UNIVERSITY OF CALIFORNIA
AT LOS ANGELES
PHILLIPS-LOOMIS MATHEMATICAL SERIES
ELEMENTS OF TRIGONOMETRY
WITH TABLES
PHILLIPS-LOOMIS MATHEMATICAL SERIES
ELEMENTS OF TRIGONOMETRY
PLANE AND SPHERICAL
BY
ANDREW W. PHILLIPS, PH.D.
AND
WENDELL M. STRONG, PH.D.
YALE UNIVERSITY
45071
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 \V. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [Hy
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 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.
Mathematical
Sciences _
Library O >
IN this work the trigonometric functions are defined as
I 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.
J The emphasis given to the formulas essential to the solu-
j tion of triangles.
v 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, 1898.
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 ( x), (180 x), (i8o-f.r), (360 x) 18
Functions of (90 y), (90+^), (270^), (2"jo-\-y) 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 78
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 i 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
_/. 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 TRIGONOMETRY
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 posi m of its sides.
Qut^ton. 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 \^J Initial line
|J RIGHT ANGLES 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 div. Jed into
four quadrants, as shown in the figure; the first quadrant
beeins at the initial line.
ii
in
IV
THE FOUR QUADRANTS
III
ANGLE IN 1ST QUADRANT
II
ANGLE IN 2D QUADRANT
ANGLE IN 3D QUADRANT
III
ANGLE IN 4TH QUADRANT
An angle is said to be in a certain quadrant if its terminal
line is in that quadrant.
EXERCISES
4. (i.) Express ^\ 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 an
angle of 73 ?
(4.) In what quadrant is an angle of 890?
DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS
5. The trigonometric functions are numbers, and are de-
fined as the ratios of lines.
Let the angle AOP be so placed that the initial line is
horizontal, and from P, 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
ANGI.K IN THH 3D QUADRANT
Denote the angle A OP by x.
SP
ANGI.B IN THH 4TH QUADRANT
-^=sine of x (written sin*).
OS
- = cosine of x (written cos*).
THE TRIGONOMETRIC FUNCTIONS
SP
tangent of x (written
OS
> = cotangent of x (written cot^r).
^1
OP
OS
- = secant of x (written sec^r).
- = cosecant of x
To the above may be added the versed sine (written versin) and coversed
sine (written coversin), svhich are defined as follows :
versiii ic \ cos x\ coversiu x = i sin a/.
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
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.
6. A line may by its length and direction represent a
number; the magnitude of the number is expressed by the
length of the line; the number is positive or negative ac-
cording to the direction of the line.
6 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.
SF SP
Thus, sin.r=-y^= = 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
C cot
C Cot B
FIG 4
THE TRIGONOMETRIC FUNCTIONS 7
SP represents the ine of x.
OS represents the coine of x,
A T represents the tangent of x.
BC represents the cotangent of x.
O T represents the secant of x.
OC represents the coecant 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,
(i.) The sine 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 rig/it of O along the initial line,
negative if it extends to the left ; SP is positive if it extends
upward from OA, negative if it extends doivnward. 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 doivnward,
The cosine and cotangent are positive if measured to the
right from OB, and negative if measured to the left.
B Cot-f- Cot- B
FIG. 3
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
+
-
+
-
0. 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.
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
9JL x- - -2- OP
' ~d~p ~ cos * ' os ~ an * ' SP~ ' * ' os
OP
B Cot C
We have, then, from Figs. 2 and 3,
SP sinac
-rr. = tan ac = ;
OS -o* /
os_
SP~
Multiplying (i) by (2),
C09iC
or
tan x =
laii./ cot 07=1,
i _ i
cot x ' tan x
Again, from Figs. 2 and 3,
OP 1
-= = ec x = ;
OS cos x '
From Figs. 2 and 3,
or
and sin*jr= I COS'JT ; cos*j:= I sin s jr.
Also, OA*+AT*=Or, and
or 1 + tan 8 a?
1 + cot'oe = csc'ac.
FIG. 3
(I)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
THE TRIGONOMETRIC FUNCTIONS \\
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.r= i.
(2.) Prove sin-r CSC.T I.
(3.) Prove tan .r cos x sin x.
(4.) Prove sin x \/i cos'' x i cos*x.
(5.) Prove tan x + cot x = .
sm.r cos-r
(6.) Prove sin 4 x cos 4 x = i 2 cos 2 .r.
(7.) Prove = sin x.
cotx secx
(8,) Prove tan x sin x -+- cos x = sec jr.
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.
COSJT V i 4= A/3.
Since x is less than 90, we know that COSJT is positive.
12 PLANE TRIGONOMETRY
Hence cos x =
,-
= --- =y 3 ;
2
iv/3
i
CSCJT=- = 2.
2
(2.) Given tan* = and * in quadrant IV; find sin* and cos jr.
Solution,
hence
sin x
COSJT
3 sin.r =
-*:
COS X,
hence
10 sin 2 jr =
!;
cos.r= T 3 1 jV' 10.
(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.) GiveVi y in quadrant IV and sin/= $, find all the other
functions oiy.
(6.) Given cos 6o = J; find all the other functions of 60.
(7.) Given sin o = o; find coso and tano.
(8.) Given tan^r = |and z in quadrant I; find the other functions
of z.
(9.) Given cot45= I ; find all the other functions of 45.
(10.) Given tanj=i\/5 and cos^ negative; find all the other
functions of y.
(11.) Given cot 30= \/3; fid tne other functions of 30.
(12.) Given 2 sinjr=i cos* and x in quadrant II; find sin*
and cos JT.
(13.) Given tan x-*r cot* = 3 and x in quadrant I ; find sin*.
THE TRIGONOMETRIC FUNCTIONS
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,
RC a
cos A = r- = - = sin B ;
AB c
=!=! = an B.
15. From 14, for an acute angle of a right triangle, we have
side opposite angle
- ;
hypotenuse
side adjacent to angle
cosine = - T-=! - - ;
hypotenuse
side opposite angle
tangent = -^3 - -~-^- , ' '
side adjacent to angle
side adjacent to angle v
cotangent = 73 - - : .
side opposite angle
(9)
1 4 PLANE TRIGONOMETRY
FUNCTIONS OF COMPLEMENTARY ANGLES
16. From 14, we have
sin A=cos=co*(9QA);
co* A = s'm= sin (9O 4);
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.
Tha 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
J7. (i.) Express as functions of an angle less than 45:
sin 70 ; cos 89 30' ; tan 63 ;
cos66; cot47; sin 72 39'.
(2.) cos;r = sin2.r; find*.
(3.) tan x cot 3* ; find A:
- (4.) sin2jr = cos3-r ; find x.
(5.) cot(30 jr) = tan(3o + 3jr); find*.
(6.) A, B, and C are the angles of a triangle; prove that
Hint. A + B + C= 1 80.
THE TRIGONOMETRIC FUNCTIONS
18. As the angle x decreases towards o (Fig. i), sinx de-
creases and cos.* increases. When OP comes into coincidence
with OA, SP becomes o, and OS becomes OA( \).
Hence sino = o. coso = i.
FIG. 3
As the angle x increases towards 90 (Fig. 2), sin.* increases
and cos:r decreases. When OP comes into coincidence with OB,
SP becomes OB{~\) and OS becomes o.
Hence 8^90 = !, cos 90 ^o.
As the angle x decreases towards o (Fig. 3), tan* decreases
and cot* increases. When OP comes into coincidence with OA,
A T becomes o and B C has increased without limit.
Hence tano = o, coto = oo.
As the angle x increases towards 90 (Fig. 4), tan.* increases
and cot* decreases. When OP comes into coincidence with OB,
^47'has increased without limit, and BCQ.
Hence tan 90 = 00, cot9o=:o.
Remark. By coto=oo 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 oo does not denote a definite number, but
simply that the number is indefinitely great.
i6
PLANE TRIGONOMETRY
In 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 = -j- 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
+ 00 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
90
1 80
270
300
sin
o
I
I
O
cos
I
-I
I
tan
o
CO
o
CO
o
cot
oo
00
O
00
19. It may now be stated that, as an angle varies, its sine and cosine
can take on 'values from / to + / only, its tangent and cotangent all
values from oo t<> -\- oo, its secant and cosecant all values from oo
to -(- . ex. j)t those between / and -f- 1.
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'P' and OS=OS f , and the
angle A OP' (Fig. 2) is equal to the supplement of AOP
(Fig. i). Also, in the triangle AOP' (Fig. 3), angle AOP'
= angle AOP' (Fig. 2).
V
r o
FIG. a
FIG. 3
THE TRIGONOMETRIC FUNCTIONS
It follows from 5 and 8 that
sin (1O as) = sin x ;
co (1O x) = cos a? ;
tan (1O x) = tan a? ; '
cot (1O 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 6b
21. In the right triangle OSP (Fig. i) angle = angle /> = 45.
and OP = i.
Hence OS = SP = i -y/2.
Therefore sin 45 = 00545 =^-\/2; 14,16
tan 45 = cot 45= i.
P
\J \.rj- O v _L Q n
v * ^^ O
2
FIG. I FIG. 2
In equilateral triangle 0/^4 (Fig. 2) the sides are oi unit length
/* bisects angle OP A, is perpendicular to OA, and bisects 0,4.
Hence, in the right triangle OPS, OS = %, SP = %-\/'$.
Therefore sin 30 := cos 60 = ^ ; 14
tan 30 = cot 60 = i v X 3 :
cot 30 = tan 60 \/3-
1 8 PLANE TRIGONOMETRY
22. The following values should be remembered :
Angle
30
45
60
90
sin
o
i
iv/2
iv/3
i
cos
i
*V1
ii/5
i
o
EXERCISES
Prove that if x = 30,
(i.) sin 2jr = 2 sin^r cos^r;
(2.) cos 3_r = 4 cos 3 .r 3 cos x ;
(3.) cos 2jr = cos s jr sin 2 A-;
(4.) sin 3* = 3 sin.r cos 2 ;r sin s .r;
2 tan JT
(5.) tan 2;r = 3.
i tan- x
(6.) Prove that the equations of exercises i and 3 are cor-
rect if ^r = 45.
(7) Prove that the equations of exercises (2) and (4) are cor-
rect if JT= 120.
The following three articles, 23-25, are inserted for
completeness. They include the functions of (90 x] and
(180 x), which, on account of their great importance, were
treated separately in 16 and 20.
FUNCTIONS OF ( x), (l8o X\ (l8o+*), (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'/ y> ,
and OS=OS'.
THE TRIGONOMETRIC FUNCTIONS
B C C' B
FIG. 3
In Figs, i and 4, triangle OSP'=tria.ngle OSP. hence SP'=SP.
In Figs, i, 2, and 4, triangle OA 7"=triangle OA T, hence A T' - A 7.
In Figs, i, 2, and 4, triangle OC' = triangle OBC. hence C'=BC.
Therefore any function of each of the angles ( x}. (180 x),
^), (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( *) = cos*
tan ( x) = tan x
cot ( x) = cot x
sin (180 + *) = sin*
cos ( 1 80 + x) = cos x
tan(i8o-r-*) = tan*
cot (180 + *) = cot*
sin (180 x) = sin*
cos (180 x) cos*
tan(i8o *)= tan*
cot(i8o- x) -cot*
sin (360 *)=: sin *
cos (360 *) = cos*
tan (360 *) = tan *
cot (360 *) = - cot*
2O
PLANE TRIGONOMETRY
FUNCTIONS OF (90 }'), (90 +j), (270 -y\ (270 -fj)
24. The line representing the sine of each of these angles is
of the same length as the line representing the cosine of j; the
cosine, tangent, or cotangent, respectively, are of the same length
as the sine, cotangent, and tangent of y.
For
Triangle OS'P' = triangle OSP, hence S'P'=OS, and OX = S
Triangle OA T ' triangle OBC, hence A T' = RC.
Triangle OBC = triangle OA T, hencf BC = AT.
Therefore any function of each of the angles (90 }'), (90
(270 y), (2 70 +y), is equal in numerical value to the co-function
THE TRIGONOMETRIC FUNCTIONS 21
of y. Its sign, however, depends on the direction of the line repre-
senting it.
Putting in the correct sign, we obtain the following table :
sin (90 y) = cos^ sin (90 + y) = cos_y
cos (90 v) = sinj' cos (90 + y) = sinjv
tan (90 y) coty tan (90 + y) = cot v
cot (90 y) = tany cot (90 +y) = tany
sin (270 y) = cos_v sin (270 +y) = cosy
cos (270 y) smy cos (270 +jy) = sinj
tan (270 y) = coty tan (270 + y) = cotjv
cot (270 y) = tany 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 212 =sin(i8o+ 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 x less than 720 is sin;r = ^^/3?
(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, find sin 196.
(9.) Simplify cos(9o + jr)cos(27o .r) sin(i8o x)s'\n(^6o .r).
0-> Simplify;!; / ( ^I^tan(90 + ^) + , ;M . ' _ rt -
sin 5 (270
(H.) Express the functions of (x 90) in terms of functions of x.
CHAPTER II
THE RIGHT TRIANGLE
2V. 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 =r s - ; (2) cosine =-^- ; 14
(3) tangent =
hypotenuse
opposite side
adjacent side
(4) cotangent =
hypotenuse
adjacent side
(5) c'=a*
16
opposite side
(6) B = (go A).
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."
a 21.34
sm/J =-= 7
c 47.653
190612
227880
190612
372680
333571
391090
381224
9866
sin ^ = .4478
^=26 36'
&=<: cos A
=47-653 x -8942
47-653
.8942
95306
190612
428877
381224
42.6113126
=42.61 f
B =(90 -26 36)^63=
= x 21.34 X42.
21-34
42.61
2134
12804
4268
8536
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
log sin A =log a log c
Iog2i.34 =1.32919
log 47.653 = I- 67809
sub.
log sin ^4=9.6511010
^=26 36' 14"
cos A=-
c
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
=(90 -26 36' I4")=63 23' 46"
area = %ab
log area = log + log a + log b
log =9.69897 10
Iog2i. 34=1. 32919
log 42. 608 = 1.629-19
log area=2. 65765
2)909.2974
454.6487
area=454.6
* In this solution the five-place table of the " Logarithms of Functions" is
used.
f No more decimal places are retained, because the figures in them are not
accurate ; this is due to the fnct that the table of " Natural Functions" is only
iour-place.
24
PLANE TRIGONOMETRY
CHECK ON THE CORRECTNESS OK THE WORK
= 90.263 x 5.043
90.263
5-043
270789
361052
4513150
' = 455-
Extracting the square root, a =
21.34, which proves the solution cor-
rect.
= 90.261 x 5.045
log 90.261 1.95550
log 5.045 = 0.70286
2)2.65836
Iog2i.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 28.453,3 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' 11 ", ^ = . 01793; find b.
(7.) In a right triangle B = f\ 34' 53", = 896.33; find a.
(8.) In a right triangle <: = 3729.4. = 2869.1 ; find A.
(9.) In a right triangle a= 1247, b= 1988 ; find c.
Co.) In a right triangle a = 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
*"Yfrus 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 depressipn of P if O is the
point of observation.
(11.) 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.) Fronft|gr 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 ft. high 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 r 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. ('Y\aX\W<\><* : ^
(i 8.) The angle of elevation of the top of a tower observed from a
point at a horizontal distaitce 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.
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.
(21.) A mark on a flag-pole is known to be 53 ft. 7 in. above the
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.
Hint. Drnw the perpendicular BD. BD bisects the base, and also the
angle ABC.
In the right triangle ABD, AB-J 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.
(25.) If a chord of a circle is 18 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.
(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.
Hint. The pentagon is divided into 5 equal isosceles triangles by its radii.
Let A OB 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.
(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 TRIGONOME TR Y
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 a = 3.72, ^ = 47 52', C= 109 10'; find
the remaining parts.
The given parts are a side and two angles.
C
Hint. /t=[i8o-(S+ 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, &= 39 8'; find the
remaining parts.
The given parts are two sides and the included angle.
= 125.3
THE RIGHT TRIANGLE 29
Siint.Dra.v/ 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, = 5.81, A = 27 23'; find the
remaining parts.
The given parts are two sides and an anglt opposite one of
them.
C
A B' D 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 a= 34.2, = 38.6, - = 55.12;
find the angles.
The given parts are the three sides.
C =66.12
* A discussion of this case is contained in a later chapter on the solution
of oblique triangles.
Hint.
PLANE TRIGONOMETRY
Let DB=x,
Hence
In each of the right triangles ACD and BCD the hypotenuse and a side
are now known ; hence these triangles can be solved.
(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
864.4 ft., the angle ACB 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, BC, $00 ft. long, is measured on shore ; the angles, ABC
and ACB, are found to be 67 43' and 74 21' 16" respectively. What
is the distance of the ship from the point Bt
(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 li ft. above the river; observed from this point the angle
of depression of the farther ban k is i 1 2'. 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 2i J ; find the other angles of the quadrilateral.
THE RIGHT TRIANGLE 31
Hint. Draw the diagonal DB.
In the triangle A BD two sides and an included angle are given, hence the
triangle can be solved.
The solution of triangle ABD gives DB.
Having found DB, there are three sides of the triangle DBC known, hence
the triangle can be solved.
(n.) In a quadrilateral ABCD, AB= 12.1, AD = 9.7, angle A
17 1 8', angle # = 64 49', angle Z)= 100; find the remaining sides,
Hint. Solve triangle ABD to find BD.
CHAPTER III
TRIGONOMETRIC ANALYSIS
30. In this chapter we shall prove the following funda-
mental formulas, and shall derive other important formulas
from them :
sin (05 + j/) = sin x cos y + cos x sin /,
-in / //sin./- cost/ -./ sin?/,
cos(a? + ?/) = cos a? cosy sin aj slniy,
cos(x j/) = cos x cosy + sin aj sinj/;
PROOF OF FORMULAS (l l)-(l4)
31. Let angle y2<9>=;tr, angle QOP=y\ then angle A OP
-(*+?)-
The angles x and j are each acute and positive, and in Fig. i
(x-\-y) is less than 90, in Fig. 2 (x+y) is greater than 90.
Q
In both figures the circle is a unit circle, and SP is perpendicular to
OA ; hence $/>= sin (JT +j), C>6'= cos (jc +.
TRIGONOMETRIC ANALYSIS 33
Draw DP perpendicular to OQ ;
then DP=s\ny, OD = cosy,
angle SPD = angle AOQ = x.
(Their sides being perpendicular.)
Draw DE perpendicular to OA, DH perpendicular to SP.
(s,'mx} x OD = s\nx cosy.
ED
(For OED being a right triangle, = sin jr.)
=(cosx] x DP=cosx sinj.
HP
(For HPD being a right triangle, = cos x.~)
Therefore, sin(x + j/) = sinic cos?/ + cosa? sin?/. (ii)
Cos O + j) = O5 = OE - HD*
OE = (cos x] x 0Z? cos x
OE
(For OED being a right triangle, - = cos x.)
snx sn/.
Ff D
(For PHD being a right triangle, -~- =sin x.)
Therefore, cos (x + y] = cosx co?/ sin a? sin?/. (13)
3. 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 y be acute, and let x be an angle in the second quad-
rant ; then ;r (90 + ^') where x' is acute.
sin (x +y] sin (90 + x' +y)
= cos(V-f-j) 24
= cos x' cosj sin x' sin y
= sin (90 + x'} cos/ + cos (90 + x') sin _y 24
cosj-|-cos;tr sinj/.
If (jc + 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 (x+y 1 ). 9
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
sin (x y) = s\v\x cos( y) + cosx sin {y).
But cos( y) = cosjy, and sin ( y)= sin/. 23
Therefore, sin (as y) = s\nx cosy cosx sin y. (12)
Substituting (y) for y in formula (13), it becomes
cos(x y)=- cosx cos( y) sin^r sin ( j/),
cosx cosy + sinx sinjj/.
Therefore, co*(x y) = cosx cosy + sinx slny.* (14)
EXERCISES
34. (r.) Prove geometrically where x and y are acute and positive :
sin(.r ^/) = sin x cos/ COSJT siny,
cos(;r _y) = cos^r cosy -f sin^- sin/.
* Formulas (12) and (14) are proved geometrically in 34. The geometric
proof is complicated by the fact that OD and DP are functions of /, while
the functions of y are what we use.
TRIGONOMETRIC ANAL YSTS
35
Hint. Angle AOQx, angle POQ=y, and angle AOP-(x-y).
Draw PD perpendicular to OQ.
Then Z?/'=sin( ;(')=: sin v; but DP is negative, therefore PD taken
as positive is equal to sin y :
OD = cos ( y) = cos y,
Angle HPD=a.ng\e AOQ=x. their sides being perpendicular.
Draw DH perpendicular v.o SP, DE perpendicular to OA.
s(x-y)=SP=ED-PH.
P'rom right triangle OED, ED.= (s'm x)x 0Z>=sin x cos/.
From right triangle DHP, PS/=(cosx)xPZ)=cosx siny.
Therefore, sin (x j)=sin x cosy cos x siny.
Cos (.r - v) = O -S = OE + DH.
From right triangle OED, OE=(cosx)x OD=cosx cosy.
From right triangle DHP, Z>//=(sin x) x / > Z> = sin x siny.
Therefore, cos (xy)=cos x cos j + sin x sin_y.
(2.) Find the sine and cosine of (45+^), (30 .r), (6o+.r), in terms
of sin x and cos.r.
(3.) Given sin.r=f, slt\}>=.^, .r and y acute; find sin(-tr+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(^r+j) and cos(jr y).
(7.) By means of the above formulas express the sine and cosine of
(i 80 x), (i8o+jr), (270 ,r), (270-}- x), in terms of sin x and cosx.
(8.) Prove sin (6o+ 45) + cos (60 + 45) = cos 45.
(9.) Given sin 45 = -^\/ 2 > COS 45 i "V/ 2 ! find sin 90 and cos 90.
(10.) Prove that sin (60 -f- x) sin (60 .r):=sin.r.
36 PLANE TRIGONOMETRY
TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES
?- Tan/'*- ,A _ sin (x+y)_ sin* cosj+cos^r sinjy
r>. 1 clll i;t -p y I
cos(;r+j) cos* cosj
Dividing each term of both numerator and denominator
of the right-hand side of this equation by cos* cosj, and
sin
remembering that = tan, we have
cos
tan x + tan y
*"(+*)=!_ tang tBn V (15)
In a similar way, dividing formula (12) by formula (14), we
obtain
tan x tan //
FUNCTIONS OF TWICE AN ANGLE
36. An important special case of formulas (11), (13), and
(15) is when y = x', we then obtain the functions of 2x in
terms of the functions of x.
From (11), s\n(x-\-x} s\\\x C.OSX + CQSX sin^r.
Hence sin2ic = 2 sinoc cosx. (17)
From (13), cos2a; = co8 2 iK-8ln 2 a;. (18)
Since cos s ^=: I sin 2 ;r, and sin"^r= I cos";r,
we derive from equation (18),
cos2x= i 2sinV, (19)
and cos2^r=2 cosV I. (20)
From (i 5), t** = c ' (21)
FUNCTIONS OF HALF AN ANGLE
,77. Equations (19) and (20) are true for any angle; there-
fore for the angle \x.
From(i9), cosx=i 2 sin 9 ^;
TRIGONOMETRIC ANALYSIS
I cos;tr
or sin $x
/I cos a? , N
therefore, sm-x = \/ -- - (22)
From (20), cos.* = 2 cos^x i ;
1 + cosx
or cos \x -- 5
/I + cos a?
therefore, cosfx = y gj
Dividing (22) by (23), we obtain
/I cos a; / x
tan \x = \ / (24)
V 1 + cosx
FORMULAS FOR SUMS AND DIFFERENCES OF FUNCTIONS
38, From formulas (n)-(i4), we obtain
sin (x + j) + sin (x y) = 2sinx cosj ;
sin (x-\-y) sin (x y} = 2cosx siny;
cos (x -\-y) + cos (xy) = 2 cosx cosy ;
cos(^4-jv) cos(^r y) 2sin^r siny.
Let u (x +y) and v (x y) ;
Substituting in the above equations, we obtain
siiiw + sin v = 2 sin-J(? +v)cos-|(u v); (25)
sin n - sin v = 2 cos-J (i + v) sin (le - v) ; (26)
cos M 4- cos v= 2 cofrj (e 4- v) cos^- (ie v) ; (27)
cos? cos^= 2sin-^-(w + v) sin^(u v). (28)
Dividing (25) by (26),
slnu+slnv _tan^(u+v)^ , .
EXERCISES
39. Express in terms of functions of x, by means of the formulas
of this chapter,
45071
38 PLANE TRIGONOMETRY
(i.) Tan(i8o .r); tan (i8o + .r).
(2.) The functions of (x 180).
(3.) Sin (x 90) and cos (x 90).
(4.) Sin(jr 270), and cos (^ 270).
(5.) The sine and cosine of (45*); of (45 -f- x).
(6.) Given tan 45= i, tan 30 = 1/3; find tan 75; tan 15.
cot a? cot?/ 1
(7.) Provecot (x + y) = . (30)
cot y -i cot x
Hint. Divide formula (13) by formula (n).
COtiC C0t?/ + l
(8. Prove cot,x-y ) = -. (31)
cot //-cot x
(9.) Prove cos (3o+j) cos (30 y) = sin/.
(10.) Prove sin 3^ = 3 sin x 4 sin'jr.
Hint. Sin 3jr = sin (x+2x).
(n.) Prove cos ^x = 4 cos s 3 cos x.
(12.) If x and / are acute and tan .* = , tan/ = J, prove that
(*+/) = 45.
(13.) Prove that tan (.*-}- 45) =
i tan^r
(14.) Given sin/ = f and/ acute; find sin /, cos^y, and tan^/.
(15.) Given cos;r= | and ^r in quadrant II; find sin 2x and
cos 2.r.
(i 6.) Given cos 45 = \ \/2 ; find the functions of 22^.
(17.) Given tan x = 2 and jr acute ; find tan \x.
(i 8.) Given cos 30 = -y/3 : find the functions of 15.
(19.) Given cos9o = o; find the functions of 45.
(20.) Find sin^-r in terms of sin^r.
(21.) Find cos5,r in terms of cos x.
(22.) Prove sin (x -\-y -j- ?) = sin x cos/ cos .z+cos x sin y cos s+cos x
cosy sinz sin x slny sinz.
Hint. Sin (x+y+z)- sin (x+y) c
(23.) Given tan 2.*- = 3 tan x; find jr.
(24.) Prove sin 32 + sin 28 = cos 2.
(25.) Prove tan x -f cot x = 2 esc 2x.
(26.) Prove (sin^-r + cosfr)^ i -4-sin x.
(27.) Prove (sin %x cos %x)* \ sin x.
TRIGONOMETRIC ANALYSIS 39
(28.) Prove cos 2x = cos 4 .r sin 4 .r.
(29.) Prove tan (45 -+- x) + tan (45 x) = 2 sec 2x.
