9
.1 2.8
2.7
.1
0.6 0.5
.2 6.8
66
5-8
2 5-6
54
.2
1.2 1.0
•3 10.2
9-9
8.7
•3 8-4
8.1
•3
1.8 1.5
•4 13-6
13.2
11.6
.4 II. 2
10.8
•4
2.4 2.0
.5 17.0
16.5 '
14-5
.5 14.0
J3-5
•5
3.0 2.5
.6 20.4
19.8
17.4
.6 16.8
16.2
.6
3-6- 3-o
•7 238
23.1
20.3
.7 19.6
18.9
7
42 3-5
.8 27.2
26.4
23.2
.8 22.4
21.6
.8
4.8 4.0
.9 30.6
29.7 26.1
• 9 25.2
24-3
5-4 4-5
78
24° 3D'
>
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
61 773
27
9.65 870
34
o. 34 i3o
9.95 902
30
3i
9
61 800
28
9.65 90^
33
o.34 o96
9.95 897
6
29
32
9
61 828
28
9.65 937
o.34 o63
9. 95 891
28
33
9
61 856
27
9.65 97i
33
O.34 O29
9.95885
6
27
34
9.6i 883
28
9.66 oo4
34
0.33 996
9.95 879
26
35
9
61 9n
28
9.66 o38
o.33 962
9.95873
25
36
9
61 939
27
9.66 071
33
o.33 929
9.95 868
6
24
37
9
61 966
28
9.66 10^
34
o.33 896
9.95 862
6
23
38
9
61 994
9. 66 1 38
o.33 862
9.95 856
22
39
9
62 021
27
28
9.66 171
33
o.33 829
9.95 85o
5
21
40
9
62 o49
9.66 204
o.33 796
9.95 844
20
4i
9
62 076
28
9. 66 238
34
o.33 762
9.95 839
6
,9
42
9
62 io4
9.66 271
o.33 729
9.95 833
18
43
9
62 i3i
28
9.66 3o4
33
33
o.33 696
9.95 827
6
17
44
9
62 i59
27
9.66 337
0.33663
9.95 821
g
16
45
9
62 186
28
9.66 371
o.33 629
9.95 8i5
i5
46
9
62 214
9.66 4o4
33
o.33 596
9.95 810
5
i4
27
33
47
9
62 241
27
9. 66 437
o.33 563
9.95 8o4
f,
i3
48
9
62 268
9.66 470
33
o.33 53o
9.95 798
12
49
9
62 296
9.665o3
33
0.33497
9.95792
6
II
50
9
62 323
9.66 537
34
0.33463
9.95 786
6
10
5i
9
62 35o
27
9.66 570
33
33
o.3343o
9.95 780
s
9
52
9
62377
9.66 6o3
o.33 397
9-95 775
g
8
53
9
62 405
9. 66 636
33
o.33 364
9.95 769
7
27
33
54
9
62432
9.66 669
o.33 33i
9.95763
6
6
55
9
62 459
9.66 702
o.33 298
9.95 757
6
5
56
9
62486
27
9.66 735
33
o.33 265
9.95 75i
4
57
9
62 5i3
27
28
9.66 768
33
o.33 232
9.95 745
6
3
58
9
62 54 1
9.66 801
o.33 199
9-9
5 739
5
2
59
9
62 568
27
9. 66 834
33
o.33 166
9.95733
I
60
9
62 595
27
9.66 867
33
o.33 1 33
9.95 728
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
65°.
PP
34
33
28
27
6 5
,
3-4
3-3 -i
2.8
2-7
.,
0.6 0.5
.2
6.8
6.6 .2
5.6
5-4
.2
1-2 I.O
•3
IO. 2
9-9 3
8.4
•3
1.8 1.5
-4
I3.6
13-2 -4
II 2
0.8
•4
2-4 2.0
-5
17.0
16.5 5
14.0
3-5
. tj
3.0 2.5
.6
20-4
19.8 .6
16.8
6.2
6
3-6 3-°
• 7
23-8
23-1 -7
19,6
8.9
.7
4-2 3-5
.8
27.2
26.4 8
22.4
1.6
8
4.8 4.0
25.2 4.3
•9
5-4 4-5
79
25°.
/
L. Sin.
d.
L. Tang.
d.
L.Cotg.
L.
Cos. d.
0
9
.62695
9.66 867
o.33 i33
9.96 728
g
60
I
9
.62 622
27
9.66 900
33
o.33 100
9.96 722
6
59
2
9
.62 649
9.66933
0.33067
9.96 716
58
3
9
.62676
27
9.66 966
33
o.33 o34
9.96 710
6
57
4
9
.62 703
27
9.66999
33
o.33 ooi
9.96 704
6
56
5
9
.62 730
9.67 o32
0.32 968
9.96 698
55
6
9
.62767
9.67 065
o.32 935
9.96 692
54
27
33
6
7
9
.62 784
9.67 098
33
o.32 902
9.96 686
6
53
8
9
.62 811
9.67i3
[
0.32 869
9.96 680
52
9
9
.62838
27
9.67 i63
32
o.32837
9.96 674
5i
10
9
.62 865
9.67 196
33
o.32 8o4
9.96 668
50
1 1
9
.62 892
26
9.67 229
33
o.32 771
9.96 663
6
49
12
9
.62 918
9.67 262
o.32 738
9.96 667
48
i3
9
.62 945
27
9.67295
32
o.32 706
9.96 65i
6
47
i4
9
.62 972
27
9.67327
33
o.32 673
9-95645
6
46
16
9
.62 999
9.67 36o
o.32 64o
9.96 639
45
16
9
.63 026
27
26
9.6739
3
33
o.32 607
9.96 633
6
44
17
9
.63 062
27
9.67 42<
5
32
o.32 674
9.96 627
fi
43
18
9
.63 079
9.67458
0.32 542
9.96 621
42
'9
9
.63 106
27
9.67 491
33
o.32 609
9.96615
4i
20
9
.63 i33
„/:
9.67 624
o.32 476
9.96 609
40
21
9
.63 169
27
9.67 556
33
o.32 444
9.96 6o3
6
39
22
9
.63 186
9.67 689
o.32 4
[ I
9.96 697
38
23
9
.63 2i3
27
26
9.67 622
33
32
o.32 378
9.96691
6
37
24
9
.63 239
9.67 654
33
o.32 346
9.96685
36
26
9
.63266
9.67 687
o.32 3i3
9.96 679
35
26
9
.63 292
9.67 719
32
0.32 28l
9.96673
34
27
33
6
27
9
.633i9
26
9.67 762
0.32 248
9.96667
6
33
28
9
.63345
9.67785
0.32 2l5
9.96 56i
32
29
9
.63 372
27
9.67 817
32
o.32 i83
9.95555
3i
30
9
.63 398
9.67 850
33
o.32 i5o
9.96 549
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
64° 30 .
PP
33
32
37
26
6
5
i
3-3
3-2
. .1
2-7
2.6
, I
0.6
o-5
2
6.6
6.4
.2
5-4
5-2
.2
1.2
I.O
3
9-9
9.6
•3
8.1
7.8
•3
1.8
i-5
4
13.2
12.8
•4
10.8
10.4
•4
2-4
2.O
5
16.5
16.0
• 5
13-5
13.0
• 5
3-°
2-5
6
19.8
19.2
6
16.2
15.6
.6
3-6
3-°
7
23-1
22.4
.7
18.9
18.2
•7
4.2
3-5
8
26.4
25-6
.8
21.6
20.8
.8
4.8
4.0
Q
24.3 23.4
S-4 4-5
80
25° 3D
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
63 398
9.67 850
o.32 i5o
9.95 549
30
3i
9
63425
26
9.67 882
32
o.32 118
9.95543
29
32
9
6345i
9.67 915
33
o.32 o85
9.95537
28
33
9
63478
27
9.67947
32
o.32 o53
9.9553i
6
27
34
35
9
9
63 53i
26
27
9.67-980
9.68 012
33
32
O.32 O2O
o.3i 988
9.95525
9.95 5i9
6
6
26
25
36
9
63 557
9.68044
32
o.3i 956
3 5i3
6
24
3?
9
63583
26
9.68 077
33
o.3i 923
9.95 507
6
23
38
9
636io
9.68 109
o.3i 891
9.95 5oo
22
39
9
63636
26
9.68 142
33
o.3i 858
9.95 494
6
21
40
9
63 662
9.68 174
32
o.3i 826
9.95488
20
4i
9
63 689
26
9.68 206
33
o.3 1 794
9.95 482
6
I9
42
9
63 715
26
9.68 239
o.3i 761
9.95 476
18
43
9
6374i
9.68 271
o.3i 729
9.95470
17
26
32
6
44
9
63 767
27
9.68 3o3
o.3i 697
9.95464
6
16
45
9
63794
9.68 336
o.3i 664
9.95458
i5
46
9
63820
26
9.68 368
32
o.3i 632
9.95 452
6
i4
47
9
63846
26
9.68 4oo
o.3i 600
9.95 446
6
i3
48
9
63872
9.68 432
32
o.3i 568
9.95 44o
12
49
9
63898
9.68 465
33
o.3i 535
9.95434
II
50
9
63 924
26
9.68 497
32
o.3i 5o3
9.95427
10
5i
9
6395o
26
9.68 529
o.3i 471
9.95 421
6
9
52
9. 63 976
26
9. 6856i
o.3i 439
9.95 4i5
8
53
9.64 002
26
9.68593
32
33
o.3i 407
9.95 4oQ
6
7
54
9
64 028
26
9.68 626
o.3i 374
9.95 4o3
6
6
55
9
64o54
9.68 658
o.3i 342
9.95 397
5
56
9
.64 080
9.68 690
32
o.3i 3io
9.95 39i
4
5?
9
.64 1 06
26
26
9.68 722
32
o.3i 278
9.95 384
7
6
3
58
9
.64 1 32
9.68 754
o.3i 246
9.95378
2
59
9
.64 i58
9.68 786
32
o.3i 214
9.95372
g
I
60
9
.64 1 84
9.68 818
32
o.3i 182
9.95 366
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
64°.
PP
33
32
27
26
7
6
.1
3-3
3-2
.,
2.7
2.6 .1
0.7
0.6
.2
6.6
6.4
.2
5-4
5-2 -2
M
1.2
•3
9.9
9.6
•3
8.1
7-8 -3
2.1
1.8
•4
13.2
12.8
•4
10.8
10.4 .4
2.8
2.4
.5
16.5
16.0
!3-5
i3-° 5
3-5
3.0
.6
19.8
19.2
.6
16.2
15.6 .6
4.2
3-6
.7
23.1
22.4
•7
18.9
18.2 .7
4-9
4.2
.8
26.4
25.6
.8
21.6
20.8 .8
5-6
4.8
•9
29.7 28.8
•9
24-3 23-4 -9
6.3 5-4
81
26°.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
0
9.64 1 84
26
9.68 818
o.3i 182
9.95 366
60
I
9.64 210
26
9.68 85o
32
o.3i 150
9.95 36o
6
59
2
9. 64 236
26
9.68 882
o.3i 118
9.95354
58
3
9.64 262
26
9.68 914
32
32
o.3i 086
9.95 348
7
57
4
9.64 288
25
9.68 946
o.3i o54
9.95 34i
6
56
5
9.64 3i3
26
9.68 978
o.3i 022
9.95335
55
6
9.64 339
26
9.69 oio
32
32
o.3o 990
9.95 329
6
54
7
9
.64365
26
9.69 042
0.30958
9.95 323
6
53
8
9
.6439i
26
9.69074
o.3o 926
9.95 3i7
52
9
9
.64417
9.69 106
32
o.3o 894
9.95 3io
7
5i
10
9
.64442
26
9.69 1 38
32
o. 3o862
9.95 3o4
50
ii
9
.64468
26
9.69 170
32
o.3o 83o
9.95 298
6
49
12
9
.64494
9.69 202
o.3o 798
9.95 292
48
i3
9
.645i9
25
9.69 234
32
o.3o 766
9.95 286
47
i4
9
.64545
26
26
9.69 266
32
o.3o 734
9.95 279
7
6
46
i5
9
.6457i
9.69 298
o.3o 702
9.95 273
45
16
9
.64 596
25
9.69 329
31
o.3o 671
9.95 267
44
17
9
.64622
26
9.69 36i
32
o.3o 6
39
9.95 261
b
43
18
9
.64647
9.6939
3
32
o.3o 607
9.95 254
42
'9
9
.64673
26
9.69425
32
o.3o 575
9.95 248
6
4i
20
9
.64 698
25
9.69 457
32
o.3o 543
9.95 242
40
21
9
.64 724
9.69 488
31
o.3o 5i2
9.95 236
39
22
9
.64 749
9.69 52O
o.3o48o
9.95 229
38
23
9
•64775
9.69 552
32
o.3o448
9.95 223
37
25
32
h
24
9
.64 800
26
9.69 584
o.3o4i6
9.95 2I7
36
25
9
.64826
9.69 6i5
o.3o385
9.95 211
35
26
9
.6485i
25
26
9.69 647
32
o.3o 353
9.95 2O4
7
f,
34
27
9
.64877
9.69679
o.3o 32i
9.95 198
6
33
28
9
.64 902
9.69 710
o.3o 290
9.95 192
32
29
9
.64 927
25
9.69 742
32
o.3o 258
9.95 i85
7
3i
30
9
.64953
9.69 774
32
o.3o 226
9.95 179
30
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
d.
'
63° 3D'.
PP
32
31
26
25
7
6
i
3-2
i'1
!
2.6
2-5
.!
0.7
0.6
2
6.4
6.2
2
5-2
5-°
2
1.4
1.2
3
9.6
9-3
3
7.8
7-5
3
2.1
1.8
4
12.8
12.4
4
10.4
10. 0
4
2.8
2.4
5
16.0
5
13-0
12.5
3.5
3-°
6
19.2
18.6
.6
15-6
15.0
6
4.2
3-6
7
22.4
21.7
•7
18.2
17-5
•7
4-9
4.2
8
'5-6
24.8
.8
20.8
20.0
.8
5-6
4.8
9
23.4 22.5
6.3 5-4
82
26° 3O .
L. Sin. d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9.64 953
9.69
774
o.
3o 226
9.95
179
g
30
3i
9.64 978
9.65 oo3
25
9.69
9.69
8o5
837
31
32
o.
0.
3o 195
3o i63
9.95
9.95
178
167
6
29
28
33
9-65 029
9.69
868
3i
o.
3o 1 32
9.95
1 60
7
27
34
9.65o54
25
9.69
32
9°° «
o.
3o 100
9.95
1 54
f>
26
35
9.65 079
9.69
932
0.
3o 068
9.95
i48
25
36
9-65 io4
25
26
9.69
963
31
32
o.
3oo37
9.95
i4i
6
24
37
9.65 i3o
25
9.699^5
0.
3o oo5
9.95
1 35
6
23
38
9.65i55
9.70 026
o.
29974
9.95
I29
22
39
9.65 i8o»
25
9.70 o58
32
0.
29 942
9.95
122
7
g
2*
40
9-65 2o5
25
9.70 089
31
o.
29 91 1
9.95
116
g
20
4i
42
9.65 23o
9. 65 255
25
25
9.70
9.70
121
i5a
32
31
o.
o.
29 879
29 848
9.95
9.95
I IO
io3
7
19
18
43
9-65 281
9.70
1 84
32
o.
29 816
9.95
°97
17
44
9.653o6
25
9.70
2l5
o.
29785
9.95
o9o
7
6
16
45
9.65 33i
9.70247
o.
29 753
9.95
084
i5
46
9-65 356
25
9.70
278
3*
0.
29 722
9.95
078
i4
47
9.65 38i
25
9.70
3o9
0.
29691
9.95
071
6
i3
48
9. 65 4o6
9.70
34i
o.
29 659
9.95
065
12
49
9.6543i
25
9.70
372
31
0.
29 628
9.95
o59
I I
50
9. 65 456
9.70 4o4
0.
29 596
9.95
052
10
5i
9.6548i
25
9.70
435
31
0.
29 565
9.95
o46
7
9
52
9.65 5o6
9. 70 466
o.
29 534
9.95
o39
g
8
53
9. 6553i
25
9.70498
32
o.
29 5o2
9.95
o33
7
54
9.65 556
25
9.70
529
o.
29471
9.95
027
7
6
55
9.65 58o
9.70
56o
o.
29440
9.95
020
g
5
56
9.65 6o5
25
9.70
592
32
0.
29 4o8
9.95
oi4
4
57
9.65 63o
25
9.70 623
o.
29 377
9.95
007
7
6
3
58
9.65 655
9.70 654
0.
29 346
9.95
OOI
2
59
9.65 680
25
9.70 685
31
0.
29315
9.94
995
I
60
9.65 705
25
9.70 717
32
o.
29 283
9.94988
0
L. Cos.
d.
L. Cotg. d.
L.
Tang.
L. Sin.
d.
f
63°.
PP
32
31
26
25
24
7 6
x
3.2
3-1
2.6
.1 2.5
2.4
.1
o. 7 o. 6
. 2
6.4
6.2
5-2
2 5-0
4.8
.2
1.4 1.2
•3
9.6
9-3
7.8
•3 7-5
7-2
•3
2.1 1.8
•4
12.8
12.4
10.4
.4 10.0
9.6
•4
2.8 2.4
.5
16.0
13.0
r 12. 5
12.0
5
3-5 3-o
.6
19.2
18.6
15.6
.6 15.0
14.4
.6
4.2 3.6
-?
22.4
21.7
18.2
•7 17-5
16.8
•7
4-9 4-2
.8
25-6
24.8
20.8
8 20.0
19.2
.8
5.6 4.8
•9
27.9 23.4
6-3 5-4
83
27°.
/
L. Sin.
d.
L. Tang.
d.
L
. Cotg.
L. Cos.
d.
0
9.66 705
9.70717
o
29283
9.94
988
60
I
9.65 729
25
9.70 748
31
o
29 252
9.94
982
59
2
9.65754
9.70779
o
29 221
9.94
Q75
58
3
9.66779
9.70 810
31
o
29 190
9.94
969
57
25
31
7
4
9.65 8o4
24
9.70 84i
32
o
29 1 59
9.94
962
6
56
5
9.66828
9.70873
o
29 127
9.94
g56
55
6
9.65853
25
9.70904
31
o
29 096
9.94949
7
54
25
31
6
7
9.60878
24
9.70935
31
o
29065
9.94943
53
8
9.65 902
9.70 966
o
29 o34
9.94 936
52
9
9.65 927
9.70997
31
o
29 oo3
9.94930
5i
10
9.65 962
24
9.71
028
31
o
28 972
9-94
92 3
g
50
1 1
9.65 976
25
9.71
o5g
31
o
28941
9.94917
6
49
12
9.66 ooi
9.71
090
o
28 910
9.94 911
48
i3
9.66026
25
9.71
121
32
o.
28 879
9.94904
7
6
47
i4
9.66 o5o
25
9.71
i53
31
o.
28847
9.94
898
46
i5
9.66075
9.71
1 84
o,
28816
9.94
891
45
16
9.66 099
9.71
215
o.
28 785
9.94
885
44
25
3»
7
J7
9.66 124
24
9.71
246
31
o.
28754
9-94
878
43
18
9.66 i48
9.71
277
o.
28 723
9-94
871
42
'9
9.66 173
9.71
3o8
o.
28 692
9.94
865
4i
20
9.66 197
9.71
339
o.
28661
9.94 858
40
21
9.66 221
25
9.71
37o
31
o.
2863o
9.94852
39
22
9.66 246
9.71
4oi
0.
28 599
9.94
845
38
23
9.66 270
9.71
43i
0.
28569
9.94
839
37
25
31
7
24
9.66 295
24
9.71
462
31
o.
28538
9-94
832
6
36
25
9.66 319
9.71
49^
o.
28 507
9.94
826
35
26
9.66343
24
9.71
524
o.
28476
9.94
819
7
34
25
31
6
27
9.66368
9.71
555
31
o.
28445
9.94
8i3
33
28
9.66 392
9.71
686
o.
284i4
9-94
806
32
29
9.66416
24
9.71
617
31
o.
28 383
9.94
799
7
3i
30
9.66 44i
9.71
648
o.
28352
9.94
793
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin. ! d.
62° 3O .
PP
32
31
30
25
24
7 6
.!
3-*
3-i
3-o
•i 2.5
2.4
.x
o. 7 o. 6
.2
6.4
6.2
6.0
.2 5.0
4.8
2
1-4 1.2
•3
9.6
9-3
9.0
•3 7-5
7.2
3
2.1 1.8
•4
12.8
12.4
12.0
.4 xo.o
9.6
4
2.8 2.4
•5
16.0
iS-5
15.0
•5 12.5
12.0
3-5 3-o
.6
19.2
18.6
18.0
.6 15.0
14.4
6
4.2 3.6
•7
22.4
21.7
21. 0
•7 17-5
16.8
•7
4.9 4.2
.8
25-6
24.8
24.0
.8 20.0
19.2
.8
5-6 4-8
•9
27.0 27.0
.9 22. S
2r.6
6.3 5-4
84
27° 30
'
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9.66441
24
24
24
24
25
24
24
24
24
24
24
25
24
24
24
24
24
24
24
23
24
24
24
24
24
24
24
23
24
24
9.71
648
31
30
31
31
31
31
30
31
31
30
31
31
3*
3°
3°
31
30
3°
3°
30
30
3°
0.28 352
9.94
793
7
6
7
6
7
7
6
7
6
7
7
7
7
t
3
d
3
(.
3
(.
3
3
e
3
3
'i
30
3i
32
33
34
35
36
37
38
39
9.66465
9.66489
9.665i3
9.66537
9.66 562
9.66586
9.66 610
9.66634
9.66658
\O vo "O VO vO VO vo VO vO
679
709
740
771
802
833
863
894
925
0.28 32i
0.28 291
0.28 260
0.28 229
0.28 198
0.28 167
0.28 137
0.28 106
0.28 075
9.94 786
9.94 780
9.94773
9.94767
9.94 760
9.94753
9.94747
9.94 740
9.94734
29
28
27
26
25
24
23
22
21
40
9.66682
9-71
955
o.
28045
9.94
727
20
4i
42
43
44
45
46
47
48
49
9.66 706
9.6673i
9.66 755
9.66779
9.66 8o3
9.66 827
9.6685r
9.66 875
9.66 899
9-71
9-72
9-72
9.72
9.72
9.72
9.72
9.72
9-72
986
017
o48
078
109
i4o
170
2OI
23l
0.
o.
o.
o.
o.
0.
o.
o.
0.
28014
27983
27 952
27 922
27891
27 860
27830
27799
27769
9.94 720
9.94 714
9-94707
9.94 700
9.94694
9.94687
9.94680
9.94674
9.94 667
'9
1 8
17
16
i5
i4
i3
12
I I
50
9.66 922
9-72
262
o.
27738
9.94
660
10
5i
52
53
54
55
56
57
58
59
9.66 946
9.66 970
9.66994
9.67 018
9.67 042
9.67 066
9.67 090
9.67 1 13
9.67i37
9.72 293
9.72 323
9.72 354
9. 72 384
9.72 415
9.72 445
9.72476
9.72 5o6
9.72 537
o.
0.
o.
o.
o.
o.
o.
o.
o.
27 707
27677
27 646
27 616
27585
27555
27 524
27494
27463
9-94654
9.94647
9.94640
9.94 634
9.94 627
9.94 620
9.94614
9.94607
9.94 600
9
8
7
6
5
4
3
2
I
60
9.6-7 161
9.72
567
o.
27433
9.94
593
0
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
62°.
PP
.1
2
3
4
31
30
35
34
23
7
6
3-1
6.2
9-3
12.4
i8!6
21.7
24.8
0.6
1.2
1.8
2-4
3-°
3-6
4-2
4.8
5-4
6.0
9.0
12.0
I5.0
18.0
21.0
24.0
27.0
5-o
7-5
10. 0
12.5
15-0
17-5
20.0
22. =;
.3 4.8
•3 7-2
.4 9.6
• 5 I2.O
.6 14.4
.7 16.8
.8 19.2
4-6
6.9
9-2
"•5
13-8
16.1
18.4
.2
•3
•4
1.4
2.1
2.8
3-5
4.2
4-9
5-6
6-3
85
28°.
'
L.Sin.
d.
L. Tang
. d.
L. Cotg.
L. Cos. d.
0
9
67 161
9.72 567
3i
30
3i
30
3i
30
30
3i
3°
3i
30
30
3i
30
3°
3i
30
3°
30
3i
30
3°
30
30
3i
3°
3°
3°
3°
3°
0.27 433
9.94 593
6
60
I
2
3
4
5
6
7
8
9
9.67 185
9.67 208
9.67 232
9.67 256
9.67 280
9.67 3o3
9.67327
9.67 35o
9-67 374
23
24
24
24
23
24
23
24
9.72 598
9.72 628
9.72 65g
9.72 689
9.72 720
9.72 750
9.72 780
9.72 811
9.72 84i
0.27 402
0.27 372
0.27 34i
0.27 3n
0.27 280
0.27 250
O.27 220
O.27 189
0.27 iSg
9-94 587
9.94 58o
9.94573
9.94 567
9.94 56o
9.94553
9.94 546
9.94 54o
9.94533
7
7
6
7
7
7
6
7
7
7
6
7
7
7
7
6
7
7
7
7
6
7
7
7
7
7
6
7
7
59
58
57
56
55
54
53
52
5i
10
9
67 398
9.72 872
0.27 128
9.94 526
50
1 1
12
i3
i4
i5
16
*7
18
[9
9.67 4ai
9.67 445
9.67468
9.67 492
9.675i5
9.67539
9.67 562
9.67 586
9.67 609
24
23
24
23
24
23
24
23
9.72 902
9.72 932
9.72 963
9.72 993
9.73 023
9.73054
9.73 o84
9.73 n4
9.73 i44
0.27 098
0.27 068
0.27 037
0.27 007
0.26 977
0.26 946
0.26 916
0.26 886
0.26 856
9.94 5 19
9.94 5i3
9.94 5o6
9.94499
9.94 492
9.94485
9.94479
9.94 472
9.94 465
49
48
47
46
45
44
43
42
4i
20
9
.67 633
9.73 175
0.26 825
9-94458
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.67656
.67 680
.677o3
.67 726
.67 750
.67773
.67 796
.67 820
.67843
24
23
23
24
23
23
24
23
9.73 205
9.73 235
9.73 265
9.73 295
9.73 326
9. 73356
9.73 386
9.73416
9.73446
0.26 795
0.26 765
0.26735
0.26 705
0.26 674
0.26 644
0.26 6i4
0.26 584
0.26 554
9.94 45 1
9.94445
9. 94438
9-94431
9.94 424
9.94417
9.94 4 10
9.94 4o4
9.94 397
39
38
37
36
35
34
33
32
3i
30
9
.67866
9.73476
o. 26 524
9.94 390
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
61° 30 .
PP
.1
.2
•3
•4
•5
.6
3
9
3*
30
24
23
.1
.2
•3
•4
:i
•9
7
6
11
9-3
12.4
I5-S
18.6
21.7
24.8
27.9
3-°
6.0
9.0
12.0
15.0
18.0
21. 0
24.0
.2
-3
•4
:!
•7
.8
•9
:i
7-2
9.6
I2.O
14.4
16.8
19.2
a
6.9
9.2
i3.8
16.1
18.4
0.7
1.4
2. I
2.8
3-5
4-2
4-9
5-6
6.3
0.6
1.2
1.8
2-4
30
3.6
4.2
4.8
5-4
86
28° 3O
L. Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.67866
9.73476
3»
30
30
3°
3°
3°
30
3<>
30
30
3°
30
3°
3°
30
30
30
3°
3°
3°
30
30
29
30
3°
3°
3°
3«>
29
3°
0.26
524
9.94
39o
7
7
7
7
7
6
7
7
7
7
7
7
7
7
7
7
6
7
7
7
7
7
7
7
7
7
7
7
7
7
30
3i
32
33
34
35
36
3?
38
39
9.67 890
9.67913
9.67 936
9.67959
9.67 982
9.68 006
9.68 029
9.68 o52
9.68075
23
23
23
23
24
23
23
23
9.735o7
9.73537
9.73567
9-73597
9.73627
9.73657
9.73 687
9.73 717
9.73 747
0.26 493
0.26 463
0.26433
o.264o3
0.26 373
0.26343
0.26 3i3
0.26283
0.26 253
9.94383
9.94376
9.94 369
9.94 362
9.94355
9 . 94 349
9.94342
9.94335
9.94 328
29
28
27
26
25
24
23
22
21
40
9.68098
23
9.73777
0.26
223
9.94
321
20
4i
42
43
44
45
46
4?
48
49
9.68 121
9.68 i44
9.68 167
9.68 190
9.68 21 3
9.68 237
9.68 260
9.68283
9.68 3o5
23
23
23
23
23
24
23
23
22
9.73807
9.73837
9.73 867
9-73897
9.73927
9.73957
9.73987
9-74oi7
9.74 047
0.26 193
0.26 i63
0.26 i33
0.26 io3
0.26073
0.26 o43
0. 26013
0.25983
0.25 953
9.94 3i4
9.94 3o7
9.94 3oo
9.94 293
9.94 286
9.94 279
9.94 273
9.94 266
9.94 259
*9
18
'7
16
i5
i4
i3
12
II
50
9.68 328
23
23
23
23
23
23
23
23
23
22
23
9.74077
0.25
923
9.94
252
10
5i
52
53
54
55
56
5?
58
59
9.68 35i
9.68374
9.68397
9.68 420
9.68443
9. 68 466
9.68489
9.68 5i2
9.68 534
9.74 107
9.74 i37
9.74 166
9.74 196
9.74 226
9.74 256
9.74 286
9.74 3i6
9-74345
0.26 893
0.25863
0.25834
o.25 8o4
o.25 774
o.25 744
o.25 714
o.25 684
0.25655
9.94
9.94
9.94
9.94
9-94
9.94
9.94
9-94
9,94
245
238
23l
224
2I7
2IO
203
i96
i89
9
8
7
6
5
4
3
2
I
60
9.68 557
9.74375
0.25
625
9.94
182
0
L. Cos.
d.
L.
Cotg.
d. L. Tang.
L. Sin. d.
'
61°.
PP
.2
•3
•4
:!
•9
31
30 29
.1
.2
•3
•4
:!
:i
•9
24
33
22
7
6
1:1
9-3
12.4
;i:i
21.7
24.8
27.4
3-o 2.9
o.o 5.8
9.0 8.7
12.0 II. 6
15.0 14.5
18.0 17.4
21. 0 20.3
24.0 23.2
27.0 26.1
2.4
4.8
7-2
9.6
12.0
14.4
16.8
19.2
::i
6.9
9.2
11.5
13.8
16.1
18.4
20.7
2.2
4-4
6.6
8.8
II. 0
13-2
5*
.2
•3
•4
:I
:I
__
0.7
1.4
2.1
2.8
3-5
4.2
4.9
5-6
6-3
0.6
1.2
1.8
2.4
rs
ti
5-4
87
-
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
0
9
.68 557
9.74375
3°
3°
3°
29
3°
3°
29
3°
3°
30
29
3<>
30
29
30
30
29
30
•29
30
29
30
3°
29
3°
29
3°
29
3°
29
o.25 625
9.94 182
7
7
7
7
7
7
7
7
7
7
7
7
8
7
7
7
7
7
7
60
I
2
3
4
5
6
8
9
9
9
9
9
9
9
9
9
9
.68 58o
.686o3
.68625
.68648
.68671
.68694
.68 716
.68 739
.68 762
23
22
23
23
23
22
23
23
9.74405
9.74435
9.74465
9.74 494
9.74 524
9-74554
9. 74 583
9.74 6 1 3
9-74643
o.25 5g5
o.25 565
0.26 535
o.25 5o6
o.25 476
o.25 446
o.25 417
o.25 387
0.25 357
9.94 175
9.94 1 68
9.94 161
9.94 1 54
9.94 147
9.94 i4o
9.94 i33
9.94 126
9.94 119
59
58
57
56
55
54
53
52
5i
10
9
.68 784
9.74673
0.25 327
9.94 112
50
1 1
12
i3
i4
i5
16
17
id
'9
9
9
9
9
9
9
9
9
9
.68 807
.68 829
.68852
.68875
.68 897
.68 920
.68 942
.68 965
.68 987
22
23
23
22
23
22
23
22
9.74 702
9.74732
9.74 762
9.74 791
9.74 821
9.7485i
9.74880
9.74 910
9.74939
o.25 298
0.25 268
o.25 238
o.25 209
o.25 179
0.2D l49
O.25 I2O
o.25 090
o.25 061
9.94 105
9.94 098
9.94 090
9.94083
9.94 076
9.94 069
9.94 062
9.94 o55
9.94 o48
49
48
47
46
45
44
43
42
4i
20
9
.69 oio
9.74969
o.25 o3i
9.94 o4i
7
40
21
22
23
24
25
26
27
28
29
9.69 o32
9.69 o55
9.69077
9.69 100
9.69 122
9.69 i44
9.69 167
9.69 189
9.69212
23
22
23
22
22
23
22
23
9.74998
9.75 028
9.75 o58
9.76087
9.75 117
9.75 i46
9.75 176
9.75 2o5
9.75235
O.25 OO2
0.24 972
0.24 942
0.24 9i3
0.24883
0.24854
0.24824
0.24795
0.24 765
9.94 o34
9.94 027
9.94 020
9.94 OI2
9.94 oo5
9.93998
9.93991
9.93 984
9.93 977
7
7
8
7
7
7
7
7
7
39
38
37
36
35
34
33
32
3i
30
9
.69234
9.75 264
0.24 736
9-93 970
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
60° 30 .
PP
.2
•3
• 4
:i
•Q
30
39
23
22
.1
.2
• 3
•4
:I
J
8
7
3-°
6.0
9.0
12.0
15-0
18.0
21.0
24.0
11
8.7
ii. 6
i4-5
17.4
20.3
23-2
. i
.2
•3
•4
•5
.6
•7
.8
• q
:i
6.9
9-2
"•5
13.8
16.1
18.4
20.7
2.2
4-4
6.6
8.8
II. O
13-2
15-4
17.6
iq. 8
0.8
1.6
2.4
3-2
4.0
4.8
5-6
6.4
0.7
1.4
2.1
2.8
3-5
4-2
4.9
*'
88
29° 30 .
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
69 234
22
23
22
22
22
23
22
22
22
9.76 264
30
29
3°
29
29
3°
29
3°
29
29
3°
29
3°
29
29
30
29
29
29
3°
29
29
29
3°
29
29
29
30
29
29
0.24 736
9.93970
30
3i
32
33
34
35
36
37
38
39
9
9
9
9
9
9
9
9
9
69 256
69 279
69 3oi
69 323
69 345
69 368
69 Sgo
69 412
69434
9.75 294
9.75 323
9.75 353
9.75382
9.75 4n
9.75441
9.75470
9.75 500
9.75 529
0.24 706
0.24 677
0.24 647
0.24 618
0.24 589
0.24 559
0.24 53o
0.24 5oo
0.24 471
9.93 963
9.93955
9.93 948
9.93 941
9.93 934
9.93927
9.93 920
9.939i2
9.93 9o5
8
7
7
7
7
7
8
7
7
7
7
8
7
7
7
8
7
7
7
7
8
7
7
8
7
7
7
8
7
29
28
27
26
25
24
23
22
21
40
9
69456
9.75558
0.24 442
9.93 898
20
4i
42
43
44
45
46
47
48
49
9
9
9
9
9
9
9
9
9
69479
69 5oi
69 523
69 545
69 567
69 589
69 61 1
69 633
69 655
22
22
22
22
22
22
22
22
9.75 588
9.75617
9.75 647
9.75676
9.75 705
9-75735
9.75 764
9<75 793
9.75 822
0.24 4i2
0.24383
0.24353
0.24 324
0.24 295
0.24 265
0.24 236
0.24 207
0.24 178
9.93 891
9.93 884
9.93 876
9.93 869
9.93 862
9. 93855
9-93847
9.93 84o
9.93 833
'9
18
17
16
i5
i4
i3
12
II
50
9
69677
9.75 852
0.24 i48
9.93 826
10
5i
52
53
54
55
56
57
58
59
9.69699
9.69721
9.69 743
9.69 765
9.69787
9.69 809
9.69 83i
9.69853
9.69875
22
22
22
22
22
22
22
22
9.75 881
9.75 910
9.75 939
9.75969
9.75998
9.76027
9.76 o56
9. 76 086
9.76 115
0.24 119
0.24 090
0.24 061
0.24 o3i
0.24 002
o.23 973
0.23 944
o.23 914
0.23885
9.93 819
9.93 81 1
9.93 8o4
9.93 797
9.93 789
9.93 782
9-93775
9.93 768
9.93 760
9
8
7
6
5
4
3
2
I
60
9
.69897
9.76 i44
0.23856
9.93 753
0
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
6O°.
PP
• 2
•3
•4
:I
30
29
23
22
.i
.2
•3
•4
:I
•7
.8
8
7
3-°
6.0
9.0
I2.O
15-0
18.0
21.0
24.0
I*
8.7
ii. 6
M-5
17-4
20.3
.1
.2
•3
•4
:i
:1
•9
ii
6.9
9.2
SI
16.1
18.4
20.7
2.2
4-4
6.6
8.8
II. O
13.2
J5-4
17.6
19.8
0.8
1.6
a-4
3-2
4'R
4.8
5-6
6-4
7.2
0.7
i-4
2,1
2.8
3-5
4-2
4-9
I'6
6-3
89
3O°.
'
L.Sin
d.
L. Tang.
d.
L
Cotg.
L.Cos. |d.
0
9.69 897
9.76
1 44
o.
23856
9.93
753
60
I
9.69919
22
9.76
178
29
o.
23 827
9.93
746
a
59
2
9.69 941
9.76
202
0.
23798
9-93
738
58
3
9.69 963
21
9-76
23l
30
0.
23769
9.93
73i
7
7
57
4
9.69 984
22
9-76
261
29
o.
23739
9.93
•724
7
56
5
9.70 006
9.76
29O
0.
23 7IO
9.93
7i7
55
6
9.70 028
9.76
3i9
o.
2368i
9.93
7°9
54
22
29
7
7
9. 70 050
9.76
348
29
o.
23652
9.93
702
53
8
9.70 072
9.76
377
o.
23623
9.93
695
52
9
9.7o 093
9.76 4o6
29
o.
23594
9.93
687
5i
10
9.7o n5
9.76435
0.
23 565
9.93
680
50
ii
9.7o 1 3
7
9-76464
29
o.
23 536
9.93
673
8
49
12
9.70 i5g
9.76
493
0.
23 507
9.93 665
48
i3
9.70 180
9.76
522
29
o.
23478
9.93 658
7
47
i4
9.70 202
22
9.76
55i
29
0.
23449
9.93 65o
8
46
i5
9 .70 224
22
9.76
58o
o.
23 42O
9.93 643
45
16
9.70 245
21
9.76
609
29
o.
23 Sgi
9.93636
7
44
17
9.70 267
22
9.76 639
3°
o.
2336i
9.93
628
43
18
9.70 288
9.76
668
o.
23332
9.93
621
42
X9
9.70 3io
22
9.76697
29
0.
23 3o3
9-93
6i4
7
g
4i
20
9.70 332
22
9.76
725
0.
23275
9.93 606
40
21
9.70 353
9.76
754
29
o.
23 246
9.93
599
7
8
39
22
9.70 375
9-76
783
0.
23 217
9.93
5oi
38
23
9.70 396
9-76
812
29
o.
23 188
9-93
584
7
37
22
29
7
24
9.70 4i8
9-76
84 1
29
o.
23 i5g
9-93
577
8
36
25
9.7o439
9-76
87o
o.
23 i3o
9-93
569
35
26
9.7o 46i
9.76899
o.
23 IOI
9.93
562
7
34
21
29
K
27
9.70 482
9<76 928
29
o.
23 072
9.93
554
33
28
9.7o 5o4
9.76
957
o.
23o43
9.93
547
32
29
9.7o 525
9.76 986
29
o.
23 Ol4
9.93
539
3i
30
9.7o 547
.9-77
015
0.
22 985
9.93
532
7
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
59° 3O .
PP 30
29
28
22
21
8 7
• i 3-0
2.9
2.8
.1 2.2
2.1
.1
0. 8 0. 7
.2 6.0
5-8
5-6
.2 4.4
4.2
.2
1.6 1.4
•3 9-o
8.7
8.4
.3 6.6
6.3
• 3
2.4 2.1
• 4 12.0
n.6
II. 2
.4 8.8
8.4
•4
3.2 2.8
•5 15-0
14-5
14.0
• 5 ii-o
10.5
•5
4-° 3-5
.6 18.0
17.4
16.8
.6 13.2
12.6
.6
4.8 4.2
.7 210
20.3
19.6
•7 15-4
14-7
•7
5.6 4.9
.8 24.0
23.2
22.4
.8 17.6
16.8
.8
6.4 5.6
•9 27.0
.9 19.8
18.9
.9
9°
30° 30
t
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
3i
32
33
34
35
36
37
38
39
9-
70547
9-77 015
29
29
28
29
29
29
29
29
28
29
29
29
29
28
29
29
29
28
29
29
28
29
29
29
28
29
28
29
29
28
0.22 985
9.93 532
30
9-
9-
9-
9-
9-
9-
9-
9-
9-
70 568
70 5go
70 61 1
70 633
70654
70 675
70697
70 718
70 739
22
21
22
21
21
22
21
21
9.77 o44
9.77073
9.77 101
9.77 i3o
9.77 i59
9.77 1 88
9.77217
9.77 246
9-77 274
O. 22 956
0.22 927
O.22 899
0.22 870
O.22 84l
O.22 8l2
0.22 783
0.22 754
O.22 726
9.93525
9.93517
9.93 5io
9.93 5o2
9.93495
9.93487
9.93480
9.93 472
9.93 465
8
7
8
7
8
7
8
7
29
28
27
26
25
24
23
22
21
40
9-
70 761
9.773o3
O.22 697
9.93457
7
8
7
8
7
8
7
8
7
8
7
8
7
8
8
7
8
7
8
7
20
4i
42
43
44
45
46
47
48
49
9-
9-
9
9-
9
9
9
9
9
70 782
70 8o3
70824
70 846
70 867
70 888
70909
7093i
70 952
21
21
22
21
21
21
22
21
9. 77 332
9.77 36i
9.77 390
9.77418
9.77447
9.77476
9.77 505
9.77533
9.77 562
O.22 668
O.22 639
0.22 6lO
O.22 582
0.22 553
0.22 524
O.22 4g5
0.22 467
0.22 438
9.93450
9.93 442
9.93435
9.93427
9.93 420
9.93 412
9.93405
9.93 397
9.93 390
1 8
16
i5
i4
i3
12
II
50
9
70973
21
21
21
22
21
21
21
21
21
21
9-77 59i
0.22 409
9.93 382
10
5i
52
53
54
55
56
57
58
59
9
9
9
9
9
9
9
9
70994
71 oi5
71 o36
71 o58
71 079
71 100
71 121
71 142
71 i63
9-77 619
9.77 648
9.77 677
9.77 706
9.77 734
9.77763
9.77 791
9.77820
9.77 849
O.22 38l
0.22 352
0.22 323
O.22 294
O.22 266
0.22 237
O.22 209
O.22 l8o
O.22 l5l
9.93375
9.93367
9.93 36o
9.93 352
9.93344
9.93337
9.93329
9.93 322
9.93 3i4
9
8
7
6
5
4
3
2
I
60
9
.71 1 84
9.77 877
0.22 123
9-93 3o7
0
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
d.
59°.
PP
.1
2
3
4
• 7
.8
•9
39
28
22
21
.1
2
•3
•4
•7
.8
•9
8 7
2.9
5-8
8.7
ii. 6
"4-5
17.4
20.3
23.2
26.1
2.8 .1
5-6
8.4 .3
II. 2 4
14.0 .5
16.8 .6
19.6 .7
22.4
25.2 .9
2.2
4-4
6.6
8.8
II. O
13-2
\7.6
19.8
2. I
4-2
6-3
8.4
10.5
12.6
14.7
16.8
18.9
o. 8 o. 7
1.6 1.4
2 4 2.1
3.2 2.8
4-o 3-5
4.8 4.2
5-6 4-9
6.4 5.6
31°.
/
L. Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.71 1 84
9.77877
O.22 123
9.93 307
60
I
.71 205
9.77906
29
0.22 094
9.93 299
g
5p
2
9.71 226
9-77 935
28
0.22 o65
9.93 291
58
3
9.71 247
9.77963
0.22 037
9.93 284
'
57
29
K
4
9.71 268
21
9.77992
28
0.22 008
9.93 276
56
5
9.71 289
9.78 020
O.2I 980
9.93 269
55
6
9.71 3io
21
9.78 049
28
O.2I 95 I
9.93 261
8
54
7
9.71 33i
21
9.78077
0.21 923
9.93 253
53
8
9.71 352
9.78 106
0.21 894
9.93 246
52
9
9.71 373
9.78135
29
28
0.21 865
9.93 238
5i
10
9
.71 393
9.78 i63
20
0.21 837
9.93 23o
50
ii
9
.71 4i4
21
9.78 192
28
0.21 808
9.93 223
7
g
49
12
9
.7i435
9.78 220
O.2I 780
9.93 21 5
48
i3
9
.71 456
21
9.78 249
28
0.21 751
9.93 207
8
47
i4
9
.71 477
21
9.78277
29
O.2I 723
9.93 2OO
7
g
46
i5
9
.71498
9.78 3o6
28
0.21 694
9.93 192
45
16
9
.71 5i9
2O
9.78334
29
0.21 666
9.93 184
44
17
9
.71 539
21
9.78 363
28
0.21 637
9.93 177
g
43
18
9
.71 56o
9.7839i
?8
0.21 609
9.93 169
42
'9
9
.71 58i
9.78419
0.21 58i
9.93 161
4i
20
9
.71 602
9.78448
28
O.2I 552
9.93 i54
7
40
- 21
9.71 622
21
9.78476
29
O.2I 524
9.93 i46
g
39
22
9
.71 643
9-78505
28
O.2I 495
9.93 i38
38
23
9
.71 664
9.78533
O.2I 467
9.93 i3i
7
37
21
29
X
24
9
.71685
9.78 562
28
0.21 438
9.93 123
g
36
25
9
.71 705
9.78 590
28
0.21 4
to
9.93 n5
35
26
9
.71 726
9.78 618
0.21 382
9.93 108
7
34
21
29
H
27
9
.71 747
9.78 647
28
0.21 353
9.93 100
g
33
28
9.71 767
9.78675
O.2I 325
9 . 93 092
32
29
9.71 788
9.78 704
28
O.2I 296
9.93084
3r
30
9
.71 809
9.78732
O.2I 268
9.93077
7
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
58° 30 .
PP
29
28
21
20
8
7
,
2.9
2.8
j
2.1
2.0
.1
0.8
0.7
2
5.1
5-6
.2
4.2
4.0
.2
1.6
1.4
•3
8-7
8.4
• 3
6.3
6.0
•3
2.4
2. 1
•4
ii. 6
II. 2
•4
8.4
8.0
•4
3«2
2.8
•5
*4* 5
14.0
•5
10.5
10. 0
• 5
4.0
3-5
.6
17-4
1 6. 8
.6
12.6
12.0
.6
4.8
4.2
•7
20.3
19.6
.7
14.7
14.0
.7
5-6
4.9
.8
23.2
22.4
.8
16.8
16.0
.8
6.4
5-6
•9
18.9 18.0
•9
7.2 6.3
92
31° 3O .
,
L. Sin. d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
.71 809
9.78732
28
29
28
28
29
28
28
29
28
28
28
29
28
28
28
29
28
28
28
28
29
28
28
28
28
28
29
28
28
28
O.2I 268
9.93077
8
8
8
7
8
8
8
8
7
8
8
8
7
8
8
8
8
8
7
g
30
3i
32
33
34
35
36
37
38
39
ooo ooo ooo
.71 829
.71850
.71 870
.71 891
.71 9n
.71932
.71 952
.71 973
.71 994
21
20
21
2O
21
20
21
21
9.78 760
9.78789
9.78817
9.78845
9.78874
9.78 902
9.78 930
9.78959
9.78987
O.2I 24O
O.2I 211
0.21 183
0.21 155
0.21 126
O.2I 098
0.21 070
O.2I o4l
0.21 Ol3
9.93 069
9.93 061
9.93 o53
9.93 o46
9.93 o3o
9.93 022
9.93 oi4
9.93 007
29
28
27
26
25
24
23
22
21
40
9
.72 014
9.79015
O.2O 985
9.92999
20
4i
42
43
44
45
46
47
48
49
ooo ooo ooo
.72 o34
.72055
.72 o75
.72 096
.72 1 16
.72l37
.72 157
.72 177
.72 198
21
20
21
2O
21
20
20
21
9.79 o43
9.79072
9.79 100
9.79 128
9.79 i56
9.79185
9.79 2i3
9.79 241
9.79269
O.2O 957
O.2O 928
O.2O 900
O.2O 872
O.2O 844
O.2O 8l5
0.20 787
O.2O 759
O.2O 781
9.92 991
9.92 983
9.92976
9.92 968
9.92 960
9.92 952
9.92 944
9.92 936
9.92929
18
17
i5
i4
i3
12
II
50
9
.72 218
9.79297
0.20 703
9.92 921
8
8
8
8
8
7
8
8
8
g
10
9
8
7
6
5
4
3
2
I
5i
52
53
54
55
56
57
58
59
9
9
9
9
9
9
9
9
9
.72 238
.72 259
.72 279
.72 299
.72 32o
.72 34o
.72 36o
.72 38i
.72 4oi
21
20
20
21
20
20
21
20
9.79 326
9.79354
9.79382
9.79410
9.79438
9.79 466
9.79495
9.79 523
9.79 55i
O.2O 674
0.20 646
0.20 618
o • 20 5oo
0.20 562
0.20 534
0.20 5o5
O.2O 477
O.2O 449
9.92 913
9.92 905
9.92897
9.92 889
9.92 881
9.92 874
9.92 866
9.92 858
9.92 850
60
9
.72 421
9-79 579
0.2O 421
9.92 842
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
t
58°.
PP
.1
.2
•3
•4
• 5
.6
•9
29
28
21
20
2
3
4
1
9
8
7
ii. 6
14.5
17.4
20.3
23.2
26.1
2.8
8.4
II. 2
14.0
16.8
19.6
22.4
25.2
.1
.2
• 3
•4
J
• 7
.8
•9
2.1
4-2
6-3
8.4
10-5
12.6
14.7
16.8
18.9
2.0
4.0
6.0
8.0
IO.O
12.0
14.0
16.0
18.0
0.8
1.6
2.4
3-2
ts
0.7
1.4
2.1
2.8
3-5
4.2
4.9
y
93
32°.
,
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
0
9.72 421
9.79 579
28
O.2O
421
9-92
842
D
60
I
9.72441
20
9-79 6°7
28
0.20
393
9.92
834
8
59
2
9.72 46i
9.79635
0.20
365
9.92
826
r>
58
3
9.72 482
2O
9.79 663
28
0.2O
337
9.92
818
8
57
4
9.72 602
20
9.79691
O.2O
3o9
9.92
810
7
56
5
9.72 522
9.79 719
28
0.20 28l
9.92
8o3
|
55
6
9.72 542
20
9.79 747
29
O.2O
253
9.92
795
8
54
7
9.72 562
2O
9.79776
28
O.2O
224
9.92
787
8
53
8
9.72 582
9.79 8o4
O.2O
196
9.92
779
0
52
9
9.72 602
9.79 832
0.20
168
9.92
771
8
5i
10
9.72 622
9.79 860
O.2O l4o
9.92
763
0
50
1 1
9.72 643
2O
9.79888
28
0.20
112
9.92
7*5
t
49
12
9.72 663
9.79916
0.20 o84
9.92
747
48
i3
9.72 683
9.79944
28
O.2O
o56
9.92
47
20
28
i4
9.72 703
9.79972
28
O.2O
028
9.92
73i
I
46
i5
9.72 723
9.80 ooo
O.2O
ooo
9.92
723
45
16
9.72 743
9.80 028
28
0.19 972
9.92
7i5
44
17
9.72763
2O
9.80 o56
28
28
o. 19
944
9.92
7°7
I
43
18
9.72783
9.80084
o. 19
916
9.92
699
f
42
'9
9.72 8o3
9.80 112
28
o. 19
888
9.92
691
5
4i
20
9.72 823
9.80 i4o
o. 19
860
9.92
683
40
21
9.72843
9.80 1 68
o. 19
832
9.92
675
|
39
22
9.72 863
9.80 195
o. 19
805
9.92
667
38
23
9.72 883
19
9.80 223
28
0.19
777
9.92
659
1
37
24
9.72 902
9.80 25i
28
o. 19
749
9.92
65i
|
36
25
9.72 922
9.80 279
o. 19
•721
9.92
643
35
26
9.72 942
20
9.80 307
28
0.19 693
9.92
635
1
34
27
9.72 962
9. 80 335
28
o. 19
665
9.92
627
|
33
28
9.72 982
9. 80 363
o. 19
637
9.92
6i9
32
29
9.73 002
20
9.80 391
28
o. 19
6o9
9.92
611
3i
30
9.73 022
9.80 419
o. 19
58i
9.92
6o3
30
L. Cos.
d.
L.
Cotg. d.
L. Tang.
L. Sin.
d.
57° 30.
PP
29
28 27
21
20
19
8
7
.1
2.9
2.8 2.7
.1
2.1
2.0
1.9
.1
0.8
0.7
.2
5-8
5-6 5-4
.2
4-2
4.0
3
.2
1.6
1.4
•3
8.7
8.4 8.1
•3
6.3
6.0
5-7
•3
2.4
2. I
•4
n.6
II. 2 10.8
•4
8.4 ' 8.0
7.6
•4
3-2
2.8
• 5
14.5
14.0 13.5
•5
10.5 10.0
9-5
. 5
4.0
3-5
.6
i7-4
16.8 16.2
.6
12.6
12.0
11.4
; -6
4.8
4-2
•7
20.3
19.6 18.9
•7
14.7
14.0
»3*3
! .7
5-6
4-9
.8
22.4 21.6
.8
16.8
16.0
15.2
.8
6.4
.9 26.1
25.2 24.3
18.9 18.0
17.1
•9
7.2
94
32° 3D
-
L. Sin. d.
L. Tang.
d. L. Cotg.
L. Cos.
d.
30
9
73 022
20
20
2O
20
»9
20
2O
20
'9
9.80419
28
27
28
28
28
28
28
28
27
28
28
28
28
27
28
28
28
27
28
28
28
27
28
28
27
28
28
27
28
28
0.19 58i
9.92 6o3
30
3i
32
33
34
35
36
37
38
39
9
9
9
9
9
9
9
9
9
73o4i
73 061
73 081
73 101
73 121
73 i4o
73 160
73 180
73 200
9.80447
9.80 4?4
9.80 5o2
9.8o53o
9. 80 558
9.80 586
9.80 6i4
9.80642
9.80 669
0.19553
o. 19 526
0.19498
0.19470
0.19 442
o. 19 4i4
0.19386
0.19358
0.19331
9.92595
9.92 587
9-92579
9.92571
9.92 563
9.92555
9.92 546
9.92 538
9.92 53o
8
8
8
8
8
9
8
8
g
29
28
27
26
25
24
23
22
21
40
9
73 219
9.80 697
0.19303
9.92 522
8
8
8
8
8
9
8
8
8
8
8
8
9
8
8
8
8
8
9
8
20
4i
42
43
44
45
46
47
48
49
ON ON ON ON ON ON ON ON ON
73239
73259
73278
73298
73 3i8
73337
73357
73377
73396
20
'9
20
20
19
20
20
»9
2O
'9
20
20
'9
20
2O
19
2O
9.80725
9.80 753
9.80 781
9.80808
9.80 836
9.80864
9.80 892
9.80 919
9.80 947
o. 19 275
o. 19 247
o. 19 219
0.19 192
0.19 1 64
0.19 i 36
0.19 108
0.19081
o. 19 o53
9.92 5i4
9.92 5o6
9.92498
9.92 490
9.92 482
9.92473
9.92 465
9.92457
9.92 449
18
16
i5
i4
i3
12
I I
50
9
734i6
9.80975
O. 19 O25
9.92441
10
5i
52
53
54
55
56
57
58
59
9
9
9
9
9
9
9
9
9
73435
73455
73474
73494
735i3
73533
.73552
.73572
.73 591
9.81 oo3
9.81 o3o
9.81 o58
9.81 086
9.81 ii3
9.81 i4i
9.81 169
9.81 196
9.81 224
0.18997
o. 1 8 970
o.i 8 942
0.18 9i4
0.18 887
0.18 859
0.18 83i
o.i 8 8o4
o. 18 776
9.92433
9.92425
9.92 4i6
9.92 4o8
9.92 4oo
9.92 392
9.92 384
9.92 376
9.92 367
9
8
7
6
5
4
3
2
I
60
9
.73611
9.81 252
o.i 8 748
9.92 359
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
57°.
PP
.2
•3
•4
•5
.6
•9
28
27
20
19
.2
•3
•4
•7
.8
9 8
2.8
5-6
8.4
II 2
I4.0
16.8
19.6
22.4
2.7 .1
5-4 -2
8.1 .3
10.8 .4
ll'.l '.I
18.9 .7
21.6 .8
24-3 -9
2.O
4-0
6.0
8.0
10.0
12. 0
14.0
16.0
18.0
1.9
3-8
5-7
7-6
9-5
11.4
»5-2
17.1
o. o o. 8
1.8 1.6
2.7 2.4
3.6 3.2
4.5 4.0
5-4 4-8
6.3 5-6
7.2 6.4
8. i 7. 2
33°.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
0
9.73 611
9.81 252
27
0.18 748
9.92359
60
I
9.73 63o
20
9.81 279
28
o. 18 721
9.92 35i
8
59
2
9.73 650
9.81 307
28
0.18 693
9.92 343
58
3
9.73 669
20
9.81 335
27
o.i8665
9.92335
9
57
4
9.73 689
19
9.81 362
28
0.18 638
9.92 326
8
56
5
9.73708
9.81 390
28
o. 18 610
9.92 3i8
55
6
9.73 727
I9
9.81 4i8
0.18 582
9.92 3io
54
20
27
8
7
9
.73747
9.81 445
28
0.18 555
9.92 3o2
53
8
9
.73766
9 81 473
0.18 527
9.92 293
52
9
9
.73785
19
9.81 5oo
27
28
o. 18 500
9.92 285
g
5i
10
9
.73805
9.81 628
28
o. 18 472
9.92 277
g
50
1 1
9
.73824
19
9.8i 556
27
o.i 8 444
9.92 269
Q
49
12
9
.73 843
9.8i 583
0.18 417
9.92 260
48
i3
9
.73863
19
9.8i 61
I
27
0.18 389
9.92 252
8
47
i4
9
.73882
19
9.8i 638
28
0.18 362
9.92 244
Q
46
i5
9
.73901
9.8i 666
0.18 334
9.92 235
45
16
9
.73921
9.8i 693
27
28
0.18 307
9.92 227
8
44
17
9
.7394o
19
9.81 721
27
o. 18 279
9.92 219
8
43
18
9
.73959
9.81 748
_Q
O. l8 252
9.92 211
42
19
9
.73978
9.81 776
o. 18 224
9.92 2O2
9
4i
20
9
-73997
9.81 8o3
0.18 197
9.92 194
g
40
21
9
.74017
9.81 83i
o. 18 169
9.92 186
9
39
22
9
.74o36
9.81 858
0.18 142
9.92 177
38
23
9
.74o55
19
9.81 886
0.18 u4
9.92 169
37
24
9
. 74 074
19
9.81 913
27
28
0.18 087
9.92 161
36
25
9
.74 o93
9.81 g4
I
o. 18 o59
9.92 i52
35
26
9
.74n3
9.81 968
27
28
0.18 o32
9.92 1 44
8
34
27
9
.74 1 32
19
9.81 996
o. 1 8 oo4
9.92 i36
9
33
28
9
.74i5i
9.82 023
0.17 977
9.92 127
g
32
29
9
• 74 17°
19
9.82 o5i
0.17 949
9.92 119
8
3i
30
9
• 74 189
9.82 078
27
o. 17 922
9.92 in
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
56° 30 .
PP
28
27
2O
19
9
8
.1
2.8
2.7
.1
2.0
1.9
i
0.9
0.8
.2
5-6
5-4
.2
4.0
3-8
2
1.8
1.6
•3
8.4
8.1
•3
6.0
5-7
3
2.7
2.4
•4
II.2
10.8
•4
8.0
7-6
4
3-6
3.2
•5
14.0
*3»5
• 5
10.0
9-5
5
4-5
4.0
.6
16.8
16.2
.6
12. 0
KM
6
5.4
4-8
•7
19.6
18.9
•7
I4.0
13-3
7
6.3
5-6
.8
22.4
•21.6
.8
10. 0
15-2
8
7-2
6.4
•9
25.2 24.3
•9
1 8.0 17. I
9
8.1 7.2
96
33° 3O .
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
• 74 189
9.82078
28
0.17 922
9.92 in
30
3i
9
.74 208
9.82 106
27
0.17 894
9.92 1 02
9
8
29
32
9
.74227
9.82 i33
28
0.17 867
9.92 094
28
33
9
.74 246
19
9.82 161
27
0.17 839
9.92 086
9
27
34
9
.74265
19
9.82 188
27
0.17 812
9.92077
s
26
35
9
.74284
9.82 2i5
28
0.17785
9.92 069
25
36
9
.743o3
X9
9.82243
0.17 757
9.92 060
9
24
19
27
8
37
9
.74322
19
9.82 270
28
0.17 780
9.92 o52
8
23
38
9
.74341
9.82 298
o. 17 702
9.92 o44
22
39
9
7436o
19
9.82 325
0.17675
9.92 o35
9
8
21
40
9
.74379
9.82 352
28
0.17 648
9.92 027
20
4i
9
.74398
9.82 38o
o. 17 620
9.92 018
8
'9
42
9
.744i7
9.82 407
_0
0.175^
3
9.92 oio
18
43
9
. 74436
*9
9.82435
o. 17 565
9.92 002
'7
jg
27
<)
44
9
.74455
9.82462
0.17 538
9.91 993
8
16
45
9
.74474
9.82489
Oo
0.17 5u
9.91 985
i5
46
9.74493
'9
9.82 517
27
0.17483
9.91 976
9
i4
47
9
74 5i2
9.82 544
o. 17 456
9.91 968
i3
48
9
7453i
9.82 671
o. 17 429
9.91 959
(i
12
49
9
74549
9.82 599
o. 17 4oi
9.91 951
I I
50
9
74568
19
9.82 626
0.17 374
9.91 942
10
5i
9
74587
19
9.82 653
27
28
0.17 347
9.91 934
9
52
9
74 606
*9
9.82 681
0.17 319
9.91 925
8
53
9
74625
'9
9.82 708
27
0.17 292
9.91 917
7
54
9-74644
'9
9.82 735
27
0.17 265
9.91 908
9
8
6
55
9
74662
9.82 762
0.17 238
9.91 900
b
56
9
.74681
19
9.82 790
O.I7 2IO
9.91 891
9
4
57
9
.74 700
i9
9.82 817
27
0.17 i83
9.91 883
0
3
58
9
•74 7*9
J9
9.82 844
o. 17 i56
9.91 874
2
59
9
.74737
18
9.82 871
27
o. 17 129
9.91 866
I
60
9
.74756
19
9.82 899
0.17 101
9.91 857
9
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
56°.
PP
28
27
19
18
9
8
.1
2.8
2.7
.1
1.9
1.8
.1
0.9
0.8
.2
5.6
5-4
.2
3.8
3-6
.2
1.8
1.6
•3
8.4
8.1
•3
5-7
5-4
•3
2.7
2.4
•4
II. 2
10.8
•4
7.6
7.2
•4
3-6
3-2
, e
14.0
t3-5
• 5
9-5
9.0
,c
4-5
4.0
.6
16.8
16.2
.6
11.4
10.8
.6
5-4
4.8
• 7
19.6
18.9
• 7
13-3
12.6
•7
6-3
5-6
.8
22.4
21.6
.8
15-2
14.4
.8
7.2
6.4
25.2 24.3
17.1 16.2
•9
8.1
7.2
97
,
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
0
9
• 74756
9.82 899
o. 17 101
9.91 857
g
60
!
9
•74775
19
9.82 926
27
0.17 074
9.91 849
59
2
9
.74794
18
9.82 953
0.17 047
9.91 84o
g
58
3
9
.74812
19
9.82 980
28
0.17 020
9.91 832
q
^>7
4
9
.7483i
19
9.83 008
27
o. 16 992
9.91 823
8
56
5
9
.74850
18
9.83o3<
o. 16 965
9.91 815
55
6
9
,74868
9.83 062
0.16 938
9.91 806
9
54
19
27
8
7
9
• 74887
19
9.83 089
28
0.16 911
9.91 798
53
8
9
.74 906
18
9-83 117
o.i6883
9.91 789
g
52
9
9
• 74 924
9.83 1 44
27
o.i6856
9.91 781
5i
10
9
.74943
18
9.83 171
0.16 829
9.91 772
50
1 1
9
.74 961
9.83 198
27
o. 16 802
9.91 763
8
49
12
9
.74 980
9.83 225
0.16 775
9.91 755
48
i3
9
.74999
18
9. 83 252
28
0.16 748
9.91 746
9
8
47
i4
9
.75 017
9-83 280
27
o. 16 720
9.91 738
46
i5
9
.75o36
18
9.83 307
0.16 693
9.91 729
45
16
9
.75o54
9.83334
27
o. 16 666
9.91 720
9
8
44
ll
9
.75 o73
18
9.83 36i
27
0.16^39
9.91 712
43
18
9
.75 091
9. 83 388
o. 16 612
9.91 703
42
J9
9
.75 1 10
18
9.834i5
0.16 585
9.91 695
4i
20
9.75 128
9-83 442
28
o.i6558
9.91 686
9
40
21
9
.75 i47
18
9.83470
0.16 53o
9.91 677
g
39
22
9
.75 i65
9-83497
0.16 5o3
9.91 669
38
23
9
.75 1 84
r9
18
9.83 524
27
27
0.16476
9.91 660
y
9
37
24
9
.75 202
9.8355i
o. 1 6 449
9.91 65i
g
36
25
9
.75 221
9.83578
0.16 422
9.91 643
35
26
9
.75 23g
9.836o5
27
0.16395
9.91 634
y
34
27
.75 258
'9
18
9. 83 632
27
o.i6368
9.91 625
9
g
33
28
9
.75 276
9.83659
0.16 34i
9.91 617
32
29
9
.75 294
9. 83 686
27
0.16 3i4
9.91 608
y
3i
30
9
.76 3i3
I9
9.83 713
27
0.16 287
9.91 599
y
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
55° 30 .