2 tan x
(30.) Prove sin2jr =
(31.) Prove cos2.r =
i+tan 2 .*-'
i tanlr
i -f- tan a .f '
I 4- sin 2.r /tan x-\- iV
(32.; Prove- =(- -)
i sin 2.r \tanjr i/
(33-) Prove
(34.) Prove
I +COS X
sin x
i cos x
. , cos x cosy
(35.) Express as a product -.*
cos x -j- cosy
COSJT cos r _ 2 sini^+r) sin^(jr y)
COSJT + COSJ 2 co
tan x 4- tan_y
(36.) Express as a product .
cot x + cot_y
cos (x -\-y)
(37.) Prove i tan-r tan y= -^-.
cosj
THE INVERSE TRIGONOMETRIC FUNCTIONS
40. Dcf. The expressions sin-'tf, cos-'tf, tan-', etc., de-
note respectively an angle whose sine is a, an angle whose
cosine is a, an angle whose tangent is a, etc. They are
called the inverse sine of a, the inverse cosine of a, the
inverse tangent of a, etc., and are the inverse trigono-
metric functions.
Sin l a 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*=i, *=3O, 150, (30 + 360), etc.,
and sin~ '^=30, 150, (3O + 36o ), etc.
Remark. The sine or cosine of an angle cannot be less than i
or greater than -f- 1; hence sin- 1 ** and cos~'a have no meaning unless
a is between i and + i. In a similar manner we see that sec~'a
and csc-'tf have no meaning if a is between i and -j- i.
EXERCISES
41. (i.) Find the following angles in degrees:
s\n- l %\/2, tan-'( i), sin-'( i).
cos-'^, cos" 1 1,
(2.) If x cot~4, find tan x.
(3.) If A- = sin- I |, find cos^r and tan x.
(4.) Find sin(tan-'^ -v/3)-
(5.) Find sin (cos -I |).
(6.) Find cot (tan- 1 ^f).
(7.) Given sin~'a = 2 cos-'rt, and both angles acute; find a.
(8.) Given sin-'a = cos~ I ; find the values of sin-'a less than 360.
(9.) Given tan~'i = \ tan~'o, and both angles less than 360; find
the angles.
(10.) Given sin-'rt r^cos-'rt and sin-'a + cos-'a =450; find sin-'rt.
(11.) Prove sin (cos -I rt):=: V 1 <**
Hint. Let JT=COS '<7 ; then a = cbsjc,
sin x= \/i cos' 2 * = \/i a 9 .
(12.) Prove tan^an-'a-l-tan- 1 ^)^ _ ,
a b
( H-) Prove tan (tan-'rt tan~'^) = -
i -\- ao
(14.) Prove cos (2 cos- 1 ^ ) = 2 rt s i.
(15.) Prove sin(2cos-'rt)=2rt y/i a*.
(i 6.) Prove tan(2tan- r rt)= : -
(17.) Prove cos (2tan~'a)= j-
(18.) Prove sin(sin~ 'a + cos~"'^) = rt V
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.
r. C C
FIG. 3
Draw the perpendicular CD. Let CD=h,
h
Then T :
(In Fig. 2 -=sin(i8o-//)=sin/0
and
.
- = sm B.
/T T-' tl
(In Fig. 3 -=rs
a
By division we obtain,
sin A
b sin i;
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,
PLANE TRIGONOMETRY
b _ sin B
c sinC
Repeating the process" we obtain
c sin C
- = .
rt sin ^4
The same procedure may be applied to all the formulas for the solution of
oblique triangles. Henceforth only one expression of each formula will be 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/i sin .5
a + b ~ sin A + sin B '
By formula (29), denoting the angles by A and B, in-
stead of u and v,
Therefore,
a b
tan |(/2 -{-/?)'
^(^l B)
(33)
This formula is used for Hie solution of triangles in which
two sides and the included angle are given.
44. Whether A is acute or obtuse, we have
C C
(If A is acute (Fig. i),AD bcosA,DB = AB AD c bcosA, CD
. If A is obtuse (Fig. 2), AD = cos (iBoP-A) = -b cos A, DB-AB
, CD b sin(i8o- A}-=
THE OBLIQUE TRIANGLE 43
a y = (e-bcosA}' > + (dsmA)\
=e*-2 be cos A + 2 (cosM +sinM).
Therefore, a 9 6 2 -f tf-zbc cos A. (34)
Tliis formula is used in deriving formula (37).
It is also used in the solution without logarithms of tri-
angles of which tivo sides and the included angle or three
sides are given.
2L2 | 2 2
5. From formula (34), cos A = ;
2bc
From formula (22), 37,
P^- 1
2 sin A = i cos A = i
2 be
Hence 2 sin 2 ^A= ; ,
2 be
2bc
2 be
Let s = , then (a b + c} = 2(s b], and
2 (S-C).
,_2 ( S -b}(s-c]
be
Substituting, 2 sin 8 i
Hence sin^ = x /^ y- C J" (35)
From formula (23), 37,
2 cos 2 1 A =
2 be
2s(s- -a}
~~bc~
In extracting the root the plus sign is chosen because it is known that
A is positive.
44
PLANE TRIGONOMETRY
Hence cos^A =.
Dividing (35) by (36), we obtain
(s a)
(s-a)(s-b](s-c)
"
sa
(36)
(37)
K
Let
js
s a
Formulas (37) and (38) are used to find the angles of a tri-
angle when the three sides are given.
FORMULAS FOR THE AREA OF A TRIANGLE
46. Denote the area by 5.
(In Fig. I, CD=asinB; in Fig. 2, CD = a sin (180-^) = a sin Z?.)
In Figs, i and 2, S=\c.CD.
Hence S = $acsinB. (39)
From formula (17),
sin B2 sin^.5 cos^Z?.
THE OBLIQUE TRIANGLE 45
Substituting for s'm^H and cos^-^ the values found in
formulas (35) and (36), we obtain
sin> = \/s(s a)(s b)(s c}.
ac*
Therefore, S=*/s(sa)(sb)(sc). (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 known; 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 ACB', 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=bs\i\A t there is no solution when a < l>s\nA ; there is one
solution, the right triangle A CD when a=bs\nA; there are two solutions
when a < l> and
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-\-B)=. 180 109 r 5o"m7o58' 10".
To find c.
To find b.
I > _sm
a sin A
log fl=l. 56512
log sin ^=9.97845
cologsin .,4 =0.22 1 23
log =1.76480
^ = 58.184
sn
a sin A
=r. 56512
logsinC=9-97559-io
colog sin A =o. 22123
log=i. 76194
^=57.80
Determine from c, C, and .5 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 = n 15', B = 47 12'.
(2.) Given a = 783.29, A = 81 52', B = 42 27'.
(3.) Given c= 1125.2, A = 79 15', -5=55 n'.
(4.) Given =15.346, ^=15 51', C=58 10'.
(5.) Given a = 5301. 5, ^=69 44', =41 18'.
(6.) Given =1002.1, ^ = 48 59', = 76 3'.
4,9. CASE II. Given two sides of a triangle and tJie 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 =1.54556
log sin ,4=9. 77064 10
colog=8. 63445 10
log sin /?=g. 95065 10
^=63 12'
C and C'.
C =i8o-(A +)=8o 39' 50'
6" ' = 180 -(/* + ')= 27 3' 50"
7't> find c and c .
c sin C
a sin ^4
log 0=1. 36555
log sin C= 9. 99421 10
colog sin /f =0.22936
log f=i. 58912
^=38.825
log a = i. 36555
log sin C' =9.6580010
colog sin A =0.22936
log f'=i. 25291
^' = 17.902
Check.
Determine b from c, C, and j? by the formula
This check is long, but is quite certain to reveal an error. A check which is
shorter, but less sure, is
c sin C
(i.) How many solutions are there in each of the following?
(i.) ^ = 30, a= 15, b 20;
(2.) A = 30, a = 10, b = 20 ;
(4.) # = 37 23', a = g.i, 6 = 7.5.
48
PLANE TRIGONOMETRY
Solve the following triangles, finding all possible solutions:
(2.) 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, b= 56.723.
(5.)Given^4= 61 16', a = 9.5124, = 12.752.
(6.) Given C= 22 32', # =0.78727, <: = 0.473 1 i.
?0. CASE III. Given two sides and the included angle.
EXAMPLE
Given a = 41. 003, = 48.718, C 68 33' 58" ; find the remaining
parts and the area.
To find A and B.
n(# - A) _b - a
b-a 7.715
b + a = 89.721
log (b a) =0.88734
colog (b + a) = 8.04710 10
log tan %(B + A) = o.i 6639
log ia.n^(B A) = 9.10083 10
l(B - A)= 7 11' 20"
^ = 62 54' 2l"
^=48 31' 41"
To find c.
c __ sin C
a sin/4
logrt = 1.61281
log sin C= 9.96888 -10
colog sin// =0.12535
log * = 1.70704
<= 50.938
To find the area.
S=\ab sin C
log ^ = 9.69897 10
logrt = 1.61281
log<= 1.68769
log sin (7=9.96888 10
log 6'= 2.96835
5= 929.72
sin C
Check.
c
~b
sin B
log sin 2? = 9.94951 10
log c = 1 . 70704
colog b 8.31231 10
log sin C 9.96886 10
THE OBLIQUE TRIANGLE
49
Solve the following triangles, and also find their areas :
(i.) Given A= 41 15', =0.14726, c=. 0.10971.
(2.) Given C= 58 47', =i 1.726, ^=16.147.
(3.) Given B 49 50', ^ = 103 74, ^-=99.975.
(4.) Given A= 33 31', =0.32041, ^=0.9203.
(5.) Given C=i28 7', =17.738, 0=60.571.
TJf. CASE IV. Given the three sides.
EXAMPLE
Given = 32.456, = 41.724, c =53.987 ; find the angles and area.
^ = 64.084
(s a) 31.628
(s b} = 22.360
(s c) 10.097
/ (s - a)(s - />)(s - f)
A \ /
v s
log (s a) = 1.50007
log (s-t) =1.34947
log (s r)=i.oo4ig
colog .r = 8. 19325 10
2)2.O46q8
log K 1.02349
To find A.
K
tanl// =
s a
log A"= i. 02 349
log (s-a) = i. 50007
sub.
log tan|^=g. 52342 10
|^ = i8 27' 23"
^=36 54' 46"
To find B.
K"
tan | .#= .
j <v
log /T= i. 02349
log (j-^) = 1.34947
sub.
log tan ^^=9. 67402 10
i^=2 5 16' 16"
^=50 32' 32"
To find C*
sub.
^^=1.02349
log (j f)=i. 00419
Check.
log tanC=o. 01930
iC=46 16' 22"
C=92 32' 44"
Find the angles and areas of the following triangles:
(i.) Given ^ = 38.516, =44.873, c=i4.5i7.
(2.) Given ^ = 2.1158, =3.5854, ^=3.5679.
. * C could be found from (A + J 5)=(i8o C), but for the sake of the check it
is worked out independently.
4
50 PLANE TRIGONOMETRY
(3.) Given = 82.818, =99.871, ^=36.363.
(4.) Given (7=36.789, t>= 11.698, ^=33.328.
(5.) Given a = ii3.oS, =131.17, ^=114.29.
(6.) Given a= .9763, =1.2489, c= 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 AC 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 16 13'. How far is the boat from the farther buoy? Are
there two solutions?
(7.) Given a =64.256, r= 19.278, C= 16 19' u"; 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.
THE OBLIQUE TRIANGLE 5!
Hint. By Geometry, angle A OJ3= 2 C.
Draw OD perpendicular to AB.
Angle DOB-\AOB=C.
DBr sin DOB=r sin C.
Hence c=2rsmC,
** x-i -
smC
(ii.) The distances AB, BC, and AC, between three cities, A, B,
and Care 12 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
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.) TAVO 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 ?
(18.) Two stakes, yf and ft, are on opposite sides of a stream; a
third stake, C, is set 62 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 CDS is 64 27', the angle DCB is 87 9'.
What is the distance?
(20.) Two islands, B and C, are distant 5 and 3 miles respectively
from a light-house, A, and the angle BAC is 33 11'; 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.
(22.) Of three mountains, A, B, and C, B is directly north of C 5
miles, A is 8 miles from C and 11 from B. How far is A south of /??
(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 the 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 ;
3 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 = 59 31'. 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 79 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 26 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', BDA = \% 17', and ADC$\ 23'. Find the distance from
AloB.
(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 50 miles 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 BCA = 108 34' and CBA =65 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^J5 sln^C sec^A.
Hint. Draw OB, OC, and the perpendicular OD.
OB and 0C bisect the angles B and C respectively, and ODr.
cosj^l
Hence
sin-^ Li sin \ C sin -J- H sin -J C
sin ^ j9 sin ^ C
= a sin A^ sin A Csec -J/4.
CHAPTER V
CIRCULAR MEASURE GRAPHICAL REPRESENTATION
CIRCULAR MEASURE
53. The length of the semicircumference of a circle is
TrR (^ = 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
T 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.
7T 7T
Thus, if the length of an arc is , it subtends the angle - radians.
C' A'
bmce one radian =
I8
7T
V,
, we have
90 radians,
i8o = 7r radians,
56
PLANE TRIGONOMETRY
270= -- radians,
360 27r radians, etc.
The value of a radian in degrees and of a degree in radians are :
i radian = 57.29578,
= 57 i 7' 45".
i .0174533 radian.
In the use of the circular measure it is customary to omit the word radian ;
thus we write - , it. etc., denoting - radians, ir radians, etc. On the other
2 to 2
hand, the symbols ' " are always printed if an angle is measured in degrees,
minutes, and seconds ; hence there is no confusion between the systems.
EXERCISES
(i.) Express in circular measure 30, 45, 60, 120, 135, 720, 990.
(Take 77=3.1416.)
(2.) Express in degrees, minutes, and seconds the angles , ,-,-.
(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. ?
54. The following important relations exist between the
circular measure x of an angle and the sine and tangent of
the angle.
7T
(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 s\nx<x< tan^r.
CIRCULAR MEASURE 57
sin x tan x
(2.) As x approaches the limit o, ana approach
Jv JC
the limit I.
Dividing sin x < x < tan x by sin x, we obtain
x i
K- <
cos^r
sin x cos^r
Inverting, i> >- .
As x approaches the limit o, cos^r approaches the length
of the radius, that is, I, as a limit.
Therefore, approaches the limit I.
x
sinje
Dividing i > > cos^r by cos^r, we obtain
X
i tan;ir
"^ -^. T
COS X X
As x approaches the limit o, cos^r approaches the limit I ;
hence approaches the limit I.
cos*
tcLtl X
Therefore, - approaches the limit I.
PERIODICITY OF THE TRIGONOMETRIC FUNCTIONS
55. The sine of an angle x is the same as the sine of
(^+360), (>-|-720 ), etc. that is, of (x+2mr\ 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 x, 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 /(*).
58 PLANE TRIGONOMETRY
The tangent of an angle x is the same as the tangent of
(,r-f 1 80), (^-+360), etc. that is, of (x+mr\ where n is any
integer.
The tangent is therefore a periodic function, having the
period 180, or TT.
The same is true of the cotangent.
GRAPHICAL REPRESENTATION
56. 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
Cosine Curve
GRAPHICAL REPRESENTATION 59
1 I f I '
Tangent Curve
f
Cotangent Curve
6o
PLANE TRIGONOMETRY
V.JT
SECANT CURVE
If the distances on OX are measured from O' instead of
(9, 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 2ir and 477-, 477 and 677, 2?r 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 REPRESENTA TION
61
-1 +1
INVERSE SINE CUKVE
INVERSE COSINE CUKVH
J I
-3 -2 -1
+2 +3
INVERSE TANGENT CURVE
62
PLANE TRIGONOMETRY
-3
-1
t-1 +2
+ 3
INVERSE SECANT
CHAPTER VI
COMPUTATION OF LOGARITHMS AND OF THE TRIG-
ONOMETRIC FUNCTIONS-DE MOIVRE'S THEOREM
HYPERBOLIC FUNCTIONS
57. A convenient method of calculating logarithms and
the trigonometric functions is to use infinite series. In
works on the Differential Calculus it is shown that
/-f2 /y3 /y4
(1 +)=*-+-- + . . . (I)
- -
a? 2 , a? 1 x 6
Another development which we shall use later is
e * = l + Ti + & + . + *< + - <4>
where ^=2.7182818 ... is the base of the Naperian system
of logarithms.
58. The series (i) converges only for values of x which satisfy the
inequality i<;r^i. The series (2), (3), and (4) converge for all
finite values of x.
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.
* 31 denotes 1x2x3; 41 denotes I X 2x3x4, 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 x
which satisfies the equation,
a* = m.
This is written x = \og a m.
The logarithm of the product of two numbers is equal to the sum
of the logarithms of the numbers.
Thus \og a mn = log a m + ]og a n.
The logarithm of the quotient of two numbers is equal to the log-
arithm of the dividend minus the logarithm of the divisor.
fflf
Thus log a - = log a m log a .
The logarithm of the power of a number is equal to the logarithm
of the number multiplied by the exponent.
Thus log a m^p log a m.
To obtain the logarithm of a number to any base a from its Na-
perian logarithm, we have
log, m
log a m = - = M a log, m,
log, a
where M. = , - ; M is called the modulus of the system.
log,tf
60. 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.
2
Substitute - for x in (i):
2
2/22 2* 3 2 4 2
If the result is to be correct to five places of decimals, we nyist 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
i
.5OOOOOO
2
I
j
.0416667
3
2
i
I
.0062500
5
2
I ,
,
.OOIIl6l
7
2 1
i
I
.0002170
9
2
i
I
77
.OOOO444
i
I
~T<!
.0000094
13
2
.5493036
Neative terms
I
I
2
?^ C
). 1 250000
f
I
4
^ =
.0156250
.--,
I
.OO26O42
6
2
I
I
i
.0004883
8
2 8
; I
I
10
2
.OOOO977
I
I
.
~~Ts =
.O000203
12
2
I
I
.OOOOO44
J4
2 14
.1438399
Subtracting the sum of the negative from the sum of the positive terms, v
obtain
i 3
^-=.4054637.
Denote the sum of the remaining terms of the series by J?.
bra,
15 2 1
The error caused by retainir
less than .0000006. Hence
the result is correct to five d
61. As remarke
calculate directly '
but it can be read
us the logarithm o
Replacing x by
5
66 PLANE TRIGONOMETRY
x* x 3 x*
1 / \ "V vV *V
log, (i x)= x -- --- .
2 34
This series converges for i<x<i.
Subtracting this from (i), we obtain
log,
which converges for i < x < i .
Putting y=l- -I, we see that y passes from o to oo as x
\i x)
passes from i to +i ; hence, if we make this substitution in
(5), we get a series
which converges for all positive values of y, and therefore enables
us to compute the Naperian logarithm of any number.
From (5) we can get another series which is useful : put
.1 (5) gh
'
'ierc M^s*- Hence,
r
.). (7)
directly
icrs can
.TIT'- .-
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=2' 2 , 6 = 2 . 3, 8 = 2 3 , 9=3*, 10=2 . 5, and log 1=0.)
In this case we are computing the logarithms of successive integers, and
should therefore use (7).
63. Example. Compute the Naperian logarithms of 2, 3, 4, and 5.
/i.i T , i 1,1 i , T i , \
. . . .
3 3 3 3 5 3 s 7 3' 9 3"
-=3333333
T-i=' OI2 3457
- . =.0008230
5 3 5
i i
- . -, = .0000653
7 3
i i
- . =.0000056
9 3
.3465729
2
Denote the sum of the remaining
terms of this series by R.
Then, by Algebra,
i i i
or R<. .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.
logr 3=5.6931458
Remark. We should get the same series if we were to use (6).
log* 3 = log* 2 + 2
- = .2000000
I + M + I. 1+M+"- V
5 3 5 5 5 s 7 5 1 /
- = .0026667
i
3
- -: = . OOOO64O
5 5 s
- . = .0000018
7 5 7
.2027325
2
.4054650
Add log* 2= .6931458
log, 3=1.0986108
9 5" i ft
or R< .00000006.
Noting the errors in the pre-
ceding column and in log* 2, we
see that the total error is less than
.00000217.
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.
lg,5 = lg,4+2(-H ---- H ---- s + . . . ),
\9 3 9 3 5 9 s /
or lg# 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:
lg r = -ooooo
log, 2= .69315
lg, 3 = 1.09861
log, 4 =1.38629
lo g* 5 = 1.60944
log, 6 = 1.791 76
log, 7 = I-9459 1
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 M 10 =. 43429448. 59
Thus Iog 10 5 = log, 5 X .43429448 = .69897.
fo. 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
06. Since tan^r= - , cot;r= - , etc., the computa-
cos^r
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
rrr rrr
~^ , it is never necessary to take x greater than in the
<-4 4
series (2) and (3). 16
o
Since - =0.785398 ...<, these series converge rapidly ; in fact,
4 10
= .000003 does not affect the fifth decimal place, and the
9! ii !
seventh.
<?7. 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
s'mnx = 2 cos .r sin (n i) x 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> = cos_y sin (30 /),
cos (yP-\-y) = cos (30 y) sin/.
08. 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".
12 = .209439516
15' =.004363323
39" = .000189076
12 15' 39" = .213991915 in circular measure.
PLANE TRIGONOMETRY
* i *
sm;r = x 1 -
3! 5!
x=. 2139919
x*
=.0000037
.2139956
subtract =.0016332
sin jr=. 2 1 23624
Correct to five decimal places.
COS*=I --- ---
2 ! 4 !
1 = 1.0000000
X*
== .0000874
4! _ _
1.0000874
jt 2
subtract -= .0228963
COSJT= .9771911
Correct to five decimal places.
DE MOIVRE'S THEOREM
09. 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 (3 and a re-
spectively.
This geometrical representation shows at once that we
can also write a in the form
= r(cos + t sin 5). (9)
70. From Algebra we recall the definition of the sum of the
complex numbers a = a + ifi and b = y + fi>\ namely
Subtraction is defined as the inverse of addition, so that
DE MO IV RE'S THEOREM 71
Multiplication is most conveniently defined when a and b are
written in form (9). If
a r (cos^+/ sin ?) and bs (cos^+/ sin0),
their product is defined by the equation
ab rs [cos (& + ^>) -f- * sin ($-\- 0)]. (i o)
Division is defined as the inverse of multiplication, so that
- = - [cos (3 - 0) 4- * sin (> - f)].
Finally, we recall that in an equation between complex numbers,
a + //3 = y + a,
we have = y, 13 = $. (n)
7 1. Consider the different powers of the complex number
*:=cos $ + / sin.
By (10) we have
x* = (cos -f * sin ^) (cos 3+t sin -&),
cos 2^ + 1 'sin 2$.
x* = x* . jc = (cos2$+/ sin 2$) (cos &-!-/' sin ^),
= cos3^ + / sin 3.?.
And, in general, for any integer n,
^ = (cos ^+/ sin -&)* = cos n$-\-i sin n.
From this equation we have De Moivre's Theorem, which
is expressed by the formula
nw-&). (12)
72. An interesting application of De Moivre's Theorem
is the expansion of sin nx and cosnx in terms of sin x and
cos^r. Expanding the left-hand side of (12) by the bino-
mial theorem, and substituting x for S, we have
"~ 2
cosnx+t sin ^ = cos** x-\-n cos""" 1 x (/sin x) -{-
3 !
)' -f .
72 PLANE TRIGONOMETRY
or
cosnx + i sinw-x^fcos*.* -- - - - cos*" 2 * sin 8 *-}- . . .)
\ 2 I )
n ( i) (n 2) 1
+ t \n cos"" 1 x sin* -- -i cos*" 3 * sin'^-f ....
Equating real and imaginary parts, as in (u), we have
cos n ~ 2 x sin'
ft (fi I ) (fi
cosnx=cos"x -- - j : cos n ~ 2 x sin'^-j- . . . (13)
3 , : + ...<-4>
Example. n = 5.
cos 5 x = cos 6 x 10 cos s .r sin".r-|-5 COSJT sin* jr.
sin 5^-= 5 COS*JT sin.r 10 cos 2 x sin 3 .r + sin* x.
THE ROOTS OF UNITY
75. We find another application of De Moivre's Theorem
in obtaining the roots of unity. The w th roots of unity are
by definition the roots of the equation
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 )i numbers
2-rrr . . 2irf
x r = cos (-; 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),
/ 2irr 2irr\ n
-*":=( cos f-z sin - ] .
\ H n I
THE ROOTS OF UNITY 73
/ 27rr\ . . / 2irr\
=cos I n . -|4-< Sin ( . ),
V / V /
= cos 27rr+/sin 2:rr,
= 1 4- z . o = i .
Therefore ,t' r 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 =
that is, the circumference is di-
vided into three equal parts by the points A, P, and Q. Then OD = ,
and DP = DQ = $\/3. Hence we see from the method of represent-
ing a complex number given above that A represents -\-i,P represents
i + z 'lA/3. Q represents / ^-y/3-
P^
= 120, angle AOQ = = 240
3 3
EXERCISES
74. (i.) Express sin 4* and cos 4* in terms of sin* and cos*.
(2.) Express sin 6* and cos 6* in terms of sin* and cos*.
(3.) Find the six 6 th roots of unity.
(4.) Find the five 5 th roots of unity.
THE HYPERBOLIC FUNCTIONS
75. The hyperbolic functions are defined by the equations
e*-e-*
siiili a? =
cosh x
(16)
in which sinh^r and cosh;r 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 cosirte 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 *,
cos*, and e* for every real value of x. These series also
serve to define sin*, cos*, and ^*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\n x cos^ + cos* sin_y, (17)
cos (x+y) = cosx cosy s'\nx siny, (18)
e*+'=f*e 3 ', (19)
hold unchanged when the variable is complex.
This fact enables us to calculate with ease sin*, cos*, and
e x 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
iy, we get
sin (x+iy) = s'\nx costy+cosx 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
Y (ix) 3 ()
3 ! 5 ! 7 !
A comparison with series (2) and (3) shows that these two
series are cos^r and sin x respectively; hence the important
formula due to Euler
e'*=cosx-\-i siii.r. (20)
This enables us to calculated 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~*=cos ix-\-i 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
- = cosh x, (23)
2
= i si nh x. (24)
J
If we compute the value of e* by the aid of series (4) for
a succession of values of x, we find that sinh-ar and cosh A*
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-
7 6
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
e ix +e~ ix
COS JC =
sill x
2
e ix_ e -ix
(25)
(26)
which are formulas frequently used.
Example. sinh (.r -\-y) = / sin i(x -\-y),
= i [sin ix cos iy -\- cos ix sin z'y],
= i \i sinh x cosh y-\-i cosh x sinh j],
= sinh x cosh y -\- cosh x sinh y.
Example. sinh x-\- sinh^ = /(sin tx 4- sin iy),
=. i 2 sin i(x-\-y) cos /(.r y\
= 2 sinh ^ (x -\-y) cosh ^ (-r y).
sinh
HYPERBOLIC FUNCTIONS 77
EXERCISES
76. (i.) Prove sinh 0=0, cosho=i.
(2.) Prove sinh ^7J7=/, cosh ^TT/=O.
(3.) Prove sinh TTI' O, coshyrz^ i.
Prove that
(4.) sin ( ix) = sin ix.
(5.) cos ( ix) = cos ix.
(6.) sinh( x) = sinh .r.
(7.) cosh( .r) = cosher.
Remark. The hyperbolic tangent, cotangent, secant, and cosecant
are defined by
sinh.jtr cosher
tanh Y > coth x >
cosher sinh.r
i i
seen x ^^ : > csch x ^^ . ;
coshjf smh^r
Prove
that
(8.)
tan (ix) = i tanh x.