PP
23
27
19
18
9
8
.1
2.8
2.7
.,
1.0
1.8
.,
0.9
0.8
.2
5-6
5-4
.2
3-8
3-6
.2
1.8
1.6
•3
8.4
8.1
•3
5-7
5-4
•3
2.7
2.4
•4
II. 2
10.8
•4
7-6
7.2
•4
3-6
3-2
1
14.0
16.8
'I'5
16.2
• 5
.6
9-5
11.4
9.0
10.8
:i
4-5
5-4
4.0
4.8
•7
19.6
i8.q
7
13-3
12.6
.7
6-3
5-6
.8
22.4
21.6
.8
15-2
14.4
. 8
7.2
6.4
• 0
25.2 24.3
17.1 16.2
8.1 7.2
98
34° 30'
!
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
.75 3i3
18
9.83
7i3
27
28
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
26
27
27
27
27
27
27
27
27
27
0.
i6287
9.91
599
30
3i
32
33
34
35
36
37
38
39
9
9
9
9
9
9
9
9
9
.7533i
.7535o
.75 368
.75386
.7544i
.75459
18
18
'9
18
18
18
19
18
18
if
18
if
if
18
18
if
18
if
if
18
if
•f
18
if
18
9.83 740
9.83 768
9. 83 795
9-83 822
9.83 849
9.83 876
9.83 903
9.83 930
9.83957
0.
0.
o.
o.
o.
0.
o.
o.
o.
16 260
16 232
16 2o5
16 178
16 i5i
1 6 124
16 097
1 6 070
i6o43
9.91
9.91
9.91
9.91
9.91
9.91
9.91
9.91
9.91
59i
582
573
565
556
547
538
53o
521
9
9
8
9
9
9
8
9
29
28
27
26
25
24
23
22
21
40
9.75496
9.83
984
o.
16 016
9.91
5l2
8
9
9
9
8
9
9
'9
9
8
9
9
9
9
8
9
9
9
9
9
20
4i
42
43
44
45
46
47
48
49
9.755i4
9.75533
9.7555i
9. 75 569
9.75587
9.75 6o5
9.75 624
9.75 642
9.75 660
9.84 01 1
9.84o38
9.84 065
9.84 092
9.84 119
9.84 i46
9.84 i73
9.84 200
9.84 227
o.
0.
0.
o.
0.
0.
0.
0.
o.
i5 989
1 5 962
i5935
1 5 908
i588i
1 5 854
i5 827
1 5 800
i5773
9.91
9.9i
9.91
9.91
9.91
9.91
9.91
9.91
9.91
5o4
495
486
477
469
46o
45i
442
433
19
18
17
16
i5
i4
i3
12
I I
50
9
.75 678
9-84
254
o.
i5746
9.91
425
10
5i
52
53
54
55
56
57
58
59
9.70 696
9.75 714
9.75 733
9.7575i
9.75 769
9.75787
9.7^805
9.75 823
9.75 84 1
9.84 280
9.84 3o7
9.84334
9.8436i
9.84388
9-844i 5
9.84442
9.84469
9.84496
0.
0.
0.
0.
0.
0.
o.
o.
0.
1 5 720
i5693
1 5 666
i5639
i5 612
i5558
i553i
i55o4
9.91
9.91
9.91
9.91
9.91
9.91
9.91
9.91
9.91
4i6
407
398
389
38i
372
363
354
345
9
8
7
6
5
4
3
2
I
60
9.75859
9-84
523
0.
i5 477
9.91
336
0
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
55°.
PP
.2
•3
•4
•9
28
27
26
19
if
.2
•3
•4
9 8
2.8
5-6
8.4
II. 2
14.0
1 6. 8
19.6
22-4
2.7
10.8
II; 2
18.9
21.6
24-3
2.6
10.4
,3.0
15-6
18.2
20.8
23-4
.1 1.9
.2 3-8
•3 5-7
•4 7-6
•5 9-5
.6 11.4
•7 J3-3
.8 15-2
.q 17.1
1.8
3-6
5-4
7.2
9.0
10.8
12.6
14.4
16.2
0.9 0.8
1.8 1.6
2-7 2.4
3-6 3-2
4-5 4-o
5-4 4-8
6-3 5.6
7.2 6.4
8.1 7.2
99
35°.
/
L. Sin. d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d*
0
9
.75859
18
9.84 5-23
o.i5 477
9.91 336
60
I
9
•75877
18
9.84 550
26
o. i5 45o
9.91 328
59
2
9
.75895
18
9.84576
o. i5 424
9.91 3i9
58
3
9.75913
9.846o3
o.i 5 397
9.91 3io
9
57
27
9
4
9
.7593i
18
9.8463o
27
o. i5 37o
9.91 3oi
56
5
9
.75949
18
9.84657
o.i5343
9.91 292
55
6
9
•75967
18
9.84684
27
o.i5 3i6
9.91 283
y
9
54
7
9
.75985
18
9.84 7n
27
o.i5 289
9.91 274
8
53
8
9
.76 oo3
9.84738
o.i5 262
9.91 266
52
9
9
.76 O2I
9-84 764
o.i5 236
9.91 257
9
5i
10
9
.76039
18
9.84 79i
27
o.i 5 209
9.91 248
50
ii
9
.76057
18
9.84818
27
o.i5 182
9.91 239
g
49
12
9
.76075
18
9.84845
o.i5 i55
9.91 23o
48
i3
9
.76 o93
18
9.84872
27
o.i5 128
9.91 221
Q
47
i4
9
.76 in
18
9.84899
26
o. i5 101
9.91 212
9
46
i5
9
.76 I29
9.84 925
o.i5 or
75
9.91 2o3
45
16
9
.76l46
18
9.84 952
27
o . 1 5 o48
9-91 194
9
44
'7
9
.76164
18
9.84979
27
o. 1 5 02 1
9.91 i85
9
43
18
9
.76 l82
18
9.85 006
0.14994
9.91 176
42
19
9
.76 200
9.85o33
26
o. 1 4 967
9.91 167
4i
20
9
.76 218
9.85 059
o . 1 4 94 1
9.91 i58
40
21
9
.76 236
17
9.85 086
27
o. i4 914
9.91 149
8
39
22
9
.76253
9.85 n3
0.14887
9.91 i4i
38
23
9
.76271
18
9.85 i4o
26
o. i4 860
9.91 i32
9
37
24
9
.76 289
18
9.85 166
27
o.i4834
9.91 123
9
36
25
9
.763o7
9.85 193
o. i4 807
9.91 n4
35
26
9
. 76 324
18
9.85 220
27
o. 14 780
9.91 105
9
34
27
9
.76342
18
9.85 247
26
o.i4 7^
>3
9.91 096
9
33
28
9
.7636o
9.85 273
o.i4 727
9.91 087
32
29
9
.76 378
9.85 3oo
o. i4 700
9.91 078
9
3i
30
9
.76395
I7
9. 85 327
o. i4 673
9.91 069
30
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
54° 3O .
PP
27
26
18
17
9 8
.1
2-7
2.6
.1
1.8
r-7
.1
0.9 0.8
.2
5-4
5-2
.2
3-6
3-4
.2
1.8 1.6
•3
8.1
7.8
3
5-4
5-1
•3
2.7 2.4
•4
10.8
10.4
•4
7.2
6.8
•4
3-6 3-2
[11
13.0
15-6
9.0
10.8
8-5
10.2
•5
.6
4-5 4-°
5-4 4-8
• 7
18.9
18.2
-7
12.6
ii. 9
•7
6.3 5-6
.8
21.6
20.8
.8
14.4
13-6
.8
7.2 6.4
-9
24--? 2}. 4
•9
16.2 15.3
•
8.1 7.2
35° 3O
r
L. Sin. d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
.76 395
9.85 327
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
26
27
27
26
27
26
27
27
26
o. 14 673
9.91 069
9
9
9
9
ro
9
9
9
9
30
3i
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
.764i3
.7643i
.76448
.76466
.76 484
.76 5oi
.76 519
.76537
.76554
18
17
18
18
'7
18
18
17
9. 85 354
9. 85 38o
9.85407
9.85434
9.85 46o
9. 85 487
9.85 5i4
9.85 54o
9.85567
o. i4 646
o. i4 620
o. i4 5g3
o.i4566
o. i4 54o
o.i4 5i3
o.i4486
0.14 46o
o.i4433
9.91 060
9.91 o5i
9.91 042
9.91 o33
9.91 oa3
9.91 oi4
9.91 oo5
9.90996
9.90987
29
28
27
26
25
24
23
22
21
40
9
.76572
9.85 594
o.i4 4o6
9.90978
9
9
9
9
9
9
9
9
10
9
9
9
9
9
9
10
9
9
9
9
20
4i
42
43
44
45
46
47
48
49
9.76 590
9.76 607
9.76 625
9.76 642
9.76 660
9.76677
9.76695
9.76 712
9.76 73o
17
18
'7
18
17
18
»7
18
9.85 620
9.85 647
9.85 674
9-85 700
9.85 727
9.85754
9-85 780
9.85 807
9.85 834
o.i4 38o
o.i4353
o.i4 326
o. 1 4 3oo
o.i4 273
o.i4 246
0. l4 220
o.i4 193
0.14 166
9.90969
9.90 960
9.9o95i
9.90 942
9.90 933
9.90 924
9.90915
9.90 906
9.90 896
J9
18
'7
16
i5
i4
i3
12
II
50
9
.76 747
9-85 860
o. 1 4 i4o
9.90 887
10
5i
52
53
54
55
56
5?
58
59
9
9
9
9
9
9
9
9
9
76765
76 782
76 800
.76817
.76835
.76852
.76 870
76 887
.76 904
'7
18
17
18
17
18
'7
17
0
9.85 887
9.85 913
9-85 940
9.85 967
9-85 993
9.86 020
9.86 o46
9.86 o73
9.86 joo
o.i4 n3
o. i4 087
o. i4 060
o. i4 o33
o. i4 007
o.i 3 980
o.i3 954
o. i3 927
o. 1 3 900
9.90 878
9.90 869
9.90 860
9.90 85i
9.90 842
9.90 832
9.90 823
9.90 8i4
9.90 805
9
8
7
6
5
4
3
2
I
~o~
60
9
.76 922
9.86 126
o.i 3 874
9.90796
L. Cos.
d.
L. Cotg. d.
L. Tang.
L. Sin.
d.
t
54°.
PP
.2
•3
•4
:5
27
26
18
17
.2
•3
•4
:!
:J
-.9
10
9
2.7
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
a* T,
2.6
5-2
7-8
10.4
13.0
15-6
18.2
20.8
23.4
.2
•3
•4
:i
:i
•9
1.8
3-6
5-4
7.2
9.0
10.8
12.6
sy
i-7
3-4
5-i
6.8
8-5
IO.2
II-9
I3.6
jia
I.O
2.0
3-o
4.0
|-°
6.0
7.0
8.0
9.0
ti
2.7
3-6
4-5
5-4
6-3
tj
101
36C
1
L.Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. i d.
0
9
.76922
9.86 126
27
26
27
26
27
26
27
26
27
27
26
27
26
27
26
27
26
26
27
26
27
26
27
26
27
26
27
26
26
27
o.i3 874
9.90 796
9
10
9
9
9
9
10
9
9
9
10
9
9
9
ro
9
9
9
10
9
60
I
2
3
4
5
6
8
9
9
9
9
9
9
9
9
9
9
.76939
.76 957
.76974
.76991
.77009
.77026
• 77043
.77 061
.77078
18
«7
17
18
17
*7
18
17
9.86 i53
9.86 179
9.86 206
9.86 232
9.86 259
9.86285
9.86 3i2
9.86338
9.86 365
o. i3 847
o.i3 821
o.i 3 794
o.i3 768
o.i3 74i
o.i3 715
o.i3688
o.i3 662
o,i3 635
9.90787
9.90777
9 .90 768
9.90759
9.90750
9.90 741
9.90731
9.90 722
9.90713
59
58
57
56
55
54
53
52
5i
10
9
.77095
J7
18
*7
»7
»7
18
*7
17
*7
9.86 392
o.i 3 608
9.90 704
50
ii
12
i3
i4
i5
16
'7
18
'9
9
9
9
9
9
9
9
9
9
•77 112
.77i3o
.77147
• 77 i64
.77181
•77 199
.77216
.77233
.77 25o
9.86418
9-86445
9.86471
9.86498
9. 86 524
9.8655i
9. 86577
9.86 6o3
9.86 63o
o.i3582
o.i3555
o.i3 529
o.i3 5o2
0.13476
o . 1 3 449
o.i3423
o.i3 397
o.i3 370
9.90 694
9.90 685
9.90 676
9.90 667
9.90 657
9.90 648
9.90 639
9.90 63o
9.90 620
49
48
47
46
45
44
43
42
4r
20
9
.77 268
9.86656
o.i3 344
9.90 61 1
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.77285
.77 3o2
.77319
.77336
.77353
.7737o
•77387
•77405
.77 422
*7
*7
17
*7
*7
V
18
»7
*7
9.86683
9.86 709
9.86 736
9.86 762
9.86 789
9.86 8i5
9.86842
9.86868
9.86 894
o.i3 317
o.i3 291
o.i 3 264
o.i3238
O.l3 211
o.i3 185
o.i3i58
o.i3 i32
o. 1 3 1 06
9.90 602
9.90 592
9.90 583
9.90 574
9. 90 565
9.90 555
9.90 546
9.90 537
9.90 527
10
9
9
9
10
9
9
10
39
38
37
36
35
34
33
32
3i
30
9
•77439
9.86 921
o. 1 3 079
9.90 5i8
y
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
53° 3O .
PP
.2
•3
4
:I
:i
9
27
26
18
17
.1
2
•3
•4
:l
•7
.8
•9
10
9
2.7
I:J
10.8
111
18.9
21.6
24.3
2.6
5-2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
. i
.2
3
•4
•5
.6
i
•9
1.8
3-6
5-4
7.2
9.0
10.8
12.6
XJ
i-7
3-4
5-1
6.8
8-5
10.2
II.Q
I3.6
15.3
I.O
2.0
3-°
4.0
5-°
6.0
7.0
8.0
9.0
0.9
1.8
2.7
3-6
4-5
5-4
6-3
g
102
30° 30 .
!
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
77439
9.86 921
26
o. 1 3 079
9.90 5i8
30
3i
9
77456
17 .
9.86 947
27
o.i3o53
9.90 509
29'
32
9
77473
9.86 974
26
o.i 3 026
9.9
0499
28
33
9
77490
17
9.87 ooo
o. 1 3 ooo
9.90490
9
27
17
27
IO
34
9
77 5o7
9.87027
26
0.12 973
9.90 48o
•26
35
9
77524
9.87 o53
26
0.12 947
9.90471
25
36
9
7754i
17
9.87079
O. 12 921
9.90 462
9
24
37
9
77558
9.87 106
27.
O. 12 894
9.90 452
IO
23-
38
9
77575
9.87 i32
26
0,12 868
9.90443
22
39
9
77 592
17
9.87 i58
0.12 842
9.90434
9
21
40
9
77609
9.87185
26
0.12 8i5
9.90424
20
4i
42
9
9
77626
77643
17
17
9.87 211
9.87 238
27
0.12 789
0.12 762
9.90415
9.90 4o5
y
10
'9
18
43
9
77 660
^
9.87 264
26
o. 12 736
9.90 396
9
10
17
44
9
.77677
9.87 290
0.12 710
9.90 386
16
45
9
77694
7
9.873i7
0.12 683
9.90377
i5
46
9
.77711
*7
9.87343
26
o. 12 657
9.90 368
y
10
i4
47
9
77728
16
9.87 369
o. 12 63i
9-90358
i3
48
9
77744
9.87 396
o. 12 6o4
9.90 349
12
49
9
77 761
*7
9.87 422
O. 12 578
9.90 339
II
50
9
77778
17
9. 87 448
0. 12 552
9.90 33o
9
10
5i
9
77 795
'7
9.87475
27
26
O. 12 525
9. 90 320
9
52
9
.77812
17
9.87 5oi
O. 12 499
9.90 3 1 1
8
53
9
.77829
17
9.87527
26
0.12473
9.90 3oi
10
7
54
9
.77846
17
9.87554
27
26
0.12 446
9.90 292
9
6
55
9
.77862
9.87 58o
O. 12 42O
9.90 282
5
56
9
.77879
17
9.87 606
0.12 394
9.90 273
y
4
57
9
.77896
17
9.87633
27
26
O. 12 367
9.90 263
10
3
58
9
.779i3
*7
9.87 659
0.12 34l
9.90 254
2
59
9
.7793o
17
9.87685
0.12 315
9.90 244
I
60
9
• 77946
16
9.87 711
0. 12 289
9.90 285
9
0
L. Cos.
d.
L. Cotg.
d. L. Tang.
L.
Sin.
d.
'
53°.
PP
27
26
17
,6
IO
9
.1
•2-7
2.6 .1
i.7
1.6
.,
1.0
O.Q
2
5-4
5-2 .2
3-4
3-2
.2
2.0
1.8
•3
8.1
7-8 -3
4.8
•3
3-o
2.7
•4
10.8
10.4 .4
6.8
6.4
•4
4.0
3-6
• 5
13-5
13.0 .5
8-5
8.0
4-5
6
16.2
15.6 .6
IO.2
9.6
.6
6.0
5-4
• 7
18.9
18.2 .7
ii 9
II. 2
-7
7.0
6-3 '
.8
21.6
20.8 .8
136
12.8
.8
8.0
7-2
•9
24-3 23.4 .9
15.3 14.4
•9
9.0
io3
37
,
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9
77946
9.87 711
0.12 289
9.90235
60
I
9
77963
17
9.87738
26
O. 12 262
9.9O 225
9
59
2
9
77980
9.87 764
O. 12 236
9.9o 216
58
3
9
77997
16
9.87 790
27
0.12 210
9.9o 206
9
$7
4
9
.78 oi3
17
9.87817
26
0.12 l83
9.9oi97
IO
56
5
9
.78 o3o
9.87843
O. 12 l57
9.90 187
55
6
9
.78047
17
16
9.87 869
26
0.12 l2
tl
9.90 178
9
10
54
7
9
.78 o63
17
9.87 895
O. 12 105
9.90 168
9
53
8
9
9
9
.78 080
.78097
»7
9.87 922
9.87948
26
26
0.12 078
O. 12 o52
9.9oi59
9.90 i49
10
52
5i
10
9
.78 n3
9-87 9?4
O. 12 026
9.9o i39
50
1 1
9
.78 i3o
9.88 ooo
O. 12 OOO
9.90 i3o
9
IO
49
12
9
.78147
16
9.88 027
o.ii 973
9.90 120
48
i3
9.78 i63
9.88 o53
o.ii 9.47
9.90 in
y
47
26
IO
i4
9
.78 180
17
9.88 079
26
o.ii 921
9.90 101
46
i5
9
.78 197
16
9.88 io5
o. 1 1 895
9.90 091
45
16
9
.78213
9.88 i3i
26
o.ii 869
9.90 082
y
44
"7
9
.78 23o
16
9.88 i58
27
26
o.ii 842
9.90 072
10
43
18
9
.78246
9.88 184
o.ii 816
9.90 o63
42
'9
9
.78263
9.88 210
26
o.ii 790
9.90 o53
4i
20
9
.78 280
16
9.88 236
o.ii 764
9.90 o43
40
21
9
.78 296
17
9.88 262
o.ii 738
9.90 o34
y
39
22
9
.78 3i3
16
9.88 289
o.ii 711
9.90 024
38
23
9
.78329
9.88 315
o.ii 685
9.90 oi4
37
17
26
9
24
9
.78 346
16
9. 8834i
26
o.ii 6v
>9
9.90005
36
25
9
.78 362
9.88 367
o.ii 633
9.89995
35
26
9
.78 379
9.88 393
o.ii 607
9.89 985
34
27
9
.78395
17
9.88420
27
26
o.ii 58o
9.89976
9
33
28
9
.78412
16
9.88446
o.ii 554
9.89 966
32
29
9
.78428
9.88 472
O.II 528
9.8
9956
3i
30
9
.78445
9.88 49>
O.II 502
9.8
994?
y
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
52° 30 .
PP
27
26
17
16
IO
9
.!
2.7
2.6
!
i-7
1.6
.1
1.0
0.9
2
5-4
5-2
2
3-4
3-2
.2
2.O
1.8
•3
8.1
7.8
3
4-8
•3
3-o
2-7
•4
10.8
10.4
4
6.8
6.4
•4
4.0
3-6
•5
»3-5
13.0
5
8-5
8.0
, e
5-O
45
.6
1 6. 2
15.6
6
IO.2
9.6
.6
6.0
5-4
• 7
18.9
18.2
7
II-9
II. 2
.7
7.0
6-3
.8
21.6
20.8
8
13-6
12.8
.8
8.0
z-2
•9
24-3 23.4
•9
15.} 14.4
•9
9.0 8.1
104
37° 30
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.73445
16
9.88 498
26
O. I I 5O2
9.89947
30
3i
9
78461
9.88 524
26
o.u 476
9.89937
29
32
9
78478
9.88 55o
O. I I 45°
9.89927
28
33
9
78494
16
9.88577
27
26
o. 1 1 423
9.89 918
y
IO
27
34
9
78 5io
9.88 6o3
26
o.u 397
9.89 908
26
35
78 527
9.88 629
o.u 371
9.8<
? 898
25
36
Q
78543
9.88655
o.u 345
9.8
?888
10
24
37
9
78 56o
»7
16
9.88 681
26
26
o.u 3i
9
9.89879
9
23
38
9
78576
9.88 707
o.u 293
9.89 869
22
39
9.78592
9.88 733
26
o. 1 1 267
9.89 859
10
21
40
9
78 609
16
9.88 759
O.U 24 I
9.89849
20
4i
9
.78625
17
9.88 786
26
O.U 2l4
9.89 84o
'9
42
9
.78 642
16
9.88 812
26
o.u 188
9.89 83o
18
43
9
78 658
16
9. 88 838
26
o.u 162
9.89 820
IO
17
44
9
.78674
9. 88 864
26
o.u i36
9.89 810
16
45
9
.78691
16
9.8889o
O.U 110
9.89 801
i5
46
9
.78 707
9.88 9i6
o.u 084
9.89791
i4
16
26
IO
47
9
.78723
16
9.88 942
26
o.u o58
9.89 781
i3
48
9
.78739
9.88 968
O.U 032
9.89 771
12
49
9
.78 756
9.88 994
26
26
o.u 006
9.89 761
I I
50
9
.78772
9.89 020
26
o. 10 980
9.89 752-
y
10
5i
9
.78 788
17
9.89 o46
27
o.io 954
9.89 742
9
52
9
.78 805
9.89 073
26
o.io 927
9.89732
8
53
9
.78821
16
9.89o99
26
o. 10 901
9.89 722
10
7
54
9
.78837
16
9.89 125
26
o.io 875
9.89 712
6
55
9
.78853
9.89 i5i
26
o.io 849
9.89 702
5
56
9.78869
17
9.89 177
26
0. 10 823
9.8
9693
9
IO
4
57
9.78 886
16
9.89 2o3
26
0.10797
9.89 683
3
58
9
.78 902
9.89 229
26
o.io 771
9.89 673
2
59
9
.78 918
9.89255
26
o. 10 745
9.89 663
I
60
9
,78934
9.89 281
o.io 719
9.8
9 653
0
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
52°.
PP
27
26
17
16
10
9
.1
2.7
2.6
.1
i.7
1.6
.1
I 0
0.9
.2
5-4
5-2
.2
3-4
3.2
.2
2 O
1.8
•3
8.1
7.8
•3
4.8
•3
3°
2.7
•4
10.8
10.4
•4
6.8
6.4
•4
40
3-6
13-5
13.0
•5
85
8.0
• 5
5 °
4-5
.6
16.2
15-6
.6
10 2
9.6
.6
6 o
5-4
.7
18.9
18.2
• 7
ii 9
II. 2
• 7
7.0
6-3
.8
21.6
20.8
.8
136
12.8
.8
8.0
7.2
24-3 23-4 -9
*5 3 *4-4
•9
io5
38°.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9.78 934
16
9.89
28l
26
0.
10719
9. 89 653
60
I
9.78 <)5o
17
9.89
3o7
26
o.
10 693
9.89
643
59
2
9.78967
16
9.89
333
26
o.
IO 667
9.89633
58
3
9.78 983
16
9.89
359
26
o.
10 64 1
9.89 624
IO
$7
4
9.78999
16
9.89
385
26
o.
10 615
9.89
6i4
IO
56
5
9.79015
16
9.89
4n
26
o.
10 589
9.89
6o4
55
6
9.79 o3i
16
9.89
437
o.
10 563
9.89
594
IO
54
7
9.79047
16
9.89
463
26
0.
io537
9.89
584
x
0
53
8
9.79 o63
9.89
489
26
o.
10 5n
9.89
574
52
9
9.79079
16
9.89
5i5
26
o.
10485
9.89
564
bi
10
9.79095
16
9.89
54i
26
o.
10 459
9.89
554
50
1 1
9-79 "
i
17
9.89
567
26
o.
io433
9.89
544
IO
49
12
9.79 128
16
9.89
593
26
o.
10 4o7
9.89
534
48
i3
9.79 i44
16
9.89
619
26
0.
10 38i
9.89
524
IO
47
i4
9.79 160
16
9.89 645
26
0.
10 355
9.89
5i4
IO
46
i5
9-79 176
16
9.89
671
0.
10 329
9.89
5o4
45
16
9-79 *92
16
9.89
697
26
0.
10 3o3
9.89
495
y
10
44
"7
9.79 208
16
9.89
?23
26
o.
I0277
9.89485
43
18
9.79 224
16
9.89
749
o.
10 25l
9.89
475
42
19
9.79 240
16
9.89
775
o.
10 225
9.89
465
4i
20
9.79 256
16
9.89
801
o.
10 199
9.89
45s
40
21
9.79272
16
9.89
827
26
o.
10 i73
9.89
445
IO
39
22
9.79 288
9.89
853
o.
10 147
9.89435
38
23
9.79 3o4
15
9.89
879
26
o.
IO 121
9.89425
IO
37
24
9-79 3i
9
16
9.89
9°5
26
o.
10 095
9.89
415
IO
36
25
9.79 335
9.89931
o.
10 069
9.89405
35
26
9.79 35i
16
9.89957
26
o.
10043
9.89
395
10
34
27
9-79367
16
9.89 983
26
o.
10 017
9.89
385
10
33
28
29
9. 79 383
9-79 399
16
16
9.90 009
9.90 035
26
o.
o.
09991
09 965
9.89
9.89
375
364
ii
32
3i
30
9.79415
9.90 06 1
0.
09 939
9.89
354
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
51° 30.
pp 26
17
16
15
ii
10
9
.1 2.6
*-7
1.6
i i-5
i.i
.j
I.O
0.9
.2 5.2
3-4
3.2
2 3.0
2.2
.2
2.0
1.8
•3 7-8
4.8
3 4-5
3-3
•3
3-°
2.7
.4 10.4
6.8
6.4
4 6.0
4.4
•4
4.0
3-6
•5 13-0
.6 15.6
8.5
10.2
8.0
9.6
5 7-5
6 9.0
11
.1
5-°
6.0
4-5
5-4
.7 18.2
II.9
II. 2
•7 10.5
7-7
• 7
7.0
6-3
.8 20.8
13-6
12.8
.8 12.0
8.8
.8
8.0
7.2
•9 23-4
15.3 14.4
•9 J3-5
•9
9.0
8.1
1 06
38° 3O'
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9-79415
16
9.90 061
0.09 939
9.89 354
30
3i
9
• 7943i
16
9.90 086
26
0.09 914
9.8
9344
29
32
9
.79447
16
9.90 112
26
0.09888
9.89 334
28
33
9
.79 463
15
9.90 1 38
26
0.09 862
9.89 324
IO
27
34
9
.79478
16
9.90 164
26
0.09836
9.8
93i4
26
35
9
.79494
16
9.90 190
26
o . 09 8 1 o
9.89 3o4
25
36
9
.79510
16
9.90 216
26
0.09 784
9.89 294
IO
24
37
9
.79 526
16
9.90 242
26
0.09 758
9.89 284
23
38
9
.79 542
9.90 268
26
0.09 732
9.89 274
22
39
9
.79 558
9.90 294
26
0.09 706
9.89 264
21
40
9
.79573
16
9. 90 320
26
0.09 680
9.89 254
20
4i
9.79589
16
9.90 346
25
0.09 654
9.8
9 244
19
42
9
.79605
16
9.90 371
_£
0.09 629
9.89 233
1 8
43
9
.79 621
15
9.90397
26
0.09 6o3
9.89 223
17
44
9
.79 636
16
9.90423
26
0.09577
9.89 2i3
16
45
9
.79652
16
9.90449
0.09 55 1
9.89 2o3
i5
46
9
.79 668
16
9.90475
26
0.09 525
9.89 193
10
i4
47
9
.79684
15
9.90 5oi
26
o . 09 499
9.89 i83
i3
48
9
.79699
9.90 527
0.09 473
9.89173
12
49
9
.79 716
9.90 553
0.09447
9.89 162
I I
50
9
•79 73'
9.90 578
25
26
0.09422
9.89 i52
10
5i
9
• 79 746
16
9.90 6o4
26
o . 09 396
9.89 142
9
52
9
.79 762
9.90 63o
26
0.09 370
9.89 i32
8
53
9
•79 77s
15
9.90 656
26
0.09344
9.89 122
10
7
54
9
.79793
16
9.90 682
26
0.09318
9.89 112
II
6
55
9
.79809
9.90 708
26
0.09 292
9.89 101
5
56
9
.79825
9.90 734
0.09 266
9.89 091
4
57
9
.79 84o
»5
16
9-90759
25
26
0.09 24 1
9.89 081
IO
3
58
9
.79856
9.90785
o . 09 2 1 5
9.89 071
2
59
9
.79872
9.90 811
0.09 189
9.89 060
I
60
9
.79887
9.90 837
0.09 1 63
9.8
9 o5o
0
L. Cos.
d.
L. Cotg
. d.
L. Tang.
L.
Sin.
d.
51°.
PP
26
25
16
'5
ii
IO
.1
2.6
2-5
.1
1.6
i.s
.,
i.i
1.0
.2
5-2
5-o
.2
3-2
3-°
.2
2.2
2.O
•3
7.8
7-5
•3
4.8
4-5
•3
3-3
3-°
•4
10.4
IO.O
•4
6.4
6.0
•4
4.4
4.0
j
13.0
15-6
12.5
15-0
1
8.0
9.6
7-5
9.0
:J
Ii
C
• 7
18.2
17-5
•7
II. 2
10.5
•7
7-7
7.0
.8
20.8
20. o
.8
12.8
12. 0
.8
8.8
8.0
23.4 22.5 .9
14.4 13.5
•9
9.9 9.0
107
39°.
!
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
0
9
.79887
16
9.90 837
26
0.09 i63
9 . 89 o5o
60
i
9
.79903
15
9.90 863
26
0.09 1 37
9.89 o4o
IO
59
2
9
•799'8
16
9.90 889
0.09 in
9.89 o3o
58
3
9
.79934
16
9.90 914
26
0.09 086
9.89 020
ii
D7
4
9
.79950
15
9.90 940
26
0.09 060
9.89 009
JO
56
5
9
.79965
16
9.90 96^
)
26
0.09 o34
0.88999
55
6
9
.79981
15
9.90992
26
o . 09 008
9.88989
II
54
7
9
.79996
16
9.91 018
25
0.08 982
9.8
8978
10
53
8
9
.80 012
9.91 o4-
26
0.08 957
9.88 968
52
9
9
.80 O27
16
9.91 069
26
0.08 9:
h
9.88 958
5i
10
9
.80043
15
9.91 ogS
26
0.08 905
9.88948
50
ii
9
.8oo58
16
9.91 121
26
0.08 879
9.88 937
49
12
9
.80 074
9.91 147
0.08 853
9.88 927
48
i3
9
.80089
16
9.91 172
26
0.08 828
9.88917
II
47
i4
9
.80 105
15
9.91 198
26
0.08 802
9.88 906
46
i5
9
.80 I2O
16
9.91 224
26
0.08 776
9.88 896
45
16
9
.80 i36
15
9.91 250
26
0.08 750
9.8
8 886
II
44
17
9
.80 i5i
15
9.91 276
25
0.08 724
9.8
8 875
43
18
9
.80 166
16
9.91 3oi
26
0.08 699
9.88 865
42
19
9
.80 182
9.91 32^
r
26
0.08 673
9.88 85s
4i
20
9
.80 197
16
9.91 353
26
0.08 647
9.88844
40
21
9
.80 2i3
15
9.91 379
25
0.08 621
9.8
8834
39
22
9.80228
16
9.91 4o^
i
26
0.08 596
9.88 824
38
23
9
.80 244
15
9.91 43o
26
0.08 570
9.88 8i3
IO
37
24
9
.80259
15
9.91 456
26
0.08 544
9.88 8o3
36
25
9
.80274
9.91 482
0.08 5i8
9.88 793
35
26
9
.80 290
9.91 507
25
0.08 493
9.88 782
34
15
26
IO
27
9
.80 3o5
15
9.91 533
26
0.08 467
9.88 772
33
28
9
.80 320
9.91 559
0.08 44 1
9.88 761
32
29
9
.80 336
9.91 58s
0.08 4i5
9.8
875i
3 1
30
9
.8o35i
9.91 610
25
0.08 390
9.8
74i
30
L. Cos.
d.
L. Cotg. | d.
L. Tang.
L.
Sin.
d.
50° 30 .
PP
26
25
16
15
ii
10
.1
2.6
2.5
.1
1.6
'•5
.1
i.i
I.O
>2
5-2
5.0
.2
3-2
3-°
.2
2.2
2.O
•3
7.8
7-5
•3
4.8
4-5
•3
3-3
3-°
•4
10.4
IO.O
-4
6.4
6.0
•4
4-4
4.0
.5
13.0
12.5
• 5
8.0
7' 5
5-5
5«o
.6
15.6
15.0
.6
9.6
9.0
.6
6.6
6.0
•I
18.2
20.8
17-5
20. 0
:l
II. 2
12.8
10.5
12.0
:!
11
7.0
8.0
•Q
23.4 22.5
•9
14.4
13.5
1 08
39° 3O .
«
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
.8o35i
9.91 610
26
26
26
25
26
26
26
25
26
26
25
26
26
26
25
26
26
25
26
26
25
26
26
25
26
26
25
26
26
25
0.08 390
9.88 741
30
3i
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
.80 366
.80 382
.80 397
.80412
.80428
.8o443
.80 458
.8o473
.80489
16
i5
15
16
15
i5
15
16
15
9.91 636
9.91 662
9.91 688
9.91 713
9.91 739
9.91 765
9.91 791
9.91 816
9.91 842
0.08 364
0.08 338
0.08 3i2
0.08 287
0.08 261
0.08 235
0.08 209
0.08 i84
0.08 i58
9.88 730
9.88 720
9.88 709
9.88 699
9.88688
9.88678
9.88668
9.88657
9.88 647
10
II
IO
II
10
IO
II
10
29
28
27
26
25
24
23
22
21
40
9
80 5o4
9.91 868
0.08 i32
9.88636
20
4i
42
43
44
45
46
4?
48
49
9
9
9
9
9
9
9
9
9
.80 519
.80 534
.80 550
.80565
.80 58o
.80595
.80 610
.80625
.8o64i
15
i5
16
15
15
15
15
IS
16
15
i5
15
15
»5
»5
15
16
15
15
IS
9.91 893
9.91919
9.91 945
9.91971
9.91 996
9.92 022
9.92 o48
9.92 073
9.92099
0.08 107
0.08 081
o.o8o55
0.08 029
o . 08 oo4
0.07 978
0.07 952
0.07 927
0.07 901
9.88 626
9.886i5
9.88 605
9.88 594
9.88 584
9.88673
9.88563
9.88 552
9.88 542
II
10
II
IO
II
IO
11
10
II
IO
II
II
IO
II
IO
II
10
II
II
i9
18
!?
16
i5
i4
i3
12
II
50
9
.80 656
9.92 125
0.07 875
9.88 53i
10
9
8
7
6
5
4
3
2
I
5i
52
53
54
55
56
57
58
59
9.80671
9.80686
9.80 701
9.80 716
9.80 731
9.80 746
9.80 762
9.80777
9.80 792
9.92 i5o
9.92 176
9.92 202
9.92 227
9.92 253
9.92279
9.92 3o4
9.92 33o
9.92 356
0.07 850
0.07 824
0.07 798
0.07 773
0.07747
0.07 721
0.07 696
0.07 670
0.07 644
9.88 521
9.88 5io
9.88499
9.88489
9.88478
9.88468
9.88457
9-88447
9.88436
60
0
.80807
9.92 38i
0.07 619
9.88425
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
5O°.
PP
.2
•3
•4
•5
.6
• 7
.8
26
25
16
15
.1
.2
•3
•4
9
II 10
2.6
5-2
7.8
10.4
13.0
156
18.2
20.8
2V4
2.5 .1
5.0 .2
7-5 -3
10.0 .4
12-5 .5
15.0 .6
17-5 -7
20.0 .8
22.5 .9
1.6
43:s
6.4
8.0
9.6
II. 2
12.8
M-4
1 5
3-°
45
6.0
7-5
9.0
10.5
12. 0
13-5
I.I 1.0
2.2 2.0
3-3 3-o
4-4 4-°
5-5 5-°
6.6 6.0
7-7 7-o
8.8 8.0
9.9 9.0
109
40<
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
0
9
.80807
15
15
15
15
15
15
15
15
15
9.92 38i
26
26
25
26
26
25
26
26
25
26
25
26
26
25
26
26
25
26
25
26
26
25
26
25
26
26
25
26
25
26
0.07 619
9.88425
10
ii
10
II
II
10
II
II
10
II
II
10
II
II
10
II
II
10
II
60
I
2
3
4
5
6
8
9
9
9
9
9
9
9
9
9
9
.80 822
.8o837
.80 852
.80 867
.80882
.80897
.80 912
.80 927
.80 942
9.92 407
9.92433
9.92458
9.92484
9.92 5io
9.92 535
9.92 56i
9.92 587
9.92 612
0.07 5g3
0.07 567
0.07 542
0.07 5i6
0.07 490
0.07465
0.07 439
0.07 4i3
0.07 388
9.88415
9.88404
9.88 394
9.88 383
9.88 372
9.88 362
9.8835i
9.88 34o
9.8833o
59
58
57
56
55
54
53
52
5i
10
9
.80957
J5
9.92 638
0.07 362
9.8
8 3i9
50
1 1
12
i3
i4
i5
16
i?
18
J9
9
9
9
9
9
9
9
9
9
.80 972
.80 987
.8l 002
.8l 017
.8l 032
.81 047
.81 061
.81 076
.81 091
15
i5
15
'5
15
*4
15
15
15
15
i5
15
i5
14
i5
15
i5
15
'4
9.92 663
9.92 689
9.92715
9.92 740
9.92 766
9.92792
9.92817
9.92 843
9.92 868
0.07 337
0.07 3i i
0.07 285
0.07 260
0.07 234
0.07 208
0.07 i83
0.07 157
0.07 i32
9.88 3o8
9.88298
9.88 287
9.88 276
9.88266
9.88 255
9.88 244
9.88 234
9.88 223
49
48
47
46
45
44
43
42
4i
20
9
.81 1 06
9.92 89^
1
0.07 106
9.88 212
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.8l 121
.81 i36
.81 i5i
.81 166
.81 1 80
.81 i95
.8l 210
.81 225
.81 240
9.92 920
9.92 945
9.92971
9.92996
9.93 022
9.93048
9.93o73
9.93099
9.93 124
0.07 080
0.07 o55
0.07 029
0.07 oo4
0.06 978
0.06 952
0.06 927
0.06 901
0.06 876
9.88 201
9.88 191
9.88 l8o
9.88 169
9.88 i58
9.88 i48
9.88 137
9.88 126
9.88 u5
IO
II
II
II
10
II
II
II
39
38
37
36
35
34
33
32
3i
30
9
.81 254
9.93 150
0.06 85o
9.8
8 105
30
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin. }d.
r
49° 30 .
PP
.1
.2
•3
•4
:l
• 7
.8
•9
26
25
15
M
.1
.2
•3
•4
•5
.6
• 7
.8
ii
IO
2.6
5-2
7.8
10.4
13.0
15.6
18.2
20.8
23-4
2.5
5-0
7-5
10.0
12.5
15-0
'7-5
20.0
22.5
.1
.2
• 3
•4
• 7
.8
r.S
3-o
4-5
6.0
7-5
9.0
10.5
12.0
*3-5
M
2.8
4-2
5.6
7.0
8.4
9-8
II. 2
i.i
2.2
3-3
4-4
5-5
6.6
7-7
8.8
9-9
1.0
2.O
3-0
4.0
5-°
6.0
7.0
8.0
9.0
40° 30 '
/
L. Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.81 254
9.93 150
25
26
26
25
26
25
26
25
26
26
25
26
25
26
25
26
25
26
26
. 0.06 85o
9.88 105
ii
ii
ii
ii
IO
II
II
II
II
II
II
10
II
II
II
II
II
II
II
II
30
3i
32
33
34
35
36
37
38
39
9.81 269
9.81 284
9.81 299
9.81 3i4
9.81 328
9.81 343
9.81 358
9.81 372
9.81 387
15
15
15
15
M
15
15
*4
15
9.93 i75
9.93 2OI
9.93 227
9.93 252
9.93278
9.93 3o3
9.93 329
9.93354
9.93 38o
0.06 825
0.06 799
0.06 773
0.06 748
0.06 722
0.06 697
0.06 671
0.06 646
0.06 620
9.88 094
9.88o83
9.88 072
9.88 061
9.88 o5i
9.88 o4o
9.88 029
9.88 018
9.88 007
29
28
27
26
25
24
23
22
21
40
9
.81 402
9.93 4o6
0.06 5g4
9.87 996
20
4i
42
43
44
45
46
4?
48
49
9
9
9
9
9
9
9
9
9
.81 417
.81 43i
.81 446
.81 46 1
.8i4?5
.81 490
.81 505
.81 519
81 534
M
15
15
M
15
15
'4
15
9.9343i
9.93457
9.93 482
9.93 5o8
9.93 533
9.93559
9.93 584
9.93 610
9.93636
0.06 569
o.o6543
0.06 5i8
0.06 492
0.06 467
0.06 44 1
0.06 4i6
0.06 390
0.06 364
9.87 985
9-87975
9.87 964
9.87953
9.87 942
9.8793i
9.87 920
9.87909
9.87 898
J9
18
!7
16
i5
i4
i3
12
II
50
9.81 549
9.93 661
26
25
26
25
26
25
26
25
26
25
0.06 339
9.87 887
10
5i
52
53
54
55
56
5?
58
59
9
9
9
9
9
9
9
9
9
81 563
81 578
81 592
81 607
81 622
.81 636
.81 65i
81 665
.81 680
15
14
IS
IS
14
»5
14
15
9.93 687
9.93 712
9.93738
9.93763
9.93789
9.93 8i4
9.93 84o
9.93865
9.93 891
0.06 3 1 3
0.06 288
0.06 262
0.06 237
0 . 06 2 I I
0.06 186
0.06 1 60
0.06 135
0.06 109
9.87877
9.87 866
9.87855
9.87844
9.87833
9.87 822
9.87 811
9.87 800
9.87789
II
II
II
II
II
II
II
II
II
9
8
7
6
5
4
3
2
I
60
.81 694
9.93 916
0.06 o84
9.87778
0
L. Cos. d.
L. Cotg
d. L. Tang.
L.
Sin.
d.
f
49°.
PP
.1
.2
•3
4
:J
26
«5
15
14
.2
•3
•4
•5
.6
:5
ii
10
2.6
5. 2
7.8
10.4
13.0
'5.6
18.2
20.8
23-4
2-5
5-o
7-5
IO.O
12.5
15-0
17-5
20. o
.1
.2
•3
•4
• 5
.6
:i
i-5
3-o
4-5
6.0
7-5
9.0
10.5
12.0
n-5
1.4
2.8
4.2
5-6
7.0
8.4
9.8
II. 2
1. 1
2.2
3-3
4-4
5-5
6.6
7-7
8.8
9.9
J.O
2.0
3-o
4.0
5-o
6.0
7.0
8.0
41°.
>
L. Sin. d.
L. Tang. d.
L. Cotg.
L. Cos.
d.
0
9.81 694
9.93 916
26
0.06 084
9.87778
60
I
9.81 709
14
9.93 942
25
0.06 o58
9.87767
5Q
2
9.81 723
9.93967
26
0.06 o33
9.87756
58
3
9.81 738
9.93993
25
0.06 007
9.87745
57
4
9.81 752
15
9 . 94 o 1 8
26
o.o5 982
9.87734
56
5
9.81 767
9.94 o44
0. 05956
9.87723
55
6
9.81 781
15
9.94069
26
o.o5 931
9.87712
54
7
9.81796
14
9.94095
25
o.o5 905
9.87 701
53
8
9.81 810
9.94 I2O
26
o.o5 880
9.87 690
52
9
9.81 825
9.94 i46
o.o5854
9.87679
5i
10
9.81 839
15
9-94 171
25
26
o.o5 829
9.87668
50
ii
9.81 854
9.94197
25
o.o5 8o3
9.87657
49
12
9.81 868
9.94 222
26
o.o5 778
9.87 646
48
i3
9.81 882
15
9.94 248
25
o.o5 752
9.87635
M
47
i4
9
.81 897
14
9.94 273
26
o.o5 727
9.87624
46
i5
9.81 911
9-94299
o.o5 701
9.87613
45
16
9
.81 926
9.94324
26
o.o5 676
9.87 601
44
17
9
.81 940
15
9.94350
25
o.o5 65o
9.87590
TI
43
18
9
.81955
9.94375
26
o.o5 625
9-87579
42
'9
9
.81 969
9 . 94 4o i
o.o5 599
9.87 568
4i
20
9
.81 983
9.94426
26
o.o5 574
9.87557
40
21
9.81998
14
9.94 452
25
o.o5 548
9.87 546
39
22
9
.82 OI2
9.94477
26
o.o5 523
9.87535
38
23
9
.82 026
9.94 5o3
o.o5 497
9.87 524
37
15
25
24
9
.82 o4i
9.94 528
26
o.o5 472
9.87613
12
36
25
9
.82 065
9.94554
o.o5 446
9.87 5oi
35
26
9
.82 069
I4
9.94579
o.o5 421
9.87490
34
15
25
27
9
.82 o84
9 . 94 6o4
26
o.o5 396
9.87479
II
33
28
9
.82 098
9.94 63o
o.o5 370
9.87468
32
29
9
.82 112
14
9.94655
26
o.o5 34s
9-87457
3i
30
9
.82 126
9.94 681
o.o5 319
9.87446
30
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
48° 30 .
PP
26
35
15
M
12
ii
.x
2.6
25
.!
1.5
1.4
.1
1.2
i.i
.2
5-2
5-o
.2
3-o
2.8
.2
2.4
2.2
•3
7.8
7-5
3
45
4-2
•3
3-6
3-3
•4
10.4
IO.O
•4
6.0
5-6
•4
4.8
4-4
13.0
12.5
7-5
7.0
.5
6.0
5-5
.6
15.6
15-0
.6
9.0
8.4
.6
7.2
6-. 6
• 7
18.2
I7-5
•7
10.5
9.8
•7
8.4
7-7
.8
20.8
20. o
.8
12.0
II. 2
.8
9.6
8-8
0
23.4 22.5
•9
13.5 12.6
•9
10.8 9.9
41° 3D .
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.82 126
9.94 68 1
o.o5 319
9- 87 446
30
3i
9.82 i4i
9.94 706
25
26
o.o5 294
9.87434
12
29
32
9.82 155
9.94732
o.o5 268
9.87423
28
33
9.82 169
15
9.94757
25
26
o.o5 243
9.87412
27
34
9.82 1 84
9.94 783
o.o5 217
9.87 4oi
26
35
9.82 198
9.94 808
o.o5 192
9.87 390
25
36
9.82 212'
14
9. 94834
o.o5 1 66
9.87 378
12
24
37
9.82 226
14
9-9485
9
25
o.o5 i
4i
9.87367
II
23
38
9.82 240
9.94884
o.o5 116
9.87 356
22
39
9.82 255
15
9.94 910
o.o5 090
9.87345
II
21
40
9.82 269
9.94935
26
o.o5 065
9.87334
II
20
4i
9.82 283
14
9.94961
25
o.o5 039
9.87 322
'9
42
9.82 297
9.94 986
26
o.o5 oi4
9.87 3n
18
43
9
.823n
9.95 OI2
o.o4 9
88
9.87 3oo
ll
15
25
12
44
9
.82 326
9.95 037
o.o4 963
9.87 288
16
45
9
.82 34o
9.95 062
o.o4 938
9.87 277
i5
46
9
.82 354
9.95 088
o.o4 912
9.87 266
i4
25
II
47
9
.82 368
9.95 n3
26
0.04887
9.87255
i3
48
9
.82 382
9.95 i3
)
o.o4 861
9.87 243
12
49
9
.82 396
14
9.95 1 64
25
26
o.o4836
9.87 232
II
I I
50
9
.82410
9.95 190
o.o4 810
9.87 221
10
5i
9
.82424
15
9.95 2i5
25
o.o4 785
9.87 209
9
52
9
.82439
9-95 24<
i
26
o.o4 760
9.87 I98
8
53
9
.82453
14
9 .95 266
o.o4 734
9.87187
11
7
*4
25
12
54
9
.82467
9.95 291
26
o.o4 709
9.8
7 J75
6
55
9
.82 48 1
9.953i7
o.o4 683
9.87 1 64
5
56
9
.82495
9.95 342
26
o.o4658
9.87 i53
12
4
57
9
.82 509
9.95 368
25
o.o4 632
9.87 i4i
3
58
9
.82 523
9.95 393
o.o4 607
9.87 i3o
2
59
9
.82 537
»4
9.95 4x8
25
26
o.o4 582
9.87 119
I
60
9
.8255i
14
9.95 444
o.o4556
9.87 107
0
L. Cos.
d.
L. Cotg.
d.
L, Tang.
L.
Sin.
d.
'
48°.
PP
26
35
'5
M
12
ii
.1
2.6
2.5
i
i.s
1.4
.i
1.2
i.i
.2
5-2
S-O
2
3-O
2.8
.2
2. 4
2.2
3
7.8
7-5
•3
4-5
4-2
• 3
3-6
3-3
4
10.4
IO.O
4
6.0
5.6
•4
4.8
4-4
• 5
13-0
12.5
^
7-5
7.0
-5
6.0
5-5
.6
15-6
15-0
6
9.0
8.4
.6
7.2
6.6
•7
18.2
17-5
7
10.5
9.8
.7
8.4
7-7
.8
20.8
20. o
8
I2.O
II. 2
.8
9.6
8.8
23.4 22.5
9
13.5 12.6
•9
10 8
n3
42°.
!
L. Sin.
d.
L. Tang1, d.
L. Cotg.
L. Cos.
d.
0
9.82 55i
14
*4
*4
M
J4
*4
J4
14
14
9.95 444
25
26
25
25
26
25
26
25
25
26
25
25
26
25
26
25
25
26
25
26
25
25
26
25
25
26
25
26
25
25
0.04 556
9.87107
ii
ii
12
II
12
II
II
12
II
60
i
2
3
4
5
6
8
9
9.82 565
9.82679
9.82 593
9.82 607
9.82 621
9.82635
9.82 649
9.82663
9.82 677
9.95 469
9.95495
9.95 52O
9.95545
9.9557i
9.95 596
9.95 622
9-95 647
9.95 672
o.o4 53i
o.o4 5o5
o.o4 48o
0.04455
o.o4 429
o.o4 4o4
o.o4 378
o.o4353
o.o4 328
9.87 096
9.87 085
9.87073
9.87062
9.87 o5o
9.87 039
9.87028
9.87 016
9.87 005
59
58
57
56
55
54
53
52
5i
10
9
.82 691
J4
14
'4
M
J4
14
13
M
X4
9.95 698
o.o4 3o2
9.86993
50
1 1
12
i3
i4
i5
16
r?
18
19
9
9
9
9
9
9
9
9
9
.82705
.82719
.82733
.82747
.82 761
.82775
.82 788
.82 802
.82 816
9.95 723
9.95 748
9<95 774
9.95 799
9.95825
9.95 85o
9.95875
9.95 901
9.95 926
o.o4 277
0.04 252
o.o4 226
O.O4 2OI
o.o4 175
o.o4 150
o.o4 125
o.o4 099
o.o4 074
9.86 982
9.86 970
9.86959
9.86 947
9.86936
9.86 924
9.86 913
9.86 902
9.86 890
12
II
12
II
12
II
II
12
49
48
47
46
45
44
43
42
4i
20
9
.82 83o
9-95952
o.o4 o48
9.86 879
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.82 844
.82858
.82872
.82885
.82 899
.82 913
.82 927
.82941
.82955
14
*4
13
»4
«4
J4
14
M
9-95977
9.96 002
9.96 028
9.96 o53
9.96 078
9.96 io4
9.96 129
9.96155
9.96 180
o.o4 023
o.o3 998
o.o3 972
o.o3 947
o.o3 922
o.o3 896
o.o3 871
o.o3845
o.o3 820
9.86 867
9.86855
9.86844
9.86832
9.86 821
9.86 809
9.86 798
9.86 786
9.86775
12
12
II
12
II
12
II
39
38
37
36
35
34
33
32
3i
30
9
.82 968
9.96 2o5
o.o3 795
9.86 763
30
L. COS.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
-
47° 3D .
PP
.2
•3
•4
•5
.6
:J
26
25
M
13
.2
•3
•4
•5
.6
•7
.8
•9
12
ii
2.6
Si
10.4
13.0
15.6
18.2
20.8
23.4
2-5
50
75
10.0
I2-5
15-0
i7-5
20.0
2
•3
•4
-5
.6
1
•9
3
4.2
5-6
7.0
8.4
9.8
II. 2
"•3
2.6
3.9
5-2
6-5
7.8
9.1
10.4
11.7
1.2
2-4
3-6
4.8
6.0
7.2
b
i.i
2. 2
3-3
4-4
1:1
U
u4
42° 3O
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
.82 968
14
14
14
13
14
*4
*4
*3
J4
M
M
13
14
*4
»3
*4
14
»3
14
9.96 2o5
26
25
25
26
25
25
26
25
25
26
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
o.o3795
9.86 763
30
3i
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
.82 982
.82 996
.83oio
.83023
.83o37
.83o5i
.83065
.83078
.83 092
9.96 23i
9.96 256
9.96 281
9.96 307
9.96 332
9.96357
9.96383
9.96408
9.96 433
o.o3 769
o.o3 744
o.o3 719
o.o3 693
o.o3 668
o.o3643
o.o3 617
o.o3 592
o.o3 567
9.86 762
9.86 740
9.86 728
9.86 717
9.86 705
9.86694
9.86 682
9.86 670
9.86 659
12
12
II
12
II
12
12
II
12
29
28
27
26
25
24
23
22
21
40
9
.83 106
9.96459
o.o3 54i
9.86 647
20
4i
42
43
44
45
46
4?
48
49
9
9
9
9
9
9
9
9
9
.83 120
.83 i33
.83 147
.83 161
.83 i74
.83 188
.83 202
.832i5
.83 229
9.96484
9.96 5io
9.96535
9.96 56o
9.96 586
9.96611
9.96 636
9.96662
9.96 687
o.o3 5i6
o.o3 490
o.o3465
o.o3 44o
o.o34i4
o.o3 389
o.o3 364
o.o3338
o.o3 3i3
9.86635
9.86624
9.86 612
9.86 600
9.86589
9.86577
9.86565
9. 86 554
9. 86 542
II
12
12
II
12
12
II
12
i9
'7
16
i5
i4
i3
12
II
50
9
.83 242
9.96 712
o.o3 288
9.86 53o
10
5i
52
53
54
55
56
5?
58
59
9. 83 256
9.83 270
9.83283
9. 83 297
9.83 3io
9.83324
9.83338
9.8335i
9.83 365
H
i3
14
»3
H
M
13
*4
»3
9.96 738
9.96 763
9.96 788
9.96 8i4
9.96 839
9.96 864
9.96 890
9.96915
9.96 940
o.o3 262
o.o3 237
O.o3 212
o.o3 186
o.o3 161
o.o3 i36
o.o3 1 10
o.o3o85
o.o3 060
9.865i8
9.86 507
9.86 495
9. 86 483
9. 86 472
9.8646o
9. 86 448
9. 86 436
9. 86 425
II
12
12
II
12
12
12
II
9
8
7
6
5
4
3
2
60
9
.83378
9.96 966
o.o3 o34
9.864i3
0
L. Cos. d.
L. Cotg. d.
L. Tang.
L.
Sin.
d.
'
47°.
PP
.1
2
•3
• 4
:l
.7
.8
•9
26
25
14
13
.1
.2
<3
•4
:i
12 II
-2.6
5-2
7.8
10.4
13.0
15-6
18.2
20.8
2:5.4
2.5 .1
5-O .2
7-5 -3
10.0 .4
12.5 .5
15.0 .6
17-5 -7
20.0 8
LI
4.2
5-6
7-o
8.4
9.8
II. 2
12.6
1:1
3-9
Is
7.8
9.1
10.4
11.7
1.2 I.I
2.4 2.2
3-6 3-3
4-8 4.4
6'° I'i
7.2 6.6
8.4 7.7
9.6 8.8
10.8 9.9
i5
43C
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
0
9
,83378
9.96 966
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
o.o3 o34
9.864i3
60
2
3
4
5
6
8
9
9.83 392
9.834o5
9.83419
9.83432
9.83446
9.83459
9.83473
9.83486
9.83 500
*3
*4
i3
14
13
H
13
J4
9.96991
9.97 016
9.97 o4a
9.97067
9.97 092
9.97 118
9-97 '43
9.97 168
9.97 i93
o.o3 009
0.02 984
0.02 958
O.O2 933
O.O2 908
0.02 882
0.02 857
0.02 832
O.O2 807
9.86 4oi
9.86 389
9.86377
9.86 366
9.86 354
9.86 342
9.86 33o
9.86 3i8
9.86 3o6
12
12
II
12
12
12
12
12
59
58
57
56
«
53
52
5i
10
9
.83 5i3
J3
9.97219
O.O2 781
9.86 295
50
1 1
12
i3
i4
i5
16
17
18
«9
9.83 527
9.83 54o
9.83554
9.83567
9. 8358i
9.83594
9.83 608
9.83 621
9.83634
i3
N
13
H
*3
H
13
13
9.97 244
9.97 269
9.97295
9.97 320
9.97345
9.97371
9.97 396
9-97 42i
9,97447
0.02 756
0.02 731
0.02 705
O.O2 680
O.O2 65g
O.O2 629
0.02 6o4
0.02 579
0.02 553
9. 86 283
9.86 271
9.86 259
9.86 247
9.86235
9.86223
9.86 211
9.86 2OO
9.86 188
12
12
12
12
12
12
II
12
12
49
48
47
46
45
44
43
42
4,i
20
9
.83648
9.97472
O.02 528
9.86 176
40
21
22
23
a4
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.8366i
.83674
.83688
.83 701
.83 715
.83 728
.8374i
.83 755
.83 768
»3
*4
»3
r4
*3
»3
H
*3
13
9.97497
9.97 523
9-97548
9.97573
9.97 598
9.97 624
9.97649
9.97 674
9-97 7<>o
0.02 5o3
O.O2 477
O.O2 452
O.O2 427
O.O2 402
O.O2 376
0.02 35i
O.O2 326
0.02 3oo
9.86 1 64
9.86 162
9.86 i4o
9.86 128
9.86 116
9.86 io4
9.86 092
9.86 080
9.86 068
12
12
12
12
12
12
12
12
12
39
38
37
36
35
34
33
32
3r
30
9
.83 781
9-97 725
O.02 275
9.86 o56
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
46° 30 .
PP
i
2
3
4
5
6
• 7
.8
•9
26
25
14
13
.1
.2
•3
•4
• 5
.6
.7
.8
12
ii
2.6
%
10.4
13.0
15-6
18.2
20.8
23.4
2-5
5-o
7-5
IO.O
12.5
15-0
17-5
20. o
.2
•3
•4
:I
•7
.8
•9
a
4.2
5.6
7.0
8.4
9.8
II. 2
12.6
1.3
2.6
3-9
5-2
6.5
7.8
9.1
10.4
11.7
1.2
2-4
3-6
4.8
6.0
7-2
8.4
9.6
10.8
1. 1
2.2
3-3
4-4
II
7-7
8.8
Q.Q
n6
43° 3O
I
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
83 781
9-97725
25
26
25
25
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
25
26
25
25
25
26
O.O2 275
9-
86o56
30
3i
32
33
34
35
36
37
38
39
9. 83 795
9. 83 808
9.83 821
9.83834
9.83 848
9-83 861
9.83 874
9.83 887
9-83 901
13
13
13
13
13
9.97750
9.97 776
9.97 801
9.97 826
9.97 85i
9.97877
9.97902
9.97927
9.97953
0.02 250
O.O2 224
O.O2 199
O.O2 174
O.O2 iJcj
O.O2 123
O.O2 098
O.O2 O73
0.02 o47
000 000 OOO
86o44
86o32
86 020
86 008
85 996
85 984
85 972
85 960
85 948
12
12
12
12
12
12
12
12
29
28
27
26
25
24
23
22
21
40
9
83 9i4
13
9.97 978
0.02 022
9
85936
20
4i
42
43
44
45
46
47
48
49
9
9
9
9
9
9
9
9
9
83 927
83 940
83 954
83 967
83 980
83 993
84 006
84 020
84o33
13
14
13
13
9.98 oo3
9.98 029
9.98 o54
9.98079
9.98 io4
9.98 i3o
9.98 i55
9.98 180
9.98 206
o.oi 997
o.oi 971
o.oi 946
o.oi 921
o.oi 896
o.oi 870
o.oi 845
o.oi 820
o.oi 794
ooo ooo ooo
85 924
85 912
85 900
85888
85 876
85 864
8585i
85839
85827
12
12
12
12
12
'3
12
12
12
12
12
12
13
12
12
12
12
12
13
'9
18
17
16
i5
i4
i3
12
I I
50
9
84 o46
9.98 23i
o.oi 769
9
85 8i5
10
9
8
7
6
5
4
3
2
I
5i
52
53
54
55
56
57
58
59
9.84059
9.84 072
9.84 o85
9.84 098
9.84 112
9.84 125
9.84 1 38
9.84 1 5 1
9.84 1 64
13
'3
'3
'3
'3
9.98 256
9.98 281
9.98 307
9.98 332
9.98357
9. 98 383
9.98408
9.98433
9.98 458
o.oi 744
o.oi 719
o.oi 693
o.oi 668
o.oi 643
o.oi 617
o.oi 592
o.oi 567
O.OI 542
9
9
9
9
9
9
9
9
858o3
8579i
85779
85 766
85754
85 742
8573o
85 718
85 706
60
9
.84177
9. 98 484
o.oi 5i6
9
85693
0
L. Cos. d.
L. Cotg. d.
L. Tang.
L. Sin.
d.
f
46°.
PP
.1
.2
•3
•4
•5
.6
•7
.8
26
25
.2
•3
•4
14
13
12
2.6
S-2
7.8
10.4
13.0
15-6
18.2
20.8
23-4
2-5
5-o
7-5
10.0
12.5
15.0
17-5
20. o
1.4
2.8
4-2
5.6
7.0
8.4
9.8
II. 2
1.3
2.6
3-9
5-2
6-5
7.8
9.1
10.4
11.7
.1
.2
•3
•4
•7
.8
1.2
2-4
3-6
4.8
6.0
7-2
8.4
9.6
10.8
117
44°.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
0
9
84i?7
13
9.98 484
o.oi 5i6
9-
85 693
60
!
9
84 190
13
9.9
8 509
25
o.oi 491
9
8568i
12
59
2
9
842o3
9.98534
o.oi 466
9
85 669
58
3
9
84216
9.98 56o
o.oi 44o
9
85 657
57
J3
25
4
9
84 229
13
9.9
858s
25
o.oi 4i5
9
85 645
13
56
5
9
84242
9.98 610
o.oi 390
9
85 632
55
6
9
.84255
M
9. 98 635
25
26
o.oi 365
9
85 620
12
54
7
9.84 269
1-3
9.98 661
25
O.OI 339
9
85 608
12
53
8
9
.84282
9.98 686
o.oi 3i4
9
85 596
52
9
9
.84295
J3
9.98 711
25
_fC
o.oi 289
9
85583
J3
5i
10
9
.843o8
9.98 737
o.oi 263
9
8557i
50
ii
9
.84 3ai
13
9.98 762
25
o.oi 238
9
85 559
49
12
9
.84334
9.98787
O.OI 2l3
9
85 547
48
i3
9
.84347
13
9.98 812
25
26
o.oi 188
9
85534
I3
12
47
i4
9
.8436o
13
9.98838
o.oi 162
9
85 522
46
i5
9
.84373
9.9
8 863
25
o.oi 137
9
85 5io
45
16
9
.84 385
9.98 888
25
O.OI 112
9
.85497
*3
44
i?
9
.84398
*3
13
9.98 913
25
26
o.oi 087
9
.85485
12
43
18
9
.844ii
9.98939
o.oi 06 i
9
.85473
42
'9
9
.84424
9.98 964
25
o.oi o36
9
.85 46o
J3
4i
20
9
.84437
9.98989
25
„/:
O.OI OI I
9
85448
40
21
9
.8445o
I3
9.99015
25
o.oo 985
9
.85436
39
22
9
.84463
9.99 o4o
o.oo 960
9
.85423
38
23
9
.84476
9.99 o65
25
o.oo 935
9
.854ii
37
X3
25
12
24
25
9
9
.84489
.84 5o2
13
9.99090
9.99 1 1 6
26
o . oo 910
0.00884
9
9
.85 399
.85386
13
36
35
26
9
.845i5
9.99 i4i
25
o.oo 859
9
.85374
34
27
9
.84528
*3
9.99 1 66
25
o.oo 834
9
.8536i
*3
33
28
9
.8454o
9.99 191
o.oo 809
9
.85349
32
29
9
.84553
'3
9.99 217
o.oo 783
9
.85 337
12
3f
30
9
.84566
9.99 242
25
o.oo 758
9
.85 324
J3
30
L. Cos. d.
L. Cotgr.
d.