(9-)
coth ( x) = coth x.
(10.)
sech ( x) = sech x.
(ii.)
cosh ! .r sinh a ^r=:j.
(12.)
sech 2 ^- + tanh z .r = i.
(13-)
cot hlr csch 2 ^ = I .
(14.)
sinh (x y) = sinh x coshj/ cosh x sinhj/.
(15-)
cosh (,r y) cosh x cosh^j/ sinh^r sinhj/.
(16.) coshj^r= t /
1-f-cosh.r
(17.) sinh s'\nhv = 2 cosh| (u -\- v) sinh ^ (u v).
(i 8.) cosh u-\- cosh?/ = 2 cosh %(u-{-v) cosh \(uv).
(19.) cosh u cosh v = 2 sinh^(-fz/)sinh|( v).
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) sin x cos x = I .
(4.) (sec/ tan/) (sec/ 4- tan/) = I.
(5.) (esc 2 cot s) (esc z 4- cot z) = i .
(6.) cos 2 / 4- (tan/ cot/) sin/ cosy = sin 2 ^
(7.) cos'-r 5!^^+ i =2 cos" a-.
(8.) (sin_y cos/) 4 = 1 2 sin/ cosj.
(9.) sin 3 jr-f-cos 8 jr = (sin jr-j-C 08 -*") ( l s ' n x
. y
(10.) - - =cot.rtany.
tan x-\-co\.y
(n.) cos*/ sin"/ = 2 cos"/ i.
(12.) i tan 4 jr = 2 sec a -r sec*.*.
COS.T
(13.) - - j = tan jr.
v J/ sm.r cot a .r
(14.) sec"/ esc*/ = tan 2 / -(-cot 5 / +2.
(15.) cot/ esc/ sec/ (i 2 sin 2 /) = tan/.
N / I Y I COS.?
(i 6.) I . cot2J =- --
\sin 2 / i -f- cosr
sec/ tan/ sin/
(I7-)
i-j-cos/ sin 3 /
(i8.) i+?-^lf=(sin.r4-cos.r) 2 .
SGC X
(i 9-) ; sin 3 jr = (cos.r sin x] (i 4- sin x COS.T).
y t sec 8 *
(20.) (sin^r cos/ 4- cos x si n/) 2 4- (cos x cos y sin. v sin /)" = !
MISCELLANEOUS EXERCISES 79
(21.) (a cos.r b sin xf -\-(a sin x-\-b cosxf a"-\-^.
i 4 tan 2 /
' ' /33^2 7, ^T^Ta"7.\s ~r~ 7T~
(cos 2 / sin 2 /) 2 (i tan 2 /) 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/ = csc/ f.
(25.) -\/2sin.r tan.i' = o.
(26.) 2 cos A- "V/3 cot.r = o.
(27.) tanj + cotj" 2 = -
(28.) 2 sin 2 / 2 = \/2 cos_y.
(29.) 3 tan 2 ^ i =4 sin 2 .r.
(30.) cos 2 x-{- 2 sin 2 .r |sin^r = o.
(31.) csc.r = tan^-.
(32.) sec a- -f- tan x ^/y.
(33.) tan x -\- 2 -y/3 cos x = o.
(34.) 3sin.r 2cos 2 .t'=o.
Express the following in terms of the functions of angles less
than 45:
(35.) sin 92.
(36.) cos 127.
(37.) tan 320.
(38.) cot 350.
(39.) sin 265.
(40.) tan 171.
(41.) Given sin x = and x in quadrant II; find all the other
functions of x.
(42.) Given cos.r = | and x in quadrant III; find all the other
functions of x.
(43.) Given tan^" = f 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 x cos x = \ \/3.
(46.) sec-r tan .r = 2 -y/3.
(47.) ta.ny-{- -y/2o cosj= o.
(48.) cos x cot x = $ .
Find all the values of y less than 360 which will satisfy the fol-
lowing equations:
(49.) tan^ + 2 sin^ = o.
(50.) (i + tan*) (i 2 sin .*):= o.
(51.) sin x cos x (i -\- 2 cos x) = o.
Prove the following:
(52.) cos78o = .
(53.) sin i48s = ^\/2.
(54,) cos 2 5 50 = </3-
(55.) sin ( 3000) = cos 30.
(56.) cos 1 300 = cos 40.
(57.) Find the value of a sin 90 -|- tano-\-a cos-i8o.
(58.) Find the value of a sin 30 + ^ tan45-|-rt cos 6o-f- tan 135.
(59.) Find the value of (a b) tan 225 -\-b cos 180 a sin 270.
(60.) Find the value of (a sin 45-}- cos 45) (a sin 135 + ^ sin 225).
RIGHT TRIANGLES
18. 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
ii 21 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 17' 37". Find the breadth of the
river.
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 ' s 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 bank the angle of elevation
of the house is 14 30' 21". Find the width of the river.
(11.) 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 11 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' 16", 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 i72i' 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
79. (i.) The area of a regular dodecagon is 37.52 ft. ; find its
apothem.
(2.) The perimeter of a regular polygon of n 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.
(11.) 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
(16.) 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.
(i 8.) 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.
(21.) Find the perimeter of a polygon of n 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 $ycos \y=.'\/\ siny.
. sin 2x 4- sin A.r
(3.) - = tan3^r.
cos 2x + cos 4*
(4.) cos 2 / tan 2 y -+- sin 9 y cot"/ = I.
. cos (x 4-y -4- z)
(5.) . . = cot x coty cotz cot x cot y cotz.
s\nx sin/ sin z
(6.) cos" (x y) sin" (x-\-y) = cos 2x cos 2y.
= _ co
COS X COS/
. COSJT sec^r
(8.)- =4cos s i;r(cos a i;r l).
sec.r
(9-)
(10.)
I COS 2 X
_ i cos 2y
I -f- cos 2y
(u.) cotx tan x = 2 co\.2x.
MISCELLANEOUS EXERCISES 85
(i 2.) tan \x-\-2 sin 2 \ x cot x = sin x.
. tan .Titan/
(13.) - = zfcsm x sec.r tany.
(14.) sin.r 2 sin 3 x = sin JT cos2x.
(i 5.) 4 sin j sin (60 y) sin (60 -\-y) = sin
, sin y(i tan 2 y) / i
(I 6.) - f --- y ( - - : --
sec 2 / \cos_y sin/
(17.) i -|- tan/ tan -| j=
(18.) sin 4^- = 4 sin .r cos 3 x 4 cos ^r sin'^r.
2
(19.) sec 2r+ tan 2^-f- i = --
i tan x
(20.) tan 50 -f- cot 50 = 2 sec 10.
(21.) cos(-r + 45 ) + sin(.r 45) = o.
tan x
(22.)
i cot 2x tan x
(23.) (i tan 2 x) sin x cos x = cos 2x \/-
cos 2x
I -j- COS
. cosy + siny
(24.) r-^ = tan 2 y + sec 2 y.
cos_y s\n_y
(25.) sin (^+/) cos^r cos (x-\-y) ein x-=
(26.) cos (-r y) siny -(- sin (j: _y) cosj/ = sin jr.
sin (xy) t sin (jy z) l sin(z x)_
(27.) - -- h" ~T~ O.
cos.r cosy cosy cosz cos 2 COS.T
sin jr+si" 2X
(28.) - - = coti^r.
cos x cos 2x
(29.) 2 sin 2 .* sin 2 j+ 2 cos* x cos 2 _y = i + cos 2x cos zy.
(30.) sin 6o+sin 30 = 2 sin 45 cos 15.
tanfr-jQ + tan,
i tan (JT ; ;/) tany
(32.) - - -
u ; sm_y tan
(33.) sin 4Jtr-f sin 2x = 2 sin
sin.r
^ 4 cos^r cos_y sin_y sn*
(35.) sin 7 5 =
.
2 \/2
(36.) 2 tan 2_y = tan(45+_y) tan (45 y).
86 PLANE TRIGONOMETRY
tan 2 .r-}- tan .r sin3.r
(37-) ~ - : -
tan2jr tan^r sin^r
3 tan y tan'y
(38.) tan 3v - - ^- - 5^.
i 3 tan>
(39.) sin6o-f- sin 20 = 2 sin 40 cos 20.
(40.) sin 40 sin io = 2cos25 sin 15.
(41.) COS2J: cos4-r = 2 sin3~vsin.r.
(42.) tan 1 5 = 2 y/3~.
(43.) (A/I -f-sin^r \/i sin^) 2 = 4 sin ! ;r.
(44.) \/i -f si
. v
(45.) - -^ ^-' 2 cos (x +y) = -r
sm.r
, ., .
(46.) -: ^- = 2 COS 2X.
sm2jr
(47.) sin 50 sin 70 -4- sin io = o.
, o \ IT 1T fyt IT
(40.) cos -- cos - = 2 sin sin
32 12 12
cos 7 5 -f cos 1 5
(Si.) tan'^
= ft
V
(52.)
(53.) sin3J--f sin 5^ = 2 sin4JT cos^r.
(54.) cos 5-r + cos gx = 2 cos jx cos 2x.
(55.) sinis^
2V 2
/
(56.) - - = tan jr.
cos 3_r + cos JT
(57.) sin $y= 5 sinj 20 sin 3 /+ 16 si
(58.) cos 57 = 5 cos_y 20 cos 3 j-f- 16
4 tan _r(i tan 2 .r)
(60.)
(6 1.) cos 3^ -\- cos 5 JT -J- cos jx + cos 1 5 x = 4 cos 4^ cos 5^ cos 6^.
MISCELLANEOUS EXERCISES 87
(62.) sin 2 ^r(cotf.r i) 2 = i sin.r.
cos 3.r 4- 3 cos x ~
(64.) sin x(\. 4- tan .r) 4- cos .r( i 4-cot.r) = csc.
(65.) cosV- S in^ = 24-sin2.r <
cosx smjr 2
(66.) cos j + cos (i 20 y) + cos (i 20 -\-y) = o.
. sin ~\x
(67.) . =2 COS 2^T 4- I.
sin JT
(cosj/ cos 3/)(sin 8_y + sin 2y) _ ^
(sin y/ sin_y)(cos4^/ cos6/)
(69.)
cos.r
. sin3
(70.) -r-2
sm.r COSJT
' COS.T
^ - ' -
(73-) = tan 4- r -
cos .i- + cos 3-tr -|- cos 5_r -|- cos "jx
If ^f, B, and C are the angles of a triangle, prove the following :
(74.) sin iA 4- sin 7.B -\- sin 2C = 4 sin A sin^ sin C.
(75.) sin 2^4 + sin 2B sin 2(7 = 4 cos .4 cos B sin C.
(76.) sinM-fsin2# + sin 2 C= 2 _^_ 2 cos ^ cos 5 cos C
(77.) tan ^4 + tan B + tan C = tan A tan ^ tan C.
Solve the following equations for values of x less than 360.
(78.) cos2^r+ cos - ;1: ' = i.
(79.) sin x-\-?>\njx = sin4_r.
(80.) cos.r sin 2x co&^x o.
'(81.) COS.T sin3;r cos2.r = o.
(82.) sin4_r 2sin2.r:=:o.
(83.) sin 2x cos2.r sin jr + cos.r = o.
(84.) sin (60 x) si n (60 + x) = + \ y^.
(85.) sin (30 + x) cos (60 + x) = \
(86.) csc-r = i -4-cot-r.
(87.) cos *x cos *x.
(88.) 2 siny=s\n2y.
(89.) sin3/-|-sin2y-|-sin
(90.) sin"jr-f 5 cosV = 3.
(91.) tan(45
OBLIQUE TRIANGLES
81. (i.) It is required to find the distance between two points, A
and B, on opposite sides of a river. A line, AC, and the angles BAC
and ACB are measured and found to be 2483 ft., 61 25', and 52 17'
respectively.
(2.) A straight road leads from a town ^4 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 ?
(3.) In order to determine the distance of a fort, A, from a battery,
/?, a line, /?C, one-half mile long, is measured, and the angles ABC
and ACB are observed to be 75 18' and 78 21' respectively. Find
the distanced/?.
(4.) Two houses, A and B, are 1728 ft. apart. Find the distance of
a third house, C, from A \i BACM Q 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' u".
(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, ,4 and B, are u miles apart. A ship, C, is
observed from them to make the angles J5AC=^i 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 js 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.
AB At miles, BC=$ 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 \/^. 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
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 foot 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 18' 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 21 1 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 B, 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.
(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 u' 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 ; 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 m.iSand 12 1.21, 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, A OB, and BOC
with the surface of the sphere. Suppose the planes DAC"
and EEC 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 #, b, c have the
same numerical measure as BOC, AOC, and AOB respec-
94 SPHERICAL TRIGONOMETRY
tively, then, AD = ta.nc, BE sine, C' = sina, OC' = cosa,
cos&, OE cosc, AC"= sin b.
In the two similar triangles OEC' and OAC",
cos c cos c cos a
OA ' i cosb
In the triangle BC'E,
, or cos a = cost? cose. (i)
~ BE . ~ sin c
sine = -757^, or sin 6=
sin
In the triangle DAC" ,
DA r tan c
tan C -.,, . , or tan 6 = - (3)
Combining formulas (2) and (3) with (i),
-_ tan b , v
Again, if AB were made the base of the right spherical
triangle ABC, we should have
z? sin ^ ^\
Sm slrTrt' ^ 5 ^
.-. Ltlll //r\
tan^^-^-^.
cos^= (7)
tatirt
From the foregoing equations we may also obtain by
combinations,
cos^=sin C cosb. (8)
cos7 sin# cos^r. (9)
cotC (10)
NAPIER'S RULES OF CIRCULAR PARTS
83. 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 a
comp G<
Thus there are five circular parts namely, />, c, comprt, comp B, 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 B and b are adjacent parts, and
comprt and comp C are opposite parts.
The ten formulas may be written and grouped as follows :
1st Group.
sin comp C = tan comprt tan b.
sin comp ,5=tan comp a tan c.
sin comp a =tan comp.Z? tan comp C.
sin c =tan comp B la.nl>.
sin b =tan comp C tanr.
"id Group.
sin comp a= cos/' cos c.
sin <=cos comp a cos comp B.
sin f=cos comprt cos comp C.
sin comp j9=cos comp C cos b.
sin comp C =cos comp B cos c.
Napier's rules may be stated :
I. The sine of the middle part is equal to tlie product of
tJie 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 a = 63 56', 6 = 40 o', to find c, B, and C.
To find c.
comp a is the middle part.
c and b are the opposite parts.
sin comprt=cos cos<r,
or cos rt=cos b cos c.
cos a
COSf = -
cos b
log cos ^7=9.64288
colog cos b 0.11575
log cos r=g. 75863
f=54 59' 47"
To find C.
comp C is the middle part.
comp a, and b are adjacent parts.
sin comp C;=tan comprt tan b,
cos C=cot a tan b.
log cot a =9 68946
log tan =9 92381
9 61327
C=6 5 45' 58"
To find B.
b is the middle part.
comp a and comp B are the opposite
parts.
sin />=cos comprt cos comp.Z?,
or sin /;=sin a sin B.
sin b
sin B--
smrt
log sin />=9.8o8o7
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 .5 and c are opposite parts.
sin comp C=cos^ cos comp B,
or cos C=cos c sin B.
log cos c =9. 75863
log sin #=9.85466
log cos "=9.61329
C=6 5 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
180 a, 1 8o , 180 B.
Given c = 26 4'.
C=36o'.
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,
3r cos C=sin B cos c,
cos C
cos c
log cos =9.90796
colog cos c =0.04659
sin =
log sin = 9.95455
B= 64 14' 30"
' = i8o-=iis 45' 30"
To find b and b'.
sin =tan c tan comp (7,
sin =tan c cot C.
log tan c= 9. 68946
log cot (7=0.13874
log sin =9.82820
b= 42 19' 17"
'=180 =137 40' 43"
To find a and a'.
sin c =cos comp a cos Comp C,
>r sin c =sin a sin C,
sin c
ir sin = - - .
sin C
log sin c=<) 64288
colog sin (7=0.23078
log sin 0=9.87366
a= 48 22' 55"
a' = i8o 0=131 37' 5"+
(Discrepancy due to omitted decimals.)
Check.
sin =cos comp <? cos comp />,
r sin =sina
log sin a or rt'=g. 87366
log sinZ? or '=9.95455
log sin =9.82821
= 42 19' 21"
'=180 =137 40' 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
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 = 140 o'. a = 90 o'.
The corresponding parts of the polar triangle are
rt' = 6356', ' = 4oo', ^' = 90.
By Napier's rules we find
#' = 45 41 '28", C' = 6s45' 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=i25o' 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 B = <)5 6'. Required the remaining parts.
(3.) In the right-angled triangle ABC, the side = 26 4', and the
angle = 36. Required the remaining parts.
(4.) In the right-angled spherical triangle ABC, the side c=S4 30',
and the angle .5 = 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 = 6g
20', and the angle C = 58 16'. Required the remaining parts.
(7.) In the spherical triangle ABC, the side a = 90, the angle C=
42 10', and the angle A -=.115 20'. Required the remaining parts.
Hint. The angled 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 (7=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 A, ED,
and AD respectively. Then AD=tan b, AE tan c, OD
sec b, OE=secc.
In the triangle ROD,
ED 1 = sec 8 ^ -f sec V 2 sec b sec c cos a.
In the triangle AED,
ED 1 tan 8 ^ -f- tan V 2 tan (5 tan <: cos A.
Subtracting these two equations and remembering that
sec a tan"=i, we have
= 2 2 sec secf cosrt + 2 tan tan^r cos A.
Reducing, we have
cosa=cos& co8c+sin6 sine cos .4. (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 b = cos c cos a -\- sin c sin a cos B,
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.
00. By making use of the polar triangle where
c=i%o-C C=i8o-c'
we may obtain a second group of formulas.
Substituting these values of a, b, c, and A in (i), and remembering
that cos(i8o A) = cos A and sin (180 A) = sin A, we have
cos^4' = cos/?' cosC'-f-sin-Z?' sin C' cosa'.
Since this is true for any triangle, we may omit the accents and
write,
cos 4= cosU cosC+sinU sin C cos a. (2)
FORMULAS FOR LOGARITHMIC COMPUTATION
91 Formula (i), cos a cos b cose + sin b sin c cos A,
cosa cos$ cos*:
gives cos A = : r : .
sm# sin c
By 36, cos^4:=i 2 sin 2 ^
cos# cos^ cose
Whence i 2s\rr$A= . . ,
sin<? sin c
cos#cos*:+sin^ sin c cosa
or sin 2 i A : ,
2 sin b smc
102 SPHERICAL TRIGONOMETRY
_cos(6c}cosa
2 sin b sin c
. a + b c , a b-\-c
ci n ^^^_______ ci n _
olil oiii
sn sn c
Putting
iU + #<: ,
=s, then - =^ <:, and
2
/sm (5
we have sm$A=\/
V
,
sin b si
Since, also, cos A I + 2 cos a ^4,
we have, similarly,
/sin ^ sin(.y )
= V/- T^ i -
v sin b sin f
Hence ~ & ) 8ln ( 8 ~
si us sin (s a)
By a like process, formula (2) reduces to
cosScos(S-A) ' .
i rn' \ /
^. If, in formula I, we advance one letter, we have
sin ^
And dividing tan ^^4 by tan^j5, and reducing, we obtain
tan \A sin (s b)
tan ^^ sin{f )'
By composition and division,
sin j
tan-fa A tan^Z? sin (s b) sin (j a)'
By 30, 38, this becomes
tanic (HI)
- B) taii4( 6)
OBLIQUE-ANGLED TRIANGLES 103
Multiplying tan^A by tan^B, and reducing, we obtain
tan -J A tan B _ sin (s c)
i sins
By division and composition, and by 30, 38, this be-
comes
co*$(A + B) tan^c ,
co$(A B) ~ tan ( + &)'
Proceeding in a similar way with formula II, we obtain
sin -| (a + 6) cot^C
And
sin(a 6) tan%(A .B)'
cot^C
(V)
cos ^ (a 6) "~ tan ^ (A
9?. In the spherical triangle ABC, suppose CD drawn per-
pendicularly to AB, then, by the formulas for right spher-
ical triangles,
In triangle A CD, sin /=sin b sin A.
In triangle BCD, sin / sin a sin B.
Whence sin a sin ^=sin b sin A,
sin a sin 6
/TTTT\
( VIT )
Remark. If (A+B)>i8o, then (a-J-)>i8oP, and if (A + B)<
, then (
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 VI, and VI 7. Check : apply III or I V.
(4.) Given two angles and included side.
Apply III and I V, and VII. Check : apply V or VI.
(5.) Given two angles and an opposite side.
Apply VII, V, and III. Check : apply I V.
(6.) Given two sides and an opposite angle.
Apply VII, V, and I V. Check : apply III.
EXAMPLE CASE (l)
95. Given a = 8i 10' = 6o2o' r=ii225'
To find A, B, and C.
a= 81 10'
f =112 25
j = i26 57' 30"
s-a=4S 47' 30"
s 6=66 37' 30"
j c = i4 32' 30"
2)19 60464
log sin ,r=9. 90259
log tan \A= 9.80232
log sin (s-a)=g. 85540
log sin(.-*)=n ofi^r i//=32 2 3 ' 19"
log sin (.?-<)=
To find A.
sin s sin (s a)
log sin (s ^=9.96281
log sin (s ^=9.39982
colog sin s=o. 14460
colog
OBLIQUE-ANGLED TRIANGLES
10:
To find B.
sins sm(j b)
log sin (.r rt) = g. 85540
log sin (sf) = g. 39982
colog sin.r =0.09741
colog sin (5 /') =0.03719
2)19.36982
log tan 5= 9.69491
Jfl=262l' 6"
.5=52 42' 12"
tan i C= , /si"('-)si('-l
V sill j sin(j f)
log sin (j <z)=g. 85540
log sin (j ^=9.96281
colog sin s=o. 09741
colog sin(j- r)=o.6ooi8
2)20.51580
logtanJC= 10.25790
JC= 61 5' 32"
C=I22 II' 4"
Check.
_, ...
Formula V, coti C=
fl=8i 10'
<J =60 20'
i4i 30' ; ^(a+<5)=
ab= 20 50'; ( ^)=
,4 =64 4' 38"
.5=52 42' 12"
A-B=i2 4' 26"
B) 6 2' 13"
log tan \(A -5)=g. 02430
log sin ^(<n + />)=9 97501
1 45' colog sin (rt - b) =0.74279
25' cot (7=9.74210
C= 61 5' 32'
C=I22 II' 4"
EXAMPLE CASE (3)
96. Given a = 78 1 5' = 56 20'
To find
(0 + ^=67 17' 30"
(*-*)= 10 57' 30"
^ C=6o
Formula V may be written
cosA( /^) cot A 6"
2 2
cos ( + /;)
log cos(rt )= 9.99201
log cot (7= 9.76144
colog cos ^ (a + />)=. 0.41337
log tan (A + B)= io. 16682
) = 5544' 36"-
f-^?)= 6 47' 4"
^4=62 31' 40"
^=48 57' 32"-
?, and c.
log sin
C=I20
(rt +^=9.96498
log cos %(a + ^=9.58663
log sin J(rt ^=9.27897
lg cos J(<z ^=9.99201
log cot =9.76 144
Tofnd^(A-B).
Formula VI may be written
.,/. m sin$(a-6)
tan * ( A J3) = - : j-p:
log sin J (a ^=9.27897
log cot ^ C=g. 76144
colog sin ^(rt +6) =0.03 502
106
SPHERICAL TRIGONOMETRY
To find c.
_ . , TTT . sin b sin C
r rom rormulfi VII sin c
Formula III may be written
j sin^(/f ^r E) tan( K)
sin .5
log sin b =9.92027
log sin (7=9.93753
colog sin .5=0.12249
log sin$(A +)= 9.91725
log tan (a l>)= 9. 28696
colog sin %(Ah)= 0.92762
log sin c =9.98029
log tan c= 10. 13183
\f= 53 33' 5"-
^=107 i 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 and 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 A sin b
sin b sin A
1 0*1 M- . n
sm a sin B
the numerator of the fraction is greater than the denominator.
OBLIQUE-ANGLED TRIANGLES
107
EXAMPLE
98. Given 0=40 16' d=
To find B, B',
To find B and B'.
Formula VII may be written
sin B=
sin A sin b
sin a
log sin A=g. 89947
log sin =9. 86924
colog sin =o. 18953
log sin ^=9.95824
= 65 1 6' 30"
^' = 114 43' 30"
To find c.
Formula IV may be written
_ cosl(A+) tanjfri + J)
cosi(^-2?)
log cos $(A + fi)=(). 71326
log tan(-f ^=9.98484
COlog COS (.4 .5) = O.O02 70
log tan ^f=g. 70080
^=26 39' 42"
'=53 19' 24"
log c
log tan $(
colog cos^(^4
)=<). 98484
log tan ^'=9.09860
\c'= 7 9' 9"
<r'=l4 1 8' 1 8"
CASE (6)
=4744' ^=52 30'
C, C, and c, c'.
To find C.
Formula V may be written
. _ sini( + ^) tanA(^4 B)
cot L= : j-
sin ^ (a b)
log sin ^(a + b)=. 9.84177
log tan^M )= 9.04901 n
colog sin ^(a b)-= i.i8633n
log cot^ C= 10.077 1 1
C= 39 56' 24"
C=7 9 52' 48"
To find C.
log sin(rt + )= 9.84177
log tan ^(^4 B')= 9.78153 n
colog sin \ (a b)= 1. 18633 n
log cot C" = 10.80963
JC'= 8 48' 41"
C"=i7 37' 22"
Formula III may be written
sin B sin r
sin b= - . .
sine
log sin .5=9.95824
log sin c=g. 90418
colog sin C=o.oo682
log sin ^=9.86924
<J=47 44'
EXERCISES
99. (i.) In the spherical triangle ABC, the side a = 124 53', the
side b = 31 19', and the angle .<4 = 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 B 68 50'. Find the other parts.
* The letter "n" indicates that these quantities are negative.
Io8 SPHERICAL TRIGONOMETRY
(3.) In the spherical triangle ABC, the side c = 78 15', =56 20',
and A = 120. Required the other parts.
(4.) In the spherical triangle ABC, the angle A 125 20', the an-
gle C=4& 30', and the side = 83 13'. Required the remaining
parts.
(5.) In the spherical triangle ABC, the side c = 40 35', = 39 10',
and a 71 15'. Required the angles.
(6.) In the spherical triangle ABC, the angle ^ = 109 55', Z?=n6
38', and C= 120 43'. Required the sides.
(7.) In the spherical triangle ABC, the angle A-=. 130 5' 22", the
angle (7=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", 22 =
31 34' 38", and C= 161 25' 17". Required the sides.
(9.) In the spherical triangle ABC, the side <r=ii2 22' 58", b-=.
52 39' 4", and a = 89 16' 53". Required the angles.
(10.) In the spherical triangle ABC, the side ^ = 76 35' 36", t> =
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-\--{-C2rt. angles) X area of the tri-rectangular triangle,
where A, B, and C are the angles of the spherical triangle.
Hence
area __ A+-\-CiSo
surface of sphere ~ 720
The surface of the sphere is 47rft*, therefore
A + B+C 18OA
The following formula, called Lhuilier's theorem, simpli-
fies the derivation of (A + B+C 180) where the three
OBLIQUE-ANGLED TRIANGLES 109
sides of the spherical triangle are given ; in it a, b t and c
denote the sides of the triangle, and 2s = a + & + c.
tan /d+g+C-180\ _ ytmni s tan i(s-a) tan i(s-6) tan i (s-c).
EXERCISES
(i.) The angles of a spherical triangle are, ^ = 63, 5=84 21',
(7=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', B = yf 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, 7 = 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 declina-
tion is 4-(no rtn ) 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 ?
In these figures P is the pole of the equator, Z the zenith, Q the celes-
tial equator. AS is the declination of the sun, ZS=go 5Q 1 , PS= 9O + dec-
lination, PZ=go 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.
114 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 io 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
1 22 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,
\V., 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 106 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
103. 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 180 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 spJierical 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^ and
C shall rest on the rim, draw the line AC=b, and in the
same way draw BCa.
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.
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
a', 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, B, 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, F, 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
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.)
Il8 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' y 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* cos*
tan x = - , cot x = . ,
cos* sin*
r i
sec * = - , esc * = .
cos* sin*
tan * cot * = i ,
sin 2 * + cos 2 * = i,
i + tan 2 * = sec' 1 *,
i + cot 2 * = csc a *.
RIGHT TRIANGLES ( 14 AND 2j}
b
sm .? = -,
c
b a
cos A = - , cos B = - ,
c c
a b
tan A=-r, tan B = -,
b a
-b a
cot A = - , cot B = i,
a b
where c = hypotenuse, a and b sides about the right angle; A and B
the acute angles opposite a and b.
FUNCTIONS OF TWO ANGLES ( 30-34)
sin (*+/) = sin* cosj + cos* sin^,
sin (* j)r=sin* cos y cos * sin y,
cos (* -\-y) = cos * COS^K sin * sin^,
cos (* /) = cos* cos/ + sin* siny.
120 RECAPITULATION OF FORMULAS
tan^r+tan y
tan (x-V-y)
i tan x tany
tan.r tan y
tan (x y) =
i+tan.r
cot x cot^ i
cot (x y) =
cotj + cot^r '
cot x cotj-f- 1
cotj cot^r
FUNCTIONS OF TWICE AN ANGLE ( 36)
sin 2^r = 2 sm^r cosx,
cos 2 JT = cos" x sin" JT
=. i 2 sin a ^r,
= 2 cos 2 jr i,
2 tan x
tan 2.r=:
cot 2.r =
i tan a jr
cot a .r i
2 cot JT
FUNCTIONS OF HALF AN ANGLE ( 37)
tan
= \/-
cot *jc=*
I COS X
SUMS AND DIFFERENCES OF FUNCTIONS ( 38)
sin u + sin z> = 2 sin (w + -v) cos ^ ( v),
sin a sinw=:2 cosj (u-\-v) sin ^( f),
cos w -f- cos v = 2 cos i ( -f- 7 ') cos i ( f),
cos cost/ = 2 sin \(u-\-v) sin ^( ^).
sin u + sin z> tan ^ ( -|- v)
sin sin v~ tan ( z*)'
RECAPITULATION OF FORMULAS 121
OBLIQUE TRIANGLES ( 42-45)
a sin A a sin A b sin B
b sin B ' c sin C ' c sin C '
= r 2 -\- a? 2ca cos B,
where s=
tan \A =
2
AT. -AT .AT
5-^*1 , Ldll IF'-* 7 t
.$ a .y b
,. ^ l(sa) (sb) (sc)
where K \
*
s
AREA OF A TRIANGLE ( 46)
S~\ac sin B. S=ba sin C. S=cb sin ^4.
LOGARITHMIC, COSINE, SINE, AND EXPONENTIAL SERIES
(58)
X X 3C^
=JT J + -J- + etc -
122 RECAPITULATION OF FORMULAS
X X* X 7
sm;r = .r 1
~i ~"~ 71 ~~ T~ """' etc<
X X" X
-! + 3 - + 4 -l+- etc -
DE MOIVRE'S THEOREM ( 71)
(cos;tr-f- \/~ * 8ittjr)*=cos*Jr-f*V ' sin.r.
( l)( 2)
sm #.r = n cos" '.r sm.r --- : - - cos"~ 3 .r sin 3 .r-f, etc.
n (n i )
HYPERBOLIC FUNCTIONS ( 75)
e* e~ x
sinh x = ,
sn JT =
2
' = cos x-\-i sinx
e** e~ tx
2
/(** *-*) "
sin ix = - L t sinh ^r,
cos t'x = =cosh x.
2
SPHERICAL TRIANGLES
RIGHT AND QUADRANTAL TRIANGLES _( 83, 87)
Use Napier's rules.
OBLIQUE TRIANGLES ( 89-93)
cosrt = cosl> cos +sin b sinr cos A.
cos A cos B cos C-\- sin B sin C cos a.
.
sin s sin (j a)
RECAPITULATION OF FORMULAS 123
/
~V
-cos S cos (S A)
cos (S) cos(S-C)'
sin (A
cos *,(A ff) tan $
sin ( + ) cot j C
sin i ( ^)~tan | (A B)
cos ^ (# + <5) _ cot | C
cos (a b) ~tan '
sin a sin b
sin A sin^'
AREA OF SPHERICAL TRIANGLES ( lOl)
tan
\ _
'
_^ tan j _
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 tlie 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
. ~ sine
sin C=
sin a
. c
sin-
^-
becomes sinC=
. 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
r r r
the trigonometric.
Thus, corresponding to the above formula of spherical
trigonometry, is the formula
sinh-
sinC=
u a
smh-
r
of pseudo-spherical trigonometry.
PSEUDO-SPHERE
The pseudo-sphere is generated by revolving the curve whose equation is
r-\- -J^^7 l . '.
y=r log * yV-.r'
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 b c
cos = cos- cos-' (i)
r r r
cosh - = cosh - cosh - , (2)
r r r
reduce to the corresponding formula for a plane right triangle; that
is, to
. a*=y+c\ (3)
Substituting the series cos -, etc., in equation (i), we obtain
( r (\i ^ ( i W-i. \ ( r (<\. \
( I -- ,(- ) +...) = ( I -- -(-) + . . . ) I I -- -(-) +...),
\ 2 !W / \ 2 \\rj J \ 2\\rJ J
1 2 , I rt 4 , I 6* i c 2 . I l> 4 .
Or I -- ; -. H --- 3 -h ... =5 I -- : -s -- , -H --- 2"<"''' (4'
2 ! r 2 4 i r 4 2 ! r* 2 ! r 2 4 1 r 4
Substituting in equation (2) the series for cosh - , etc. , which we obtain from
x ~ x
, we have
. i ,i , f
or i + :-n + -j+... = i+-7 -; + :T + -- 1+... (5)
2 ! ;-* 4 ! r 4 2 ! r* 2 ! ; J 4 i r*
Cancelling I in equations (4) and (5), multiplying by r 2 , 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 Bright angle.
(I.) Plane, sin C=--
c . . ^ sin c
Spherical, sin C=- --
sin a
Pseudo-spherical, sin C=-r-r
Oblique Triangles.
(2.) Plane, a 7 b 1 + ? 2 be cos A
Spherical, cos a = cos cos c-\- sin 3 sin cos A
Pseudo-spherical, cosh a = cosh b cosh r + sinh b sinhc cos A.
(3.) Plane,
Spherical,
---r- , , t i 1
:an v - -^ -? = tani-tani - tan $ -- tan i
Pseudo-spherical,
(iSoO-^-H^+C 1 ) */ T^~~ TU-") , (*-*) .
tan s -=Y tanh ^ - tanh ^ ' tanh J S ' tanh |
ANSWERS TO EXERCISES
4 (page 3).
(5.)cos/ = 4, tan/ = f,
(i.) 192 51' 25^".
cot y = J, sec/ = f ,
Quadrant III.
csc/ = .
(2.) 2 5 .
(6.) sin 60 = ^ \/3,
(3.) 2870, 647.
tan 60 = \/3,
(4.) Quadrant III.
cot 60 = ^ \/3>
sec 60 = 2,
9 (page 9).
esc 60 = f -v/3.
tan 1000 is negative.
(7.) cos o = i, tan o = o.
cos 810 is o.
(8.) sin^ = f, cos 2 = |,
sin 760 is positive.
cot 2 = |, sec .2 = f ,
cot 70 is negative.
CSC2- = |.
cos 550 is negative,
tan 560 is negative.
(9.) sin 45 = cos 45 = i -v/2,
tan 45= i,
sec 300 is positive,
cot 1 560 is negative.
sec 45 = esc 45 = ^'2.
.
( T Q \ Clrji/ __ 1 -t/ ^ f*dz 1/
sin 130 is positive.
^ 5^y 3, -ub_y
cos 260 is negative.
cot/ = f v/5, sec/ = |,
tan 310 is negative.
csc/ = ! y'5.
(11.) sin 30 = ^ cos3o = |-v/
13 (page n).
tan 30 = | -v/3,
(3.) cos 30 = -v/3.
sec ^ O o _ a ^~
tan 3 o = v/3,
CSC 30 = 2.
cot 30 = -v/3,
(12.) sin J r = f, cos^ = --|.
sec 30 = | -v/3,
(13.) J\.\ Vs-
CSC 30 = 2.
v
(4.) cos.r= \/2,
17 (page 14).
tan .r = ^ \/2,
(i.) sin 70 = cos 20,
cot .r = 2 \/2,
cos 60 = sin 30,
sec .r | -v/2,
cos 89 31'= sin 29',
CSC X ~ 3.
cot 47= tan 43,
9
ANSWERS TO EXERCISES
(2.)
(3.)
(4.)
(5.)
(i.)
(2.)
(3.)
tan 63= cot 27,
sin 72 39'= cos 17 21'
.r = 3 o.
-r = 22 30'.
.r=i8.
(4.)
(5.
25 (page 21).
225 and 315,
60 and 240.
60, 120, 420, 480.
sin 30= -|,
cos 30=^ -v/3.
sin 765= cos 765 = \ -\/2,
sin 1 20= \ i/3-
cos 120 = ,
sin 210= \,
cos 210= \ \/3.
The functions of 405 are
equal to the functions of 45.
sin 6oo= \ \/$,
cos 600 = ,
tan 600= -y/3,
cot 600 = \/3,
sec 600 = 2,
esc 600 = | -y/3.
The functions of 1125 are
equal to the functions of 45. i
sin 45 = -/I,
cos 45= | "v/2.
tan 45= cot 45= i ,
sec 45=-v/2,
CSC 45= -y/2.
sin 225= cos 225= -v/2,
tan 225= cot 225= i,
sec 225= esc 225= v/ 2 -
The functions of 120 are
the same as those of 600
given in (4).
sin 225 = ^ \/2,
cos 225 = 1\/ 2 '
tan 225= cot 225= i ,
sec 225= -\/2,
CSC 225= -\/2,
sin 420 = ^ v/3,
cos 420 ^,
tan 420 = y'T
cot 420 = i \/3,
sec 420 = 2,
esc 420 == | \/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 197 = cos 17,
tan 197 = tan 17,
cot 1 97 = cot 1 7,
sec 1 97 = sec 1 7,
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 sin 5 x.
(10.) i -{-sec 2 x.
(11.) sin (x 90)= cos. r,
cos (x 90) = sin .r,
tan (x 90) = cot x,
cot (x 90) = tan x,
sec (x 90) = esc x,
esc (.r 90) = sec x.
28 (page 24).
(I.) a =62.324,
y4 = 32 52' 40".
(2.) =21.874,
yj = 39 45' 28".
5 = 50 14' 32".
(3.) a = 300.95,
= 683.96,
# = 66 15'.
(4.) = 26.608,
c = 45- 763,
# = 35 33'-
area = 495-34-
(5-) ^ = 3-9973.
^ = 4.1537,
.4 = 15 46' 33",
area = 2. 257.
(6.) = 0.01729.
(7.) = 298.5.
(8.) ^ = 39 42' 24".
(9.) ^- = 2346.7.
(10.) # = 28 57' 8".
dr.) 444.16 ft.
(12.) 186.32 ft.
(I3-) 34 33' 44"- %
(14.) 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.) = 8o 32",
,4 = C = 49 59' 44".
(23.) =53 i6' 3 6",
= 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 = 20.13 ' 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.
^=45 27' 14",
C = 95 24' 46".
(3.) A3 = 7.674$,
^L' = 2.6435,
B = 46 43 '50',
^' = 133 16' 10",
io$ 53' 10",
' = \g 20' 50".
(4.) A = 37 53'.
# = 43 52' 25",
132
ANSWERS TO EXERCISES
C = 9 8? 14' 35"
(5-) 902.94-
(6.) 1253.2 ft.
(7.) 357-224 ft.
(8.) ^ = 44 2' 9",
= $i 28' u",
C = 84 29' 40",
area = 126100 sq. ft.
(9.) 407.89 ft.
(io.) B= \2\ 7' 16",
C = 92 20' 38",
D = 7\ u' 6".
(u.) BC- 6.6885,
DC 1.9915.
34 (page 34).
(2.) sin (454- -i^) =
i-y/ 2 (cos.r-f sin x),
cos (45+ f) =
4 V 2 < cos x sin ;r),
sin (yfx) =
^ (cos* -y/3 sin JT),
cos (30^ =
i ( V / 3 cos -*" + sin ^),
sin (60+*) =
^ (-V/3 cos A- 4- sin x),
cos (60+-*-) =
i(cos.r -y/3 sin.r)-
(3.) sin (.r+_v) = f.
sin (.r .y)=.
U-) sin 75 =
cos 75 =
(j.) sin 15=
cos 15=
4
V6 y/2
4
-y/6 \/2
39 (Page 37).
(5.) sin(45-.r) =
5 \/2 (cos .1 sin .V),
cos (45 .r) =
| \/ 2 (cos a- -f- sin x),
sin(45+.r) =
- ^ -y/2 (cos .i- + sin .v),
cos(45+x) =
\-\/2 (cos x sin.r).
(6.) tan 7 5 =2 4- V3.
tan 1 5 = 2 v/3-
(14.) ini7=
cos \y
(15.) sin 2.r = ff,
cos 2x = $ 5 .
(16.) sin 22^ = -K/2 -y/2,
CSC 22^ c = -y/4 + 2 \/ 2 -
07-)
(i 8.) sin I5 = K/ 2 ""V3.
ANSWERS TO EXERCISES
133
esc 15 = 2 \J 2
(20.) sin 5_r =
5 sin x 20 sin 3 x
+ 16 sin 5 .f.
(21.) cos 5-r =
5 cos x 20 cos 3 x
-f- 1 6 cos 5 x.
(23.) The values of jr<36o are,
o, 30, 1 50, 1 80, 210, 330.
(36.) tan.r tanj.
41 (page 40).
(i.) sin ' | -v/ 2 =45, 135.
45+ 360, etc.,
cos ' \ 60, 300, etc.,
tan-' ( i)= 1 3 5, 3 1 5, etc.,
cos 1 i =0, 360, etc.,
sin 1 ( 1) = 210, 330, etc.
(2.)
(3-) c
(4.) si
(5.) sin(cos-'J) = f
(6.) cot (tan > j\) = 1 7
(8.) 45, 225.
(9.) JT=:45 ,J =180.
(10.) sin"" 1 ^ = 225.
48 (page 46).
(i.) C=m33'.
^ = 2133.5,
c = 2477.8.
(2.) C=554i'-
^ = 534.05,
^ = 653.52.
(3.) C'=45 34',
a 1548.1,
^=1293.7.
(4.) ^ = 105=59',
CZ =: 54.OI8,
<: =47.738.
(5.) v9 = 6858' (
^ = 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.) ff=i6 57' 21",
C- 1 5 5' 39".
^ = 0.32122.
(3.) r= 2.5719,
^=13 15' i",
C= 142 13' 59"-
(4.) c = 93-59. ^' = 54-069,
^ = 26 52' 7", #'=133 7' 53".
C = i3i46 / 53",C'=253i'7".
(5.) No solution.
(6.) <= 1.0916, '=0.36276,
,4 =z 3937 ' 1 6", ,4 ' =: 1 4o2 2 '44",
^ .ii7 5o'44",^'=i7 5' 1 6".
50 (page 48).
(i.) a = 0.097 1,
# = 90 35' 36",
C=489'34",
5 = 0.0053261.
134
ANSIVERS TO EXERCISES
(2.) C \\.1\\,
7>=48 D 44' 32",
A = 76 20' 5",
C = 95 15' 56",
# = 44 52 55".
5 = 0.60709.
5 = 80.962.
(3.) = 85.892,
52 (page 50).
A =67 21' 42",
(i.) 1116.6 ft.
C = 6248' 1 8",
(2.) 3081.8 yards.
5=3962.8.
(3.) 638.34 ft.,
(4.) a =0.6767,
14653 sq. ft.
/>' = 1 5 9' 2 1 ",
(4.) 4.1 and 8.1.
C= 131 19' 39".
(5.) 13.27 miles.
5 = 0.08141.
(6.) 6667 ft. One solution.
(5.) r = 72.87,
(7-) 121.97.
^ = 40 50' 32".
(8.) 44 2' 56".
=ll 2' 28".
(9.) 32.151 sq. miles.
5 = 422.65.
(11.) 54 29' 12".
(12.) a 12296 ft.,
51 (page 49).
^=13055 ft.
(13.) 294.77 ft.
(i.) ,4 = 55 20' 42",
(14.) 222.1 ft.
^=106 35' 36",
C=i8 3' 42",
5 = 267.92.
(2.) A = 34 24' 26",
(i 6.) 4202.1 ft.
(17.) 72.613 miles.
(18.) 50-977 ft-
(19.) 0.85872 miles.
# = 73 14' 56",
C = 72 20' 36",
5=3.6143.
(20.) 2.98 miles.
(21.) 1 393.9 ft.
(22.) 8.2 miles.
(3.) A = 52 20 24",
=107 19' 14",
(23.) 1 87.39 ft.
(24.) 0.60 1 1.
C=20 20' 24",
(25.) 4.8112 miles.
5=1437.5.
(4.) A = 97 48',
(26.) 60 51' 8".
B= 18 21 48",
(27-) 37.365 ft-
C=63 50' 12",
(28.) 3.2103 miles.
5=193.13.
(29.) 10.532 miles.
(5.) A = 54 20' 1 6".
(30.) 851.22 yards.
^ = 70 27' 46".
(3 1 -) 9-5722 miles.
C= 54 72',
(32.) 6.1271 miles.
5 = 6090.
(33.) 280.47 ft.
(6.) A = 35 59' 30".
(34-) 1 23.33 ft.
ANSWERS TO EXERCISES
135
(35-) 4-8ii2 miles.
(4.) .(-=!, .1^ = 0.3090 +/ 0.95 n.
(36.) 2666.1 ft.
.r 2 = 0.8090 + i o. 5878,
.r 3 = 0.8090 / 0.5878.
53 (page 56).
.1-^ = 0.3090 i 0.951 1.
(i.) 30 = 0.5236,
45 = 0.7854.
60 = 1.0472,
I 20 2.0944,
135= 2.3562,
J20 12.5664,
77 (page 78).
(23.) ^ = 30.
(24.) y = 30.
(25.) x o or 45.
(26.) -r = 6o.
990= 17.2788.
(28.) / = 45-
(2.) -JT = 22 30',
(29.) jr = 45.
-=18,
10
i = 28 38' 53",
l~ = 1 00 1 6' 4".
(3-) I-35.0.54.
(30.) .r = 30.
(31.) ,r = 6o.
(32.) x = yP.
(33.) No angle < 90.
(34.)* =30.
(35.) sin 92 = cos 2.
74 (page 73).
(36.) cos 127=: sin 37.
(37.) tan 320 = tan 40.
(i.) sin 4jr = 4 cos 3 ;r sin^r
(38.) cot 350 = cot 10.
4 cos x sin 3 x.
(39.) sin 265 = cos 5.
cos 4_r = cos 4 .r
6 cos 2 x sin 2 x -}- sin 4 x.
(2.) sin 6;r = 6 cos 5 x sin .r
(40.) tan 171= tan 9.
(41.) cos.r = 1^/33'
20 cos 3 x sin 3 .r
ssV33.
+ 6 cos x si n 5 JT,
cot^ = i-v/33,
cos 6x = cos 6 or
sec x g'g -v/3~3~,
y' 1 5 cos* .* sin 2 x
CSC X = |.
+ 1 5 cos 2 x sin 4 .r sin 6 x.
(42.) sin^ = -i-/55.
(3.) * =i, ^ | = j + /^3,
tan^r^i-v/55-
/-
cot x = fg \/55,
-r 2 = | + /, .r a = i,
sec x = |,
,-
csc.r=r /?; \/SS-
x \ ? Z ^~'
(43.)^n.r = -^V^
jr 5 = \ i .
cot.r^f, sec.i- = %\/i3,
136
ANSWERS TO EXERCISES
cscx = ^13.
(21.) 71 33' 54"-
(44.) sin x = J f \/74.
(22.) 858,160 miles.
cos x =. , 7 r \/74.
(23.) 238,850 miles.
< T / T^'
tan .r = f , sec .r = \ -\/74'
i /
(24.) 2163.4 miles.
(25.) 90,824,000 miles.
esc -t = i y 74.
(26.) 432.08 ft.
(45.) Quadrant II or IV.
(27.) 60.191 ft.
(46.) Quadrant I or II.
(28.) 0.32149 mile.
(47.) Quadrant III or IV.
(29.) 193.77 ft.
(48.) Quadrant I or II.
(49.) .r = o, 120, 1 80, 240.
79 (page 83).
(50.) .r = 30, 135, 150, 315.
(i.) 3.416 ft.
(51.) .r = o, 90, 120, 180, 240,
(2.) 3.7865 ft.
270.
(3.) 20.45 ft-
(57-) o.
(4.) 36.024^.
(58.) a.
(5.) 8.6058 sq.ft.
(59.) 2(a-b\
(6.) 181.23 in.
(60.) i(fl 2 b).
(7-) 2.9943 ft-
78 (page 80).
(8.) 5.1311 in.
(9.) 25.92 ft.
(i.) 306.32 ft.
(10.) 92 i' 24".
(2.) 831.06 ft.
(11.) 1.2491.
(3.) 53 28' 14".
(12.) 33 1 2' 4".
(4-) 49-39 ft.
(13.) 11248 ft.
(5.) 0.43498 mile.
(14.) 0.60965 miles.
(6.) 209.53 ft.
(15.) 1.3764.
(7.) 7.3188 ft.
(16.) 1.9755-
(80 37 36' 30".
(17.) 19.882.
(9.) 109.28 ft.
(i 8.) 0.9397.
(10.) 502.46 ft.
(19.) 6.4984.
(u.) 6799.8 ft.
(20.) 3.4641-
(12.) 219.05 ft.
(21.) 6.1981.
(13.) 49i.76ft.
(22.) 6.9978.
(14.) 50 32' 44".
(23.) 15.25.
(15.) 49 44' 38".
80 (page 84).
(i 6.) 34.063 ft.
(78.) x 90, 1 20. 240, 270.
(17.) 32.326 ft., 29 6' 35".
(79.) .r = o, 20, 45, 90, ioo c
(18.) 5.6569 miles an hour.
135, 140, 1 80, 220 C
(19.) 56.295 ft.
225, 260, 270, 3i5 c
(20.) 103.09 ft.
340.
ANSWERS TO EXERCISES
137
(80.) ,r = o, 30, 90, 150, 1 80,
270.
(8 1.) x = o, 45, 120, 240, 225,
270.
(82.) x = o, 90, 1 80, 270.
(83.) x = cP, 90, 210, 330.
(84.) x = 240, 300.
(85.) .r = 2io, 330.
(86.) x = o, 90.
(87.) ,r = o, 1 80.
(88.) .r=zo, 1 80.
(89.) x = cP, 90, 120, 1 80, 240
270.
(90.) x = 4S, 135- 22 5. 3'5
(91.) .r = 30 , 150, 210, 330.
81 (page 88).
(i.) 2145.1 ft.
(2.) 12.458 miles.
(3.) 1.1033 miles.
(4.) 1 508.4 ft.
(5-) i7i93Y ards -
(6.) 1.2564 miles.
(7.) 1346.3^.
(8.) 387.1 yards.
(9.) 5.1083 miles.
(10.) 3791-8 ft.
(n.) 4-4152 ft-
(12.) 28 57' 20".
(13.) 115.27.
(14.) 44.358 ft.
(15.) 92.258 ft.
(16.) 101 32' 16".
(17.) 0.83732 mile.
(18.) 539.1 ft.
(19.) 1.239.
(20.) 152.31 and 238.3.
(21.) 68.673 ft.
(22.) 32.071 ft.
(23.) 137.78 ft.
(24.)
(25-)
(26.)
(27-)
(28.)
(29-)
(30.)
(31-)
(32.)
(33.)
(34.)
(35.)
(36.)
(37-)
(38.)
(39-)
(40.)
(4I-)
(42.)
(43-)
(44-)
(45-)
(46.)
(47-)
(48.)
(49-)
(50-)
55-74 ft.
247.52 ft.
556.34 ft.
465.72 ft.
109.22 ft.
2639.4 ft.
396.54 ft.
287.75 ft-
2280.6 ft.
64.62 ft.
127.98 ft.
45-183 ft-
4365.2 ft.
140.17 ft.
610.45 ft.
i56.66ft.
41 48' 39" and 125 25 57'
51,288,000.
366680.
i i 586.
947460.
9929-3-
751-62 sq. ft.
3I45.9-
855.1.
876.34.
88 (page 98).
(i.) ^=54 59' 47".
= 45 41 '28",
C=6 5 4 5' 58",
(2.) C= 7 i 36' 47".
b = 95 22',
<r= 7I 32 1 14".
(3.) C=64 1 4' 3-".
C'= 115 45' 30".
a= 48 22' 55",
rt' = i3 37' 5".
c=42 19' 17".
138
ANSWERS TO EXERCISES
c = i 37 40 43".
(4.) C = 65 49' 54",
= 63 10' 6",
^ = 38 59' 12".
(5.) <z = 75 13' i",
75-= 58 25' 46",
= 67 27' i".
(6.) a = 76 30' 37",
= 65" 28' 58,"
c = 55 47' 44".
(7.) B = 54 44' 23",
^ = 64 36' 39",
^ = 47 57' 45"-
(8.) />' = 96 13' 23".
" = 73 17' 29",
<r=:70 8' 38".
(9.) = 66 58',
flr=n 35' 49".
<r = 4 35' 26".
(10.) <r = 6i4'55",
<5 = 40 30' 22",
< = 50 30' 32".
99 (page 107).
(i.) * = i5535' 22",
B = 10 19' 34",
C=i7i 48' 22".
(2.) rf = i3i 3 6'36",
6=116 36' 38",
^ = 29 ii' 42".
(3.) a = 107 7' 45".
= 48 57' 29",
C = 62 31 '40".
(4.) = 62 54' 43",
#= 114 30' 26",
= 56 39' 10".
(5.) ^ = 130 35' 56".
B = 30 25' 34",
C = 31 26' 32".