L. Tang.
L. Sin. d.
/
45° 30 .
PP
26
«5
M
13
12
.1
2.6
2-5
.1
1-4
1.3
.1
1.2
.2
5-2
5-o
.2
2.8
2.6
.2
2.4
•3
7.8
7-5
• 3
4.2
3-9
•3
3-6
•4
10.4
10.0
•4
5-6
5-2
•4
4-8
•5
13.0
12.5
•5
7.0
6.5
.5
6.0
.6
15-6
15.0
.6
8.4
7.8
.6
7.2
•7
18.2
17.5
• 7
9-8
9-1
•7
8.4
.8
20.8
20.0
.8
II. 2
10.4
.8
9.6
•9
23.4 22.5
.9 12.6
"7
.9 10.8
118
44° 3O
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
.84 566
13
*3
i3
13
12
13
»3
13
r3
12
»3
13
'3
12
13
»3
»3
f2
*3
'3
'3
12
13
*3
12
'3
13
12
'3
9.99 242
25
26
25
25
25
26
25
25
25
26
25
25
25
26
25
25
26
25
25
25
26
25
35
25
26
25
25
25
26
25
o.oo 758
9
.85 324
30
3 1
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
.84579
.84 592
.84 605
.846i8
.8463o
.84643
.84656
.84669
.84682
9.99 267
9 99 293
9.99 3i8
9.99 343
9.99 368
9.99394
9.99419
9.99 444
9.99469
o.oo 733
o.oo 707
o.oo 682
o.oo 657
o.oo 632
o.oo 606
o.oo 58 1
o.oo 556
o.oo 53i
9
9
9
9
9
9
9
9
9
.853i2
.85 299
.85 287
.85274
.85 262
.85 250
.85 237 -
.85 225'
.85 212
*3
12
13
12
12
»3
-12
»3
29
28
27
26
25
24
23
22
21
40
9
.84 694
9.99495
o.oo 5o5
9
.85 200
20
4i
42
43
44
45
46
4?
48
49
9.84 707
9.84 720
9.84733
9-84745
9. 84 758
9-84 771
9.84784
9.84 796
9.84 809
9.99 52O
9.99 545
9.99 570
9.99 596
9.99 621
9.99 646
9.99672
9.99697
9.99722
o.oo 48o
o.oo 455
o.oo 43o
o . oo 4o4
o.oo 379
o.oo 354
O.OO 328
o.oo 3o3
o.oo 278
9
9
9
9
9
9
9
9
9
.85 187
.85175
.85 162
.85 150
.85 i37
.85 125
.85 112
.85 100
.85087
13
12
13
12
13
12
13
12
*3
;89
17
16
i5
i4
i3
12
I I
50
9
.84 822
9.99 747
o.oo 253
9
.85 074
10
5i
52
53
54
55
56
5?
58
59
9
9
9
9
9
9
9
9
9
.84835
.84847
.8486o
.84873
.84885
.84898
.84911
.84923
.84936
9-99773
9.99798
9.99 823
9.99848
9.99874
9.99899
9.99924
9.99949
9-99975
o.oo 227
O.OO 2O2
o.oo 177
O.OO I 52
o.oo 126
O.OO IOI
o.oo 076
o.oo o5i
O.OO O25
9
9
9
9
9
9
9
9
9
.85 062
.85 049
.85o37
.85o24
.85 012
.84 999
.84986
.84974
.84961
»3
12
13
12
»3
13
12
13
9
8
7
6
5
4
3
2
I
60
9
.84949
r3
0.00 OOO
0.00 OOO
9
.84949
0
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
45°.
PP
.2
•3
•4
:!
:l
•9
26
25
. i
.2
-3
•4
:I
•9
14
13
.1
.2
•3
•4
• 5
.6
•7
.8
•9
12
2.6
5-2
7.8
10.4
13.0
15-6
18.2
20.8
23-4
2.5
5-o
7-5
IO.O
12.5
15.0
17-5
20. o
22-5
a
4.2
5.6
7.0
8.4
9.8
II. 2
12.6
i-3
2.6
3-9
5-2
6-5
7.8
9.1
10.4
11.7
1.2
2-4
3-6
4.8
6.0
7-2
8.4
9.6
10.8
119
TABLE III
FIVE-PLACE LOGARITHMS
OF THE
SINE AND TANGENT OF
SMALL ANGLES
THE SINE AND TANGENT TO EVERY SECOND FROM O° TO 8' J TO EVERY
TEN SECONDS FROM O° TO 2°.
THE COSINE AND COTANGENT TO EVERY SECOND FROM 90° TO 89°
52' ; TO EVERY TEN SECONDS FROM 90° TO 88°.
0°.
FUNCTIONS OF SMALL ANGLES
LOGARITHMIC SINE AND TANGENT.
0"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
0 o
10
20
5- 68557
98660
68557
72697
*°°779
98660
76476
$02800
#16270
79952
*0473Q
,28763
83170
1*06579
#38454
86167
#08351
#46373
88969
#IO°55
*53o°7
91602
,11694
#58866
94085
#13273
#63982
96433
**4797
*68557
98660
^16270
50
40
30
30
40
5°
6. 16270
28763
38454
29836
39315
19072
30882
40158
20409
3i904
40985
21705
32903
41797
22964
33879
42594
24188
34833
43376
25378
35767
44145
26536
36682
44900
27664
37577
45643
28763
38454
46373
20
IO
o 59
t o
IO
20
6.46373
6. 5 3067
8866
7090
3683
9406
7797
4291
9939
8492
4890
#0465
9»75
548i
#0985
ft9
6064
**499
•ss
*2007
*"6s
7207
*2509
*'8f
7767
*3°o6
*2442
8320
*3496
*3°67
8866
*39g2
50
40
3°
30
40
50
6.63982
* 8F
6.72697
4462
8990
3090
4936
9418
3479
5406
9841
3865
5870
#0261
4248
633°
^0676
4627
678!
*io88
5003
7235
*i496
5376
7680
*i90Q
5746
8121
*23°o
6112
8557
#2697
6476
20
10
o 58
2 o
10
20
6476
9952
6.83170
6836
*028§
3479
7»93
*o6i§
3786
7548
*°943
4091
7900
*i268
4394
8248
*i59'
4694
8595
*i9"
4993
8938
*2230
5289
9278
*2545
5584
9616
+2859
5876
9952
*3I7°
6167
50
40
30
30
40
So
6167
896g
6.9 1602
6455
9240
1857
6742
9509
2110
7027
9776
2362
73io
#0042
2612
759i
#0306
2861
7870
*o568
3109
8l47
*0829
3355
8423
*io88
3599
8697
**346
3843
8969
^1602
4085
20
IO
o 57
3 o
10
20
4085
6433
8660
4325
6661
8877
4565
6888
9°93
4803
7113
93°7
5039
7338
9520
5275
7561
9733
55°9
7783
9944
5742
8004
*°i55
5973
8224
#0364
6204
8443
*0572
6433
8660
*0779
50
40
30
3°
40
50
7.00779
2800
4730
0986
2997
49'9
1191
3193
5106
1395
3388
5293
1599
3582
5479
1801
3776
5664
2003
3968
5849
2203
4160
6032
2403
4351
6215
2602
454i
6397
2800
473°
6579
20
IO
o 58
4 o
10
20
6579
8351
7. 1 0055
6759
8525
O222
s
7118
8870
0553
7296
9041
0718
7474
9211
0882
7651
9381
1046
7827
955i
1209
8003
9719
1371
8177
9887
1533
8351
*°°55
1694
50
4°
30
3°
40
5°
1694
3273
4797
1854
3428
4947
2014
3582
5096
2174
3736
5244
2333
3889
5392
2491
4042
5540
2648
4194
5687
2805
4346
5833
2962
4497
5979
3»8
4647
6125
3273
4797
6270
20
IO
o 55
5 o
IO
20
7.16270
7694
9072
6414
7834
9208
6558
7973
9343
6702
8112
9478
6845
8250
9612
6987
8389
9746
7130
8526
9879
7271
8663
#OOI2
7413
8800
*oi45
7553
8937
#0277
7694
9072
#0409
50
40
30
30
4°
5"
7.20409
1705
2964
0540
1833
3088
°67i
1960
3212
0802
2087
3335
0932
2213
3458
1062
2339
358o
1191
2465
3702
I320
2590
3824
1449
2715
3946
1577
2840
4067
i705
2964
4188
20
IO
o 54
6 "
10
20
4188
5378
6536
4308
5495
6650
4428
5612
6764
4548
5728
6877
4668
6991
4787
596i
7104
4906
6076
7216
5024
6192
7329
SH2
6307
744 >
5260
6421
7552
5378
6536
7664
50
40
3°
3°
40
So
7 o
IO
20
i
7775
8872
9942
7886
8980
*°°47
7997
9088
#0152
8107
9196
*0257
8217
9303
*0362
8327
9410
+0467
8437
9517
*°571
8546
9623
*o675
8655
9730
*°779
8763
9836
#0882
20
IO
o 53
7.30882
1904
2903
0986
2005
3001
1089
2106
3100
1191
2206
3*98
1294
2306
3296
1396
2406
3393
1498
2506
349 »
1600
2606
3588
1702
2705
3685
1803
2804
3782
1904
2903
3879
50
40
30
3°
40
50
•M
3879
4833
5767
««M^^M»
10"
3975
•— «•
4071
5022
5952
4167
5"6
6044
m^—m—i
4263
5209
6i35
— «^—
4359
5303
6227
4454
5396
6318
«__«
4549
5489
6409
4644
5582
6500
4739
5675
6591
— —
4833
5767
6682
— —
20
IO
o 52
•^•—
9"
8"
7"
6"
5"
4"
3"
2"
1"
0"
LOGARITHMIC COSINE AND COTANGENT.
89C
FUNCTIONS OF SMALL ANGLES.
'
L. Sin. L. Tang.
L. Sin.
L. Tang.
0 o
IO
20
3o
4o
5o
5.68557
5.98 660
6. 16 270
6.28 763
6.38454
5.68557
5.98 660
6. 16 270
6.28763
6.38454
o 60
5o
4o
3o
20
10
73o
4o
5o
7. 33 879
7.34833
7.35767
7. 33 879
7.34833
7.35767
3o
20
IO
8 o
IO
20
3o
4o
5o
7.36 682
7.37577
7.38454
7.393i4
7.40 1 58
7.40985
7.36682
7.37577
7-38455
7-393i5
7.40 i58
7.40985
o 52
5o
4o
3o
20
IO
1 o
IO
20
3o
4o
5o
6.46 373
6.53o67
6.58866
6. 63 982
6.68557
6.72 697
6.46 373
6.53o67
6.58866
6. 63 982
6.68557
6.72 697
o 59
5o
4o
3o
20
IO
9 o
10
20
3o
4o
5o
7-4i 797
7.42 594
7.43376
7.44 i4s
7.44 900
7.45643
7-4i 797
7.42 594
7.43376
7.44i45
7-44 900
7.45643
o 51
5o
4o
3o
20
IO
2 o
IO
20
3o
4o
5o
6.76476
6.79952
6.83 170
6.86 167
6.88 969
6.91 602
6.76476
6.79 952
6.83 170
6.86 167
6.88 969
6.91 602
o 58
5o
4o
3o
20
10
10 o
IO
20
3o
4o
5o
7.46373
7.47 090
7.47797
7.48491
7.49 175
7.49849
7.46373
7.47091
7.47797
7.48492
7.49 176
7.49 849
o 50
5o
4o
3o
20
10
3 o
10
20
3o
4o
5o
6.94085
6.96433
6.98 660
7.00779
7.02 800
7.04 73o
6.94085
6.96433
6.98 660
7.00779
7.02 800
7.04 73o
o 57
5o
4o
3o
20
10
11 o
IO
20
3o
4o
5o
7-5o 5i2
7-5i 165
7.5i 808
7.52 442
7.53067
7.53683
7«5o 5i2
7.5i i65
7. 5 1 809
7.52443
7.53 067
7.53683
o 49
5o
4o
3o
20
10
4 o
10
20
3o
4o
5o
7.o6579
7.o835i
7.10 o5s
7.11 694
7.i3273
7.14 797
7.06 579
7. 08 352
7.10 o55
7.11 694
7.i3273
7.14797
o 56
5o
4o
3o
20
10
12 o
10
20
3o
4o
5o
7.54 291
7.54890
7.5548i
7.56 o64
7.56639
7.57 206
7.54 291
7.54 890
7.5548i
7.56o64
7.56639
7.57207
o 48
5o
4o
3o
20
IO
5 o
IO
20
3o
4o
5o
7. 16 270
7- 1?694
7. 19 072
7.20 409
7.21 7o5
7.22 964
7. 16 270
7.17694
7.19073
7.20 409
7.21 705
7.22 964
o 55
5o
4o
3o
20
10
13 o
IO
20
3o
4o
5o
7.57767
7.58 320
7-58866
7.59 4o6
7.59939
7.60 465
7.57767
7.58 320
7.58867
7.59 4o6
7.59939
7.60466
o 47
5o
4o
3o
20
10
6 o
10
20
3o
4o
5o
7.24 188
7.25 378
7.26 536
7.27 664
7.28763
7.29 836
7.24 188
7.25 378
7.26536
7.27 664
7.28764
7.29 836
o 54
5o
4o
3o
20
10
14 o
10
20
3o
4o
5o
7.60985
7.61 499
7.62 007
7.62 5og
7.63 006
7.63496
7.60 986
7.61 500
7.62 008
7.62 5io
7. 63 006
7-63 497
o 46
5o
4o
3o
20
IO
7 o
IO
20
3o
7.3o 882
7.3i 904
7.32 go3
7. 33 879
7.30882
7. 3 1 904
7 . 32 go3
7. 33 879
o 53
5o
4o
3o52
15 o
7.63 982
7.63 982
o 45
L. Cos.
L. Cotg.
a i
L. Cos.
L. Cotg.
n ,
123
89°.
FUNCTIONS OF SMALL ANGLES.
0°.
r rt
L. Sin.
L. Tang.
, „
L. Sin.
L. Tang.
15 o
10
20
3o
4o
5o
7.63 9»2
7-64461
7.64936
7.654o6
7.65 870
7.6633o
7.63 982
7.64462
7.64937
7.65 4o6
7.65871
7.6633o
o 45
5o
4o
3o
20
10
22 3o
4o
5o
7 .81 591
7.81 911
7.82 229
7.81 59i
7.81 912
7.82 23o
3o
20
IO
23 o
IO
20
3o
4o
5o
7.82 545
7.82 859
7. 83 i7o
7.83479
7.83786
7.84 091
7.82 546
7.82860
7.83 171
7.8348o
7. 83 787
7 .84 092
o 37
5o
4o
3o
20
10
16 o
10
20
3o
4o
5o
7.66 784
7.67235
7.67 680
7.68 121
7.68557
7.68 989
7.66785
7.67235
7.67680
7.68 121
7.68558
7.68 990
o 44
5o
4o
3o
20
IO
24 o
IO
20
3o
4o
5o
7.84 393
7-84694
7.84 992
7.85 289
7.85583
7.85876
7-84394
7.84695
7.84993
7.85 290
7.85 584
7. 85 877
o 36
5o
4o
3o
20
IO
17 o
IO
20
3o
4o
5o
7-69417
7.69 84i
7. 70 261
7.70676
7.71 088
7-71 496
7.69418
7.69 842
7.70 261
7.70677
7.71 088
7-71 496
o 43
5o
4o
3o
20
10
25 o
IO
20
3o
4o
5o
7.86 166
7. 86 455
7 86 741
7.87 026
7.87 309
7.87590
7.86 167
7.86456
7.86743
7.87027
7.87310
7.8759i
o 35
5o
4o
3o
20
IO
18 o
10
20
3o
4o
5o
7.71 900
7.72 3oo
7.72697
7.73090
7.73479
7-73865
7.71 900
7,72 3oi
7.72697
7.73 090
7.7348o
7.73866
o 42
5o
4o
3o
20
10
26 o
10
20
3o
4o
5o
7.87870
7.88 147
7-88423
7.88697
7.88 969
7.89 24o
7.8787i
7.88 i48
7.88424
7.88 698
7.88 970
7.89 241
o 34
5o
4o
3o
20
IO
19 o
10
20
3o
4o
5o
7.74248
7-74627
7.75 oo3
7.75376
7.75745
7.76 112
7.74248
7.74628
7.75 oo4
7.75377
7.75746
•7.76 n3
o 41
5o
4o
3o
20
10
27 o
IO
20
3o
4o
5o
7.89 509
7.89776
7.90 o4 1
7.90 3o5
7.90 568
7.90 829
7.89 5io
7.89777
7.90043
7.90307
7.90 569
7.90 83o
o 33
5o
4o
3o
20
10
20 o
IO
20
3o
4o
5o
7.76475
7. 76 836
7-77 193
7-77548
7.77899
7.78248
7.76476
7.76837
7«77 i94
7.77549
7.77900
7.78 249
o 40
5o
4o
3o
20
IO
28 o
IO
20
3o
4o
5o
7.91 088
7.91 346
7.91 602
7.9i857
•7.92 1 10
7.92 362
7.91 o89
7.91347
7.9,1 6o3
7.91 858
7.92 in
7.92 363
o 32
5o
4o
3o
20
10
21 o
10
20
3o
4o
5o
7.78594
7.78938
7.79278
7.79616
7.79952
7.80284
7.78595
7.78938
7.79279
7.79617
7.79952
7.80285
o 39
5o
4o
3o
20
10
29 o
IO
20
3o
4o
5o
7.92 612
7.92 861
7.93 108
7.93354
7.93599
7.93 842
7.92 6i3
7.92 862
7.93 no
7.93 356
7.93 601
7-93844
o 31
5o
4o
3o
20
10
22 o
IO
20
3o
7.80615
7.80 942
7.81 268
.81 591
7.8o6i5
7.80943
7.81 269
7.81 591
o 38
5o
4o
3o37
30 o
7.94 o84
7.94 086
o 30
L. Cos. L. Cotg.
" '
L. Cos. i L. Cotg.
// /
89°.
FUNCTIONS OP SMALL ANGLES.
0°.
, "
L. Sin.
L. Tang.
, „
L. Sin.
L. Tang.
30 o
10
20
7.94 o84
7.94325
7.94564
7.94086
7.94 326
7.94 566
o 30
5o
4o
37 3o
4o
5o
8.o3775
8.o3 967
8.o4 1 59
8.o3777
8.o3 97o
8.o4 162
3o
20
10
3o
4o
5o
7.94 802
7.96 o39
7.96 274
7.94804
7.96 o4o
7.95 276
3o
20
10
38 o
IO
20
8.o435o
8.o4 54o
8.o4 729
8.o4353
8.o4543
8.o4732
o 22
5o
4o
31 o
10
20
7.96 5o8
7.9574i
7.95 973
7.95 5io
7.95743
7.95974
o 29
5o
4o
3o
4o
5o
8.o4 918
8.o5 io5
8.o5 292
8.o4 921
8.o5 108
8.o5 295
3o
20
IO
3o
4o
5o
7.96 2o3
7.96 432
7.96 660
7.96 205
7.96434
7.96 662
3o
20
10
39 o
IO
20
8.05478
8.o5663
8.o5848
8.o548i
8.o5666
8.o585i
o 21
5o
4o
32 o
10
20
7.96887
7.97 n3
7.97337
7.96889
7-97 n4
7.97339
o 28
5o
4o
3o
4o
5o
8.o6o3i
8.06214
8.06 396
8.o6o34
8.06 217
3.o6 399
3o
20
10
3o
4o
5o
7.97660
7-97 782
7.98 oo3
7.97662
7.97 784
7.98 oo5
3o
20
10
40 o
IO
20
8.06678
8.o6758
8.o6938
8.o658i
8.06 761
8.06 94 1
o 20
5o
4o
33 o
10
20
7.98 223
7.98442
7.98 660
7.98 225
7.98444
7.98 662
o 27
5o
4o
3o
4o
5o
8.07 117
8.07 296
8.07473
8.07 I2O
8.07 298
8.o7476
3o
20
IO
3o
4o
5o
7.98 876
7.99 092
7.99 3o6
7.98 878
7.99 094
7.99 3o8
3o
20
IO
41 o
IO
20
8.07 650
8.07 826
8.08 002
8. 07 653
8.07 829
8.08 005
o 19
5o
4o
34 o
10
20
7.99 620
7.99 732
7.99943
7.99 522
7.99 734
7.99 946
o 26
5o
4o
3o
4o
5o
8.08 176
8.o835o
8.08 624
8.08 1 80
8.08 354
8.08 627
3o
20
IO
3o
4o
5o
8.00 1 54
8.00 363
8.00 571
8.00 1 56
8.oo365
8.00 574
3o
20
10
42 o
IO
20
8.08696
8.08868
8.09 o4o
8 .08 700
8.08 872
8.09043
o 18
5o
4o
35 o
10
20
8.00 779
8.00 985
8.01 190
8.00 781
8.00987
8.01 193
o 25
5o
4o
3o
4o
5o
8.09 2IO
8.09 38o
8.09 650
8.09 214
8.09 384
8. 09 553
3o
20
10
3o
4o
5o
8.01 395
8.01 598
8.01 801
8.01 397
8.01 600
8.01 8o3
3o
20
10
43 o
IO
20
8.09 718
8.09 886
8.ioo54
8.09 722
8.09 890
8. 10 067
o 17
5o
4o
36 o
10
20
8.02 OO2
8.02 203
8. 02 402
8. 02 oo4
8. 02 2o5
8.02 405
o 24
5o
4o
3o
4o
5o
8. 10 220
8.io386
8.10 552
8. 10 224
8.10 390
8.io555
3o
20
IO
3o
4o
5o
8. 02 601
8. 02 799
8.02 996
8.02 6o4
8.02 801
8.02 998
3o
20
10
44 o
10
20
8.10 717
8.10 881
8 . 1 1 o44
8. 10 720
8.io884
8. 1 1 o48
o 16
5o
4o
37 o
10
20
8.o3 192
8.o3 387
8.o3 58i
8.o3 194
8.o339o
8.o3 584
o 23
5o
4o
3o
4o
5o
8. 1 1 207
8. 1 1 370
8. ii 53i
8. I I 211
8. ii 373
8. ii 535
3o
20
IO
3o
8.03775
8.o3777
3o22
45 o
8. 1 1 693
8. ii 696
3 15
L. Cos.
L. Cotg.
" '
L. Cos.
L. Cotg.
" '
125
89°.
FUNCTIONS OP SMALL ANGLES.
/ //
L. Sin.
L. Tang.
/ tr
L. Sin. L.Tang.
45 o
10
20
3o
4o
5o
8.11 693
8. ii 853
8.i2oi3
8. 12 172
8.i233i
8.12489
8. n 696
8. ii 857
8.12 017
8.12 176
8.12335
8.12493
o 15
5o
4o
3o
20
10
52 3o
4o
5o
8.18 387
8.18 524
8.18662
8.18 392
8.18 53o
8.18667
3o
20
IO
53 o
10
20
3o
4o
5o
8.18 798
8.18 935
8.19 071
8.19 206
8.i934i
8.19476
8.18 8o4
8.18 940
8.19076
8. 19 212
8.19347
8.19481
o 7
5o
4o
3o
20
IO
46 o
10
20
3o
4o
5o
8.12 647
8.12 8o4
8.12 961
8.i3 117
8.i3 272
8.13427
8.12 65i
8.12 808
8.12 965
8.i3 121
8.i3276
8.i343i
o 14
5o
4o
3o
20
IO
54 o
10
20
3o
4o
5o
8.19610
8.19744
8.19877
8.20 OIO
8.20 i43
8. 20 275
8.19616
8.19 749
8.19 883
8. 20 016
8. 20 149
8.20 281
o 6
5o
4o
3o
20
IO
47 o
10
20
3o
4o
5o
8.i3 58i
8.i3735
8.13888
8.i4o4i
8.i4 193
8.i4344
8.i3 585
8.i3 739
8.i3 892
8.14045
8.i4 197
8.i4348
o 13
5o
4o
3o
20
IO
55 o
IO
20
3o
4o
5o
8.20 407
8.20538
8.20 669
8.20 800
8.20 930
8.21 060
8.20 4i3
8.20 544
8.20675
8.20806
8.20936
8.21 066
o 5
5o
4o
3o
20
10
48 o
10
20
3o
4o
5o
8.14496
8.i4646
8.i4 796
8.i4945
8. 1 5 094
8.i5243
8.i4 500
8.i465o
8.i48oo
8. i4 950
8. 1 5 099
8.i5 247
o 12
5o
4o
3o
20
IO
56 o
IO
20
3o
4o
5o
8.21 189
8.21 319
8.21 447
8.21 576
8.21 7o3
8.21 83i
8.21 195
8.21 324
8.21 453
8.21 58i
8.21 709
8.21 837
o 4
5o
4o
3o
20
IO
49 o
10
20
3o
4o
5o
8.i5 39i
8.15538
8.15685
8.i5832
8.i5978
8.l6 123
8.i5 395
8.15543
8.i5 690
8.i5836
8.i5 982
8.16 128
o 11
5o
4o
3o
20
IO
57 o
IO
20
3o
4o
5o
8.21 958
8.22 085
8.22 211
8.22 337
8.22463
8.22 588
8.21 964
8.22 091
8.22 217
8.22 343
8.22 469
8.22 595
o 3
5o
4o
3o
20
IO
50 o
10
20
3o
4o
5o
8.16 268
8.i64i3
8.i6557
8.16 700
8.i6843
8.16986
8.16 273
8.16417
8.i656i
8.16 705
8.i6848
8. 16 991
o 10
5o
4o
3o
20
10
58 o
IO
20
3o
4o
5o
8.22 713
8.22838
8.22 962
8. 23 086
8.23 210
8.23 333
8.22 720
8.22844
8.22968
8.23 092
8.232i6
8.23 339
o 2
5o
4o
3o
20
IO
51 o
10
20
3o
4o
5o
8.17 128
8.17 270
8. 17 4i i
8.17 552
8. 17 692
8.17882
8.17 i33
8.17275
8.17416
8.17557
8.17697
8.i7837
o 9
5o
4o
3o
20
IO
59 o
10
20
3o
4o
5o
8.23456
8.23578
8.23 700
8.23 822
8.23944
8.24065
8.23462
8.23 585
8.23 707
8.23 829
8.23 950
8.24 071
o 1
5o
4o
3o
20
10
52 o
10
20
3o
8.17971
8.18 no
8.18 249
8. T* 3s7
8.17976
8.18 ii5
8.18254
8.18 392
o 8
5o
4o
3o 7
60 o
8.24 186
8.24 192
o 0
L. Cos. L. Cotg1.
" '
L. Cos.
L. Cotg".
" '
126
89°.
FUNCTIONS OF SMALL ANGLES.
1°.
/ II
L. Sin.
L.Tang.
r ti
L. Sin.
L. Tang.
0 o
10
20
8.24 186
8.243o6
8.24426
8.24 192
8.243i3
8.24433
o 60
5o
4o
7 Jo
4o
5o
8.29 3oo
8.29 407
8.29 5i4
8.29 309
8.29416
8.29 523
3o
20
IO
3o
4o
5o
8.24546
8.24665
8.24785
8.24553
8.24 672
8.24 791
3o
20
10
8 o
IO
20
8.29 621
8.29727
8. 29 833
8.29 629
8.29 736
8.29842
o 52
5o
4o
1 o
10
20
8.24903
8.25 022
8.25 i4o
8.24 910
8.25 029
8.25 i47
o 59
5o
4o
3o
4o
5o
8.29939
8-.3oo44
8.3o 150
8.29 947
8.3oo53
8.3o i58
3o
20
10
3o
4o
5o
8.25258
8.25 375
8.25493
8.25265
8.25 382
8.25 500
3o
20
10
9 o
IO
20
8.30255
8.3o359
8.3o464
8.3o263
8.3o368
8.3o473
o 51
5o
4o
2 o
10
20
8.25 609
8.25 726
8.25842
8.25 616
8.25 733
8.25849
o 58
5o
4o
3o
4o
5o
8.3o568
8.30672
8.30776
8.3o577
8.3o68i
8.30785
3o
20
IO
3o
4o
5o
8.25958
8.26074
8.26 189
8.25965
8.26081
8.26 196
3o
20
10
10 o
IO
20
8.30879
8.3o983
8.3i 086
8.3o888
8.3o 992
8.3i 095
o 50
5o
4o
3 o
10
20
8.263o4
8.26419
8.26533
8.26 3i2
8.26426
8.2654i
0 57
5o
4o
3o
4o
5o
8.3i 188
8.3i 291
8.3i 393
8. 3 1 198
8. 3 1 3oo
8. 3 1 4o3
3o
20
10
3o
4o
5o
8.26648
8.26 761
8.26875
8.26655
8.26 769
8.26882
3o
20
10
11 o
10
20
8.3i 4g5
8.3i 597
8.3i 699
8.3i 505
8.3i 606
8.3i 708
o 49
5o
4o
4 o
10
20
8.26988
8.27 101
8.27 214
8.26 996
8.27 109
8.27 221
o 56
5o
4o
3o
4o
5o
8.3i 800
8.3i 901
8.32 002
8. 3 1 809
8.3i 911
8.32 012
3o
20
IO
3o
4o
5o
8.27 326
8. 27 438
8.27 550
8.27334
8.27446
8.27558
3o
20
IO
12 o
10
20
8.32 io3
8.32 2o3
8.32 3o3
8.32 112
8.32213
8.32 3i3
o 48
5o
4o
5 o
10
20
8.27 661
8.27773
8.27883
8.27 669
8.27 780
8.27 891
o 55
5o
4o
3o
4o
5o
8.324o3
8.325o3
8.32 602
8.324i3
8.32 5i3
8.32612
3o
20
1C
3o
4o
5o
8.27 994
8.28 104
8.28 215
8.28002
8.28 112
8.28 223
3o
20
IO
13 o
10
20
8.32 702
8.32 801
8.32 899
8.32 711
8.32811
8.32 909
o 47
5o
4o
6 o
10
20
8.28324
8.28434
8.28 543
8.28 332
8.28442
8.2855i
o 54
5o
4o
3o
4o
5o
8.32 998
8. 33 096
8.33 195
8.33 008
8.33 106
8.33205
3o
20
IO
3o
4o
5o
8.28652
8.28 761
8.28 869
8.28660
8.28 769
8.28 877
3o
20
10
14 o
10
20
8.33292
8. 3339o
8.33488
8.33 3o2
8.334oo
8.33498
o 46
5o
4o
7 o
10
20
8.28977
8.29085
8.29 193
8.28 986
8.29 094
8.29 2OI
o 53
5o
4o
3o
4o
5o
8.33 585
8.33682
8.33779
8.33 595
8. 33 692
8.33789
3o
20
10
3o
8 . 29 3oo
8.29 3og
3o52
15 o
33875
8.33886
o 45
L. Cos.
L. cotg.
" '
L. Cos.
L. Cotg.
FUNCTIONS OP SMALL ANGLES
1°.
/ tr
L. Sin.
L. Tang.
, „
L. Sin.
L. Tang.
15 o
10
20
8.33876
8.33972
8.34068
8.33 886
8.33982
8.34078
o 45
5o
4o
22 3o
4o
5o
8.38 oi4
8.38 101
8.38 189
8.38 026
8.38 n4
8.38 202
3o
20
10
3o
4o
5o
8.34 i64
8.34260
8.34355
8.34174
8.34270
8.34366
3o
20
10
23 o
10
20
8.38 276
8.38363
8.3845o
8.38289
8.38376
8.38463
o 37
5o
4o
16 o
10
eo
8.3445o
8.34546
8.3464o
8.3446i
8.34556
8.3465i
o 44
5o
4o
3o
4o
5o
8.38 537
8.38 624
8.38 710
8.38 550
8.38636
8.38 723
3o
20
10
3o
4o
5o
8.34735
8.3483o
8.34924
8.34746
8.3484o
8.34935
3o
20
10
24 o
10
20
8.38 796
8.38882
8.38,968
8.38 809
8.38895
8.38981
o 36
5o
4o
17 o
I O
20
8.35oi8
8.35 112
8.35206
8.35 029
8.35 123
8.35217
o 43
5o
4o
3o
4o
5o
8.39 o54
8.39 1 39
8.39 225
8.39067
8.39 i53
8.39238
3o
20
IO
3o
4o
5o
8.35 299
8. 35392
8.35485
8.353io
8.354o3
8.35497
3o
20
10
25 o
10
20
8.39 3io
8.39395
8.39480
8.39323
8.39408
8.39493
o 35
5o
4o
18 o
10
20
8.35578
8. 35 671
8.35764
8.35 590
8.35682
8.35775
o 42
5o
4o
3o
4o
5o
8.39565
8.39 649
8.39734
8.39578
8.39663
8.39747
3o
20
IO
3o
4o
5o
8.35856
8.35948
8.36 o4o
8.35867
8.35959
8.36o5i
3o
20
10
26 o
IO
20
8.39818
8.3g 902
8.39986
8.39832
8.39 916
8.4o ooo
o 34
5o
4o
19 o
10
20
8.36 i3i
8.36223
8.363i4
8.36 i43
8.3623s
8.36326
o 41
5o
4o
3o
4o
5o
8.4o 070
8.4o i53
8.40237
8.4oo83
8.4o i67
8.4o25i
3o
20
IO
3o
4o
5o
8.364o5
8.36496
8.36587
8.36417
8.365o8
8.36 599
3o
20
IO
27 o
10
20
8.4o 320
8.4o4o3
8.4o486
8.4o334
8.4o4i7
8.4o 500
o 33
5o
4o
20 o
10
20
8.36678
8.36768
8.36858
8.36 689
8.36 780
8.36870
o 40
5o
4o
3o
4o
5o
8.40669
8.4o65i
8.40734
8.4o583
8.4o665
8.4o 748
3o
20
IO
3o
4o
5o
8.36948
8.37038
8.37 128
8.36 960
8.37050
8.37 i4o
3o
20
IO
28 o
IO
20
8.40816
8.40898
8.40980
8.4o83o
8.40913
8.4o 995
o 32
5o
4o
21 o
10
20
8.37217
8.37 3o6
8.37395
8.37 229
8.373i8
8.374o8
o 39
5o
4o
3o
4o
5o
8.4i 062
8.4i 1 44
8.4i 225
8.4i o77
8.4i i58
8.4i 240
3o
20
10
3o
4o
5o
8.37484
8.37 573
8.37662
8. 37497
8.37585
8.37674
3o
20
10
29 o
IO
20
8.4i 307
8.4i 388
8.4i 469
8.4i 32i
8.4i4o3
8.4i484
o 31
5o
4o
22 o
10
20
8.37750
8.37838
8.37 926
8.37762
8.3785o
8.37938
o 38
5o
4o
3o
4o
5o
8.4i 55o
8.4i 63i
8.4i 711
8.4i 565
8.4i 646
8.4i 726
3o
20
IO
3o
8.38oi4
8.38026
3o37
30 o
Mi 792
Mi 8o7
o'30
L. Cos.
L. Cotg.
n ,
L. Cos.
L. Cotg.
' "
128
88°.
FUNCTIONS OF SMALL ANGLES.
1°.
/ "
L. Sin. L.Tang.
/ ft
L. Sin. L.Tang.
30 o
10
20
3o
4o
5o
8.4i 79-
8.4i 872
8.4i 962
8.42 o32
8.42 112
8.42 192
8.4i 807
8.4i 887
8.4i 967
8.42 o48
8.42 127
8.42 207
o 30
5o
4o
3o
20
10
37 3o
4o
5o
8.45 267
8.4534i
8.454i5
8.45 285
8.45 359
8.45433
3o
20
IO
38 o
IO
20
3o
4o
5o
8.45489
8.45563
8.45637
8. 45 710
8. 45 784
8.45857
8. 455o7
8.4558i
8.45655
8.45728
8.45 802
8.45875
o 22
5o
4o
3o
20
IO
31 o
10
20
3o
4o
5o
8.42 272
8.4235i
8.4243o
8.42 5io
8.42689
8.42667
8.42 287
8.42 366
8.42446
8.42625
8.42 6o4
8.42683
o 29
5o
4o
3o
20
10
39 o
IO
20
3o
4o
5o
8.45 930
8.46oo3
8.46076
8.46 149
8.46 222
8.46 294
8.45948
8.46 021
8.46094
8.46 167
8.46 240
8.46 3i2
oSl
5o
4o
3o
20
10
32 o
10
20
3o
4o
5o
8.42 746
8.42825
8.42903
8.42982
8.43 060
8.43 i38
8.42 762
8.42 84o
8.42 919
8.42 997
8.43075
8.43x54
o 28
5o
4o
3o
20
10
40 o
10
20
3o
4o
5o
8.46 366
8.46439
8.465n
8.46583
8.4665s
8.46727
8.46 385
8.46457
8.46529
8.46602
8.46674
8. 46 745
o 20
5o
4o
3o
20
10
33 o
10
20
3o
4o
5o
8.43 2ltJ
8.43293
8. 4337i
8.43448
8.43526
8.436o3
8.43232
8.43 3o9
8. 43387
8.43464
8.43542
8.436i9
o 27
5o
4o
3o
20
10
41 o
IO
20
3o
4o
5o
8.46 799
8.46870
8.46942
8.47oi3
8.47084
8.47 i55
8.46817
8.46889
8.46960
8.47o3a
8.47 io3
8.47 i?4
o 19
5o
4o
3o
20
10
34 o
10
20
3o
4o
5o
8.4^680
8.43757
8.43834
8.43910
8.43987
8.44o63
8.43 696
8.43773
8.4385o
8.43 927
8.44oo3
8. 44 080
o 26
5o
4o
3o
20
10
42 o
IO
20
3o
4o
5o
8.47226
8.47297
8.47368
8.47439
8.47 509
8.47 58o
8.47245
8.473i6
8.47387
8.47458
8.47528
8.47 599
o 18
5o
4o
3o
20
10
35 o
10
20
3o
4o
5o
8.44 139
8.44216
8.44292
8.44367
8.44443
8.445i9
8.44 1 56
8.44232
8.443o8
8.44384
8.4446o
8.44536
o 25
5o
4o
3o
20
IO
43 o
IO
20
3o
4o
5o
8.47 650
8.47 720
8.47 790
8.47 860
8.47 930
8.48 ooo
8.47 669
8.47 74o
8.47810
8.47 880
8.4795°
8.48 020
o 17
5o
4o
3o
20
10
36 o
10
20
3o
4o
5o
8.44594
8.44669
8.44745
8.44820
8.44895
8.44969
8.446n
8.44686
8.44762
8.44837
8.44912
8.44987
o 24
5o
4o
3o
20
IO
44 o
IO
20
3o
4o
5o
8.48 069
8.48 i39
8.48 208
8.48 278
8.48 347
8.484i6
8.48090
8.48 i59
8.48 228
8.48 298
8.48 367
8.48436
o 16
5o
4o
3o
20
10
37 o
10
20
3o
8.45o44
8.45 119
8.45 193
8.45 -j-;-
8.45 061
8.45 i36
8.45 210
8.45 285
o 23
5o
4o
3o22
45 o
J. 48 485
M8 5o5
o 15
L. Cos. L. Cotg.
" '
L. Cos.
L. Cotg.
" '
88°.
129
FUNCTIONS OF SMALL ANGLES
1°.
, „
L. Sin.
L. Tang.
, „
L.Sin.
L. Tang.
45 o
10
20
8.48 485
8.48 554
8.48622
8.48 5o5
8.48574
8.48643
o 15
5o
4o
52 3d
4o
5o
8.5i 48o
8.5i 544
8. 5 1 609
8.5i 5o3
8.5i 568
8.5i 632
3o
20
IO
3o
4o
5o
8.48 691
8.48 760
8.48828
8.48 711
8.48 780
8.48849
3o
20
IO
53 o
10
20
8.5i 673
8.5i 737
8.5i 801
8.5i 696
8.5i 760
8;5l 824
o 7
5o
4o
46 o
10
20
8.. 48 896
8.48 965
8.49033
8.48917
8.48985
8.49.053
o 14
5o
4o
3o
4o
5o
8.5; 864
8.5i 928
8.5i 992
8.5i 888
8.5i 952
8.52 oi5
3o
20
IO
3o
4o
5o
8.49 101
8.49 169
8.49236
8.49 121
8.49 189
8.49257
3o
20
10
54 o
IO
20
8.52o55
8.52 119
8.52 182
8 . 52 079
8.52 i43
8.52 206
o 6
5o
4o
47 o
10
20
8.49 3o4
8.49372
8.49439
8.49325
8.49 393
8.49460
o 13
5o
4o
3o
4o
5o
8.52245
8.523o8
8.5237i
8.52 269
8.52 332
8.52 396
3o
20
10
3o
4o
5o
8.49 5o6
8.49574
8.49641
8.49528
8.49 595
8.49 662
3o
20
10
55 o
IO
20
8.52434
8.52 497
8.52 56o
8.52 459
8.52 522
8.52 584
o 5
5o
4o
48 o
10
20
8.49 708
8.49775
8.49842
8.49 729
8.49 796
8.49863
o 12
5o
4o
3o
4o
5o
8.52623
8.52685
8.52748
8.52647
8.52 710
8.52 772
3o
20
10
3o
4o
5o
8.49908
8.49975
8.5oo42
8.49930
8.49997
8.5oo63
3o
20
10
56 o
IO
20
8.52 810
8.52872
8.52935
8.52 835
8.52897
8.52 960
o 4
5o
4o
49 o
10
20
8.5o 108
8.5o i74
8.5o24i
8.5o i3o
8.5o 196
8.50263
o 11
5o
4o
3o
4o
5o
8.52 997
8.53o59
8.53 121
8.53022
8.53o84
8.53 i46
3o
20
10
3o
4o
5o
8.5o 307
8.5o373
8.5o439
8.5o329
8.50395
8.5o46i
3o
20
IO
57 o
IO
20
8.53 i83
8.53245
8.533o6
8.53 208
8.53 270
8.53 332
o 3
5o
4o
50 o
10
20
8.5o 5o4
8.5o 570
8.5o636
8.5o527
8.5o593
8.5o658
o 10
5o
4o
3o
4o
5o
8.53368
8.53429
8.53491
8.53 393
8.53455
8.535i6
3o
20
IO
3o
4o
5o
8.5o 701
8.5o 767
8.5o832
8.5o 724
8.5o 789
8.5o855
3o
20
10
58 o
IO
20
8.53552
8.536i4
8. 53 675
8.53578
8.53639
8.53 700
o 2
5o
4o
51 o
10
20
8.50897
8.5o963
8.5i 028
8.5o 920
8.5o985
8.5i o5o
o 9
5o
4o
3o
4o
5o
8.53736
8.53797
8.53858
8.53762
8.53823
8.53884
3o
20
10
3o
4o
5o
8.5i 092
8.5i i57
8.5l 222
8.5i u5
8.5i 180
8.5i 245
3o
20
10
59 o
10
20
8.53919
8. 53 979
8 . 54 o4o
8.53945
8.54oo5
8. 54 066
o 1
5o
4o
52 o
10
20
8.5i 287
8.5i 35i
8.5i 4i6
8.5i 3io
8.5i 374
8.5i 439
o 8
5o
4o
3o
4o
5o
8.54 ioi
8.54 161
8.54 222
8.54 127
8.54 187
8. 54 s48
3o
20
10
3o
8.5i 48o
.5i 5o3
3o 7
60 o
8.542*
8.54 3o8
> 0
L. Cos.
L. Cotg.
,, *
L.Cos.
L. Cotg.
'
130
88°.
TABLE IV
FOUR-PLACE
NAPERIAN LOGARITHMS
NAPERIAN LOGARITHMS.
LOGARITHMS OF POWERS OF 10.
Num.
Log.
Num.
Log.
10
2.3O26
. i
3~.6974
IOO
4.6o52
.01
5.3948
1000
6.9078
.001
7.0922
IOOOO
9.2103
.0001
^.7897
IOOOOO
1 1 .5129
.00001
72.4871
IOOOOOO
i3.8i55
.00000 I
a. 1845
IOOOOOOO
16.1181
.000000 I
17.8819
IOOOOOOOO
18.4207
.00000001
19.5793
I OOOOOOOOO
20.7233
.00000000 1
21 .2767
Num.
Log.
Num.
Log.
LOGARITHMS OF NUMBERS FROM i TO 10.
N
0
1
2
3
4
5
6
7
8
9
1.0
o.oooo
OIOO
0198
0296
0392
o488
o583
o677
0770
0862
.1
.2
.3
0.0953
0.1823
0.2624
io44
1906
2700
n33
1989
2776
1222
2070
2852
i3io
2l5l
2927
1398
223l
3ooi
i484
23ll
3o75
i57o
2390
3i48
i655
2469
3221
i74o
2546
3293
.4
.5
.6
0.3365
o.4o55
0.4700
3436
4l2I
4762
35o7
4i87
4824
3577
4253
4886
3646
43i8
4947
37i6
4383
5oo8
3784
4447
5o68
3853
45n
5i28
392O
4574
5i88
3988
4637
5247
• 7
.8
•9
o.53o6
o.5878
0.6419
5365
5933
6471
5423
5988
6523
548i
6o43
6575
5539
6098
6627
5596
6i52
6678
5653
6206
6-729
57io
6259
678o
5766
63i3
683i
5822
6366
6881
2.0
0.6931
6981
7o3i
7080
7129
7178
•722-7
7275
7324
7372
N
0
1
2
3
4
5
6
7
8
9
NAPERIAN LOGARITHMS.
N
0
1
2
3
4
5
6
7
8
9
2.0
2.1
2.2
2.3
0.6931
6981
7o3i
7080
7129
7178
7227
7275
7324
7372
0.7419
o.7885
0.8329
7467
793°
8372
75i4
7975
84i6
756i
8020
8459
7608
8o65
85o2
7655
8109
8544
77OI
8i54
8587
7747
8198
8629
7793
8242
8671
7839
8286
87i3
2.4
2.5
2.6
o.8755
0.9163
o.9555
8796
9203
9^94
8838
9243
9632
8879
9282
967o
8920
9822
9708
8961
936i
9746
9002
9400
9783
9042
9439
982I
9o83
9478
9858
9I23
95i7
9895
2.7
2.8
2.9
3.0
3.i
3.2
3.3
0.9933
1.0296
1.0647
9969
o332
0682
6006
0367
0716
6o43
o4o3
0750
6080
o438
0784
61 16
o473
0818
Ol52
o5o8
o852
018,8
o543
0886
0225
o578
o9i9
6260
o6i3
o953
1.0986
1019
io53
1086
1119
n5i
n84
1217
I249
1282
I.i3i4
i.i632
i.i939
1 346
i663
1969
i378
1694
2000
i4io
1725
2o3o
1 442
i756
2060
i474
i787
2090
i5o6
181-7
21 19
i537
1 848
2l49
i569
1878
2179
1600
I9o9
2208
3.4
3.5
3.6
1.2238
1.2528
1.2809
2267
2556
2837
2296
2585
2865
2326
26i3
2892
2355
2641
2920
2384
2669
2947
24i3
2698
29-75
2442
2726
3002
2470
2754
3o29
2499
2782
3o56
3.7
3.8
3.9
4.0
4.i
4.2
4.3
i.3o83
i.335o
i.36io
3uo
3376
3635
3i37
34o3
366i
3i64
3429
3686
3i9i
3455
3712
3218
348 1
3737
3244
35o7
3762
3271
3533
3788
3297
3558
38i3
3324
3584
3838
1.3863
3888
3913
3938
3962
3987
4012
4o36
4o6i
4o85
i.4no
i.435i
1.4586
4i34
4375
4609
4i59
4398
4633
4i83
4422
4656
4207
4446
4679
423i
4469
4702
4255
4493
472D
4279
45i6
4748
43o3
454o
477<>
4327
4563
4793
4.4
4.5
4.6
i.48i6
i.5o4i
1.5261
4839
5o63
5282
486i
5o85
53o4
4884
5107
5326
4907
5129
5347
4929
5i5i
5369
49^1
5i73
5390
4974
5i95
54i2
4996
5217
5433
5oi9
5239
5454
4-7
4.8
4-9
5.0
5.i
5.2
5.3
1.5476
1.5686
1.5892
5497
57o7
SgiS
55i8
5728
5933
5539
5748
5953
556o
5769
5974
558i
579o
5994
56o2
58io
6oi4
5623
583i
6o34
5644
585i
6o54
5665
5872
6o74
1.609^
6n4
6i34
6i54
6i74
6194
6214
6233
6253
6273
1.6292
i.6487
1.6677
63i2
65o6
6696
6332
6525
67i5
635i
6544
6734
637i
6563
6752
63go
6582
6771
6409
6601
6-790
6429
6620
6808
6448
6639
6827
6467
6658
6845
5.4
5.5
5.6
1.6864
1.7047
1.7228
6882
7066
7246
6901
7o84
7263
6919
7102
7281
6938
7120
7299
6956
7i38
73i7
6974
7i56
7334
6993
7i74
7352
7011
7I92
737o
•7029
721O
7387
5-7
5.8
5.9
6.0
i.74o5
1.7579
i.775o
7422
7596
7766
744o
76i3
7783
7457
763o
7800
7475
7647
7817
74g2
7664
7834
75o9
768i
785i
7527
7699
7867
7544
77i6
7884
756i
7733
79oi
1.7918
7934
7951
7967
7984
8001
8017
8o34
8o5o
8066
0
1
2 3
4
5
6
7
8
9
i33
NAPERIAN LOGARITHMS.
N
0
1
2
3
4
5
6
7
8
9
6.0
1.7918
7934
795i
7967
79»4
8001
8017
8o34
8o5o
8066
6.1
6.2
6.3
i.8o83
.8245
.84o5
8099
8262
8421
8116
8278
8437
8i32
8294
8453
8i48
83io
8469
8i65
8326
8485
8181
8342
85oo
8i97
8358
85i6
82i3
8374
8532
8229
839o
8547
6.4
6.5
6.6
.8563
.8718
.8871
8579
8733
8886
8594
8749
89oi
8610
8764
89i6
8625
8779
8931
864i
8795
8946
8656
8810
896i
8672
8825
8976
8687
884o
899i
87o3
8856
9oo6
6.7
6.8
6.9
.9021
.9169
.93i5
9o36
9184
933o
9o5i
9i99
9344
9o66
92l3
9359
9081
9228
9373
9o95
9242
9387
9i 10
9257
9402
9I25
9272
94i6
9i4o
9286
943o
9i55
93oi
9445
7.0
.9459
9473
9488
95o2
95i6
953o
9544
9559
9573
9587
7-i
7.2
7.3
.9601
.974i
.9879
96i5
9755
9892
9629
9769
99o6
9643
978a
9920
9657
9796
9933
967i
98io
9947
9685
9824
996i
9699
9838
9974
97i3
985i
9988
9727
9865
OOOI
7-4
7.5
7-6
2. 001 5
2.0149
2.0281
0028
0162
0295
0042
0176
o3o8
oo55
oi89
0321
0069
O2O2
o334
0082
02 1 5
o347
oo96
O229
o36o
oio9
0242
o373
OI22
0255
o386
oi36
0268
o399
7-7
7.8
7-9
2.0412
2.o54i
2.0669
o425
o554
0681
o438
0567
o694
o45i
o58o
0707
o464
0592
0719
o477
o6o5
0732
o49o
0618
o744
o5o3
o63i
o757
o5i6
o643
o769
o528
o656
0-782
8.0
2.0794
0807
o8i9
o832
o844
o857
o869
0882
o894
o9o6
8.1
8.2
8.3
2 .0919
2.I04I
2.u63
ogSi
io54
1175
o943
1066
1187
o956
1078
1199
0968
1090
I2II
o98o
1 1 02
1223
0992
1 1 14
1235
ioo5
1126
I247
IOI7
n38
1258
IO29
i i5o
I27O
8.4
8.5
8.6
2.1282
2.l4oi
2.i5i8
1294
1412
1529
i3o6
1424
i54i
i3i8
i436
i552
i33o
1 448
1 564
1 342
i459
1576
i353
i47i
i587
i365
i483
i599
i377
i494
1610
i389
i5o6
1622
8.7
8.8
8.9
2.i633
2.1748
2.1861
1 645
1759
1872
i656
1770
i883
1668
1782
1894
1679
i793
I9o5
i69i
1804
I9i7
1702
i8i5
I928
I7i3
182-7
i939
I725
i838
i95o
i736
1849
1961
9.0
2. 1972
i983
i994
2006
2017
2028
2o39
2o5o
2061
2072
9.1
9.2
9.3
2.2083
2.2192
2.23OO
2094
22O3
23ll
2IO5
22l4
2322
21 16
2225
2332
2127
2235
2343
2i38
2246
2354
2148
2257
2364
2l59
2268
2375
2I7O
2279
2386
2181
2289
2396
9.4
9.5
9.6
2.24O7
2.25l3
2.2618
2418
2523
2628
2428
2534
2638
2439
2544
2649
245o
2555
2659
2460
2565
2670
247r
2576
2680
2481
2586
269O
2492
2597
2701
2502
2607
2711
9-7
9.8
9.9
2.2721
2.2824
2.2925
2732
2834
2935
2742
2844
2946
2752
2854
2g56
2762
2865
2966
2773
2875
2976
2783
2885
2986
2?93
2895
2996
28o3
29o5
3oo6
28l4
29l5
3oi6
10.0
2.3026
3i26
3224
3322
34i8
35i4
36o9
37o3
3796
3888
N
0
1
2
3
4
5
6
7
8
9
1 34
TABLE V
FOUR-PLACE LOGARITHMS
OF NUMBERS
FOUR-PLACE LOGARITHMS.
N
0
1
2
3
4
5
6
7
8
9
"IT
oooo
o43
086
128
170
212
•B^^^^B
253
•BBBMH
294
334
~
1 1
4i4
453
492
53i
569
607
645
682
719
755
12
792
828
864
899
934
969
100^
ro38
1072
1 106
i3
ii39
I73
206
239
271
3o3
335
367
399
43o
i4
46i
492
523
553
584
6i4
644
673
7o3
732
i5
1761
79°
818
847
875
903
93i
959
987
2OI,
16
20
4i
068
o95
122
1 48
i75
2OI
227
253
279
ll
3o4
33o
355
38o
4o5
43o
455
48o
5o4
529
18
553
577
60 1
625
648
672
695
718
742
765
'9
788
810
833
856
878
900
923
945
967
989
20
3oio
032
o54
o75
o96
118
i39
1 60
181
201
21
222
243
263
284
3o4
324
345
365
385
4o4
22
424
444
464
483
5o2
522
54 1
56o
579
598
23
617
636
655
674
692
7'
I
729
747
766
784
24
802
820
838
856
874
892
9°9
927
945
962
25
3979
997
4or4
4o3i
4o48
4o65
4082
4o99
4n6
4i33
26
4i5o
1 66
i83
200
216
232
249
265
281
298
27
3i4
33o
346
362
378
393
4o9
425
44o
456
28
472
487
502
5i8
533
548
564
579
594
6o9
29
624
639
654
669
683
698
7i3
728
742
757
30
477i
786
800
8i4
829
843
857
87,
886
9oo
3i
9i4
928
942
955
969
983
997
5oi i
5o24
5o38
32
5o5i
o65
o79
092
io5
n9
i3a
i45
i59
I72
33
1 85
198
21 I
224
237
25o
263
276
289
302
34
3i5
328
34o
353
366
378
39i
4o3
4i6
428
35
544 1
453
465
478
490
502
5i4
527
539
55i
36
563
575
587
599
611
623
635
647
658
67o
37
682
694
7o5
717
729
74o
752
763
775
786
38
798
8o9
821
832
843
855
866
877
888
899
39
911
922
933
944
955
966
977
988
999
6010
40
••••••i
6021
•MM^^^M
o3i
••••••
042
••HM^M
o53
••••Mi
o64
o75
o85
o96
IO7
117
N
0
1
2
3
4
5
6
7
8
9
PP
38
32
28
35
22
21
19
18
17
16
.1
.2
7-6
1!
> 5-6
5-o
.2
4-4
4-2
3-8
.2
3-6
3-4
3-2
•3
11.4
9.e
8.4
7-5
•3
6.6
6-3
5-7
•3
5-4
4.8
•4
»5-2
I2.i
II.2
IO.O
•4
8.8
8.4
7.6
•4
7.2
6.8
6.4
• 5
19.0
j6.c
> 14.0
12.5
•5
I.O
10.5
9-5
•5
9.0
8-5
8.0
6
22.8
19.2
16.8
15.0
.6
3-2
12.6
11.4
.6
10.8
10.2
9.6
•7
26.6
22.4
19.6
J7-5
• 7
5-4
'4-7
'3-3
• 7
12.6
ii. 9
II. 2
.8
30-4
25.6
22.4
20.0
.8
7.6
1 6.' 8
'5-2
.8
14.4
13-6
12.8
•9 34-2
28.8 25.2 22.5
•9
9.8 18.9
J5- 3 '4-4
1 36
FOUR-PLACE LOGARITHMS.
N
0
1
2
3
4
5
6
7
8
9
40
6021
o3i
042
o53
064
M^M
07
HIM
5
o85
••— ^—
096
I07
117
4i
128
1 38
i49
160
I70
180
191
201
212
222
42
232
243
253
263
274
284
294
3o4
3i4
325
43
335
345
355
365
375
385
395
4o5
4i5
425
44
435
444
454
464
474
484
493
5o3
5x3
522
45
6532
542
55i
56i
57i
58o
59o
599
6o9
618
46
628
637
646
656
665
675
684
693
702
712
47
721
73o
739
749
758
767
776
785
794
8o3
48
812
821
83o
839
848
857
866
875
884
893
49
902
911
92O
928
937
946
955
964
972
981
50
6990
998
7oo7
7016
7024
7o33
7042
7o5o
7°59
7o67
5i
7076
o84
o93
IOI
I IO
118
126
i35
i43
152
52
1 60
168
I77
i85
i93
202
210
218
226
235
53
243
25l
269
267
275
284
292
3oo
3o8
3i6
54
324
332
34o
348
356
364
372
38o
388
396
55
74o4
4l2
4i9
427
435
443
45i
459
466
474
56
482
490
497
5o5
5i3
52O
528
536
543
55i
57
559
566
574
582
589
597
6o4
612
6i9
627
58
634
642
649
657
664
672
679
686
694
701
59
709
716
723
73i
738
745
752
760
767
774
60
7782
789
796
8o3
810
818
825
832
839
846
61
853
860
868
875
882
88
9
896
9o3
9io
917
62
924
93i
938
945
952
959
966
973
98o
987
63
993
8000
8oo7
8oi4
802 1
8028
8o35
8o4i
8o48
8o55
64
8062
o69
o75
082
089
o96
102
io9
116
122
65
8l29
i36
149
i56
162
i69
176
182
189
66
i95
202
209
2l5
222
228
235
2
4r
248
254
67
261
267
274
280
287
293
299
3o6
3l2
319
68
325
33i
338
344
35i
357
363
37o
376
382
69
388
395
4oi
4o7
4i4
420
426
432
439
445
70
45i
457
463
4?o
476
482
488
••^•••v
494
•i^^M 1
5oo
••^^•V
5o6
N
0
2
3
4
5
6
7
8
9
PP
15
14 13
12
IO
9
8
7
6
.,
1.5
1.4 1.3
1.2
.!
i.i
I.O
0.9
.1
0.8
0.7
0.6
.2
3-o
2.8 2.6
2.4
.2
2.2
2.0
1.8
.2
1.6
1.4
1.2
•3
4-5
4-2 3-9
3-6
• 3
3-3
3-°
2.7
•3
2.4
2,1
1.8
•4
6.0
5-6 5-2
4-8
• 4
4-4
4.0
3-6
•4
3-2
2.8
2.4
•5
7-5
7.0 6.5
6.0
•5
5-o
4-5
• 5
4.0
3-5
3-o
.6
9.0
8.4 7-8
7.2
.6
6.6
6.0
5-4
.6
4.8
4.2
3-6
• 7
10.5
9.8 9.1
8.4
•7
7-7
7.0
6-3
•7
5-6
4-9
4.2
.8
I2.O
1 1. 2 10.4
9.6
.8
8.8
8.0
7.2
8
6.4
5-6
4.8
•9 i3-5
126 11.7 10.8 .9 9.9
i37
FOUR-PLACE LOGARITHMS.
N
0
1
2
3
4
5
6
7
8
9
70
845i
~
463
470
476
482
488
494
5oo
5o6
71
5i3
5i9
525
53i
537
543
549
555
56i
567
72
573
579
585
5oi
597
6o3
6o9
6i5
621
627
73
633
639
645
65i
657
663
669
675
681
686
74
692
698
704
710
716
722
727
733
739
745
75
875!
756
762
768
774
779
785
79i
797
802
76
808
8i4
820
825
83i
837
842
848
854
859
77
865
871
876
882
887
893
899
9o4
9io
9i5
78
92I
927
932
938
943
949
954
96o
965
971
79
976
982
987
993
998
9oo4
9oo9
9oi5
9O2O
9025
80
9o3i
o36
042
047
o53
o58
o63
o69
o74
079
81
o85
o9o
o96
IOI
106
112
117
122
128
i33
82
1 38
i43
i49
1 54
i59
i65
170
i75
180
186
83
|9I
i96
2OI
206
212
2I7
222
227
232
238
84
243
248
253
258
263
269
274
279
284
289
85
9294
299
3o4
3o9
3i5
320
325
33o
335
34o
86
345
35o
355
36o
365
37o
375
38o
385
390
87
39!
j
4oo
4o5
4io
4i5
420
425
43o
435
44o
88
445
45o
455
46o
465
469
474
479
484
489
89
494
499
5o4
5o9
5i3
5i8
523
528
533
538
90
9542
547
552
557
562
566
57i
576
58i
586
91
59(
)
595
600
6o5
6o9
6i4
6i9
624
628
633
92
63(
J
643
647
652
657
661
666
671
675
680
93
685
689
694
699
7o3
7o8
7i3
717
722
727
94
73i
736
74 1
745
75o
754
759
763
768
773
95
9777
782
786
79i
795
800
8o5
8o9
8i4
818
96
823
827
832
836
84 1
845
85o
854
859
863
97
868
872
877
881
886
89o
894
899
-9o3
9o8
98
9I2
9i7
081
926
9
3o
934
939
943
948
952
99
956
96i
965
969
974
978
983
987
99i
996
100
oooo
oo4
oo9
oi3
OI7
022
026
o3o
o35
o4o
N
0
1
2
3
4
5
6
7
8
9
PP
7
6
5
4
.1
0.7
0.6
.1 0.5
0.4
.2
M
12
.2 1.0
0.8
•3
2.1
i 8
•3 *-5
1.2
.4
2.8
2.4
.4 2.0
1.6
• 5
3-5
3 o
• 3 2.5
2.O
.6
4.2
3-6
.6 3.0
2.4
• 7
4-9
4.2
•7 3-5
2.8
.8
5-6
4.8
.8 4-0
3-2
-9
5-4
•9 4-5 3-6
i38
TABLE VI
FOUR-PLACE LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
TO EVERY TEN MINUTES
POUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L.Tang.
d.
L. Cotg.
L. Cos.
d.
0 o
10
20
3o
4o
5o
7
7
8
8
.4637
.7648
.9408
.o658
.1627
son
1760
1250
969
792
669
580
5"
458
4i3
578
3011
1761
i249
969
792
670
580
5"
457
4i5
378
348
322
300
281
263
249
235
223
213
202
194
l85
I78
I7I
l65
158
»54
I48
2.5363
2.2352
2.0591
1.9342
i.8373
o.oooo
0 . OOOO
o.oooo
o.oooo
o.oooo
o.oooo
0
o
o
0
0
I
0
0
o
0
I
0
I
0
o
I
0
I
I
I
o 90
5o
4o
3o
20
10
7.4637
7.7648
7-9409
8.o658
8.1627
1 o
10
20
3o
4o
5o
8
8
8
8
8
8
.2419
.3o88
.3668
.4179
.4637
.5o5o
8.2419
8.3089
8.3669
8.4i8i
8.4638
8.5o53
i.758i
i .691 i
i. 633 i
1.5819
1.5362
i .4947
9-9999
0.9999
9.9999
9.9999
9.9998
9.9998
o 89
5o
4o
3o
20
IO
2 o
10
20
3o
4o
5o
8
8
8
8
8
8
.5428
.5776
.6097
.6397
.6677
.6940
348
321
300
280
263
248
235
222
212
2O2
I92
8.543i
8.5779
8.6101
8.64oi
8.6682
8.6945
.4569
.4221
.3899
.3599
.33i8
.3o55
9-9997
9-9997
9.9996
9.9996
9.9995
9-999^
o 88
5o
4o
3o
20
IO
3 o
10
20
3o
4o
5o
8
8
8
8
8
8
.7188
.7423
.7645
.7867
.8069
.8261
8.7194
8.7429
8.7652
8.7865
8.8067
8.8261
.2806
.2571
.2348
.2i35
.i933
.1739
9-9994
9.9993
9.9993
9.9992
9.9991
9-999°
o 87
5o
4o
3o
20
IO
4 o
IO
20
3o
4o
5o
8
8
8
8
8
8
.8436
.86i3
.8783
.8946
.9104
.9266
185
I77
170
l63
158
IS2
M7
8.8446
8.8624
8.8795
8.8960
8.9118
8.92-72
.i554
.1376
.1205
. io4o
.0882
.0728
9.9989
9.9989
9.9988
9.9987
9.9986
9.9985
0
I
o 86
5o
4o
3o
20
10
5
0
8.9403
8.9420
i.o58o
9.9983
o 85
L
. Cos.
d.
L. Cotg.
d.
L. Tang.
L.Sin.
d.
' O
PP
.2
•3
•4
:I
1
348
300
263
2
3
4
I
I
235
213
185
.i
.2
•3
•4
J
:!