(6.) a =98 21' 22",
b = 109 50' 8",
c = 115 13' 4".
(7.) /^ = 32 26-9",
a = 84 14' 32",
f = 51 6' 12".
(8.) a = 80 5' 8",
b = jo 10 36",
<r = i45 5' 2".
(9.) A = 70 39' 4",
# = 48 36' 2",
C=ii9 15' 2".
(IO.) flr=4O O' 12",
= 42 15' II",
Cm 21 36' 19".
loo (page 109).
(i.) 80.895 sq- in -
(2.) 26.869 sq. in.
(3,) 158.41 sq. in.
(4-) 3999 sq. miles.
ioi (page 112).
(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
FIVE- PLACE AND FOUR-PLACE
PHILLIPS-LOOMIS MATHEMATICAL SERIES
LOGARITHMIC
AND
FIVE-PLACE AND FOUR-PLACE
BY
ANDREW W. PHILLIPS, PH.D.
AND
WENDELL M. STRONG, PH.D.
YALE UNIVERSITY
NEW YORK AND LONDON
HARPER & BROTHERS PUBLISHERS
1899
THE PHILLIPS-LOOMIS MATHEMATICAL SERIES.
ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D.,
and IRVING FISHER, Ph.D. 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.]
GEOMETRY OF SPACE. By ANDREW W. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D. Crown 8vo, Cloth, $1 25. [By mail, $1 35.]
OBSERVATIONAL GEOMETRY. By WILLIAM T. CAMPBELL, A.M.
Crown 8vo, Cloth.
ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By
ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale
University. Crown 8vo, Cloth, 90 cents. [By mail, 98 cents.]
LOGARITHMIC AND TRIGONOMETRIC TABLES. Five-Place
and Four- Place. By ANDREW W. PHILLIPS, Ph.D., and WENDELL
M. STRONG, Ph.D. Crown 8vo, Cloth, $1 00. [By mail, $1 08.]
TRIGONOMETRY AND TABLES. By ANDREW W. PHILLIPS,
Ph.D., and WENDELL M. STRONG, Ph.D. In One Volume. Crown
8vo, Half Leather, $1 40. [By mail, $1 54.]
LOGARITHMS OF NUMBERS. Five-Figure Table to Accompany
the " Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and
IRVING FISHER, Ph.D. Crown 8vo, Cloth, 30 cents. [By mail, 35 cents.]
NEW YORK AND LONDON :
HARPER & BROTHERS, PUBLISHERS.
Copyright, 1898, by HARPER & BROTHERS
All rights rtstt*ved.
CONTENTS
TABLE PACK
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 100 = 2, because 100 = lo"
log i o, " . i = 10
log .1 = I, " ,l-=\Q- 1
log 3 =47712, " 3 =io- 4771!1
In general, logm = x if ; = 10*.
2. To multiply two numbers, add their logarithms. The
result is the' logarithm of the product.
Proof. Iim = io* so that log m =x,
and n = ioy " " log n y,
then mn = io x+ - y " " log mn = x-\-y.
Hence log mn = log;;* + 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.- = = ^=,0^;
n 10*
Hence log =x y = \ogm log.
4. To raise a number to a power \ multiply the logaritJim
of the number by the index of the power. The result is the
logarithm of the power.
vi INTRODUCTION TO THE TABLES.
Proof. m" (io*Y = io- ax ;
Hence logm" = a.v = a logw.
5. To extract a root of a number, divide the logarithm of
tlie number by the index of the root. The result is the loga-
ritJnn of the root.
X
Proof. * m .. Ao* = 10*.
* I x log;;z
Hence log ^ / m = - = - -
b~ b
6. Restatement of laws :
log mn = log in, + log n ;
m
log = logm-logw ;
_logm
~~
7. Most numbers are not integral powers of 10; hence
most logarithms are of decimal form.
Thus, log 2. 2 .34242, log 4 = .60206.
8. 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
910, 8 10, etc.
Thus, log. 2 1 544 = i + .33333- written 9.33333 10 ;
log .021 544 = 2 + .33333, " 8.3333310;
log .002 1 544 = 3 + .33333, " 7-33333-iQ.
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; tli e characteristic is +2; the mantissa
COMMON LOGARITHMS. vii
is -f--33333 : log .021544=8.33333 10 ; the characteristic is 8 10
= 2 ; the mantissa is -f- .33333.
0. It is evident that the larger a number the larger its logarithm.
Hence the logarithm of any number
between i and 10 is o + a mantissa,
" 10 " ico " . i + " "
.1 i "-i + "
.01 " .1 " 2 + " " 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 right 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, the 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 =9-33333io
log .021544 = 8.3333310
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 105 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 = .022844- 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 I2.3(=4i 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;
I57-035. " " 157.04.
15* 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 io log. 1345.
Find the number of which 9.12895 io is the logarithm.
The mantissa .12895 is not m tne tables, but is
between .12905 = mantissa of 1346
and .12872= " " 1345.
.00033 = tne difference.
. 1 2895 = 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.
3 = 7 nearly.
Hence 9.12895 io = 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.
log 5789.2 = 3.76262
log .01 83 1 5 =8.2628 1 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 = 0.4971 5
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. 1 41 6 = 0.497 1 5
1.41306 =log25.886
Find the value of (2.1345)'.
log 2. 1 345 =0.32930
5
i. 64650 = log 44.310
Find the value of \/ .01 021.
log .0102 1 = 8.00903 10
= 28.00903 - 30
28.00903 30
-^ =9.33634- io = log.2i694
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS, xi
17. The logarithm of is called the cologarithm of m,
and is obtained by subtracting log m from zero.
Thus, if log m = 9.76423 10, colog #2 =0.23577.
It is frequently shorter to add cologw than to subtract
logw when we wish to divide by a number m.
The following example illustrates this:
r j *u i 57-9^ x 42.24
Find the value of ^^ -
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 ID 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' = 0.32020.
xii INTRODUCTION TO THE TABLES.
10. 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, a,nd 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 15' log sin 28 14' = 23 (found in column "d.")
log sin 28 1 4' = 9.67492 10
23 X .6 = 13.8 = 14 nearly
log sin 28 14' 36" = 9.67 506 10
Find log cos 39 17' 22" (=log cos 39 i7.3')-
log cos 39 1 7' = 9.8887 5 10
iox.3=_ 4
log cos 39 17' 22" = 9.8887 1 10
Find log tan 51 27' 44" (=log tan 51 27.7^')-
log tan 51 27 ' = .09862
26x.7i=_ 19
log tan 51 27' 44" = .0988 1
i
Find log cot 67 18' 46".
log cot 67 18' =9.62150 10
36 X .7$ =_ 28
Hence log cot 67 1 8' 46" = 9.62 122 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 ^ = 9.67433 10; find x,.
log sin 28 n' = 9.6742 1 10
log sin x log sin 28 ii' = 12
and log sin 28 12' log sin 28 ii' = 24
Hence .r = 28 1 1' 30" (f of i r being 30' ).
Find the angle whose log 005 = 9.88231 10.
log cos 40 1 8' = 9.88234 10.
6o"X^=i6".
Hence log cos 40 18' 16" = 9.88231 10.
Find the angle whose log tan =0.17844.
log tan 56 27 =0.17839.
60" x &= 11".
Hence log tan 56 27' n" = 0.17844.
Find the angle whose log 001 = 9.87432 10.
log cot 53 10' = 9.87448 10.
60" X ^ = 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
xiv 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.
25. 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 -PL ACE LOGARITHMS
OF NUMBERS
100-13O
N
1
2
3
4
5
O
7
8
9
100
oo ooo
o43
087
i3o
i 7 3
217
260
3o3
346
389
IOI
432
4?5
5i8
56i
6o4
64?
689
732
* 775
8i 7
1 02
860
903
945
988
*o3o
"072
*i 15
*i 57
*242
io3
OI
284
^26
368
4io
452
494
536
5 7 8
620
662
io4
7o3
745
787
828
870
912
9 53
995
*o36
*o 7 a
io5
02
119
1 60
202
243
284
3 2 5
366
407
449
490
1 06
53i
572
612
653
6 9 4
735
776
816
85?
898
107
9 38
979
9
*o6o
*IOO
*i4i
*i8i
*222
*262
*302
1 08
o3
342
383
423
463
5o3
543
583
623
663
7 o3
109
743
782
822
862
902
94i
981
*02I
*o6o
*IOO
110
o4 i3g
179
218
258
297
336
376
415
454
4y3
1 1 1
532
5 7 i
610
650
689
727
766
8o5
844
883
I 12
922
961
999
*o38
*77
*n5
*i54
*I92
*23l
*26g
n3
o5
3o8
346
385
4'23
46 1
500
538
576
6i4
652
"4
690
729
767
805
843
88 1
918
956
994
*032
n5
06 070
1 08
i45
i83
221
258
296
333
3 7 i
4o8
116
446
483
521
558
5 9 5
633
670
707
744
781
117
819
856
8 9 3
93o
967
*oo4
*o4i
+078
*"5
*i5i
118
07
188
225
262
298
335
3 7 2
4o8
445
482
5i8
119
555
5gi
628
664
700
737
77 3
809
846
882
120
918
954
990
*02 7
*o63
*99
*i3s
*i 7 i
*20 7
* 2 43
121
08
279
3i4
35o
386
422
458
493
529
565
600
122
636
672
707
743
778
8i4
84g
884
92O
955
123
991
*O26
*o6i
*096
*l32
*i67
*2O2
*23 7
*2 7 2
*3o 7
124
09
342
377
4l2
44?
482
5i 7
552
58 7
621
656
125
691
726
760
79 5
83o
864
899
934
968
*oo3
126
10 037
072
1 06
140
'75
209
243
278
3l2
346
127
38o
415
449
483
5i 7
55i
585
619
653
68 7
128
721
755
789
823
85 7
890
924
958
992
*025
129
1 1
126
1 60
n;3
227
261
294
32 7
sor
130
3 9 4
428
46 1
4g4
5 2 8
56i
5 9 4
628
661
6 9 4
N
O
1
2
3
4
5
6
7
8
J>
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
11.7 3
1 1. 4
u. i
10.8
4
17.6
17.2 16.8
4
16.4
16.0
i5.6 4
15.2
i4.8
i4.4
5
22.0
21.5 21.0
5
20. 5
2O.O
19.5 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
2 7 .3 7
26.6
25.9
25.2
8
35.2
34.4 33.6
8
32.8
32.0
3i.2 8
3o.4
29.6
28.8
!g.6
38.7 37.8
g
36.9
36.o
35.i 9
34.2
32.4
130-160
JN
O
1
2
3
4
5
6
7
8 | 9
KO
1 1
3 9 4
428
46i
494
528
56i
5 9 4
628
661
6 9 4
727
760
79 3
826
860
8g3
926
9 5 9
992
*024
.
j <*
i2
o5 7
090
123
i56
189
222
254
287
320
35 2
i33
385
4i8
45o
483
5i6
548
58 1
6i3
646
6 7 8
1 34
7 IO
?43
77 5
808
84o
872
95
9 3 7
969
*OOI
1 35
i3 o33
066
098
i3o
162
194
226
258
290
322
i36
354
386
4i8
45o
48 1
5i3
545
5 77
609
64o
i3 7
6 7 2
704
7 35
767
799
83o
862
8 9 3
925
956
1 38
988
*oig
*o5i
*082
*ri4
*i4/5
*I 7 6
*208
*23 9
*2 7 O
i!
9
i4
3oi
333
364
3 9 5
426
45 7
48 9
52O
55i
582
140
6i3
644
675
7 o6
7 3 7
7 G8
799
829
860
891
i/
il
922
953
9 83
*oi4
*45
*o 7 6
*io6
*i3 7
*i68
*i 9 8
n
J2
i5
229
25c;
290
320
35i
38i
4l2
442
4 7 3
5o3
i43
534
564
5 9 4
625
655
685
7 i5
7 46
776
806
i44
836
866
8 97
927
9 5 7
987
*oi7
*o4 7
*0 77
*I0 7
i45
16
i3 7
i6 7
'97
227
256
286
3i6
346
3 7 6
4o6
i46
435
465
495
524
554
584
6i3
643
6 7 3
702
i4 7
7 32
761
791
820
850
879
909
938
967
997
i48
17 026
o56
085
n4
US
i 7 3
202
23l
260
289
149
319
348
3 7 7
4o6
435
464
4g3
522
55i
58o
150
609
638
667
696
725
?54
782
811
84o
869
it
)I
898
926
9 55
984
*oi3
*o4i
*O 7 O
*o 99
*I2 7
*i56
152
18
i84
2l3
2
4i
270
298
327
355
384
4l2
44i
i53
469
498
526
554
583
611
63g
66 7
696
724
i54
752
780
808
83 7
865
8 9 3
921
949
977
*oo5
i55
19 o33
061
*9
n 7
1 45
i 7 3
201
229
285
1 56
3l2
34o
368
3 9 6
424
45i
479
5o 7
535
562
i5 7
5go
618
645
6 7 3
700
728
7 56
7 83
811
838
1 58
866
893
921
948
976
*oo3
*o3o
*o58
*o85
*II2
if
9
20
i4o
167
i
94
222
249
276
3o3
33o
358
385
160
4l2
43 9
466
493
52O
548
575
602
629
656
N
O
1
2 ! 3
4
5
7
8
9
PP
35
34 33
32 31 30 29 28 27
i
3.5
3.4 3.3
i
3s
3.i
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 i3.2
4
12.8
12.4
T2.O
4 1 1. 6
1 1. 2
10.8
5
i 7 .5
17.0 i6.5
5
16.0
i5.5
l5.O
5 i4.5
i4.o
i3.5
6
2I.O
20.4 19.8
6
19.2
18.6
18.0
6 17.4
16.8
16.2
7
24.5
23.8 23.i
7
22.4
21.7
2I.O
7 20.3
19.6
18.9
8
28.0
27.2 26.4
8
2 5.6
24.8
24.O
8 23 X 2
22.4
21.6
9
3i.5
3o.6 29.7
\
28.8
27.9
27.0
9 26.1
25.2
24.3
160-190
N
O
1
2
3
4
5
6
7
8
9
160
20 4l2
43 9
466
493
520
548
575
602
629
t>56
161
63
710
7 3 7
7 63
79
817
844
871
898
9 2 5
162
952
978
*oo5
*032
*o5 9
*o85
*II2
*i3 9
*i65
i63
21 219
245
272
2 99
325
352
3 7 8
405
43i
458
1 64
484
5ll
53 7
564
5 9 o
6,7
643
669
696
722
i65
748
775
801
827
854
880
906
9 32
9 58
9^5
166
22 OI I
o3 7
o63
o8 9
u5
i4i
167
194
220
246
167
272
298
324
350
376
4oi
427
453
479
5o5
168
53i
55 7
583
608
634
660
686
712
7 3 7
763
169
789
8i4
84o
866
891
917
9 43
968
994
*oi 9
170
23 045
070
096
121
i4 7
172
198
223
249
274
171
3oo
325
35o
376
4oi
426
452
477
5o2
528
172
553
578
6o3
629
654
679
704
729
754
779
I 7 3
805
83o
855
880
95
9 3
955
980
*oo5
*o3o
i r
r4
24055
080
105
i3o
'55
180
204
229
254
279
I 7 5
3o4
32 9
353
378
4o3
428
452
4?7
502
176
55i
5 7 6
601
625
650
6 7 4
699
724
748
77 3
177
797
822
846
87,
8 9 5
920
944
969
99 3
*oi8
178
25 o42
066
091
n5
i3 9
1 64
188
21 2
237
261
179
285
3io
334
358
382
4o6
43i
455
479
5o3
180
52 7
55i
5 7 5
660
624
648
672
696
720
7 44
181
768
79 2
816
84o
864
888
912
9 35
9 5 9
9 83
182
26 007
o3i
055
<>79
1 02
126
i5o
i 7 4
198
221
i83
245
269
293
3i6
34o
364
387
4u
435
458
1 84
482
5o5
5-29
553
5 7 6
600
623
647
670
6 9 4
i85
717.
74 1
7 64
788
811
834
858
881
95
928
1 86
9 5,
975
998
*O2I
*o45
*o68
"091
*u4
*i38
*j6i
187
27 i84
2O 7
23l
254
277
3oo
323
346
370
3 9 3
1 88
4i6
43 9
462
485
5o8
53i
554
5 77
600
623
189
646
66 9
6
?a
7 i5
738
761
?84
8o 7
83o
852
190
875
898
921
944
967
989
*OI2
*o35
*o58
*o8i
N
O
1
ti
,'3
4
5
6
7
#
O
PP 27 26 25
24 23 22
21 20 11)
i
2.7 2.6 2.5
i
2.4
2.3
2.2
i
2.1
2.0
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 io.4 10.0
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.O
5
io.5
IO.O
9 .5
6
16.2 i5.6 i5.o
G
i4.4
i3.!s
13.2
6
12.6
12.0
7
18.9 18.2 17.5
7
16.8
16. 1
i5.4
7
i4. 7
i4.o
i3.3
8
21.6 2O.8 20.
I 9 .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
o
18.9
18.0 17.1
19O 23O
N
O
1
2
3
4
5
6
7
8
9
190
27 876
898
921
944
967
989
*OI2
*o35
*o58
*o8i
191
28 io3
126
149
171
ic;4
21 7
240
262
285
307
192
33o
353
3 7 5
3 9 8
421
443
466
488
5n
533
I 9 3
556
5 7 8
601
623
646
668
691
7 i3
7 35
758
ig4
780
8o3
825
847
870
892
914
9 3 7
9 5 9
981
igS
29 oo3
026
o48
070
092
11 K.
i3 7
i5g
181
203
196
226
248
270
292
3i4
336
358
38o
4o3
425
197
447
469
491
5i3
535
55 7
579
601
623
645
198
66 7
688
710
732
754
776
798
820
842
863
199
885
907
929
9 5i
973
994
*oi6
*o38
*o6o
*o8i
200
3o io3
125
i46
168
190
21 I
233
2 55
276
298
2OI
320
34 1
363
384
4o6
428
449
471
492
5i4
202
535
55 7
5 7 8
600
621
643
664
685
707
728
203
75
771
792
8i4
835
856
878
899
920
942
204
963
984
*oo6
*O27
*o48
*o6g
*Q9I
*I 12
*i33
*i54
205
3i 175
197
218
23g
260
281
302
323
345
366
2O6
38 7
4o8
429
450
4 7 i
492
5i3
534
555
576
207
697
618
63 9
660
681
702
723
744
765
7 85
208
806
827
848
869
890
911
g3i
952
9 7 3
994
209
32 015
o35
o56
077
098
118
i3g
1 60
181
201
210
222
243
263
284
305
325
346
366
387
4o8
21 I
4'28
44g
46 9
490
5io
53i
552
5 7 2
5 9 3
6i3
212
634
654
675
6 95
7 i5
?36
7 56
777
797
818
2l3
838
858
879
899
919
940
960
980
*OOI
*02I
2l4
33 o4r
062
082
102
122
i43
1 63
i83
203
224
2l5
244
264
284
3o4
325
34s
365
385
4o5
425
216
445
465
486
5o6
526
546
566
586
606
626
217
646
666
686
706
726
?46
766
786
806
826
218
846
866
885
goS
925
945
9 65
985
*oo5
*025
219
34o44
o64
o84
io4
124
i43
i63
i83
203
223
220
242
262
282
3oi
321
34i
36i
38o
4oo
420
221
43 9
45 9
4?9
4 9 8
5i8
53 7
55 7
5 77
5 9 6
616
222
635
655
6 7 4
6 9 4
7 i3
7 33
7 53
772
792
811
223
83o
850
869
889
908
928
947
967
986
*oo5
224
35 025
o44
o64
o83
1 02
122
i4i
160
180
199
225
218
2 38
257
276
2g5
3i5
334
353
3 7 2
3 9 2
226
4n
43o
44g
468
488
507
526
545
564
583
227
6o3
622
64 1
660
679
698
717
7 36
7 55
774
228
79 3
8i3
832
85i
870
889
908
927
946
9 6 5
229
984
*oo3
*O2I
*o4o
*o59
*078
*097
*n6
*i35
*i54
230
36 i 7 3
192
21 I
229
2^8
267
286
305
324
342
N
1
2
3
4
5
6
7
8
9
23O 26O
N
O
1
2
3
4
5
6
7
8
9
230
36 173
I 9 2
21 I
22 9
248
267
286
305
324
342
831
36 1
38o
3 99
4i8
436
455
4 7 4
4c
>3
Sii
53o
232
549
568
586
605
624
64a
66 1
680
6 9 8
717
233
7 36
7 54
773
79 I
810
829
84?
866
884
9 o3
234
922
9 4o
9 5 9
977
996
*oi4
*o33
*o5i
^070
*o88
235
3 7 io 7
125
i44
162
181
199
218
236
254
273
a36
291
3io
3a8
346
365
383
4oi
420
438
45 7
23 7
475
4g3
5n
53o
548
566
585
6o3
621
63 9
238
658
676
6 9 4
712
7 3i
749
767
7 85
8o3
822
23 9
84
Q
858
876
8 9 4
912
93i
949
9 6 7
985
*oo3
240
38 021
039
057
o 7 5
o 9 3
I 12
i3o
1 48
1 66
1 84
s4i
202
220
238
256
274
292
3io
328
346
364
242
382
399
417
435
453
471
48 9
5o 7
525
543
243
56i
578
596
6i4
632
650
668
686
7 o3
721
244
7 3 9
7 5 7
775
792
810
828
846
863
881
899
245
917
934
g52
97<>
987
*oo5
*023
*o4i
*o58
*o 7 6
246
39 094
i ii
129
i46
1 64
182
199
. 2I 7
235
252
247
270
287
305
322
34o
358
3 7 5
3 9 3
4io
428
248
445
463
48o
4 9 8
5i5
533
55o
-568
585
602
a4 9
620
63 7
655
672
690
707
7
24
7^
(2
7 5 9
777
250
794
811
82 9
846
863
88 1
898
9 i5
9 33
9 5o
5i
967
985
*002
*oi 9
*o37
*o54
*o7i
*o88
*io6
*I23
252
4o i4
i5 7
175
I 9 2
209
226
243
261'
2 7 8
295
253
3l2
32 9
346
364
38i
398
4
15
432
44 9
466
a54
483
5oo
5i8
535
552
56 9
586
6o3
620
63 7
255
654
671
688
705
722
7 3 9
756
773
79
807
256
824
84 1
858
875
892
909
926
943
9 6o
976
257
99 3
*OIO
*027
*o44
*o6i
*o 7 8
^ ( )
95
*in
"128
*i45
258
4i 162
79
196
212
229
246
263
280
296
3i3
25 9
33o
34 7
363
38o
3 97
4i4
43o
447
464
48 1
260
497
5i4
53i
547
564
58i
5 97
6i4
63i
64?
N
1
2
3
4
5
7
8
9
PP
19 18 17 16
15
14
i
1.9 1.8
1.7 i 1.6
i.5
~i4
2
3.8 3.6
3.4 a 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 .o
8.5 5 8.0
7-5
7.0
6
ii.4 10.8
10.2 6 9.6
9.0
8.4
7
i3.3 12.6
1 1.9 7 i 1 1.2
io.5
9-8
8
i5.2 i4.4
i3.6 8 12.8
12.0
I 1.2
9
17.1 16.2 i5.3 9 j i4.4
i3.5
12.6
26O-3OO
N
1
2
3
4
5
6
7
8
9
2GO
4i 497
5i4
53i
547
564
58i
5 97
614
63i
64 7
261
664
68 1
697
714
7 3i
?4?
764
780
797
8i4
262
83o
84?
863
880
896
913
929
946
963
979
263
996
*OI2
*O2g
*o45
*o62
*c>78
*o 9 5
*ui
*I27
*i44
a64
42 160
I 77
193
210
226
243
25g
275
292
3o8
265
3^5
34 1
35?
3 7 4
390
4o6
423
43 9
455
4 7 2
266
488
5o4
621
53 7
553
670
586
602
619
635
267
65i
667
684
700
716
732
749
765
781
797
268
8i3
83o
846
862
878
8 9 4
911
927
943
9 5 9
269
97 5
991
*oo8
*O24
*o4o
*o56
*072
*o88
*io4
*I2O
270
43 i36
l52
169
185
2OI
217
233
249
265
28l
271
297
3i3
329
345
36i
3?7
3 9 3
409
425
44 1
272
45 7
4?3
48 9
505
521
53?
553
56 9
584
600
2 7 3
616
632
648
664
680
696
712
727
743
7^9
274
77 5
791
807
823
838
854
870
886
902
917
2 7 5
9 33
94g
9 6 5
981
996
*OI2
*028
*o44
*o5 9
*o 7 5
276
44091
107
122
1 38
i54
I7O
i85
2OI
217
232
277
248
264
279
2 95
3u
326
342
358
3 7 3
38 9
278
4o4
420
436
45i
46 7
483
498
5i4
529
545
279
56o
5 7 6
5 9 2
607
623
638
654
669
685
7OO
280
716
7 3i
747
762
778
79 3
809
824
84o
855
281
871
886
902
917
932
948
9 63
979
994
*OIO
282
45 025
o4o
o56
071
086
102
117
i33
i48
i63
283
179
ig4
209
225
24o
255
271
286
3oi
3i 7
284
332
347
362
3 7 8
3 9 3
4o8
423
43 9
454
46 9
285
484
500
5'5
53o
545
56i
5 7 6
5gi
606
621
286
637
652
667
682
697
712
728
743
7 58
77 3
287
788
8o3
818
834
849
864
879
8 9 4
909
924
288
9 3 9
954
969
984
*ooo
*oi5
*o3o
"045
*o6o
*75
289
46 090
105
120
135
150
165
1 80
J 95
210
225
290
24o
255
270
285
3oo
315
33o
345
35 9
3?4
291
38 9
4o4
4l9
434
44g
464
479
494
5og
523
292
538
553
568
583
5g8
6i3
627
642
657
672
298
687
702
716
7 3i
746
761
776
79
8o5
820
294
8X5
850
864
879
894
909
923
g38
9 53
967
2g5
982
997
*OI2
*O26
*o4i
*o56
*O7O
*o85
*IOO
*u4
296
4? 129
1 44
I 69
173
1 88
202
217
232
246
261
297
276
290
3o5
3i 9
334
349
363
3 7 8
392
407
298
422
436
45 1
465
48o
4 9 4
5og
524
538
553
299
56 7
582
5 9 6
611
625
64o
654
669
683
698
300
7 1 2
727
74 1
7 56
770
784
799
8i3
828
842
N
O
1
2,
3
4
5
6
7
8
9
3OO 33O
N
O
1
2
3
4
5
6
7
8
9
800
47 712
727
?4i
766
77
784
799
8i3
828
42
3oi
85 7
871
885
900
914
929
-943
958
972
986
302
48 ooi
oi5
029
o44
o58
073
087
1OI
116
i3o
3o3
i44
i5 9
I 7 !