171
158 147
S3
104.4
139.2
174.0
208.8
»j
g
9o
120
150
1 80
210
240
26.3
52-6
78.9
105.2
i3i-5
157-8
184.,
210.4
•
23-5
47.0
70.5
94.0
117.5
141.0
164.5
i88.c
21.3
42. c
63-9
85.2
106.5
127.1
149.1
170.4
18.5
37-o
55-5
74.0
92-5
III.O
129.5
148.0
166.5
17.1
34-2
5i-3
68.4
85-5
102.6
119.7
136.8
15.8 14.7
31.6 29.4
47-4 44-i
63.2 58.8
79-o 73-5
94.8 88.2
no. 6 102.9
126.4 117-6
142.2 i32-3
POUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
5 o
IO
20
3o
4o
5o
8.9403
8.9545
8.9682
8.9816
8.9945
9.0070
142
137
'34
129
125
122
"5
"3
109
107
104
102
99
97
95
93
89
87
85
84
82
80
79
78
76
75
73
73
8.9420
8.9563
8.9701
8.9836
8.9966
9.0093
143
138
135
130
127
123
120
"4
in
1 08
105
104
101
98
97
94
93
89
87
86
84
82
81
80
78
77
76
74
i.o58o
1.0437
1.0299
i.oi64
i.oo34
0.9907
9.9983.
9.9982
9.998i
9.998o
9-9979
9-9977
2
I
I
2
I
I
2
I
2
2
I
2
2
o 85
5o
3o
20
IO
6 o
IO
20
3o
4o
5o
9.0192
9.081 i
9.0426
9.0539
9.0648
9.0755
9.0216
9.o336
9.o453
9.0667
9.0678
9.0786
0.9784
0.9664
0.9547
0.9433
0.9322
0.9214
9-9976
9-9975
9-9973
9.9972
9.9971
9.9969
o 84
5o
4o
3o
20
10
7 o
IO
20
3o
4o
5o
9.0859
9.0961
9. 1060
9.n57
9-1252
9.i345
9.0891
9.o995
9. 1096
9.1194
9.1291
9. i 385
0.9109
0.9005
0.8904
0.8806
0.8709
o.86i5
9.9968
9.9966
9.9964
9.9963
9.9961
9.9959
o 83
5o
4o
3o
20
IO
8 o
IO
20
3o
4o
5o
9.i436
9. 1612
9.1697
9.1781
9.i863
9.1478
9.1569
9.i658
9.1745
9.i83i
9.i9i5
0.8522
o.843i
0.8342
0.8255
0.8169
o.8o85
9.9958
9.9956
9.9954
9.9952
9.995o
9.9948
I
2
2
2
2
2
2
2
2
2
2
2
2
o 82
5o
4o
3o
20
10
9 o
IO
20
3o
4o
5o
9.1943
9.2022
9.2100
9.2176
9.2324
9-1997
9.2O78
9.2i58
9.2236
9.23i3
9.2889
o.8oo3
0.7922
0.7842
0.7764
0.7687
0.7611
9.9946
9.9944
9.9942
9.9940
9.9938
9 . 9936
o 81
5o
4o
20
IO
10 o
9.2397
9.2463
o.7537
9.9934
o 80
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' o
PP 138
"5
"7
.1
.2
•3
•4
•9
104 97
89
.1
.2
•3
•4
I
84
78 73
' I3'J
27. o
•3 4'-4
•4 55-2
• 5 69.0
.6 82.8
.7 96.6
.8 110.4
.9 124.2
12.5
25.0
37-5
50.0
62.5
75-°
87-5
1OO.O
112.5
11.7
23-4
46.8
58.5
70.2
8..g
93.6
105-3
10.4 9.7
20.8 19.4
31.2 29.1
41.6 38.8
52-0 48.5
62.4 58.2
72.8 67.9
83.2 77.6
93.6 87.3
8.9
17.8
26.7
35-6
44-5
53-4
62.3
71.2
80. i
25-2
33-6
42.0
50.4
58.8
67.2
7-8 7-3
15.0 14.6
23.4 21.9
31.2 29.2
39-o 36.5
46.8 43.8
54-6 51.1
62.4 58.4
70. 2 65. 7
i4r
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
10 o
10
20
3o
4o
5o
9
9
9
9
9
9
.2397
.2468
.2538
.2606
.2674
.2740
71
7o
68
68
66
9. 2463
9.2536
9 .2609,
9.2680
9.2750
9.2819
73
73
7i
70
69
68
66
67
65
64
63
63
61
61
61
59
59
58
57
57
56
55
55
54
53
53
53
5»
52
5i
0.7537
0.7464
o.739i
0.7320
0.7250
0.7181
9.9934
9.993i
9.9929
9.9927
9.9924
9.9922
3
2
2
3
2
o 80
5o
4o
3o
20
10
11 o
IO
20
3o
4o
5o
9
9
9
9
9
9
.2806
.2870
.2934
.2997
.3o58
.3119
64
64
63
61
61
9.2887
9.2953
9.3o2o
9.3o85
9.3i49
9.32I2
0.7113
0.7047
0.6980
o.69i5
o.685i
0.6788
9.9919
9.9917
9.9914
9.9912
9.9909
9.9907
3
2
3
2
3
2
o 79
5o
4o
3o
20
IO
12 o
10
20
3o
4o
5o
9
9
9
9
9
9
•3i79
.3238
.3296
.3353
.34io
.3466
59
58
57
57
56
9.3275
9.3336
9.3397
9.3458
9.35 i 7
9.3576
0.6725
0.6664
o.66o3
O.6542
0.6483
0.6424
9.9904
9.9901
9.9899
9.9896
9.9893
9.9890
3
3
2
3
3
3
3
3
3
3
3
3
3
3
3
4
3
3
4
o 78
5o
4o
3o
20
10
13 o
10
20
3o
4o
5o
9.352i
9.3575
9.3629
9.3682
9.3734
9.3786
55
54
54
53
52
52
5i
5°
50
49
49
48
47
9.3634
9.369i
9.3748
9.38o4
9.3859
9.39i4
0.6366
o.63o9
0.6252
o.6i96
o.6i4i
0.6086
9.9887
9.9884
9.9881
9.9878
9.9875
9.9872
o 77
5o
4o
3o
20
10
14 o
10
20
3o
4o
5o
9.3837
9.3887
9.3937
9.3986
9.4o35
9.4o83
9.3968
9.4021
9.4o74
9.4127
9.4178
9.4a3o
o.6o32
o.5979
o.5926
0.5873
0.5822
0.5770
9.9869
9.9866
9.9863
9.9859
9.9856
9.9853
o 76
5o
4o
3o
20
IO
15
0
9.4i3o
9.4281
o.57i9
9.9849
o 75
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' o
PP
.2
•3
•4
:l
7i
68
66
.1
.2
•3
•4
:!
:i
•9
64 61 58
.1
.2
•3
•4
:!
:i
55
53 5i
7-1
14.2
21.3
28.4
33
Si
6.8
13-6
20.4
27.2
34-o
40.8
47.6
54-4
61.2
6.6
!|!
26.4
33-o
39-6
46.2
52.8
59-4
6.4 6.1 5.8
12.8 12.2 II. 6
19.2 l8.3 17.4
25.6 24.4 23.2
32.0 30.5 29.0
38.4 36.6 34.8
44.8 42.7 40.6
51.2 48.8 46.4
57- 6 54-9 52-2
5-5
II. 0
16.5
22.0
27-5
33-o
38.5
44.0
49-5
5-3 5-i
10.6 10.2
iS-9 '5-3
21.2 20.4
26.5 25.5
31.8 30.6
37- * 35^7
42.4 40.8
47-7 45-9
142
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
15 o
10
20
3o
4o
5o
9.4i3o
9.4177
9.4223
9.4269'
9.43i4
9.4359
47
46
46
45
45
44
44
44
42
43
42
4i
41
4i
40
40
40
39
39
38
38
37
38
36
37
36
36
35
36
35
9.4281
9.433i
9.438i
9.443o
9.4479
9.4527
50
50
49
49
48
48
47
47
47
46
46
45
45
' 45
44
44
44
43
43
42
42
42
42
4i
4*
40
40
40
40
40
o.57i9
0.5669
0.5619
0.5570
0.5521
0.5473
9.9849
9.9846
9.9843
9.9839
9.9836
9.9832
3
3
4
3
4
o 75
5o
4o
3o
20
IO
16 o
10
20
3o
4o
5o
9.44o3
9-4447
9.4491
9.4533
9.4576
9.4618
9.4575
9.4622
9.4669
9.4716
9.4762
9.4808
o.5425
0.5378
o.533i
0.5284
0.5238
0.5192
9.9828
9.9825
9.9821
9.9817
9.9814
9.9810
3
4
4
3
4
o 74
5o
4o
3o
20
IO
17 o
IO
20
3o
4o
5o
9-4659
9.4700
9.4741
9.4781
9.4821
9.4861
9.4853
9.4898
9.4943
9.4987
9-5o3i
9.5075
o.5i47
O. 5 I 02
o.5o57
o.5oi3
0.4969
0.4925
9.9806
9.9802
9.9798
9.9794
9.9790
9.9786
4
4
4
4
4
4
o 73
5o
4o
3o
20
IO
18 o
10
20
3o
4o
5o
9.4900
9.4939
9.4977
9. 5oi5
9-5o52
9.5090
9.5u8
9«5i6i
9.52o3
9.5245
9.5287
9.5329
0.4882
0.4839
0.4797
o.4755
o.47i3
0.4671
9.9782
9.9778
9-9774
9-977°
9.9765
9-976*
4
4
4
4
5
4
o 72
5o
4o
3o
20
IO
19 o
IO
20
3o
4o
5o
9.5126
9.5i63
9.5199
9.5235
9.5270
9.53o6
9.5370
9.54n
9.545i
9.5491
9.553i
9.5571 ,
o.463o
0.4589
0.4549
0.4509
0.4469
0.4429
9.9757
9.9752
9-9748
9.9743
9.9739
9.9734
5
4
5
4
5
o 71
5o
4o
3o
20
10
20
0
9.534i
9.56n
0.4389
9.973o
o 70
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' O
PP
2
3
4
5
.6
• 7
.8
•9
49 47
45
.1
.2
«3
•4
• 5
.6
44
43
41
.1
.2
• 3
•4
5
6
!
40
38
36
4-9 4-7
9.8 9.4
14.7 14.1
19.6 18.8
24-5 23.5
29.4 28.2
34-3 32.9
39-2 37-6
44- 1 42.3
4-5
9.0
13-5
18.0
22.5
27.0
3'- 5
30.0
4°-5
ti
13-2
17.6
22. 0
26.4
30.8
35-2
tl
12.9
17.2
21.5
25.8
30.1
34-4
38.7
t:
12.3
i6.4
20.5
24.6
28.7
32.8
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
3-8
7-6
11.4
15-2
19.0
22.8
26.6
30-4
34-2
3-6
7-2
10.8
14.4
18.0
21.6
25.2
28.8
32-4
i43
POUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
20 o
10
20
3o
4o
5o
9.534i
9.5375
9.5409
9.5443
9.5477
9.55io
34
34
34
34
33
33
33
33
32
32
3»
32
3i
3»
3<>
3*
3°
3°
29
3°
29
29
29
28
28
28
28
28
27
27
9«56i i
9.565o
9. 5689
9.5727
9.5766
9.58o4
39
39
38
39
38
38
37
38
37
37
37
36
36
36
36
36
35
36
35
34
35
34
35
34
34
33
34
33
34
•
33
o.4389
o.435o
o.43n
0.4273
0.4234
o.4i96
9.973o
9.9725
9.9721
9.9716
9.9711
9.9706
5
4
5
5
5
o 70
5o
4o
3o
20
10
21 o
10
20
3o
4o
5o
9.5543
9.5576
9.5609
9.564i
9.5673
9.5704
9-5842
9.5879
9.5917
9.5954
9.599i
9.6028
o.4i58
0.4 I 21
o.4o83
o.4o46
o.4oo9
o.3972
9.9702
9.9697
9.9692
9.9687
9.9682
9.9677
5
5
5
5
5
o 69
5o
4o
3o
20
IO
22 o
10
20
3o
4o
5o
9.5736
9.5767
9.5798
9.5828
9.5859
9.5889
9.6064
9.6100
9.6i36
9.6172
9.6208
9.6243
o.3936
o.39oo
0.3864
0.3828
o.3792
o.3757
9.9672
9.9667
9.9661
9.9656
9.9651
9.9646
5
6
5
5
5
o 68
5o
4o
3o
20
IO
23 o
10
20
3o
4o
5o
9.5919
9.5948
9.5978
9.6007
9.6o36
9.6o65
9.6279
9.63i4
9.6348
9. 6383
9.6417
9. 6452
0.3721
0.3686
0.3652
0.3617
0.3583
0.3548
9.9640
9.9635
9.9629
9.9624
9.9618
9.9613
5
6
5
6
5
o 67
5o
4o
3o
20
10
24 o
IO
20
3o
4o
5o
9.6093
9.6121
9.6149
9.6177
9.6205
9.6232
9.6486
9.6520
9. 6553
9.6587
9.662o
9.6654
o.35i4
o.348o
0.3447
o.34i3
o.338o
0.3346
9.9607
9.9602
9.9596
9.9590
9.9584
9.9579
5
6
6
6
5
o 66
5o
4o
3o
20
10
25
0
9.6259
9.6687
o.33i3
9.9573
o 65
L. Cos.
d.
L.Cotg.
d.
L. Tang.
L. Sin.
d.
' 0
PP
. i
.2
•3
•4
• 7
.8
•9
39
37
35
,i
.2
•3
•4
•5
.6
1
•9
34
33
32
2
3
4
1
31
3°
29
1:1
11.7
15-6
'9-5
23-4
27-3
3' ?
35-i
3-7
7-4
n. i
14.8
18.5
22.2
25-9
29.6
3-5
7.0
10.5
14.0
17-5
21.0
24-5
28.0
3'-5
U
10.2
I3.6
17.0
20.4
23.8
27.2
30.6
U
9-9
III
19.8
23.1
26.4
20. 7
E
9.6
12.8
16.0
19.2
22.4
25.6
n
9-3
12.4
;i:i
£J
27.9
3-o
6.0
9.0
12.0
15-0
18.0
21.0
24.0
I1
8.7
ii. 6
i4-5
17-4
20.3
23.2
26.1
1 44
FOUR-PLACE LOGARITHMIC FUNCTIONS.
0 '
L. Sin.
d.
L.Tang.
d.
L. Cotg.
L.
Cos.
d.
25 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6269
.6286
.63i3
.634o
.6366
.6392
27
27
27
26
26
26
26
26
25
26
25
9.6687
9.6720
9.6762
9.6786
9.6817
9.6860
33
32
33
32
33
32
32
32
31
32
32
31
3»
30
30
3*
30
30
30
29
30
29
30
29
29
o.33i3
0.3280
0.3248
o.32i5
o.3i83
o.3i5o
9.9673
9.9667
9.9661
9.9666
9.9549
9.9643
6
6
6
6
6
o 65
5o
4o
3o
20
10
26 o
10
20
3o
4o
5o
9
9
9
9
9
9
.64i8
.6444
.6470
.6496
.6621
.6546
9.6882
9.6914
9.6946
9.6977
9.7009
9.7040
o.3n8
o.3o86
o.3o54
o.3o23
0.2991
0.2960
9.9537
9.953o
9.9524
9.9618
9.9612
9.9606
7
6
6
6
7
o 64
5o
4o
3o
20
10
27 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6670
.6696
.6620
.6644
.6668
.6692
24
25
25
24
24
24
9.7072
9.7103
9 . 7 i 34
9.7166
9.7i96
9.7226
0.2928
0.2897
0.2866
0.2835
0.2804
0.2774
9.9499
9.9492
9.9486
9.9479
9.9473
9.9466
7
6
7
6
7
o 63
5o
4°
3o
20
10
28 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6716
.6740
.6763
.6787
.6810
.6833
24
23
24
23
23
9-7257
9.7287
9.7317
9.7348
9.7408
o.2743
0.2713
0.2683
0.2662
0.2622
0.2692
9.9459
9.9453
9.9446
9.9439
9.9432
9.9426
6
7
7
7
7
o 62
5o
4o
3o
20
10
29 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6856
.6878
.6901
.6923
.6946
.6968
22
23
22
23
22
9.7438
9.7467
9.7497
9.7626
9.7666
9.7585
0.2662
0.2533
o.25o3
0.2444
0.24 i 5
9.9418
9.9411
9 . 94o4
9.9397
9.9390
9.9383
7
7
7
7
7
o 61
5o
4o
3o
20
10
30
0
9
.6990
9.7614
0.2386
9.9375
o 60
L
. Cos.
d.
L. Cotg.
d.
L.Tang.
L. Sin.
d.
' O
PP
.2
•3
•4
:J
•9
28
27
26
.1
.2
•3
•4
:i
25
24 23
.1
22
7
6
2.8
5-6
8.4
II. 2
14.0
16.8
19.6
22.4
2-7
ti
10.8
13-5
16.2
18.9
21.6
24-3
2.6
10.4
13.0
15.6
18.2
20.8
23-4
2-5
7-5
10.0
12.5
15-0
17-5
20.0
22-5
2.4 2.3
4.8 4.6
7.2 6.9
9.6 9.2
12.0 II.5
14.4 13.8 :
16.8 16.1 i
19.2 18.4
21.6 20.7
2.2
n
8.8
II.O
13-2
19.8
0.7
1.4
2.1
2.8
3-5
4.2
4.9
5-6
6-3
0.6
1.2
1.8
2-4
5-4
i45
FOUR-PLACE LOGARITHMIC FUNCTIONS
o
r
L. Sin.
d.
L. Tang.
d.
L.Cotg.
L. Cos.
d.
30 o
10
20
3o
4o
5o
9.6990
9.7OI2
9.7088
9.7O55
9.7076
9.7097
22
21
22
21
21
21
21
21
21
20
21
20
20
20
20
20
20
19
19
20
19
18
18
18
18
9-76i4
9.7644
9-7673
9-77°i
9-773o
9.7759
30
29
28
29
29
0.2886
0.2356
0.2827
0.2299
0.2270
O.224l
9.9375
9.9868
9.9861
9.9353
9.9346
9-9338
7
7
8
7
8
o 60
5o
4o
3o
20
10
31 o
10
20
3o
4o
5o
9.7II8
9.7160
9.7181
9.7201
9. 7222
9.7788
9.7816
9.7845
9.7878
9.7902
9.7980
29
28
29
28
29
28
28
28
28
28
28
27
28
28
27
28
27
28
27
27
27
27
27
27
O.22I2
0.2184
o.2i55
0.2127
0.2098
0.2070
9.933i
9.9323
9.9815
9.9808
9.9800
9.9292
7
8
8
7
8
8
o 59
5o
4o
3o
20
10
32 o
10
20
3o
4o
5o
9.7242
9.7262
9.7282
9.7802
9.7822
9.7342
9.7958
9.7986
9.8014
9.8042
9.8070
9.8097
O.2O42
O.2OI4
0.1986
0.1958
o. 1980
o. 1908
9.9284
9.9268
9.9260
9.9252
9.9244
8
8
8
8
8
o 58
5o
4o
3o
20
10
33 o
10
20
3o
4o
5o
9.736i
9.738o
9 . 74oo
9.7419
9. 7438
9.7457
9.8125
9.8i53
9.8180
9.8208
9.8235
9.8268
0.1875
0.1847
o. 1820
0.1792
o. 1765
o.i737
9.9286
9.9228
9.9219
9.9211
9.9208
9.9194
8
9
8
8
9
o 57
5o
4o
3o
20
10
34 o
10
20
3o
4o
5o
9.7476
9.7494
9.75i3
9.753i
9.755o
9.7568
9.8290
9.83i7
9. 8344
9.837i
9.8898
9.8425
0. I7IO
0.1688
o.i656
o. 1629
o. 1602
o. i575
9.9186
9.91-77
9.9169
9.9160
9.9i5i
9.9142
9
8
9
9
9
8
o 56
5o
4o
3o
20
IO
35
0
9.7586
9.8452
27
o.i548
9.9184
o 55
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.Sin.
d.
' O
PP
2
3
4
j
29 28
27
.2
•3
•4
• 5
.6
•9
22
21
20
.2
•3
•4
19
8 7
2.9 2.8
5-8 5-6
8.7 8.4
II. 6 II. 2
14-5 14-0
17.4 16.8
20.3 19.6
23.2 22.4
26.1 25.2
2.7
10.8
16.2
18.9
21.6
24-3
2.2
a
8.8
II. 0
13-2
\7'.6
IQ.8
2.1
8-4
10.5
12.6
i6!8
18.9
2.0
4-0
6.0
8.0
10.0
12. 0
14.0
16.0
18.0
'•9
3-8
5-7
7.6
9-5
11.4
13-3
15-2
0.8 0.7
1.6 1.4
2.4 2.1
3-2 2.8
4-° 3-5
4.8 4.2
5-6 4-9
6.4 5.6
i46
POUR PLACE LOGARITHMIC FUNCTIONS.
O I
L. Sin.
d.
L.Tang.
d.
L. Cotg
L. Cos.
d.
35 o
10
20
3o
4o
5o
9.7686
9.7604
9.7622
9.7640
9.7667
9.7676
18
18
18
18
18
'7
9.8462
9.8479
9.8606
9.8533
9.8669
9.8686
27
27
27
26
27
27
26
27
26
27
26
26
27
26
26
26
26
26
26
26
26
26
26
26
25
26
26
25
o.i548
o. 1621
0.1494
o. 1467
o. i44i
o. i4i4
ON ON ON ON ON ON
9134
9126
9116
9107
9098
9089
9
9
9
9
9
9
IO
9
9
10
9
o 55
5o
4o
3o
20
10
36 o
10
20
3o
4o
5o
9
9
9
9
9
9
.7692
.7710
.7727
• 7744
.7761
.7778
9.86i3
9.8639
9.8666
9.8692
9.8718
9.8746
0.1387
o. i36i
o.i334
o.i3o8
0.1282
0.1266
9.9080
9.9070
9.9061
9.9062
9.9042
9.9033
o 54
5o
4o
3o
20
10
37 o
IO
20
3o
4o
5o
9.7796
9.7811
9.7828
9.7844
9.7861
9.7877
I7
16
16
'7
16
9.8771
9.8797
9.8824
9.8860
9.8876
9.8902
o. 1229
O. I2O3
o. 1176
o. i 160
O. I 124
0.1098
9.9023
9.9014
9.9004
9.8996
9.8986
9.8976
9
10
9
10
10
o 53
5o
4o
3o
20
IO
38 o
10
20
3o
4o
5o
000 000
.7893
.7910
.7926
.7941
.7967
.7973
16
15
16
16
9.8928
9.8964
9.8980
9.9006
9.9032
9.9068
o. 1072
o . i o46
O. IO2O
0.0994
0.0968
0.0942
9.8966
9.8966
9.8946
9.8935
9.8926
9.8916
10
10
10
10
10
o 52
5o
4o
3o
20
IO
39 o
10
20
3o
4o
5o
000 000
.7989
.8oo4
.8020
.8o35
.8o5o
.8066
15
16
15
15
16
9.9084
9.9110
9.9i35
9.9161
9.9187
9.9212
0.0916
0.0890
0.0866
0.0839
o.o8i3
0.0788
9.8906
9.8896
9.8884
9.8874
9.8864
9-8853
10
II
10
10
II
o 51
5o
4o
3o
20
IO
40
0
9
.8081
15
9.9238
26
0.0762
9.8843
o 50
L
.Cos.
d.
L. Cotg.
d.
L.Tang.
L. Sin.
d.
' 0
PP
.1
.2
•3
•4
•5
.6
• 7
.8
•9
36
as
18
.2
•3
•4
•7
.8
I?
if
15
.1
.2
•3
•4
•5
.6
•7
.8
•9
ii
IO
9
2.6
5-2
7.8
10.4
13.0
15-6
18.2
20.8
23-4
2-5
5-o
7-5
IO.O
12.5
15.0
17-5
20. o
22.5
1.8
3-6
5-4
7.2
9.0
10.8
12.6
14.4
'
3-4
6.8
8-5
10.2
II-9
13-6
'5-3
1.6
tf
6.4
8.0
9.6
II. 2
12.8
14.4
3-o
4-5
6.0
7-5
9.0
10.5
12:0
i.i
2.2
3-3
4-4
1:1
1.1
1.0
2.O
4.0
6.0
7.0
8.0
9.0
0.9
1.8
2.7
3-6
4-5
5-4
6-3
7.9
8.1
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L.Tang.
d.
L. Cotgr.
L. Cos.
d.
40 o
IO
20
3o
4o
5o
9.8081
9.8096
9.8111
9.8126
9.8140
9.8i55
15
15
14
15
15
9.9238
9.9264
9.9289
9.93i5
9.9341
9.9366
26
25
26
26
25
26
25
26
25
26
25
25
26
25
26
25
25
26
25
25
25
25
25
25
26
25
25
25
26
25
0.0762
0.0736
0.071 i
o.o685
0.0669
o.o634
9.8843
9.8832
9.8821
9.8810
9.8800
9.8789
ii
ii
10
II
o 50
5o
4o
3o
20
IO
41 o
IO
20
3o
4o
5o
9.8169
9.8184
9.8198
9.8213
9.8227
9.8241
15
14
15
14
14
9.9392
9.9417
9.9443
9.9468
9.9494
9.95i9
0.0608
o.o583
o.o557
o.o532
o.o5o6
o.o48i
9
9
9
9
9
9
.8778
.8767
.8756
.8745
.8733
.8722
II
II
II
12
II
o 49
5o
4o
3o
20
IO
42 o
IO
20
3o
4o
5o
9.8255
9.8269
9.8283
9.8297
9-83ii
9.8324
«4
H
J4
14
»3
9.9544
9.9570
9.9595
9.9621
9.9646
9.9671
o.o456
o.o43o
o.o4o5
0.0379
o.o354
0.0329
9.8711
9.8699
9.8688
9.8676
9.8665
9.8653
12
II
12
II
12
o 48
5o
4o
3o
20
IO
43 o
IO
20
3o
4o
5o
9.8338
9.835i
9.8365
9.8378
9.8391
9-84o5
'3
'4
»3
13
»4
9.9697
9.9722
9.9747
9.9772
9.9798
9.9823
o.o3o3
0.0278
0.0253
0.0228
O.O2O2
0.0177
9
9
9
9
9
9
.864i
.8629
.8618
.8606
.8594
.8582
12
II
12
12
12
o 47
5o
4o
3o
20
10
44 o
IO
20
3o
4o
5o
9.8418
9-843i
9-8444
9.8457
9.8469
9.8482
»3
»3
13
12
»3
9.9848
9.9874
9.9899
9 . 9924
9.9949
9.9975
0.0152
0.0126
O.OIOI
0.0076
o.oo5i
O.OO25
9
9
9
9
9
9
.8569
.8557
.8545
.8532
.8520
.85o7
12
12
»3
12
13
o 46
5o
4o
3o
20
IO
45 o
9.8495
0,0000
o.oooo
9
.8495
o 45
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' O
PP
.1
.2
•3
•4
:i
.1
•9
26
«5
15
M
13
12
i
.2
•3
•4
:!
:l
•9
IX
IO
2.6
5-2
78
10.4
130
15-6
182
20.8
23 4
2-5
5-o
7-5
10.0
12 5
150
17 5
20.0
* 5
3-o
4-5
60
75
9.0
10.5
I2.O
13-S
.2
• 3
•4
:!
;J
•9
a
4-2
5-6
7-0
8.4
9.8
H.2
12.6
a
3-9
5-2
6-5
7.8
9.1
104
11.7
1.2
y
4.8
6.0
7.2
8-4
9.6
10.8
I.I
2.2
3-3
4.4
1:1
7-7
8.8
9-9
1.0
2.O
3-0
4-0
5-°
6.0
7.0
8.0
9.0
i48
TABLE VII
FOUR-PLACE
NATURAL TRIGONOMETRIC
FUNCTIONS
TO EVERY TEN MINUTES
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
0 o
10
20
3o
4o
5o
o.oooo
0.0029
0.0068
0.0087
o.oi 16
o.oi45
29
29
29
29
29
3»
29
29
29
29
29
o
o
o
o
0
o
.0000
.0029
.0068
.0087
.01 16
.0145
29
29
29
29
29
3°
29
29
29
29
29
29
29
29
3°
29
29
29
29
29
3°
29
29
29
30
29
29
29
30
29
infinit.
343.7737
171.8864
114.6887
86.9398
68.7601
it*
818
613
477
382
312
26c
22C
i8fi
163
H3
126
112
IOC
9c
81
7A
6i
6-.
5y
55
4C
4!
4-
3<
i
i
0
0
.0000
.0000
.0000
.0000
•9999
•9999
o
o
o
I
o
o 90
5o
4o
3o
20
10
1 o
IO
20
3o
4o
5o
0.0176
0.0204
0.0233
0.0262
0.0291
O.O320
0
o
o
o
0
o
.0176
. o£o4
.0233
.0262
.0291
032O
67.2900
49. 1039
42.9641
38.i885
34.3678
3i.24i6
61
98
56
07
62
o
o
o
0
0
0
.9998
.9998
•9997
•9997
.9996
.9996
0
i
o
I
I
I
2
I
I
o 89
5o
4o
3o
20
10
2 o
10
20
3o
4o
5o
o.o349
0.0378
0.0407
o.o436
o.o465
o.o494
29
29
29
29
29
o
0
o
0
0
0
.o349
.o378
.0407
.o437
.o466
.0496
28.6363
26.43i6
24.54i8
22.9038
21.4704
20.2066
bj
47
98
80
34
48
o
0
0
0
o
o
.9994
.9993
.9992
•999°
.9989
.9988
o 88
5o
4o
3o
20
IO
3 o
10
20
3o
4o
5o
0.0623
0.0662
o.o58i
0.0610
o .o64o
0.0669
29
29
29
30
29
0
o
0
o
o
0
.0624
.o553
.0682
.0612
.o64i
.0670
19.0811
18.0760
17.1693
16.3499
i5.6o48
14.9244
4b
61
57
94
5i
04
37
40
98
07
57
43
,61
o
0
o
0
0
0
.9986
.9986
.9983
.9981
.9980
.9978
I
2
2
I
2
o 87
5o
4o
3o
20
10
4 o
10
20
3o
4o
5o"
0.0698
0.0727
0.0766
0.0786
0.08 i 4
o.o843
29
29
29
29
29
29
0
0
0
0
0
0
.0699
.0729
.0768
.0787
.0816
.o846
i4.3oo7
13.7267
13.1969
12. 7062
I2.25o5
11.8262
0
0
0
0
0
o
.9976
•9974
.9971
.9969
.9967
.9964
2
3
2
2
3
o 86
5o
4o
3o
20
10
5 o
o.
0872
0
.0876
ii.43oi
0
.9962
o 85
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
/ 0
PP
.2
•3
•4
J
:i
•9
26053
16380
i "45
.i
.2
• 3
•4
•5
.6
•7
.8
•9
8194
6237
4907
.1
.2
•3
•4
•i
:I
•9
3961
30
29
2605
5211
7816
10421
13027
15632
18237
20842
23448
1638
3276
4914
6552
8190
9828
11466
13104
14742
1125
2249
3374
4498
5623
6747
7872
8996
IOI2I
819.4
1638.8
2458-2
3277-6
4097.0
4916.4
5735-8
6555.2
r374-6
623.7
1247.4
1871.1
2494.8
3"8.s
3742.2
4365-9
4989.6
490.7
981.4
1472.1
1962.8
2453-5
2944.2
3434-9
3925.6
4416.
396-1
792.2
1188.3
1584-4
1980.5
2376.6
2772.7
3168.8
$564-9
3-0
6.0
9.0
12.0
15-0
18.0
21.0
24.0
2.9
5-8
8-7
ii. 6
14-5
17.4
20.3
23.2
26.1
160
POUR-PLACE NATURAL FUNCTIONS.
0 '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
5 o
10
2O
0.0872
0.0901
0.0929
29
28
o.o875
0.0904
0.0934
29
3°
n.43oi
u.o594
io.7n9
3707
3475
o
0
0
.9962
•9959
•9957
3
2
o 85
5o
4o
3o
0.0968
29
29
0.0963
^9
29
10.3854
32^
5
o
.9954
3
•3
3o
4o
0.0987
0.0992
io.o78o
0
.9951
20
5o
o. 1016
29
O. IO22
3°
9.7882
0
• 9948
3
10
6 o
10
o. io45
o. 1074
29
o. io5i
o. 1080
29
9.5i44
9.2553
2738
2591
0
0
• 9945
.9942
3
o 84
5o
20
0.
no3
29
O.IIIO
3°
9.oo98
2455
o
•9939
3
4o
3o
4o
o.
o.
I I 32
1161
29
29
o. i 189
o. 1 169
*y
30
8.7769
8.5555
2329
2214
0
o
.9936
.9932
3
4
3o
20
5o
o.
1190
•'y
0.1198
^9
8.345o
0
.9929
3
IO
7 o
IO
20
o.
o.
o.
1219
1248
1276
29
28
o. 1228
o. 1267
0.1287
29
30
8.i443
7.953o
7.7704
1913
1826
0
0
0
.9926
.9922
.9918
3
4
o 83
5o
4o
3o
o.
i3o5
29
o.i3i7
30
29
7.5958
0
.9914
4
3o
4o
o.
i334
o.i346
7.4287
0
.9911
20
5o
o.
i363
^9
0.1376
7.2687
0
.9907
4
IO
8 o
IO
o.
0.
1392
1421
29
29
o.i4o5
o.i435
30
7.ii54
6.9682
J533
1472
0
0
.9903
.9899
4
o 82
5o
20
o.
1449
o.i465
3°
6.8269
I4I3
0
.9894
5
4o
3o
0.