187
202
216
2.3
244
259
273
3o4
287
302
3i6
33o
344
35 9
373
38 7
4oi
4i6
3o5
43o
444
458
4?3
48 7
5oi
5i5
53o
544
558
3o6
672
586
601
615
629
643
657
671
686
700
307
7i4
728
742
7 56
770
785
799
8i3
827
84 1
3o8
855
869
883
897
911
926
940
954
968
982
309
996
*OIO
*O24
*o38
*052
*o66
*o8o
*094
*io8
*I22
810
49 i36
i5o
1 64
178
192
206
220
234
248
262
3u
276
290
3o4
3i8
332
346
36o
374
388
402
3l2
4i5
429
443
45 7
4?i
485
499
5i3
627
54i
3i3
554
568
582
5 9 6
610
624
638
65i
665
679
3i4
6 9 3
707
721
734
748
762
776
79
8o3
817
3i5
83i
845
85g
872
886
900
914
927
941
955
3i6
969
982
996
*O1O
*024
*o37
*o5i
*o65
*079
*og2
3i7
5o 106
I2O
i33
J47
161
174
188
202
2l5
229
3i8
243
256
270
284
297
3n
325
338
352
365
319
379
393
4o6
420
433
447
46i
4?4
488
5oi
320
5'5
529
542
556
56 9
583
5 9 6
610
623
637
3ai
65i
664
678
691
75
718
732
745
759
772
322
786
799
8i3
826
84o
853
866
880
8g3
907
323
920
934
947'
961
974
987
*OOI
*oi4
*028
*o4i
324
5i 055
068
081
95
1 08
121
135
i48
162
75
325
188
202
2l5
228
242
255
268
282
295
3o8
326
322
335
348
362
3 7 5
388
4O2
415
428
44 1
32 7
455
468
48 1
495
5o8
521
534
548
56i
5 7 4
328
587
601
6i4
627
64o
654
667
680
6 9 3
706
329
720
7 33
746
?5g
772
786
799
812
825
838
330
85i
865
878
891
904
917
g3o
943
9 5 7
97
N
O
1
2
3
4
5
G
7
8
PP 15 14 13 12 11
i
i.5
i.4
1.3
I 1.2 I.I
2
3.o
2.8
2.6
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.
11.2
10.4
8 9.6 8.8
9
i3.5
12.6
it. 7
9 10.8 9.9
330 37O
N
O
1
2
3
4
5
O
7
8
9
330
5i S5i
865
878
891
904
917
9 3o
943
9 5 7
970
33i
9 83
99 6
*oo 9
*022
*o35
*o48
*o6i
*o 7 5
*o88
*IOI
332
52 1 14
127
i4o
i53
166
179
I 9 2
205
218
23l
333
244
257
270
284
297
3io
3 2 3
336
34 9
362
334
375
388
4oi
4i4
427
44o
453
466
479
4 9 2
335
5o4
5i 7
53o
543
556
56 9
682
5 9 5
608
621
336
634
647
660
673
686
6 99
711
724
73?
75o
33 7
?63
776
789
802
815
827
84o
853
866
879
338
8 9 2
95
9.17
9 3o
9 43
9 56
9 6 9
9 82
994
*oo7
33 9
53 020
o33
o46
o58
071
o84
97
I IO
122
i35
340
1 48
161
I 7 3
186
199
212
224
23 7
250
263
34i
275
288
3oi
3i4
326
33 9
352
364
3 77
3 9 o
342
4o3
4i5
428
44 1
453
466
479
4 9 i
5o4
5i 7
343
52 9
542
555
567
58o
5 9 3
6o5
618
63i
643
344
656
668
681
6 9 4
706
719
732
744
757
769
345
782
794
807
820
832
845
85 7
870
882
8 9 5
346
9 o8
9 2O
9 33
9 45
9 58
97
9 83
99 5
*oo8
*O2O
34?
54o33
o45
o58
070
o83
o 9 5
1 08
I2O
i33
i45
348
1 58
170
i83
i 9 5
208
220
233
245
258
270
349
283
2 95
307
32O
332
345
357
37O
382
3 9 4
350
407
4i 9
432
444
456
46 9
48i
4 9 4
5o6
5i8
35i
53i
543
555
568
58o
5 9 3
605
617
63o
642
352
654'
667
679
6 9 i
704
716
728
74 1
753
765
353
111
79
802
814
827
83 9
85i
864
876
888
354
9 oo
9 i3
9 2 5
9 3 7
9 4 9
9 62
974
9 86
998
*OII
355
55023
o35
o4?
060
072
o84
096
108
121
i33
356
i45
i5 7
i6 9
182
i 9 4
206
218
230
242
2 55
357
267
279
291
3o3
3i5
328
34o
352'
364
376
358
388
4oo
4i3
425
43 7
44 9
46i
4?3
485
497
35 9
5o 9
522
534
546
558
670
582
5 9 4
606
618
360
63o
642
654.
666
678
6 9 i
7o3
7'5
727
7 3 9
36i
7 5i
763
775
787
799
811
823
835
84?
85 9
36a
87.
883
895
97
919
9 3i
9 43
955
967
979
363
991
*oo3
*oi5"
*027
*o38
*o5o
*o62
*o 7 4
*o86
*o 9 8
364
56 1 10
122
1 34
i46-
1 58
170
182
i 9 4
2O5
217
365
22 9
24l
253
265
277
28 9
3oi
3l2
324
336
366
348
36o
372
384
3 9 6
407
4i 9
43i
443
455
36?
46 7
4?8
4 9 o
5O2
5i4
526
538
549
56i
573
368
585
5 9 7
608
620
632
644
656
667
679
691
36 9
703
7 i4
726
788
75
761
77 3
785
797
808
370
820
832
844
855
867
879
891
9 O2
9 i4
926
N
O
\
2
3
4
5
6
7
8
9
37O-40O
N
O
1
2
8
4
5
7
8 9
370
56 820
832
844
855
86 7
879
891
902
914
926
3 7 i
9 3 7
949
961
972
9 84
996
*oo8
*oig
*o3i
*o43
3 7 2
57054
066
078
089
101
n3
124
1 36
i48
159
3 7 3
171
1 83
194
206
217
229
24 1
252
264
276
3 7 4
287
299
3io
322
334
345
35 7
368
38o
392
3 7 5
4o3
426
438
449
46 1
4 7 3
484
496
5o 7
3 7 6
519
53o
542
553
5 6 5
576
588
600
611
623
377
634
646
65 7
669
680
692
7 o3
7*5
726
7 38
3 7 8
749
761
772
784
7 9 5
807
818
83o
84 1
852
3 79
864
8 7 5
88 7
898
910
921
933
944
9 55
9 6 7
380
978
99
*OOI
*oi3
*O24
*o35
*o4 7
*o58
*o 7 o
*o8i
38i
58 092
io4
n5
127
1 38
i4 9
161
I 7 2
1 84
'95
382
206
218
229
240
252
263
2 7 4
286
297
309
383
320
33i
343
354
365
3 77
388
3 99
4io
422
384
433
444
456
46 7
4?8
4 9 o
5oi
5l2
524
535
385
546
55 7
56 9
58o
591
602
6i4
625
636
64 7
386
65g
6 7 o
681
692
704
7i
=
726
7 3 7
749
760
38 7
77'
782
794
805
816
82 7
838
850
861
872
388
883
8 9 4
906
9'7
928
9 3 9
95o
961
973
984
389
995
"006
*OI 7
*oa8
*o4o
*o5i
*o62
*o 7 3
*o84
*og5
390
5g 106
118
129
i4o
i5i
162
I 7 3
i84
1 9 5
207
3 9 i
218
229
240
a5i
262
2 7 3
284
2 9 5
3o6
3i8
3 9 2
329
34o
35i
362
3 7 3
384
3 9 5
4o6
4i 7
428
3 9 3
43 9
45o
46 1
4 7 2
483
494
5o6
5. 7
5 2 8
539
3 9 4
55
56 f
5 7 2
583
5 9 4
605
616
627
638
649
3 9 5
660
6 7 i
682
693
7 o4
7'5
726-
737
7 48
759
3 9 6
77
7 8o
791
802
8i3
824
835
846
85.7'
868
3 97
8 79
890
901
912
923
934
945
9 56
966
977
3 9 8
988
999
*OIO
*02I
*032
*o43
*o54
*o6g
*o 7 6
*o86
3 99
60 097
1 08
119
i3o
i4i
I 52
i63
178
1 84
196
400
206
2I 7
228
2 3 9
249
260
271
282
2 9 3
3o4
N
O
1 1 2
3
4
5
6
7
8
9
PP 12 11
10 9
I
1.2 I.I I
i.o o. 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
4OO 44O
N
O
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
34 7
358
36 9
3 79
3 9 o
4oi
4l2
402
423
433
444
455
466
477
48 7
4 9 8
5o 9
52O
4o3
53i
54i
552
563
5?4
584
5 9 5
606
617
627
4o4
638
649
660
6 7 o
681
692
7 o3
7 i3
724
7 35
4o5
746
7 56
767
77 8
788
799
810
821
83i
842
4o6
853
863
8 7 4
885
8 9 5
906
917
9 2 7
9 38
949
407
9 5 9
970
981
991
*OO2
*oi3
*023
*o34
*o45
*o55
4o8
61 066
077
087
098
IO9
119
i3o
i4o
i5i
162
409
172
i83
ig4
2O4
215
225
236
247
257
268
410
2 7
289
3oo
3io
321
33i
342
352
363
3 7 4
4n
384
395
4o5
4i6
426
43 7
448
458
46 9
4 7 9
4l 2
4go
5oo
5u
521
532
542
553
563
5 7 4
584
4i3
5 9 5
606
616
627
637
648
658
66 9
679
690
4i4
700
7 n
721
7 3i
7 42
752
763
773
7 84
79 4
4i5
805
8i5
826
836
847
85 7
868
878
888
899
4i6
909
920
g3o
94 1
9 5i
962
972
982
99 3
*oo3
4i 7
62 oi4
024
o34
045
o55
066
076
086
97
107
4i8
118
128
i38
i4g
169
170
180
190
2OI
21 I
419
221
232
242
252
263
273
284
294
3o4
3i5
420
325
335
346
356
366
3 77
38 7
397
4o8
4i8
421
428
439
44g
45 9
46 9
48o
4go
5oo
5n
521
422
53i
542
552
562
572
583
5 9 3
6o3
6i3
624
423
634
644
655
665
675
685
696
706
716
726
424
7 3 7
7 4 7
7 5 7
7 6 7
778
788
79 8
808
818
829
425
83 9
849
859
8 7 o
880
890
9 oo
910
921
g3i
426
941
g5i
961
972
982
992
*002
*OI2
*022
*o33
427
63o43
o53
o63
o 7 3
08 3
og4
io4
u4
124
1 34
428
i44
'55
165
'75
185
i 9 5
2O5
2l5
225
236
429
246
256
266
2 7 6
286
296
3o6
3i 7
32 7
337
430
34 7
35 7 .
36 7
3 77
387
397
4o 7
417
428
438
43i
448
458
468
4 7 8
488
498
5o8
5i8
528
538
432
548
558
568
5 7 g
58 9
5 99
6o 9
619
629
63 9
433
649
65g
669
679
689
699
79
719
729
7 3 9
434
?49
7 5g
769
779
789
799
809
819
829
83 9
435
849
85 9
869
879
889
899
99
919
9 2 9
9 3 9
436
949
9 5 9
969
979
988
998
*oo8
*oi8
*028
*o38
43 7
64o48
o58
068
078
088
098
1 08
118
128
i3 7
438
i4 7
,5 7
167
177
187
197
2O 7
217
227
237
43 9
246
256
266
276
286
296
3o6
3i6
326
335
440
345
355
365
375
385
3 95
4o4
4i4
424
434
N
O
1
2
3
4
5
6
7
8
9
44O-47O
N
O
i
2
3
4
5
O
7
8
J)
440
64345
355
365
375
385
3 95
4o4
4i4
424
434
44 1
444
454
464
4 7 3
483
493
5o3
5i3
523
532
442
542
552
562
5 7 2
582
591
601
6n
621
63i
443
64o
65o
660
670
680
689
699
709
719
729
444
738
7 48
758
768
77*7
787
797
807
816
826
445
836
846
856
865
8 7 5
885
895
904
914
924
446
933
9 43
9 53
963
972
982
992
*OO2
*O1 I
*02I
44?
65 o3i
o4o
o5o
060
070
079
089
99
108
118
448
128
i3 7
i4 7
i5 7
167
176
186
I 9 6
205
2I 5
449
225
234
244
254
263
273
283
292
302
3l2
450
321
33i
34 1
35o
36o
36 9
3 79
38 9
3 9 8
4o8
45i
4i8
42 7
43 7
44?
456
466
4?5
485
495
5o4
452
5i4
523
533
543
552
562
5 7 i
58i
5gi
600
453
610
6,9
62 9
639
648
658
66 7
677
686
696
454
706
7 ,5
725
734
744
753
763
772
782
792
455
801
811
820
83o
83 9
'849
858
868
877
887
456
896
906
9 i6
925
935
9 44
954
963
97 3
982
45 7
992
*OOI
*OII
*O2O
*o3o
*o39
*o4 9
*o58
*o68
*o 7 7
458
66087
096
106
"5
124
1 34
i43
i53
162
172
45 9
181
191
200
2IO
219
229
238
247
25 7
266
460
276
285
2 95
3o4
3i4
323
332
342
35i
36i
46 1
370
38o
389
3 9 8
4o8
417
42 7
436
445
455
462
464
4?4
483
492
5O2
5u
521
53o
53 9
549
463
558
567
5 77
586
5 9 6
605
6i4
624
633
642
464
652
661
6 7 i
680
689
699
708
717
727
736
465
745
755
7 64
773
7 83
792
801
811
820
829
466
83 9
848
85 7
867
876
885
8 9 4
904
9i3
922
46 7
9 32
g4i
95c
>
960
969
978
987
997
*oo6
*oi5
468
67 025
o34
o43
o52
062
071
080
089
099
108
46 9
117
127
1 36
i45
i54
1 64
r ? 3
182
201
470
210
219
228
2 3 7
24 7
256
265
274
28/1
2 9 3
N
g
1
2
3
4
5
O
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.o
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
O
1
2
3
4
5
6
7
8
8
470
67 210
219
228
23 7
24?
a56
265
274
24
2 9 3
47'
3O2
3n
321
33o
33 9
348
357
367
376
385
4?2
3 9 4
4o3
4i3
422
43i
44o
44g
45 9
468
477
4?3
486
4g5
5o4
5i4
523
532
54i
55o
56o
56 9
4?4
5 7 8
58 7
596
6o5
6x4
624
633
642
65i
C6o
4 7 5
669
679
688
697
7 o6
7 i5
724
733
742
752
4?6
761
770
779
788
797
806
8i5
825
834
843
477
852
861
870
879
888
897
906
916
9 2 5
934
4 7 8
943
g52
961
970
979
988
997
*oo6
*oi5
*O24
479
68o34
o43
o52
06 1
O 7 O
079
088
097
106
n5
480
124
i33
142
i5i
1 60
169
178
187
196
2O5
48 1
215
224
233
242
a5i
260
269
278
287
296
482
805
3i4
323
332
34 1
350
35 9
368
3 77
386
483
3 95
4o4
4i3
422
43i
44o
44g
458
46 7
4?6
484
485
4g4
502
5u
520
529
538
547
556
565
485
574
583
592
601
610
619
628
63 7
646
655
486
664
673
681
690
52?
708
717
726
7 35
744
48 7
7 53
762
771
780
789
797
806
8i5
824
833
483
842
85i
860
869
8 7 8
886
8 9 5
904
gi3
922
48 9
9 3,
g4o
949
g58
966
975
984
99 3
*OO2
*OII
490
69 020
028
o3 7
o46
o55
o64
073
082
090
099
4gi
108
117
126
135
1 44
I 52
161
170
I 79
1 88
492
197
205
2l4
223
232
24 1
249
258
267
276
4g3
285
294
302
3n
32O
329
338
346
355
364
494
373
38i
3gO
3 99
4o8
4i7
425
434
443
452
495
46 1
46 9
478
48 7
496
5o4
5i3
522
53i
53 9
496
548
55 7
566
5 7 4
583
592
601
609
618
627
497
636
644
653
662
671
679
688
697
7o5
714
498
723
7 32
740
749
7 58
7 6 7
77 5
784
79 3
801
499
810
819
827
836
845
854
862
8 7 I
880
888
500
897
906
gi4
923
g32
9 4o
949
g58
966
97 5
5oi
984
992
*OOI
*OIO
*oi8
*O2 7
*o36
*o44
*o53
*o62
502
70 070
079
088
096
105
n4
122
i3i
i4o
i48
5o3
i5 7
i65
174
i83
191
200
2O 9
217
226
234
5o4
243
252
260
269
278
286
295
3o3
3l2
321
5o5
329
338
346
355
364
3 7 2
38i
389
3 9 8
4o6
5o6
4.5
424
432
44 1
44g
458
46 7
4 7 5
484
492
607
5oi
5o 9
5i8
526
535
544
552
56i
569
5 7 8
5o8
586
5 95
6o3
612
621
629
638
646
655
663
609
672
680
689
697
706
7 i4
7 23
7 3i
740
?49
510
7 5 7
766
774
783
791
800
808
817
825
834
N
O
1
2
8
4
5
6
7
8
9
13
510-540
N
O
1
2
3
4
O
7 8
o
510
70767
766
774
7 3
791
800
808
8,7
tt*5
834
5ii
842
85 1
85 9
868
876
885
8y3
902
910
919
5l2
927
935
944
962
961
969
978
986
995
*oo3
5i3
71 OI2
020
029
037
o46
o54
o63
071
079
088
5i4
096
105
ii
I
122
i3o
i3 9
i47
i55
1 64
I 7 2
5i5
181
189
19
}
206
2l4
223
23l
24O
248
25 7
5i6
265
2 7 3
282
290
299
3o 7
3i5
324
332
34i
5i 7
349
35 7
366
374
383
3gi
3 99
4o8
4i6
425
5i8
433
44 1
450
458
466
475
483
492
500
5o8
5, 9
5i 7
5 2 5
533
542
55o
56 7
5 7 5
584
692
520
600
609
617
625
634
642
65o
65 9
667
6 7 5
521
684
692
700
709
717 725
734
742
7 5o
7 5 9
522
767
775
784
792
800 809
817
825
834
842
523
85o
858
867
8 7 5
883
892
900
908
917
9 2 5
524
933
94 1
95
9 58
966 9 7 5
983
99'
999
*oo8
5 2 5
72 016
024
o32
o4i
o4g o5 7
066
074
082
090
526
099
107
u5
123
i3a i4o
i48
1 56
,65
173
527
181
189
'9
3
206
2l4 222
230
23 9
s4 7
255
528
263
272
280
288
296 3o4
3i3
321
329
33 7
529
346
354
362
370
3 7 8 38 7
3 95
4o3
4n
419
530
4a8
436
444
452
46o 469
477
485
4 9 3
5oi
53i
Sog
5i8
52<
5
534
542
55o
558
56 7
575
583
532
5gi
5 99
607
616
624
632
64Q
648
656
665
533
6 7 3
681
689
697
7 o5
7 i3
7 22
73o
7 38
7 46
534
754
762
77<
j
779
787
795
8o3
811
819
827
535
835
843
852
860
868
8 7 6
884
8 9 2
900
908
536
916
9 2 5
9 33
94 1
949
9 5 7
9 65
97 3
981
989
537
997
*oo6
*oi4
*022
*o3o
*o38
*o46
*o54
*o62
*070
538
7 3 078
086
094
102
I ! I
119
I2 7
135
i43
i5i
53 9
i Sg
167
'75
i83
igi
199
2O 7
2l5
223
23l
540
2 3 9
247
255
263
2 7 2
280
288
296
3o4
3l2
N
O
1
2
3
4
5
o
7
8
i)
PP 9
8 7
i
0.9
i
0.8
i
0.7
2
1.8
2
1.6
2
1.4
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
O
1
2
3
4
5
6
7
8
9
540
7 3 23 9
24 7
255
263
2 7 2
280
288
296
3o4
3l2
54i
320
328
336
344
352
36o
368
376
384
3 9 2
542
4oo
4o8
4i6
4a4
432
44o
448
456
464
4 7 2
543
48o
488
496
5o4
5l2
52O
5a8
536
544
552
544
56o
568
5 7 6
584
592
600
608
616
624
632
545
64o
648
656
664
672
679
687
695
7 o3
711
546
719
727
7 35
7 43
7 5i
7 5 9
767
775
7 83
79'
54?
799
807
815
823
83o
838
846
854
862
8 7 o
548
878
886
8 9 4
902
910
918
926
9 33
g4i
9 4 9
549
9 5 7
965
97 3
981
989
997
''oog
*oi3
*O2O
*028
550
7 4 o36
o44
o5a
060
068
o 7 6
o84
O 9 2
99
1O 7
55i
n5
123
i3i
1 3g
i4 7
'55
162
170
178
166
552
194
202
2IO
218
225
233
24l
24g
257
265
553
2 7 3
280
288
296
3o4
3l2
320
827
335
343
554
35i
35g
36 7
3 7 4
382
390
3 9 8
4o6
4i4
421
555
429
43 7
445
453
46 1
468
4 7 6
484
4 9 2
500
556
5o 7
sis
523
53i
53g
54 7
554
562
570
5 7 8
55 7
586
5 9 3
601
609
617
624
632
64o
648
656
558
663
671
679
687
6 95
702
710
718
726
733
55 9
7 4i
7 4 9
7 5 7
?64
772
780
7 88
79 6
8o3
811
5GO
819
82 7
834
842
850
858
865
8 7 3
881
88 9
56i
896
904
912
920
927
935
943
9 5o
9 58
9 66
562
9 7 4
981
989
997
*oo5
*OI2
*02O
*028
*o35
*o43
563
7 5 o5i
o5g
066
074
082
089
97
105
n3
I2O
564
128
i36
i43
i5i
i Sg
166
i 7 4
182
189
197
565
205
2l3
220
228
236
243
25l
25 9
266
274
566
282
289
297.
3o5
3l2
320
328
335
343
35i
56 7
358
366
3 7 4
38i
38g
397
4o4
4l2
420
427
568
435
442
45o
458
465
4 7 3
48i
488
496
5o4
569
5i'i
5ig
526
534
542
549
55 7
565
572
58o
570
58 7
5 9 5
6o3
610
618
626
633
64 1
648
656
5 7 i
664
671
679
686
6 9 4
702
709
717
724
7 32
572
?4o
74?
755
762
770
778
7 85
79 3
800
808
5 7 3
8i5
823
83i
838
846
853
861
868
876
884
5 7 4
891
899
906
gi4
921
929
9 3 7
944
952
9 5 9
5 7 5
9 6 7
974
982
989
997
*oo5
*OI2
*O2O
*O2 7
* 35
5 7 6
7 6 o4s
050
o5 7
065
O 7 2
080
087
095
io3
no
5 77
118
125
i33
i4o
1 48
i55
i63
170
178
i85
5 7 8
i 9 3
200
208
2l5
223
230
238
245
s53
260
5 79
268
2 7 5
283
290
298
3o5
3i3
320
828
335
580
343
35o
358
365
3 7 3
38o
388
3 9 5
4o3
4io
N
1
2
3
4
5
6
7
8
9
58O 61O
N
O
1
iJ
3
4
5
7
8
J)
580
76 343
35o
358
365
3 7 3
38o
388
3 9 5
4o3
4 i <>
58i
4ib
425
433
44o
448
455
462
4 7 o
477
485
582
492
500
507
5'5
522
53o
53 7
545
552
55 9
583
567
5 7 4
582
58 9
5 97
6o4
612
619
626
634
584
64i
649
656
664
671
678
686
6 9 3
701
7 o8
585
716
723
?3<
>
738
7 45
7 53
760
768
775
7 82
586
79
797
805
812
819
82 7
834
842
849
856
58 7
864
871
879
886
893
901
908
916
923
9 3o
588
9 38
945
9 53
960
9 6 7
975
982
989
997
*oo4
58 9
77 012
019
026
o34
o4i
o48
o56
oG3
070
078
590
o85
093
IOO
107
"5
122
129
i3 7
i44
i5i
5gi
i5 9
1 66
I 7 3
181
188
195
203
2IO
217
225
592
232
240
24 7
254
262
269
2 7 6
283
291
298
5 9 3
3o5
3i3
320
32 7
335
342
349
35 7
364
3 7 i
594
3 79
386
3 9 3
4oi
4o8
4i5
422
43o
43 7
444
5 9 5
452
45 9
466
4?4
48i
488
495
5o3
5io
5i 7
596
5 2 5
532
53 9
546
554
56i
568
5 7 6
583
590
5 97
5 97
605
612
619
62 7
634
64 1
648
656
663
5 9 8
6 7 o
677
685
692
699
7 o6
7 i4
721
728
7 35
5 99
743
75
7 5 7
764
772
779
786
7 9 3
801
808
600
8i5
822
83o
83 7
844
85 1
85 9
866
8 7 3
880
60 1
887
895
902
909
916
924
g3i
9 38
945
g52
602
960
967
974
981
988
996
*oo3
*OIO
*OI 7
*025
6o3
78 o32
o3g
o46
o53
061
068
075
082
089
097
6o4
io4
1 1 1
118
125
l32
i4o
i47
1 54
161
1 68
6o5
176
i83
190
197
2O4
211
219
226
233
240
606
24?
254
262
269
276
283
290
297
305
3l2
607
3io
326
333
34o
34 7
355
362
369
3 7 6
383
608
3 9 o
3 9 8
405
4 12
419
426
433
44o
44 7 .
455
609
462
46 9
4 7 6
483
490
497
5o4
5l2
5i 9
526
010
533
54o
547
554
56i
56 9
5 7 6
583
5 9 o
5 97
N
O
1
2
3
4
5
6
7
8
9
PP 8
7 t>
i
0.8
i
0.7
i 0.6
2
1.6
2
1.4
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
61O 65O
N
O
1
2
3
4
5
6
7
8
9
610
78 533
54o
54 7
554
56i
56 9
5 7 6
583
5go
5 97
611
6o4
611
618
625
633
64o
647
654
661
668
612
675
682
689
696
704
711
718
7 2 5
7 3s
7 3 9
6i3
746
753
760
767
774
781
789
796
8o3
810
6i4
817
824
83i
838
845
852
869
866
8 7 3
880
6i5
888
895
902
909
916
923
gSo
9 3 7
944
g5i
616
958
9 65
972
979
986
99 3
*ooo
*oo7
*oi4
*O2I
617
79 029
o36
o43
050
057
o64
071
078
085
092
618
099
1 06
n3
I2O
127
1 34
i4i
1 48
1 55
162
619
169
176
i83
igo
'97
204
21 I
218
225
232
620
23g
246
253
260
267
274
28l
288
295
302
621
809
3i6
323
33o
33 7
344
35i
358
365
372
622
379
386
3g3
4oo
407
4i4
421
428
435
442
623
449
456
463
470
4?7
484
491
498
55
5u
624
5i8
5 2 5
532
53 9
546
553
56o
56 7
5 7 4
58i
626
588
595
602
609
616
623
63o
63?
644
65o
626
65 7
664
671
678
685
692
699
706
7*3
720
627
727
7 34
74 1
748
?54
761
768
776
782
789
628
796
8o3
810
817
824
83i
837
844
85i
858
629
865
872
879
886
893
900
906
gi3
920
927
630
934
g4i
948
955
962
969
97 5
982
989
996
63 1
80 oo3
OIO
017
024
o3o
037
o44
o5i
o58
065
632
072
079
o85
092
099
1 06
u3
I2O
127
1 34
633
i4o
i4y
1 54
161
168
175
182
188
195
202
634
209
216
223
229
236
243
25o
257
264
2 7 I
635
277
284
291
298
305
3l2
3i8
325
33 2
33g
636
346
353
359
366
373
38o
387
3g3
4oo
407
63 7
4i4
421
428
434
44i
448
455
462
468
4?5
638
482
48 9
496
5O2
Sog
5i6
523
53o
536
543
63 9
55o
55 7
564
570
577
584
591
5 9 8
6o4
611
640
618
625
632
638
645
652
65g
665
672
679
64 1
686
6 9 3
699
706
7 i3
720
726
7 33
?4o
?4?
642
?54
760
767
774
781
787
794
801
808
8i4
643
821
828
835
84 1
848
855
862
868
875
882
644
889
8 9 5
902
909
916
922
929
936
943
949
645
g56
963
969
976
983
990
996
*oo3
*OIO
*oi7
646
81 023
o3o
037
o43
o5o
057
o64
070
077
o84
647
090
097
io4
1 1 1
117
124
i3i
i3 7
1 44
i5i
648
1 58
164
171
178
i84
191
198
2O4
211
218
649
224
23l
238
245
25l
258
265
271
278
285
650
291
298
305
3n
3i8
325
33i
338
345
35i
N
1
2
3
4
5
6
7
8
9
65O-680
N
1
a
3
4
5
O
7
8 9
650
81 291
298
305
3n
3i8
325
33i
338
345
35i
65i
358
365
3 7 i
3 7 8
385
391
3 9 8
405
4n
4i8
65 2
425
43i
438
445
45i
458
465
4 7 i
4 7 8
485
653
491
498
55
5n
5i8
5 2 5
53i
538
544
55i
654
558
564
5 7 i
5 7 8
584
591
598
6o4
611
617
655
624
63i
63 7
644
65 1
657
664
6 7 i
677
684
656
690
697
704
710
717
723
73o
7 3 7
743
75
65 7
757
763
770
776
7 83
79
796
8o3
809
816
658
823
829
836
84a
849
856
862
869
875
882
65 9
889
8 9 5
902
908
9-5
921
928
935
94 1
948
660
954
961
968
974
981
987
994
*ooo
*OO 7
*oi4
66 1
82 020
027
o33
o4o
o46
o53
060
066
o 7 3
79
662
086
092
099
io5
112
119
125
I 32
1 38
663
161
i58
1 64
171
1 7 8
184
191
19-7
204
210
664
217
223
230
236
243
249
256
263
269
276
665
282
289
295
302
3o8
315
321
3 2 8
334
34 1
666
347
354
36o
36 7
3 7 3
38o
387
3 9 3
4oo
4o6
667
4i3
419
426
432
43 9
445
45 2
458
465
4 7 i
668
478
484
491
497
5o4
5io
5i 7
523
53o
536
669
543
549
556
562
569
5 7 5
582
588
595
60 1
670
607
6i4
620
627
633
64o
646
653
65 9
666
671
672
679
685
692
698
75
711
7 i8
7 24
7 3o
672
7 3 7
743
750
756
7 63
769
776
782
789
79 5
6 7 3
802
808
8i4
821
827
834
84o
84 7
853
860
6 7 4
866
872
879
885
892
898
95
911
918
924
6 7 5
gSo
9 3 7
943
950
956
9 63
969
9 7 5
982
988
6 7 6
995
*OOI
*oo8
*oi4
"020
*027
*o33
*o4o
*o46
*052
677
83o5g
o65
072
078
085
091
097
io4
no
"7
678
123
129
1 36
142
i4<;
'55
161
1 68
i 7 4
181
679
187
ig3
200
206
2l3
219
225
232
238
245
680
a5i
25 7
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
i .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
9
6.3
9
5.4
18
680720
N
O
1
2
3
4
5
6
7 8
9
080
83 25i
257
264
2 7 O
276
283
289
296
3O2
3o8
681
315
321
327
334
34o
34 7
353
35 9
366
372
682
3 7 8
385
3gi
3 9 8
4o4
4io
417
423
429
436
683
442
448
455
46i
46 7
4 7 4
48o
48 7
493
499
684
5o6
5l2
5i8
525
53i
53 7
544
55o
556
563
685
669
5 7 5 582
588
5g4
601
6o 7
6i3
620
626
686
632
63g
645
65i
658
664
6 7 o
677
683
689
687
696
702
708
7i5
7 2I
727
7 34
74o
7 46
753
688
7 5 9
766
771
77 8
7 84
79
797
8o3
809
816
689
822
828
835
84i
84 7
853
860
866
8 7 2
879
690
885
891
897
904
910
916
923
929
9 35
942
691
948
954
960
967
97 3
979
985
992
998
*oo4
692
84 on
017
023 029
o36
042
o48
055
06 1
067
6 9 3
073
080
086
092
098
105.
i ii
117
123
i3o
6 9 4
1 36
I 42
i48
'55
161
i6 7
i 7 3
180
186
192
6 9 5
198
205
211
217
223
230
236
242
248
2 55
696
261
267
2 7 3
280
286
292
298
305
3n
3, 7
697
323
33o
336
342
348
354
36i
367
3 7 3
379
698
386
3g2
3 9 8
4o4
4io
4. 7
423
429
435
442
699
448
454
46o
466
473
4 7 g
485
491
497
5o4
700
5io
5i6
522
528
535
54i
54 7
553
55 9
566
701
672
5 7 8
584
Sgo
597
6o3
609
6i5
621
628
702
634
64o
646
652
658
665
6 7 i
677
683
689
703
696
702
708
714
720
726
7 33
7 3 9
745
7 5i
704
7 5 7
763
77
776
782
788
794
800
8o 7
8i3
706
819
825
83i
837
844
850
856
862
868
8 7 4
706
880
887
8 9 3
899
go5
911
91?
924
9 3o
936
707
942
948
g54
960
967
9 7 3
979
9 8 5
991
997
708
85 oo3
009
016
022
028
o34
o4o
o40
052
o58
709
065
071
77
o83
089
ogS
101
IO 7
n4
120
710
126
132
1 38
1 44
i5o
i56
i63
169
75
181
711
187
ig3
199
2o5
21 I
2I 7
224
23o
236
242
712
248
254
260
266
272
2 7 8
285
291
2 97
3o3
7'3
3og
3i5
321
327
333
339
345
352
358
364
714
870
3 7 6
382
388
3 9 4
4oo
4o6
4l2
4i8
425
7 ,5
43 1
43 7
443
449
455
46 1
46 7
4 7 3
4 7 9
485
716
4gi
497
5o3
5og
5i6
522
528
534
54o
546
717
552
558
564
570
5 7 6
582
588
5 9 4
600
606
718
612
618
625
63i
63?
643
649
655
661
66 7
719
673
679
685
691
697
7 o3
709
7i5
7 2I
7 2 7
720
7 33
7 3 9
7 45
7 5i
7 5 7
7 63
7 6 9
77 5
7 8l
7 88
N
O
1
2
3
4
5
O
7
8
9
72O-75O
N
1
*J
3
4
5
G
7
8
<>
720
85 733
739
745
7DI
7 5 7
?63
769
77 5
781
7 8
721
794
800
806
812
818
824
83o
836
842
848
722
854
860
866
872
8 7 8
884
890
896
902
908
7 23
9i4
920
926
932
938
944
95
956
962
968
724
974
980
986
992
998
*oo4
*OIO
*oi6
*O22
*028
7 25
86o34
o4
j
o46
o58
064
070
o 7 6
082
088
726
094
100
106
112
118
124
i3o
i36
i4i
i4 7
727
i53
1 59
i65
171
177
i83
189
, 9 5
2OI
207
728
8l3
219
225
23l
23 7
243
249
2 55
26l
267
729
273
279
285
291
297
3o3
3o8
3i4
32O
326
730
332
338
344
35o
356
362
368
3 7 4
38o
386
7 3i
3 9 2
3 9 8
4o4
4io
4i5
421
427
433
43 9
445
7 32
45i
45 7
463
46 9
475
48i
48 7
493
499
5o4
733
5io
5i6
522
528
534
54o
546
552
558
564
734
670
5 7 6
58i
58 7
5 9 3
5 99
6o5
611
617
623
7 35
629
635
64 1
646
652
658
664
670
676
682
7 36
688
6 9 4
700
7 o5
711
717
7 23
729
735
7 4i
7 3 7
74?
7 53
759
7 64
770
776
782
788
794
800
7 38
806
812
8i 7
823
829
835
84i
847
853
85 9
7 3 9
864
870
876
882
888
8 9 4
900
906
911
917
740
923
929
9 3 5
94 1
947
953
9 58
964
970
976
?4 1
982
986
994
999
*oo5
*OII
*OI 7
*023
*029
*o35
742
87 o4o
o46
o58
o64
070
o 7 5
08 1
087
093
7 43
099
105
in
116
122
128
1 34
i4o
i46
i5i
744
i5 7
i63
169
175
181
186
192
198
2f>4
2IO
745
216
221
227
233
239
245
25l
256
262
268
746
274
280
286
291
297
3o3
309
315
320
3 2 6
74?
332
338
344
349
355
36i
36 7
3?3
3 79
384
748
3go
396 4O2
4o8
4i3
419
425
43i
43 7
442
749
448
454
46o
466
471
477
483
48 9
495
5oo
750
5o6
5i2 5i8
523
529
535
54 1
547
55 2
558
5
O
1
li
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.o
9
5.4
9
4.5
750-790
N
1
2
3
4
5
6
7 8
9
750
87 5o6
5l2
5i8
523
529
535
54i
54 7
552
558
7 5i
564
570
5 7 6
58i
58 7
SgS
5 99
6o4
610
616
762
622
628
633
63g
645
65i
656
662
668
6 7 4
7 53
679
685
691
697
7o3
7 o8
7 i4
720
726
7 3i
754
73?
743
749
?54
760
7 66
772
777
783
789
7 55
795
800
806
812
818
823
829
835
84 1
846
766
852
858
864
869
8 7 5
88 1
88 7
892
898
904
7 5 7
910
gi5
921
927
933
g38
944
95
955
961
7 58
967
97 3
978
984
990
996
*OOI
*oo7
*oi3
*oi8
7 5 9
88024
o3o
o36
o4i
o4 7
o53
o58
o64
070
o 7 6
760
08 1
o8 7
og3
098
io4
I IO
116
121
127
i33
761
1 38
1 44
150
i56
161
167
i 7 3
I 7 8
1 84
190
7 62
ig5
201
207
213
218
224
230
235
24 1
24?
7 63
252
258
264
270
2 7 5
281
28 7
292
298
3o4
7 64
3og
3i5
321
826
33 2
338
343
349
3 55
36o
7 65
366
3 7 2
3 77
383
38 9
3 95
4oo
4o6
412
417
7 66
423
429
434
44o
446
45i
457
463
468
4 7 4
767
48o
485
4gi
497
502
5o8
5i3
5ig
525
53o
768
536
542
54 7
553
55 9
564
5 7 o
5 7 6
58i
58 7
769
5g3
5 9 8
6o4
610
6i5
621
62 7
632
638
643
770
649
655
660
666
6 7 2
677
683
689
6 9 4
700
771
yoS
711
717
722
728
734
7 3 9
745
75o
7 56
772
762
7 6 7
773
779
7 84
79
79 5
801
807
812
773
818
824
829
835
84o
846
852
85 7
863
868
774
8 7 4
880
885
891
897
902
908
gi3
919
9 2 5
775
93o
g36
g4i
94?
953
9 58
964
969
975
981
776
986
992
997
*oo3
*oo9
*oi4
*O2O
*025
*o3i
*o37
777
89 042
o48
o53
oSg
o64
070
o 7 6
08 1
087
092
778
098
io4
109
"5
I2O
126
i3i
i3 7
i43
i48
779
i54
i5 9
165
170
i 7 6
182
i8 7
193
198
2O4
780
209
2l5
221
226
232
23 7
243
248
254
260
781
265
271
276
282
28 7
293
298
3o4
3io
3i5
782
321
3a6
33 2
337
343
348
354
36o
365
3 7 i
7 S3
3 7 6
382
387
3g3
3 9 8
4o4
409
4J5
421
426
?34
432
437
443
448
454
45 9
465
470
476
48i
7 85
48 7
492
498
5o4
5og
5'5
52O
526
53i
53 7
786
542
548
553
55 9
564
5 7 o
5 7 5
58i
586
592
787
5 97
6o3
609
6i4
620
626
63i
636
642
64?
788
653
658
664
669
675
680
686
691
697
702
789
708
7 i3
-719
724
7 3o
7 35
7 4i
7 46
752
7 5 7
790
763
7 68
774
779
785
79<>
796
801
807
812
N
O
1
2
3
4
5
6
7
S
9
79O 82O
N
O
1
2
3
4
5
G
7
8
9
700
89 7 63
768
774
779
75
79
796
801
807
812
79 r
818
823
829
834
84o
845
85i
856
862
867
792
873
878
883
889
8 9 4
900
goS
911
916
922
79 3
927
933
9 38
9 44
9 4 9
955
960
966
97i
977
794
982
988
99 3
998
*oo4
*oog
^ ^
'5
*O2O
*O26
*o3i
79 5
90 037
042
o48
o53
o5 9
o64
069
075
080
086
796
091
097
103
108
ii3
119
124
129
135
i4o
797
i46
i 5i
i5 7
162
168
I 7 3
I7 9
1 84
189
'95
798
200
206
211
217
222
22 7
233
238
244
249
799
2 55
260
266
2 7 I
276
282
287
293
298
3o4
800
3o 9
3f4
32O
3 2 5
33l
336
342
347
352
358
801
363
36 9
3 7 4
38o
385
390
3 9 6
4oi
4o 7
4i 2
802
4i 7
423
428
434
43 9
445
450
455
46i
466
8o3
472
477
482
488
493
499
5o4
5og
5'5
52O
8o4
526
53f
536
542
547
553
558
563
56 9
5?4
8o5
58o
585
5 9 o
596
601
607
612
617
623
628
806
v 634
63 9
644
650
655
660
666
671
677
682
807
687
6 9 3
698
703
709
7 i4
720
7 25
?3o
7 36
808
7 4i
747
7 52
7 5 7
7 63
7 68
77 3
779
7 84
789
809
795
800
806
811
816
822
82 7
832
838
843
810
849
854
85 9
865
870
8 7 5
881
886
891
897
811
902
907
913
918
924
929
9 34
94o
9^5
9 5o
812
9 56
961
966
972
977
982
988
99 3
998
*oo4
8i3
91 009
"i4
020
O25
o3o
o36
o4i
o46
o52
057
8i4
062
068
o 7 3
078
o84
089
094
IOO
io5
I 10
8i5
116
I '21
126
I 32
i3 7
i4a
i48
i53
i58
164
816
169
i ?4
1 80
185
I 9 O
196
2OI
206
212
217
817
222
228
233
238
243
249
254
269
265
270
818
2 7 5
281
286
291
2 97
302
307
3l2
,3i8
323
819
328
334
33 9
344
350
355
36o
365
37'
3 7 6
820
38 1
387
3 9 2
3 9 7
4o3
4o8
4i3
4i8
424
429
N
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
2.0
5
3.o 5
2.5
6
3.6 6
3.o
7
4.a 7
3.5
8
4.8 8
4.o
9
5.4 9
4.5
82O 86O
N
O
1
a
3
4
8
6
7
8
9
820
91 38i
387
392
3 97
4o3
4 08
4i3
4i8
424
429
821
434
44o
445
45o
455
46 1
466
471
477
482
822
48 7
492
498
5o3
5o8
5i4
5i 9
524
529
5 3 5
823
54o
545
55i
556
56i
566
5 7 2
5 77
582
58 7
824
5g3
598
6o3
609
6i4
619
624
63o
635
64o
826
645
65i
656
661
666
672
677
682
68 7
6 9 3
826
698
708
709
714
719
724
73o
735
74o
7 45
827
7 5i
7 56
761
766
772
777
782
787
79 3
798
828
8o3
808
8i4
819
824
829
834
84o
845
85o
829
855
861
866
8 7 z
876
882
887
892
897
903
830
908
9i3
918
924
929
934
9 3 9
944
950
955
83i
960
g65
971
976
981
986
991
997
*OO2
*oo 7
832
92 OI2
018
023
028
o33
o38
o44
o4g
o54
o5<j
833
065
070
75
080
08 5
091
096
IOI
106
in
834
117
122
127
i3a
i3 7
i43
i48
i53
i58
i63
835
169
i?4
179
1 84
189
95
200
205
2IO
. 21 5
836
221
226
3l
236
241
247
252
25 ?
262'
26 7
83 7
2 7 3
278
283
288
293
298
3o4
3og
3i4
319
838
324
33o
335
34o
345
35o
355
36i
366
3 7 i
83 9
376
38 1
387
3g2
3 97
402
407
4l2
4i8
423
840
428
433
438
443
449
454
45 9
464
46 9
4 7 4
84i
48o
485
4go
495
5oo
5o5
5n
5i6
521
526
842
53i
536
542
54?
552
55 7
562
56 7
5 7 2
5 7 8
843
583
588
593
5 9 8
6o3
609
614
619
624
629
844
634
63 9
64$
650
655
660
665
670
6 7 5
68 1
845
686
691
696
701
7 o6
711
716
7 22
7 2 7
732
846
?3?
742
?4?
752
7 58
763
768
77 3
778
783
84?
788
793
799
8o4
809
8i4
819
824
829
834
848
84o
845
850
855
860
865
870
8 7 5
881
886
849
891
896
901
906
911
916
921
9 2 7
9 32
9 3 7
850
942
947
952
9 5 7
962
967
97 3
978
9 83
988
85i
99 3
998
*oo3
*oo8
*oi3
*oi8
*O24
*029
*o34
*o39
862
9 3o44
o4g
o54
oSg
o64
069
075
080
085
090
853
095
IOO
io5
no
u5
I2O
125
l3i
i36
i4i
854
i46
i5i
i56
161
166
171
176
181
186
192
855
197
202
207
212
217
222
227
232
23 7
242
856
24?
252
258
263
268
2 7 3
278
283
288
293
85 7
298
3o3
3o8
3i3
3i8
3 2 3
3 2 8
334
33 9
344
858
349
354
359
364
369
3 7 4
3 79
384
389
3 9 4
85g
3 99
4o4
409
4i4
420
425
43o
435
44o
445
860
45o
455
46o
465
470
4 7 5
48o
485
4go
495
N
O
1
2
3
4
5
6
7
8
9
23
800-890
N
i
2
3
4
5
6
7
8
9
800
93450
455
46o
46g
470
4?5
48o
485
490
495
861
5oo
5o5
5io
5i5
52O
526
53i
536
54 1
546
862
55i
556
56i
566
5 7 i
576
58i
586
591
5 9 6
863
601
606
611
616
621
626
63i
636
64 1
646
864
65i
656
661
666
671
676
682
687
692
697
865
702
707
712
717
722
727
732
?3?
7 42
?4?
866
752
7 5 7
762
767
772
777
782
78?
79 2
797
867
802
807
812
817
822
827
832
83 7
842
847
868
85a
857
862
867
872
877
882
887
892
897
869
902
907
912
917
922
927
932
9 3 7
942
947
870
952
9 5 7
962
967
972
977
982
987
992
997
871
94 002
007
OI2
017
022
027
O32
037
042
o4?
872
052
057
062
067
072
077
082
086
091
096
873
101
106
III
116
121
126
I
3i
i36
i4i
146
874
i5i
1 56
161
166
171
176
181
186
191
196
8 7 5
201
206
211
216
221
226
23l
236
240
245
876
25o
255
260
265
270
275
280
285
290
2 9 5
877
3oo
3os
3io
315
320
325
33o
335
34o
345
878
349
354
359
364
369
3 7 4
3 79
384
38g
3 9 4
879
3 99
4o4
409
4i4
419
424
429
433
438
443
880
448
453
458
463
468
473
478
483
488
493
881
498
5o3
507
5l2
5i 7
522
527
532
53 7
542
882
547
552
55 7
562
56?
5 7 i
5 7 6
58i
586
691
883
596
601
606
611
616
621
626
63o
635
64o
884
645
65o
655
660
665
670
675
680
685
689
885
6 9 4
699
704
709
714
719
724
729
734
y38
886
743
748
7 53
758
7 63
768
773
778
7 83
787
887
792
797
802
807
812
817
822
827
832
836
888
84 1
846
85i
856
86 1
866
87,
876
880
885
889
890
8 9 5
900
95
910
9'5
919
924
929
934
890
9 3 9
<;4 i
949
954
9 5 9
9 63
968
9?3
978
983
N
g
1 2
3
4
5
<
7
8
J)
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
2.0
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
890-93O
N
1
2
3
4
5
6
7
8
9
890
94939
9 44
949
954
9 5 9
963
968
97 3
978
9 83
891
988
993
998
*OO2
*oo7
*OI2
*oi7
*O22
*O27
*032
892
96 o36
o4i
o46
o5i
o56
061
066
071
075
080
893
o85
090
95
IOO
105
109
n4
II 9
124
129
8 9 4
1 34
1 3g
i43
i48
i53
i58
i63
168
I 7 3
I 77
8 9 5
182
187
192
197
202
2O 7
21 I
216
221
226
896
3i
236
z4o
245
25o
255
260
265
270
2 7 4
897
279
284
289
294
299
3o3
3o8
3i3
3i8
323
898
328
332
33 7
342
34?
352
357
36i
366
3 7 i
899
376
38i
386
Sgo
3 9 5
4oo
405
4io
4i5
4ig
900
424
429
434
43 9
444
448
453
458
463
468
901
4?2
4?7
482
48 7
492
497
5oi
5o6
5n
5i6
902
521
5 2 5
53o
535
54o
545
55
554
SSg
564
908
56 9
5?4
5 7 8
583
588
593
5 9 8
602
607
612
904
617
622
626
63i
636
64 1
646
65o
655
660
gb5
665
670
674
679
684
689
694
698
703
708
906
7 i3
718
722
727
732
7 3 7
742
746
7 5i
7 56
907
761
766
77
77 5
780
785
789
794
799
8o4
908
809
8i3
818
823
828
832
83 7
842
84 7
852
99
856
861
866
871
8 7 5
880
885
890
895
899
910
904
909
914
918
923
928
933
g38
942
94 7
911
952
9 5 7
961
966
971
9 7 6
980
9 85
99
995
912
999
*oo4
*oo9
*oi4
*oi9
*023
*028
*o33
*o38
*o42
918
96 047
052
o5 7
06 1
066
O 7 I
076
080
o85
090
914
95
099
io4
109
n4
118
123
128
i33
i3 7
916
i4a
147
152
i56
161
166
171
i 7 5
1 80
185
916
190
194
199
204
209
2l3
218
223
22 7
232
917
23 7
242
246
25l
2 56
261
265
270
2 75
280
918
284
289
294
298
3o3
3o8
3i3
3i 7
322
32 7
919
332
336
34 1
346
35o
355
36o
365
36 9
3 7 4
920
3 79
384
388
3 9 3
3 9 8
402
407
4l2
4i 7
421
921
426
43i
435
44o
445
450
454
45 9
464
468
922
4?3
478
483
487
4g2
497
5oi
5o6
5n
5i5
923
52O
5 2 5
53o
534
539
544
548
553
55.8
562
924
56 7
5 7 2
5 77
58i
586
5gi
5 9 5
600
605
609
926
6i4
619
624
628
633
638
642
647
652
656
926
661
666
670
6 7 5
680
685
689
6 9 4
699
7 o3
927
708
73
717
722
727
7 3i
7 36
74 1
7 45
7 5o
928
755
7 5 9
764
769
774
778
783
788
79 2
797
9 2 9
802
806
811
816
820
825
83o
834
83g
844
930
848
853
858
862
867
872
876
88 1
886
890
N
1
2
3
4
5
6
7
8
9
25
930-960
N
1
2
3
4
5
O
7
8
9
930
96 848
853
858
862
867
872
876
881
886
890
9 3i
895
900
904
909
914
918
9 23
928
932
9 3 7
g32
942
946
9 5i
956
960
9 6 5
970
974
979
9 84
9 33
988
99 3
997
*OO2
*OO 7
*OI I
*oi6
*02I
*025
*o3o
9 34
97035
o3 9
o44
049
o53
o58
o63
067
072
077
9 35
08 1
086
o 9 o
09 5
IOO
104
109
u4
118
123
9 36
128
132
i3 7
141
1 46
i5i
i55
160
165
169
9 3 7
174
179
i83
188
I 9 2
197
202
206
21 I
216
9 38
220
225
230
234
23 9
243
248
253
257
262
9 3 9
267
271
276
280
285
2 9 O
2 9 4
2 99
3o4
3o8
940
3i3
3i 7
322
32 7
33i
336
34o
345
350
354
9 4i
35 9
364
368
3 7 3
377
382
3
87
3 9 i
3 9 6
4oo
942
4o5
4io
4i4
4i 9
424
428
433
43 7
442
44?
943
45i
456
46o
465
470
4?4
479
483
488
4 9 3
944
497
502
5o6
5u
5i6
52O
525
52 9
534
53 9
945
543
548
552
55 7
562
566
5 7 i
5 7 5
58o
585
946
58 9
594
5 9 8
6o3
607
612
617
621
626
63o
94?