1478
29
o.i49
3
29
6.69i2
'357
o
.9890
4
3o
4o
5o
o.
0.
1607
i536
29
o. 1624
o.i554
30
6.56o6
6.4348
1258
0
0
.9886
.9881
5
20
IO
9
0
o.
1 564
o.i584
3°
6.3i38
,,68
0
.9877
o 81
IO
20
0.
0.
1693
1622
29
o.i6i4
o. i644
3°
6.i97o
6.o844
1126
0
0
.9872
.9868
4
5o
4o
?K
29
10
K
5
3o
o.
i65o
0.1673
3°
5.9758
0
.9863
3o
4o
o.
i67g
o. 1703
5.87o8
1050
0
.9868
20
5o
0.
1708
29
o.i733
3°
5.7694
1014
0
.9853
5
IO
10
O
o.
i736
o.i763
5.6713
901
0
.9848
o 80
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
f O
PP
2738
1533
981
30
29
28
5 4
3
tl
273.8
153-3
98.1 .1
3.0
2.9
2.8
.!
0.5 0.4
o-3
2
547-6
306.6
196.2 .2
6.0
5-8
5.6
.2
i.o 0.8
0.6
•3
821.4
459-9
294-3 -3
9.0
8-7
8.4
•3
1 5 i
1.2
0.9
•4
1095.2
6'3-2
392.4 .4
12. 0
n.6
II. 2
•4
2.0
1.6
1.2
• 5
1369.0
766.5
490.5 .5
15-0
14-5
14.0
.5
2-5
2.O
»-5
.6
1642.8
919.8
588.6 .6
18.0
J7-4
16.8
.6
3-o
2.4
1.8
.7
1916.6
1073.1
686.7 -7
21.0
20.3
19.6
.7
3-5
2.8
2.1
.8
2190.4
1226.4
784.8 .8
24.0
23.2
22.4
.8
4.0
3-2
2.4
2464.2
T.7Q.7 882.0 .9
27.0 26.1
•9
4.5 3.6 2.7
POUR-PLACE NATURAL FUNCTIONS.
0 '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
10 o
IO
20
3o
4o
5o
o.
0.
o.
o.
o.
o.
1735
i765
1794
1822
1861
1880
29
29
28
29
29
28
29
28
29
28
29
28
29
28
a8
29
28
29
28
28
28
29
28
28
28
29
28
28
28
0.1763
o.i793
0.1823
o.i853
o.i883
o. 1914
3°
3°
3°
3°
31
30
30
3°
3i
3°
3°
3i
3°
30
31
3°
3i
5.6713
5.5764
5.4845
5.3955
5.3o93
5 .2257
94
9'
8c
8(
8;
81
7*
7<
74
r<
7<
6£
6f
6<
62
61
5c
5*
5<
5!
54
52
5i
5C
45
4*
4«
45
44
43
9
9
)0
>2
6
0.9848
0.9843
0.9838
o.9833
0.9827
0.9822
5
5
5
6
5
o 80
5o
4o
3o
20
10
11 o
10
20
3o
4o
5o
o.
o.
0.
o.
o.
o.
1908
i937
1966
i994
2O22
2o5i
0.1944
o. 1974
0.2OO4
o.2o35
o.2o65
0.2095
5.i446
5.o658
4.9894
4-9i52
4.843o
4.7729
8
4
2
•2
)j
3
4
6
9
3
7
n
0.9816
0.9811
0.9805
0.9799
0.9793
0.9787
5
6
6
6
6
o 79
5o
4o
3o
20
IO
12 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
2079
2IO8
2i36
2164
2193
2221
0.2126
o.2i56
0.2186
0.2217
0.2247
0.2278
4.7046
4.6382
4.5736
4.5i07
4.4494
4.3897
0.9781
0.9775
0.9769
0.9763
0.9757
0.9750
6
6
6
6
7
o 78
5o
4o
3o
20
10
13 o
IO
20
3o
4o
5o
o.
0.
0.
o.
o.
o.
225O
2278
23o6
2334
2363
2391
0.2309
0.2339
0.2370
0.2401
0.2432
0.2462
30
31
3i
3i
30
4.33i5
4.2747
4.2193
4.i653
4. 1126
4.0611
8
4
0
7
5
3
i
i
9
9
8
9
0.9744
0.9737
0.9730
0.9724
0.9717
0.9710
7
7
6
7
7
7
7
7
8
7
7
g
o 77
5o
4o
3o
20
IO
14 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
0.
2419
244?
2476
2604
2532
256o
0.2493
0.2524
0.2555
0.2586
0.2617
0.2648
3*
3i
3i
3»
3»
4.0108
3.9617
3.9i36
3.8667
3.8208
3.7760
0.9703
0.9696
0.9689
0.9681
0.9674
0.9667
o 76
5o
4o
3o
20
IO
15
O
o.
2588
0
.2679
3.732i
0
.9659
o 75
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' o
PP
i
2
3
4
I
I
742
448
31
.1
.2
•3
•4
-5
.6
:i
30
29
28
.1
.2
•3
•4
•5
.6
• 7
.8
7
6
5
74.2
148.4
222.6
296.8
371-0
445-2
S'9-4
593-6
44-8
89.6
134-4
179.2
224.0
268.8
313-6
358.4
403.2
t:
9-3
12.4
;i:I
21.7
24.8
3-o
6.0
9.0
12.0
15-0
18.0
21.0
24.0
*«
8.7
ii. 6
M-5
17.4
20.3
2.8
5-6
8.4
II. 2
14.0
16.8
19.6
22.4
0.7
1.4
2.1
2.8
3-5
4-2
4.9
r
0.6
1.2
1.8
2-4
H
ti
5-4
o-5
I.O
i-5
2.0
2-5
3-o
3-5
4.0
4-5
152
POUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
15 o
10
20
3o
4o
5o
o.
0.
0.
o.
o.
0.
2588
2616
2644
2672
2700
2728
28
28
28
28
28
28
28
28
28
28
28
28
28
27
28
28
27
28
28
27
28
28
27
0.2679
0.271 i
0.2742
0.2773
o.28o5
0.2836
32
3»
32
32
32
3*
32
32
3»
32
32
32
32
32
3.7321
3.6891
3.6470
3. 6059
3.5656
3.526i
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3<
3
2
2
2
2
2
2
2<
2<
2.
2.
2.
3°
:i
ii
33
M
37
79
71
^5
37
50
0.9659
0.9652
0.9644
0.9636
0.9628
0.9621
7
8
8
8
7
o 75
5o
4o
3o
20
IO
16 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
2756
2784
2812
2840
2868
2896
0.2867
0.2899
0.2931
0.2962
0.2994
o.3o26
3.4874
3.4495
3.4i24
3.3759
3.34o2
3.3o52
0.9613
0.9605
0.9596
0.9588
0.9580
0.9572
8
9
8
8
8
o 74
5o
4o
3o
20
IO
17 o
IO
20
3o
4o
5o
0.2924
0.2952
0.2979
0.3007
o.3o35
o.3o62
0.3089
0. 3l2I
o.3i53
o.3i85
0.3217
3.2709
3.237i
3.2o4i
3.1716
3.i397
3.io84
13
38
5°
25
9
3
0.9563
o.9555
o.9546
o.9537
o.9528
O.952O
8
9
9
9
8
o 73
5o
4o
3o
20
IO
18 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
3090
3n8
3i45
3i73
3201
3228
0.3249
o.328i
o.33i4
0.3346
o.3378
o.34n
32
33
32
32
33
3.0777
3.o475
3.0178
2.9887
2.9600
2.93i9
Jl
m
17
H
37
h
o.95n
o.95o2
o.9492
o.9483
o.9474
o.9465
9
IO
9
9
9
o 72
5o
4o
3o
20
IO
19 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
3256
3283
33u
3338
3365
3393
27
28
27
27
28
0.3443
0.3476
o.35o8
o.354i
0.3574
0.3607
33
33
33
33
2.9042
2.877o
2.8502
2.8239
2. -7980
2. 7725
1
2
,8
>3
9
5
o.9455
o.9446
o.9436
o.9426
o.94i7
o.94o7
9
10
10
9
IO
o 71
5o
4o
3o
20
IO
20
0
o.
3420
27
o.364o
2.7475
0
o.9397
o 70
Cos.
d.
Cotg
d.
Tang.
d
•
Sin.
d.
' O
PP
.1
.2
•3
•4
:J
•9
255
33
32
31
28
27
10
9
8
25-5
51.0
76.5
102.0
127.5
i53-o
178-5
204.0
1:1
9-9
13.2
16.5
19.8
23.1
26.4
29-7
3.2
6.4 .2
9.6 .3
12.8 .4
16.0 .5
19.2 .6
22.4 .7
25.6 .8
11
9-3
12.4
S3
£i
2.8
tf
II. 2
14-0
16.8
19.6
22.4
2-7
K
10.8
'3-5
16.2
18.9
21.6
24-3
.1 I.O
.2 2.0
•3 3-°
.4 4.0
•5 5'°
.6 6.0
•7 7-°
.8 8.0
.9 9.0
a
2-7
3-6
4-5
5-4
6-3
K
0.8
1.6
2.4
3-2
4.0
48
5-6
6.4
7-2
i53
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
20 o
IO
20
3o
4o
5o
o.
o.
0.
o.
0.
o.
3420
3448
3475
35o2
3529
3557
28
27
27
27
28
27
27
27
27
27
27
o.364o
0.3673
0.3706
0.3739
0.3772
o.38o5
33
33
33
33
33
2.7475
2.7228
2.6985
2.6746
2.65n
2.6279
247
243
239
235
232
228
o.9397
0.9387
o.9377
o.9367
o.9356
o.9346
IO
IO
10
II
IO
o 70
5o
4o
3o
20
IO
21 o
IO
20
3o
4o
5o
o.
o.
0.
o.
0.
0.
3584
36n
3638
3665
3692
37i9
0
0
0
0
0
0
.3839
.3872
.3906
.3939
.3973
.4oo6
33
34
33
34
33
2.6o5i
2.5826
2.56o5
2.5386
2.5172
2.4960
225
221
2I9
214
212
o.9336
o.9325
o.93o4
o.9293
0.9283
II
IO
II
II
IO
o 69
5o
4o
3o
20
IO
22 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
3746
3773
38oo
3827
3854
388i
27
27
27
27
27
0
0
0
0
0
0
.4o4o
• 4o74
.4108
'•4i76
.4210
34
34
34
34
34
2.4545
2.4342
2.3945
2.3750
206
203
2OO
I97
195
0.9261
0.9250
0.9239
0.9228
0.9216
II
II
II
II
12
o 68
5o
4o
3o
20
IO
23 o
10
20
3o
4o
5o
o.39o7
o.3934
o.396i
o.3987
o.4oi4
o.4o4i
27
27
26
27
27
26
27
26
27
26
27
0
0
0
0
0
0
.4245
•4279
.43i4
.4348
.4383
.44i7
35
34
35
34
35
34
35
35
35
35
35
36
35
2.3559
2.3369
2.3i83
2.2998
2.2817
2.2637
I90
1 86
185
181
1 80
177
174
i73
170
168
1 66
i6<i
0.9205
o . 9 i 94
0.9182
o.9i7i
0.9159
o.9i47
II
12
II
12
12
o 67
5o
4o
3o
20
IO
24 o
10
20
3o
4o
5o
o.
o.
o.
0.
o.
o.
4o67
4094
4120
4i47
4173
4200
0
0
0
0
0
o
.4452
.4487
.4522
.4557
.4592
.4628
2.2460
2.2286
2.2II3
2.i943
2.1775
2. 1609
0.9135
0.9124
0.9112
o.9ioo
o.9o88
o.9o75
II
12
12
12
13
12
o 66
5o
4o
3o
20
IO
25
0
o.
4226
0
.4663
2.i445
0
.9o63
o 65
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' O
PP
.1
.2
•3
•4
•5
.6
177
35
34
.2
•3
•4
• 5
.6
• 7
.8
•9
33
27
26
.1
.2
•3
•4
•5
.6
12
ii
IO
17.7
35-4
70.8
88.5
106.2
123.9
141.6
3-5
7.0
10.5
14.0
21.0
24-5
28.0
3z-5
U
10.2
13-6
I7.0
20.4
23.8
27.2
1:1
9-9
13.2
16.5
19.8
III
2.7
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
24-3
2.6
10.4
13.0
15-6
18.2
20.8
23-4
1.2
2.4
3-6
6.0
7.2
8.4
9.6
10.8
2.2
3-3
4-4
ii
1.1
1.0
2.O
3-o
4.0
5-°
6.0
7.0
8.0
1 54
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
25
0
o.
4226
0.4663
06
2.i445
162
0.9063
o 65
IO
20
o.
o.
4253
4279
26
0.4699
0.4734
35
36
2.1283
2.II23
160
158
0.9051
0.9038
13
12
5o
4o
3o
o.
43o5
26
0.4770
06
2.0965
1*6
0.9026
3o
4o
o.
433i
o.48o6
2.0809
0.9013
20
5o
0.
4358
o.484i
35
2 o655
J54
0.9001
IO
26
0
0.
4384
26
0.4877
36
2.o5o3
0.8988
o 64
10
o.
44io
0.4913
2.o353
0.8975
5o
20
0.
4436
26
0.495
0
37
2.O2O4
149
0.8962
IJ
4o
fh
36
i
7
JO
3o
o.
4462
26
0.4986
16
2.0057
0.8949
3o
4o
o.
4488
O.5O22
.9912
0.8936
20
5o
o.
45i4
26
o.5o5
9
37
.9768
144
0.8923
M
IO
27
0
o.
454o
26
o.Sog
5
.9626
0.8910
o 63
10
o.
4566
o.5i32
.9486
0.8897
5o
20
o.
4592
26
0.5169
37
.9347
1
59
0.8884
1J
4o
3o
4o
0.
o.
46i7
4643
25
26
0.5206
0.5243
37
37
.9210
.9074
136
0.8870
0.8857
13
3o
20
5o
0.4669
26
0.5280
37
.8940
134
0.8843
M
IO
28
0
o.
4695
o.53i7
.8807
X33
0.8829
o 62
10
o.
4720
0.5354
.8676
0.8816
5o
20
o.
4746
20
0.5392
3^
.8546
130
0.8802
14
4o
3o
0.
4772
26
o.543o
38
.84i8
128
0.8788
14
3o
4o
5o
o.
o.
4797
4823
36
o. 5467
o.55o5
38
.8291
.8i65
126
0.8774
0.8760
20
IO
29
0
o.
4848
0.5543
38
i.8o4o
I25
0.8746
o 61
IO
o.
4874
o.558i
S8
1.7917
123
0.8732
14
5o
20
o.
4899
25
0.5619
3»
1.7796
121
0.8718
M
4o
3o
4o
o.
o.
4924
25
26
0.5658
0.5696
39
38
1.7675
1.7556
121
0.8704
0.8689
14
15
3o
20
5o
o.
4975
•3
0.5735
39
i.7437
o.8675
'4
IO
30
0
o.
5ooo
0.5774
1.7321
0.8660
o 60
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
> o
PP
149
I3i
39
38
37
36
25
M
13
. I
14.9
2Q 8
26 2
3-9 -i
78 2
3-8
3-7
3-6
.1
2.5
LI
26
•3
44-7
39-3
"•7 -3
11.4
II. I
10.8
•3
7-5
4-2
3-9
•4
59-6
52.4
15.6 .4
15.2
14-8
14.4
•4
10. 0
5-6
5-2
.5
74-5
65-5
»9-5 -5
19.0
i8.S
18.0
•5
12.5
7.0
6-5
.6
89.4
78.6
23-4 -6
22.8
22.2
21.6
.6
15-0
8.4
7.8
•7
104.3
91.7
27-3 -7
26.6
25.9
25-2
.7
I7-5
9.8
9.1
.8
119.2
104.8
31.2 .8
30.4
29.6
28.8
.8
20.0
II. 2
10.4
117.9 35-i -9
32-4
22.5
12.6 11.7
i55
FOUR-PLACE NATURAL FUNCTIONS.
o
'
s
in.
d.
1
rang
.
d.
Cot£
r.
d
.
Cos.
d.
30
i
£
i
|
0
0
0
0
^o
0
o.
o.
o.
o.
o.
o.
5ooo
5o25
5o5o
5o75
5ioo
5i25
25
25
25
25
25
0
0
0
0
0
0
.577
.58i
.585
.589
.593
.596
i
2
I
D
3
I
38
39
39
40
39
.73
I.72<
.70(
.69r
.68(
.67,
21
D5
?°
77
34
)3
ii
ii
U
ii
ii
6
5
3
3
C
0
0
0
0
o
.8660
.8646
.863i
.8616
.8601
.8587
14
15
15
15
J4
o 60
5o
4o
3o
20
IO
31
i
£
i
\
0
0
0
0
o
0
o.
o.
o.
o.
o.
o.
5i5o
5i75
52OO
5225
525o
5275
25
25
25
25
'25
0
0
o
o
o
0
.600
.6o4i
.608
.612!
.616!
.620!
)
3
3
3
3
39
40
40
40
40
.66.
.65:
.645
.63
.62:
.6ic
i3
M
26
9
2
>7
1C
1C
10
10
10
9
8
7
7
5
0
0
o
o
o
0
.8572
.8557
.8542
.8526
.85n
.8496
i5
15
16
15
15
16
o 59
5o
4o
3o
SO
IX)
32
i
£
c
/!
i
0
0
o
0
0
0
o.
o.
o.
o.
o.
o.
5299
5324
5348
5373
5398
5422
25
24
25
25
24
0
0
0
0
o
0
.6241
.628*
.633*
.637
.64i!
.645
?
?
5
[
2
3
40
41
41
4i
4i
.6o(
.59c
.57c
.56c
.55c
.54c
>3
)0
>8
>7
>7
^7
10
10
1C
10
IO
3
2
I
0
O
o
o
o
o
o
o
.848o
.8465
.845o
.8434
.84i8
.84o3
15
IS
16
16
15
ifi
o 58
5o
4o
3o
20
10
33
i
£
4
c
C)
0
0
0
0
0
o.
o.
o.
o.
o.
o.
5446
5471
5495
55i9
5544
5568
25
24
24
25
24
0
0
0
0
0
0
.649^
.6531
.657
.66i<
.666
.670
i
7
?
[
5
42
4i
42
42
42
.53c
.53c
.52(
.5i(
,5oi
• 49
>9
>i
>4
>8
3
9
y
9
9
9
9
9
8
7
6
5
4
o
o
0
o
o
0
.8387
.837i
.8355
.8339
.8323
.83o7
16
16
16
16
16
o 57
5o
4o
3o
20
10
34
i
£
C
2
f.
0
0
o
0
o
0
o.
o.
o.
o.
o.
o.
5592
56i6
564o
5664
5688
5712
24
24
24
24
24
24
0
0
0
0
0
0
.674
.678
.683<
.687
.69i(
.695«
7
5
5
I
42
43
43
43
43
.48i
.47:
.4&
.45J
• 44(
.43-
56
*3
il
)0
)0
JO
y
9
9
9
9
9
3
2
I
0
O
o
o
o
o
o
0
.829o
.8274
.8258
.8241
.8225
.8208
J7
16
16
*7
16
17
16
o 56
5o
4o
3o
20
IO
35
0
o.
5736
24
0
.700
I
43
1.42*
h
0
.8l92
o 55
c
OS.
d.
(
:otg.
d.
Tanj
?•
d
•
Sin.
d.
' 0
PP
13
42
41
40
25
2
t
17
16
15
.1
.2
•3
•4
:l
:i
I
2
2
3
3
tl
2.9
7.2
;:i
0.1
ti
t;
12.6
16.8
21. 0
25-2
29.4
33-6
t
12.
16.
20.
24.
28.
I
2
3
4
!
.1
.2
•3
•4
'
i
i
i
4.0
8.0
2.O
6.0
0.0
4.0
8.0
J2.0
(6.0
2.5
5-o
7-5
IO.O
12.5
15-0
17-5
20. o
2
4
7
9
12
H
16
"9
21
5
2
6
o
4
8
.2
.6
I
2
3
4
5
6
78
*-7
3-4
5-i
6.8
8-5
10.2
II.9
I3.6
1.6
43:I
6.4
8.0
9.6
II. 2
12.8
14.4
1.5
3-°
4-5
6.0
75
9.0
10.5
12.0
13-5
1 56
POUR PLACE NATURAL FUNCTIONS.
o /
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
35 o
10
20
3o
4o
5o
o
o
o
0.
o.
o
5736
6760
5783
58oy
583i
5854
24
23
24
24
23
0.7002
o.7o46
o.7o89
o.7i33
o.7i77
O.722I
44
43
44
44
44
44
45
45
45
45
45
46
45
46
46
47
46
.4281
.4i93
• 4io6
.4oi9
.3934
,3848
88
87
87
85
86
84
84
83
83
82
Si
0.8l92
o.8i75
o.8i58
o.8i4i
0.8124
o.8io7
*7
i7
17
J7
17
o 55
5o
4o
3o
20
IO
36 o
10
20
3o
4o
5o
o.
0.
o.
o.
o.
o.
5878
5901
5925
5948
6972
5995
23
24
23
24
23
O.7265
o.73io
o.7355
o.74oo
o.7445
o.749o
.3764
.368o
.3597
.35i4
.3432
.335i
o.8o9o
o.8o73
o.8o56
o.8o39
0.8021
o.8oo4
*7
17
*7
18
17
o 54
5o
4o
3o
20
10
37 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
6018
6o4i
6o65
6088
61 1 1
6i34
23
24
23
23
23
o.7536
o.758i
O.7627
o.7673
O.772O
o.7766
.3270
.3i9o
.3iu
.3o32
.2954
.2876
80
79
79
78
78
o.7986
o.7969
o.795i
o.7934
o.79i6
o.7898
17
18
17
.8
18
o 53
5o
4o
3o
20
10
38 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
0.
6i57
6180
6202
6225
6248
627I
23
22
23
23
23
o.78i3
o.786o
o.79o7
o.7954
0.8002
o.8o5o
47
47
47
47
48
48
.2799
.2723
.2647
.2572
• 2497
.2423
77
76
76
75
75
74
o.788o
o.7862
o.7844
o.7826
o.78o8
o.779o
18
18
18
18
18
o 52
5o
4o
3o
20
IO
39 o
10
20
3o
4o
5o
o.
0.
0.
o.
0.
0.
6293
63i6
6338
636i
6383
64o6
23
22
23
22
23
o.8o98
o.8i46
o.8i95
0.8243
O.8292
0.8342
4*
48
49
48
49
So
.2349
.2276
.2203
.2l3l
.2o59
.1988
74
73
73
72
72
7i
o.777i
o.7753
o.7735
o.77I6
o.7698
o.7679
18
18
*9
18
19
o 51
5o
4o
3o
20
IO
40
0
0.
6428
o.839i
49
i.i9i8
70
o.766o
'9
o 50
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
t O
PP
.1
.2
•3
•4
• 5
.6
:l
48
47
46
45
44
23
.1
.2
•3
•4
•5
.6
:!
22
19
18
4.8
9.6
14.4
19.2
24.0
28.8
tf
43-2
4-7
9.4
14.1
18.8
2238:2
32.9
37-6
42.3
4.6
9.2 .2
13-8 -3
18.4 .4
23-° -5
27.0 .6
32-2 .7
36.8 .8
41.4 .9
4-5
9.0
13-5
18.0
22.5
27.0
3' -5
36.0
40.5
44
8.8
»3-2
17.6
22.0
26.4
30.8
35-2
2.3
4.6
6.9
9.2
"•5
13-8
16.1
18.4
2.2
4-4
6.6
8.8
II. 0
13.2
3
19.8
i.9
3-8
5.7
7-6
9-5
11.4
13-3
15-2
17.1
1.8
3-6
5-4
7-2
9.0
10.8
12.6
14.4
16.2
i57
POUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
40 o
10
20
3o
4o
5o
0.6428
o.645o
0.6472
0.6494
0.6617
o.6539
22
22
22
23
22
0.8391
o.844i
0.8491
o.854i
0.8691
0.8642
50
5°
5°
So
51
Si
51
52
51
52
53
52
53
53
53
54
54
54
55
55
55
55
56
c.6
1.1918
1.1847
1.1778
1.1708
i.i64o
i . 1671
7*
69
70
68
69
o. 7660
0.7642
0.7623
0.7604
0.7686
0.7566
18
*9
»9
19
J9
o 50
5o
4o
3o
20
10
41 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
656i
6583
66o4
6626
6648
6670
22
21
22
22
22
0.8693
o.8744
0.8796
0.8847
0.8899
0.8962
i.i5o4
i.i436
1.1369
i.i3o3
1.1237
1.1171
67
68
67
66
66
66
0.7647
0.7628
0.7609
0.7490
0.7470
0.7461
«9
19
'9
19
20
19
o 49
5o
4o
3o
20
IO
42 o
10
20
3o
4o
5o
o.
o.
0.
o.
o.
o.
6691
67i3
6734
6756
6777
6799
22
21
22
21
22
0.9004
0.9067
0.9110
0.9163
0.9217
0.9271
i . i 106
i .io4i
1.0977
i .0913
i.o85o
1.0786
65
65
64
64
63
64
o.743i
0.7412
0.7392
o.7373
o.7353
o.7333
»9
20
19
2O
20
o 48
5o
4o
3o
20
10
43 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
6820
684i
6862
6884
6906
6926
21
21
22
21
21
0.9326
0.9380
0.9435
0.9490
0.9645
0.9601
i .0724
i .0661
i .0699
i.o538
1.0477
i . o4 i 6
63
62
61
61
61
o.73i4
o.7294
0.7274
o.7254
0.7234
0.7214
'9
20
20
2O
20
2O
o 47
5o
4o
3o
20
IO
44 o
IO
20
3o
4o
5o
0.6947
0.6967
0.6988
0.7009
0.7030
0.7060
20
21
21
ZI
20
21
0.9667
0.9713
0.9770
0.9827
0.9884
0.9942
56
57
57
57
58
i.o355
1.0296
1.0235
i .0176
1.0117
i. 0068
60
60
59
59
59
0.7193
o.7i73
o.7i53
o.7i33
O.7II2
O.7O92
20
20
20
21
20
o 46
5o
4o
3o
20
10
45
0
o.
7071
i .0000
i .0000
O.7O7I
o 45
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' o
PP
.1
.2
•3
•4
j
:i
•9
57
55
54
53
51
22 21
20
19
5-7
11.4
17.1
22.8
28.5
34-2
39-9
45.6
5 1 • .1
5-5
II. O
16.5
22.0
27-5
33-o
38.5
44.0
49-5
5-4 •'
10.8 .2
16.2 .3
21.6 .4
27.0 .5
32.4 -6
37-8 -7
43-2 .8
48.6 .9
5-3
10. 0
iS-9
21.2
26.5
31.8
37-i
42.4
47-7
5-i
10.2
»5-3
20-4
25-5
30-6
35-7
40.8
45-9
2.2 .1 2.1
4.4 .2 4.2
6.6 .3 6.3
8.8 .4 8.4
ii. o .5 10.5
13.2 .6 12.6
*5-4 -7 14-7
17.6 .8 16.8
19.8 .9 18.9
2.0
4-0
6.0
8.0
IO.O
12. 0
14.0
IO.O
18.0
3
5-7
7-6
9-5
11.4
i3-3
15.2
17.1
i58
TABLE VIII.
SQUARES AND SQUARE ROOTS OF NUMBERS.
SQUARES OF INTEGERS FROM 10 TO 100.
N
0
1
2
3
4
5
6
7
8
9
mmmumum
IO
MMHMM^
100
^^^^••M
121
1 44
mmmmm^^
169
196
226
256
289
324
36i
20
4oo
44 1
484
629
576
625
676
729
784
84 1
3o
900
961
1024
1089
n56
1225
1296
1 369
1 444
l52I
4o
1600
1681
1764
1849
i936
2025
2116
2209
23o4
2401
5o
2600
2601
2704
2809
2916
3o25
3i36
3249
3364
348 1
60
36oo
8721
3844
3969
4096
4225
4356
4489
4624
4761
70
4900
5o4i
5i84
5329
5476
56a5
5776
5929
6o84
6241
80
64oo
656i
6724
6889
7066
7225
?396
7569
7744
7921
90
8100
8281
8464
8649
8836
9025
9216
9409
9604
9801
SQUARE ROOTS OF NUMBERS FROM 0 TO 10, AT INTERVALS OF .1.
N
•••••i
0
.0
^••^•M
0
.1
.3i6
.2
.447
.3
.548
.4
.632
.5
.707
.6
"775
.7
~
.8
.894
.9
.949
i
I.OOO
i. 049
1.095
i.i4o
i.i83
1.225
1.265
i.3o4
1.342
1.378
2
i.4i4
1.449
1.483
i.5i7
1.549
i.58i
1.612
1.643
1.673
1.703
3
1.732
1.761
1.789
1.817
1.844
1.871
1.897
1.924
1.949
i.975
4
2.000
2.025
2.049
2.074
2.098
2. 121
2.i45
2.168
2.191
2.214
5
2.236
2.258
2.280
2.302
2.324
2.345
2.366
2.387
2.408
2.429
6
2.449
2.470
2.490
2.5lO
2.53o
2.55o
2.569
2.588
2.608
2.627
7
2.646
2.665
2.683
2.702
2.720
2.739
2.7^7
2.775
2.793
2.811
8
2.828
2.846
2.864
2.881
2.898
2.915
2.933
2.950
2.966
2.983
9
3.000
3.017
3.o33
3.o5o
3.o66
3.082
3.098
3.n4
3.i3o
3.i46
SQUARE ROOTS OF INTEGERS FROM 10 TO 100.
N
0
1
2
3
4
5
6
7
8
9
m^mmmm*
IO
3.162
I^T"
3.464
3.6o6
3.742
3.873
4.000
4.123
4.243
4.359
20
4.472
4.583
4.690
4. -796
4.899
S.ooo
5.099
5.196
5.292
5.385
3o
5.477
5.568
5.657
5.745
5.83i
5.916
6.000
6.o83
6.i64
6.245
4o
6.325
6.4o3
6.48i
6.557
6.633
6.708
6.782
6.856
6.928
7.000
5o
7.071
7-Ui
7.21 1
7.280
7.348
7-4i6
7-483
7.55o
7.616
7.681
60
7-746
7.810
7.874
7-9^7
8.000
8.062
8.124
8.i85
8.246
8.3o7
70
8.367
8.426
8.485
8.544
8.602
8.660
8.718
8.775
8.832
8.888
80
8.944
9.000
9.o55
9.1 10
9.i65
9.220
9.274
9.327
9.38i
9-434
90
9-487
9.539
9.592
9-644
9.695
9-747
9.798
9.849
9.899
9.950
z59
TABLE IX.
THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF
NUMBERS FROM 0 TO 2.5, AT INTERVALS OF .1.
X
cosh a?
sinha?
tanh a?
e*
e *
0
i .000
o
o
i .000
I .000
. i
i .oo5
.100
.100
i.io5
.905
.2
i .020
.201
.197
I .221
.819
.3
i.o45
.3o5
.291
i.35o
.74i
.4
1.081
.4n
.38o
1 .492
.670
.5
1.128
.521
.462
1.649
.607
.6
i.tSS
.637
.537
1.822
.549
• 7
1.255
.759
.6o4
2.014
•497
.8
i.337
.888
.664
2.226
.449
•9
1.433
1.027
.716
2 .46o
.407
1.0
i.543
1.176
.762
2.718
.368
. i
i .669
1.336
.801
3.oo4
.333
.2
1.811
i .509
.834
3.320
.3oi
.3
1.971
1.698
.862
3.669
.273
.4
2.l5l
i .904
.885
4.o55
.247
.5
2.352
2.129
.905
4.482
.223
.6
2.577
2.376
.922
4.953
,202
•7
2.828
2.646
.935
5.474
.183
.8
3.107
2.942
•947
6.o5o
.i65
•9
3.4i8
3.268
• 956
6.686
.i5o
2.0
3.762
3.627
.964
7.389
.i35
2. I
4.i44
4.022
.970
8.166
. 122
2.2
4.568
4.457
.976
9.025
.III
2.3
5.o37
4.937
.980
9.974
. IOO
2.4
5.557
5.466
.984
II .023
.091
2.5
6.i32
6.o5o
.987
12.182
.082
1 60
TABLE X
CONSTANTS
MEASURES AND WEIGHTS
AND OTHER CONSTANTS
, ....
"* '' *'
.* „ •„
•• /*' •"• «•"•• - • - -
,. /x, , .,. :„.. xx.' ., .,
"• •"
' ' ' ' '•" •' ..... ' '•
-
(
rw^rf ^r&rW^r
,-'•>•< •••' :<••
• >.. . • • '.
' •','••• ' ''
•'
<v v
B'
Si
w P-
rt P-
IT
f
, Elements
plans and
ottBtry
Y 9 194
May'47Pf
M3O6247
THE UNIVERSITY OF CALIFORNIA LIBRARY
UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
REC'D LD
APR 9 1961
30cf
LD
SEP 2 0 1962
LD 21-100m-ll,'49(B7146sl6)476
5
V
^~
' 3 ^ ul L^,
rr /
k-l>,ks-c,. ru-y*- frf 0*
,
Oa ju~r ^*.
t X
A f /MA4^e
; e,^ ot° U. T -,H*"
*fjj^cti4 * */*
v^- J^^XA-^> j Wt^
UA_
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11