635
64o
644
64 9
653
658
663
667
672
676
948
681
685
6 9 o
6 95
6 99
704
708
7 i3
717
722
949
727
7 3i
736
74o
745
74 9
?54
759
763
768
950
772
111
782
786
791
7 9 5
800
8o4
809
8i3
9 5i
818
823
827
832
836
84 1
845
85o
855
85 9
9 52
864
868
8 7 3
877
882
886
891
896
900
95
953
909
914
918
9 23
928
932
9 3 7
94 1
946
9 5o
954
955
9 5 9
964
9 68
973
978
9 82
987
99'
99 6
9 55
98 ooo
005
009
oi4
019
O23
028
o3a
037
o4i
956
o46
o5o
055
o5 9
o64
068
o 7 3
078
082
087
9 5 7
091
o 9 6
IOO
105
109
u4
118
123
127
I 32
9 58
i3 7
i4i
1 46
i5o
155
i5 9
164
168
i 7 3
177
9 5 9
182
186
191
, 9 5
200
2O4
20 9
2l4
218
223
960
227
232
236
24 1
245
250
254
2 5 9
263
268
N
1 2
3
4
5
6
7
8
9
PP 5
4
i o.5
i
o.4
2 1.0
2
0.8
3 i.5
3
1.2
4 2.0
4
1.6
5 2.5
5
2.0
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
o
1
2
3
4
5
6
7
8 9
9oo
98 227
232
236
24l
245
250
254
25g
263
268
961
272
2 77
281
286
290
2 95
299
3o4
3o8
3i3
962
3i8
322
327
33i
336
34o
345
349
354
358
963
363
367
372
3 7 6
38i
385
390
3 9 4
3 99
4o3
964
4o8
4l2
4i 7
421
426
43o
435
43 9
444
448-
965
453
457
462
466
4 7 i
4 7 5
48o
484
48 9
4 9 3
966
498
502
607
5n
5i6
52O
5 2 5
529
534
538
967
543
547
552
556
56i
565
5 7 o
5 7 4
5 79
583
9 68
588
592
5 97
601
6o5
610
6i4
619
623
628
969
632
63 7
64 1
646
65o
655
65g
664
668
6 7 3
970
677
682
686
691
6 9 5
700
7 o4
709
7 i3
717
971
722
726
7 3i
7 35
7 4o
7 44
?4g
7 53
7 58
762
972
767
771
776
780
?84
789
79 3
798
802
807
97 3
811
816
820
825
829
834
838
843
84 7
85i
974
856
860
865
869
8 7 4
8 7 8
883
887
892
896
97 5
900
95
909
914
918
923
927
932
936
941
976
945
949
954
958
9 63
9 6 7
972
976
981
9 85
977
989
994
998
*oo3
*oo 7
*OI2
*oi6
*021
*025
*O29
978
99 o34
o38
o43
o4 7
o52
o56
06 1
065
069
074
979
078
o83
087
092
096
IOO
105
109
u4
118
980
123
127
i3i
i36
i4o
i45
149
j54
1 58
162
981
167
171
176
180
185
189
193
198
202
207
982
21 I
216
220
224
229
233
238
242
24 7
a5i
983
255
260
264
269
2 7 3
2 77
282
286
291
295
984
3oo
3o4
3o8
3i3
3i 7
322
3 2 6
33o
335
33 9
986
344
348
352
35 7
36i
366
3 7 o
3 7 4
379
383
986
388
392
396
4oi
4o5
4io
4i4
419
423
42 7
987
432
436
44i
445
449
454
458
463
467
4 7 i
988
4?6
48o
484
48 9
4g3
498
5O2
5o6
5ii
5i5
989
620
524
528
533
53 7
542
546
55o
555
55 9
990
564
568
572
577
58 1
585
Sgo
5 9 4
5 99
6o3
991
607
612
616
621
625
629
634
638
642
64 7
992
65i
656
660
664
669
6 7 3
677
682
686
691
99 3
6 95
699
704
708
712
7 i 7
7 2I
726
73o
7 34
994
7 3 9
?43
747
752
7 56
7 6o
7 6 5
769
774
778
99 5
782
787
791
79 5
800
8o4
808
8i3
817
822
996
826
83o
835
83 9
843
848
852
856
861
865
997
870
8 7 4
878
883
88 7
891
896
900
904
909
998
gi3
917
922
926
93o
9 3 5
9 3 9
944
948
952
999
9 5 7
961
965
970
974
978
983
987
991
996
1000
oo ooo
oo4
009
oi3
OI 7
022
026
o3o
035-
oSg
X
O
1
2
3
4
5
6
7
8
9
TABLE II
FIVE -PL ACE LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
TO EVERY MINUTE
0.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
o . oo ooo
60
1
6.46 373
6.46 373
3.53627
o.oo ooo
5 9
2
6 . 76 4?6
6. 76 4?6
3.23 524
o.oo ooo
58
3
6.94 085
17609
6.94085
17609
3.o5 915
o.oo ooo
57
12494
I2 49
4
5
7.06 579
7. 16 270
9691
7.06 679
7.16 270
9691
2.93 421
2.83 730
o.oo ooo
o.oo ooo
56
55
6
7.24 188
7.24 188
7918
2.76 812
o.oo ooo
54
6694
669
1
7
8
7.30882
7.36682
5800
7.80882
7-36682
5800
2.69 1 1 8
2.633i8
o.oo ooo
o.oo ooo
53
52
9
7.41 797
5"5
7-4i 797
5"5
2.58 2o3
o.oo ooo
5i
10
7-46 3 7 3
7 .463 7 3
4576
2.53 627
o.oo ooo
50
1 1
7.5o 5i2
4'39
7.5o 5i2
4139
2. 4g 488
o.oo ooo
49
12
i3
7.54 291
7.57 767
3476
7.54 291
7.57 767
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
2.39 oi4
2.36oi8
o.oo ooo
o.oo ooo
46
45
16
7.66 784
7.66785
2803
2.33 2i5
o.oo ooo
44
2633
263
?
'7
7.69 417
2483
7.69418
2482
2.3o 582
9.99999
43
18
'9
7.71 900
7.74 248
2348
7.71 900
7.74 248
2348
2.28 100
2.25 762
9.99999
9-99999
42
4i
20
7.76 4?5
7.76 476
2.23 524
9-99999
40
21
7. 7 85 9 4
7.78 595
2.21 4O5
9-99999
3 9
22
7.80 615
7.8o6i5
2.19 385
9.99999
38
23
7.82545
7.82 546
193
2.17 454
9.99999
37
1848
184
1
24
25
7.843 9 3
7.86 166
1773
7-843 9 4
7.86 167
'773
2. I 5 606
2.i3 833
9.99999
9-99999
36
35
26
7.87870
1704
7.87871
2.12 129
9-99999
34
1639
163
J
27
7 .8c
) 59
7-8<
) 5io
2. I
0*490
9-99999
33
28
7.91 088
7.91 089
2.O8 911
9-99999
32
29
7.92 612
7.92 6i3
2.O7 387
9.99998
3i
30
7.94 o84
7 ,g4 086
2.o5 914
9.99998
30
L. Cos.
d.
L. Cotg.
d.
L. Ta,ng.
L
Sin.
' ~
89 3O .
PP
9691
4576
2997
2483
2119
1848
1704
1579
1472
.1
969
4S8
3 oo
.1
248
212
,85
.1
170
'S8
'47
.2
19,8
9'5
599
.2
497
424
37
.2
34'
jrf
294
3
2907
899
3
745
6 3 6
554
3
5"
474
442
4
3876
1830
1199
4
993
848
739
4
682
632
589
5
4846
2288
1498
5
1242
Io6o
924
5
852
789
736
.6
2646
1798
.6
1490
1271
1109
.6
IO22
947
883
7
6784
3203
2098
7
1738
1483
1294
7
"93
1105
1030
.8
7753
3661
2398
.8
1986
1695
,478
.8
1363
1263
1178
8722
4.18
2697
9
9 '5^4
1421
1325
3o
O 3O .
'
L. Sin.
d.
L. Tang 1 . d.
L. Cotg.
L.
Cos.
30
7.94 084
1424
1379
1336
1297
1259
1223
1190
1158
1128
IIOO
1072
1046
IO22
999
976
954
934
914
896
877
860
843
827
812
797
782
769
755
743
730
7.94 o6
1424
'379
1336
1297
1259
1223
1190
"59
1128
IIOO
1072
1047
1022
99 8
976
955
934
9'5
895
878
860
843
828
812
797
782
769
756
742
73
2.u5 914
9.99998
30
3i
32
33
34
35
36
3 7
38
3 9
7.95 5o8
7.96 887
7.98 223
7.99 620
8.00 779
8. 02 002
8.o3 192
8.o4 35o
8.o5478
7.96 5io
7.96 889
7.98 225
7.99 522
8.00 781
8 . 02 oo4
8.o3 194
8.o4 353
8.o548i
2.04 490
2.03 III
2.01 775
2.OO 4?8
I .99 219
I. 9 79 9 6
I .96 806
i .95 64?
i .94 519
9.99998
9.99998
9.99998
9.99998
9.99998
9.99998
9-99997
9.99997
9-99997
29
28
27
26
25
24
23
22
21
40
8.06
5 7 8
8.06 58i
i .93 4 1 9
9-99997
20
4i
42
43
44
45
46
4?
48
49
8.07 650
8.08 696
8.09 718
8. 10 717
8. 1 1 6 9 3
8.12 647
.8.i3 58i
8.i44g5
8.i5 3gi
8.07 653
8.08 700
8.09 722
8. 10 720
8. 1 1 696
8.i265i
8.13585
8. 1 4 500
8.i5 3 9 5
i .92 347
1.91 3oo
i .90 278
1.89 280
1.88 3o4
1.87 349
1.86415
i .85 5oo
i.84 605
9-99997
9.99997
9.99997
9.99996
9.99996
9.99996
9.99996
9.99996
9.99996
'9
18
17
16
i5
i4
i3
12
I I
50
8.16268
8.16 273
i.83 727
9.99995
10
5i
52
53
54
55
56
5?
58
5 9
8.17 128
8.17 971
8.18 798
8.19 610
8. 20 407
8.21 189
8.21 968
8.22 713
8.23456
8.17 :33
8.17 976
8.18804
8.19616
8.2o4i3
8.21 196
8.21 964
8.22 720
8.23 462
1.82 867
i .82 024
i .81 196
i. 80 384
i. 79 58 7
1.78805
1.78 o36
i .77 280
1.76 538
9.99995
9.99995
9-99995
9-99995
9-99994
9.99994
9.99994
9.99994
9.99994
9
8
7
6
5
4
3
2
I
60
8.24 186
8.24 192
1.75 808
9.99993
L. Cos.
d.
L. Cotg.
d.
L. Tang-.
L.
Sin.
'
89.
PP
.1
.2
3
4
.6
'.B
1379
1323
IIOO
no
22O
33
440
550
660
770
880
990
.1
.2
3
4
'.6
:!
9
999
914
860
812
769
730
138
276
414
552
690
827
965
1103
1241
122
245
367
489
612
734
856
978
IIOI
100
200
300
400
500
599
699
799
899
qi
'83
274
366
457
548
640
73i
823
86
172
258
344
43
5'6
602
688
774
.1
.2
3
4
.6
.B
81
162
2 44
3 ^
406
487
568
650
73'
77
'54
231
308
385
461
I 38
615
692
%
219
292
365
438
5 i'
584
657
3i
1.
/
L. Sin. d.
L. Tang.
d.
L. Cotg.
L.
Cos.
8.24 186
8.24 192
1.75 808
9.99993
60
I
8.24903
706
8.24 910
706
I .75 090
9.99 9 y3
5 9
2
8.26 609
8.25 616
1.74384
9.99993
58
3
8.26 3o4
695
8.26 3i2
1.73688
9.99993
^7
684
684
4
8.26988
671
8.26 996
6?i
i .73 oo4
9.99992
56
5
8.27 661
8.27 669
1.72 33i
9.99992
55
6
8.28 324
8.28332
663
1.71 668
9.99992
54
653
654
7
8.28 977
(,AA
8.28 986
64.1
1.71 oi4
9.99992
53
8
8.29 621
8.29 629
i .70 371
9.99 992
52
9
8.30255
634
8.3o 263
634
i .69 737
9.99991
5i
10
8.30879
t)24
616
8.3o888
6,1
I .69 112
9-99 99'
50
1 1
8.3i
4q5
608
8.3i 505
607
1.68495
9.99991
49
12
8.32 io3
8.32 112
1.67888
9.99990
48
i3
8.32 702
599
8.32 711
599
I .67 289
9.99990
47
590
59 1
i4
8.33 292
583
8.33 3o2
184
1.66 698
9.99990
46
i5
8.33875
8.33886
1.66 ii4
9.99990
45
16
8.3445o
568
8.3446i
575
568
i.6553 9
9.99989
44
17
8.35 018
560
8.35 029
cfif
i .64 971
9.99989
43
18
8.35 5?8
8.35 590
i .64 4io
9.99989
42
'9
8.36 i3i
8.36 i43
553
1.63857
9.99989
4i
20
8.36678
539
8.36689
54 6
i .63 3u
9.99 988
40
21
8.37 217
533
8.37 229
i .62 771
9.99 988
3 9
22
8.37750
r 2 6
8.37 762
1.62 :>38
9.99988
38
23
8.38276
8.38 289
527
i .61 711
9.99 987
37
5 2C
24
8.38 796
514
8.38 809
i .61 191
9.99987
36
25
8.39 3io
cnS
8.3 9 323
i .60 677
9.99987
35
26
8.39818
8. 3g 832
509
i. 60 168
9.99986
34
502
50:
27
8.40 320
406
8.4o334
i .5g 666
9.99986
33
28
8.4o8i6
8.4o 83o
i .59 170
9.99986
32
2 9
8.4i 307
491
8.4
321
49!
i.58 679
9.99 985
3i
30
8.4i 792
485
8.4
807
486
i.58 i 9 3
9.99985
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
-
88 3O
PP
706
663
634
599
575
553
533
5M
496
.1
70.6
66.3
63.4
.1
59-9
57-5
55- 3
.1
53-3
51-4
49.6
.2
141.2
132.6
126.8
.2
119.8
115.0
110.6
.2
106.6
102.8
99.2
3
211.8
198.9
190.2
3
"79-7
'72-5
165.9
3
159-9
154.2
148.8
4
282.4
265.2
2536
4
239.6
230.0
221.2
4
213.2
205.6
198.4
5
353-0
33i-5
3'7-
5
299.5
287. s
276-5
5
266. s
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
4'9-3
402.5
387-'
7
373-
359-8
347-2
.8
564.8
530-4
507.2
.8
479.2
460.0
442-4
.8
426.4
411.2
396.8
596.7 570.6
Q
539- 1 5'7-5
4<'7-7
462 6
446.4
32
13O.
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
30
8.4i
792
8.4i
807
i .58 193
9.99985
30
3i
8.42
272
8.42
287
1.57 713
9.99985
29
32
8.42
746
8.42
762
1.57 238
9-99 9 8 4
28
33
8.43
216
470
8.43 232
470
i.56 768
9.99984
27
464
464
34
8.4368o
8.43696
i .56 3o4
9-99 984
26
35
8.44
i3g
459
8.44 i56
1.55844
9.99983
25
36
8.44
5 9 4
455
8.446n
455
i.55 38g
9.99983
24
37
8.45
o44
45
8.45 061
45
i.54 939
9.99983
23
38
8.45489
445
8.45 5o 7
1.54493
9.99982
22
3 9
8.45
g3o
441
8.45 948
441
I .54 O52
9-99 982
21
40
8.46
366
436
8.46 385
437
i.53 6i5
9.99 982
20
4i
8.46
799
433
8.46 817
432
428
i.53 i83
9.9998!
'9
42
8.47
226
8.4? 245
i.52 755
9.99981
18
43
8.47
650
424
8.4? 669
424
i.52 33i
9.99981
'7
419
44
8.48069
8.48089
416
i .5i 911
9.99980
16
45
8.48485
8.48 5o5
i .5i 495
9.99980
i5
46
8.48896
411
8.48 917
412
i .5
i o83
9-99979
i4
47
8.49 3o4
408
8.49 325
i 50675
9-99979
i3
48
8.49 708
8.49 729
i .5o 271
9-99979
12
49
8.5o 1 08
8.5o i3o
401
i .4g 870
9.99978
I I
50
8.5o 5o4
396
8.5o 527
397
1.49473
9.99978
10
5!
8.5o 897
393
8.5o 920
393
i .49 080
9-99 977
9
r
8.5i
287
8.5i
3io
i .48 690
9-99977
8
53
8.5i
673
382
8.5i
696
300
383
i.48 3o4
9-99977
7
54
8.52o55
8 .52 079
180
i .47 921
9.99976
6
55
8.52434
8.52 45g
i .4? 54i
9.99976
5
56
8.52 810
37 6
8.52 835
37 6
i .47 i65
9.99975
4
373
373
5 7
8.53 i83
8.53 208
i .46 792
9-99975
3
58
8.53 552
8.53 578
i .46 422
9.99974
2
5 9
8.53 919
367
8.53 945
3 6 7
i.46o55
9.99974
I
60
8.54282
8.54 3o8
i .45 692
9.99974
L.Cos.
d. | L. Cotg.
d.
L. Tang.
L.
Sin.
88.
PP
470
455
441
424
4 o8
396
386
376
367
.1
47.0
45-5
44.1
.1
42.4
40.8
39-6
.1
38.6
37.6
36.7
.2
94.0
91 o
88.2
.2
84.8
81.6
79.2
.2
77.2
75.2
73-4-
3
141.0
'3 6 -5
132.3
3
127.2
122.4
118.8
3
115-8
II2.8
10. 1
4
188.0
182.0
176.4
4
169.6
163.2
158.4
4
'544
150.4
46.8
5
235.0
227.5
220.5
5
212.
204.0
198.0
5
193.0
188.0
183-5
6
282.0
2 73-
264.6
.6
254-4
244-8
237.6
.6
231.6
225.6
220.2
7
329.0
318-5
308.7
7
206.8
285.6
277-2
7
270.2
263.2
856.9
.8
376.0
364.0
352-8
.8
339-2
326 4
316.8
.8
308.8
300.8
293.6
.9 423.0
409. 5 396. g
9
38i.fi ^67.2
356-4
9 347-4
338.4 330.3
33
2.
/
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
8.54282
8.54 3o8
i .45 692
9.99974
60
I
8.54642
357
8.54669
M
1
1.45 33i
9.99973
5 9
2
8.54 999
8.55 02 7
i .44 9?3
9.99973
58
3
8.55 354
355
8.55382
355
1.44618
9.99972
5 7
4
8.55 7 o5
8.55 734
352
i .44 266
9.99972
56
5
8.56 o54
8.56o83
i.43 917
9.99971
55
6
8.56 4oo
346
8.56429
34 6
i.43 571
9.99971
54
7
8.56 743
343
8.66773
344
i.43 227
9.99970
53
8
8.57 o84
8.5 7 n4
1.42 886
9.99970
62
9
8.57 421
337
8.57452
338
1.42 548
9.99969
5i
10
8.5 77 5 7
33 6
8.57 788
33 6
I .42 212
9.99969
50
1 1
8.58 089
8.58 121
333
33
I ./
ii8 79
9.99968
49
12
8.58 419
8.5845i
I .1
ii 54g
9.99968
48
1 3
8.58 7 47
8.58 779
3 2
I .4l 221
9.99967
47
i4
i5
S.Sg 072
8.5 9 3 9 5
325
323
8.59 io5
8.59428
326
323
I .40 895
i .4o 572
9.99967
9.99967
46
45
16
8.5g 715
8.59 749
321
i .40 25i
9.99966
44
i?
8.60 o33
316
8.60068
319
Jyfi
i.3g 982
9.99966
43
18
8.60 349
8.60 384
i.Sg 616
9.99965
42
19
8.60 662
3'3
8.60698
3M
i .39 3o2
9.99964
4i
20
8.60 973
8. 6 1 009
3"
i .33 991
9.99964
40
21
8.61 282
307
8.61 3ig
310
i. 3868i
9.99963
3 9
22
8.61 58g
8.61 626
i.38 374
9.99963
38
23
8.61 894
8.61 9 3i
305
1.38069
9.99962
3 7
302
3
1
24
8.62 196
301
8.62 234
1.37 766
9.99962
36
25
8.62 497
8.62 535
1.37465
9.99961
35
26
8.62 795
8.62834
299
1.37 166
9.99961
34
296
29
r
27
28
8.63 091
8.63 385
294
8.63 i3i
8.63426
295
1.36 869
i.36 574
9.99960
9.99960
33
32
2 9
8.63 678
293
8.63 718
i.36 282
9-99 9 5 9
3i
30
8.63 968
8.64 009
291
1.35 991
9-99959
30
L.
Cos.
d.
L. Cotg. d.
L.
Tang.
L. Sin.
87 3O'.
PP
360
350
340
33<>
330
310
300
290
38 5
.1
,6
35
34
.!
33
32
3'
.1
3
29
28.5
.2
72
70
68
2
66
64
62
.2
60
s
57-0
3
1 08
105
IO2
3
99
96
93
3
90
8 7
85.5
4
144
140
itf
4
132
128
124
4
120
116
114.0
5
1 80
175
170
165
1 60
155
5
ISO
'45
142 5
.6
216
210
204
6
198
192
1 86
.6
1 80
'74
171.0
7
252
245
2 3 8
7
231
224
217
7
3IO
203
199-5
.8
288
280
272
8
264
256
248
.8
240
232
228.0
9 3 2 4
SJS
306
2Q7 288
279
9
270
256-5
34
2 3O .
,
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
30
8.63 968
8.64 009
1 i .35 991
9-99 9 5 9
30
3i
8.64 256
287
8.64 298
287
i .35 702
9.99958
2 9
32
8.64 543
8.64585
i.35 415
9.99958
28
33
8.64827
8.64 870
285
i .35 i3o
9-99 U 5 7
27
283
28,
34
8.65 no
8.65 i54
281
1.34846
9.99956
26
35
8.65 3 9 i
8.65435
1.34565
9.99956
25
36
8.65 670
279
8. 65 715
1.34285
9.99 955
24
277
2?
3?
8.65 9 4?
8.65 99 3
i .34 007
9.99955
23
38
8.66223
8.66 269
i.33 7 3i
9.99954
22
39
8.66497
274
8.66 543
274
1.33457
9.99 9 5 4
21
40
8.66 769
8.66816
i.33 1 84
9.99953
20
4i
8 .67 o3g
8.67 087
271
i .32 gi3
9.99952
19
42
8.67 3o8
2 9
8.6 7 356
i.32 644
9.99962
18
43
8.6
7 5 7 5
266
8.6 7 624
266
1.32 3 7 6
9.99951
17
44
8.67841
8.6 7 890
vfit
I .32 I IO
9.99951
16
45
8.68 io4
2 3
8.68 1 54
i ."i
!i 846
9.99950
i5
46
8.68367
263
260
8.68 417
263
26l
i.3i 583
9.99949
i4
47
8.68 627
8.68678
260
i .r
!l 322
9.99949
i3
48
8.68 886
8.68 g38
i .3i 062
9-99948
12
49
8.69 144
258
8.69 196
258
i.3o 8o4
9-99948
I I
50
8 .69 4oo
8.69453
257
i.3o54 7
9-9994?
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
9.99946
8
53
8.70 i5g
252
8.70 214
252
i .29 7 86
9.99945
7
250
25
54
8.70 409
8.70465
1.29 535
9-99944
6
55
8.70 658
8.70 714
i .29 286
9.99944
5
56
8.70905
247
8.70 962
2 4 a
i .29 o38
9.99943
4
240
5?
8.71 i5i
8.71 208
i .28 7 g2
9.99942
3
58
8.71 3g5
8.71 453
1.28 54 7
9.99942
2
5 9
8.71 638
243
8. 7 i 6g 7
244
1.28 3o3
9-99 94 i
I
60
8.71 880
8.71 g4o
i .:
28 060
9.99940
L.
Cos.
d.
L. Cotg.
d.
L. Tang. | L.
Sin.
f
87.
PP
280
275
270
265
260
255
250
45
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
SS-o
54-o
.2 53.0
52.0
51.0
.2
50.0
49.0
48.0
3
84.0
82.5
81.0
3 79-5
78.0
76.5
3
75-o
73-5
72.0
4
112.
IIO.O
108.0
.4 Td6.0
104.0
IO2.O
4
100.0
98.0
96.0
5
140.0
'37-5
135-0
5 132-5
130.0
127.5
5
125.0
122.5
I2O.O
.6
168.0
165.0
162.0
.6 159.0
156.0
'53-0
.6
150.0
147.0
144.0
7
196.0
192-5
189.0
7 '85.5
182.0
178.5
7
175-0
171.5
1 68.0
.8
224.0
220. o
216.0
.8 212.
208 o
204.0
.8
2OO.O
196.0
192.0
<
252.0
247-5
2 43-
9 238.5 234.0
229.5
.p 225.0
220.5
216.0
35
3.
'
L.
Sin.
d.
L.Tang. d. L. Cotg.
L.
Cos.
8.71 880
340
239
238
237
235
234
232
232
230
229
228
226
226
224
223
222
22O
22O
2ig
2I 7
216
216
214
213
212
211
2IO
209
208
208
8.71 g4o
241
239
239
237
236
234
234
232
231
329
229
227
226
225
224
222
222
220
219
2IQ
217
216
215
214
213
211
211
210
209
208
I .28 060
9.99 y 4o
60
I
2
3
4
5
6
7
8
9
8.72 120
8.72 359
8.72 697
8. 72 834
8.73 069
8. 7 3 3o3
8.73535
8.73 767
8. 7 3 997
8.72 181
8.72 420
8.72 65g
8.72 896
8.73 i32
8.73 366
8.73 600
8.73832
8. 7 4o63
I .27 819
i .27 58o
i .27 34i
i .27 104
1.26868
1.26634
i .26 4oo
1.26 168
i .25 937
9.99940
9.99 9 3 9
9.99 938
9.99938
9.99 9 3 7
9.99936
9.99936
9.99935
9.99 9 34
5 9
58
5 7
56
55
54
53
52
5i
10
8.74 226
8.74 292
i .25 708
9.99934
50
1 1
12
i3
i4
i5
16
i?
18
'9
8.74454
8. 7 468o
8.74 906
8. 7 5 i3o
8.75 353
8.75575
8. 7 5 79 5
8.76 oi5
8.76234
8.74 52i
8.74748
8.74 974
8.75 199
8.75423
8. 7 5645
8. 7 586 7
8.76 087
8.76 3o6
i .25 479
I .25 252
I .25 O26
i .24 801
1.24 577
1.24355
1.24 i33
i.23 gi3
i .23 694
9.99933
9.99932
9.99932
9.99 9 3i
9.99930
9.99929
9.99929
9.99928
9.99 927
49
48
47
46
45
44
43
42
4i
20
8.76451
8.76 525
i .23 4?5
9.99926
40
21
22
23
24
25
26
27
28
29
8.76667
8.76883
8.77097
8.77 3io
8.77 522
8.7 7 733
8.77943
8.78 i52
8.78 36o
8.76 742
8.76958
8.77173
8.77 387
8.77 600
8.77 811
8.78 022
8.78 232
8. 7 844i
i.23 258
I .23 O42
I .22 827
I .22 6l3
I .22 4<">
I .22 189
I .21 978
I .21 768
I .21 55g
9.99926
9.99925
9.99924
9.99923"
9.99923
9.99922
9.99921
9.99920
9.99920
3 9
38
3?
36
35
34
33
32
3i
30
8.78 568
8.78 64g
I .21 35l
9.99919
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
86 3O .
PP
238
34
23.4
46.8
70.2
93.6
117.0
140.4
163.8
187.2
210.6
329
.i
.2
3
4
'.6
9
335
33O
316
313
308
304
.1
.2
3
4
'.6
.8
9
23.8
47 .6
71.4
5
95.2
119.0
142.8
166.6
190.4
214.2
22.9
45.8
68.7
91.6
"45
37-4
160.3
183.2
206.1
22.5
45- o
67.5
90.0
112.5
i35-o
'57-5
180.0
52- 5
22.0
&0
88.0
IIO.O
132.0
154.0
176.0
198.0
21.6 .1
432 .2
64.8 .3
86.4 .4
108.0 .5
129.6 .6
151.2 .7
172.8
104.4 -9
21.2
42.4
63.6
84.8
I o6.0
127.2
148.4
169.6
90.8
20.8
4 ,.6
62.4
832
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 30 .
/
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos.
30
8.78 568
8.78 649
i .21 35i
9.99 919
30
3i
8.78774
8.78 855
.21 145
9.99918
29
32
8.78 979
.79 061
.20 989
9.99917
28
33
8.79 i83
8 . 79 266
.20 ?34
9-99 9 1 ?
27
203
204
34
8.79 386
8.79-470
.20 53o
9.99916
26
35
8.79 588
8.79.673
.20 32?
9.99915
25
36
8.79 789
8.79 875
.20 125
9.99914
24
2OI
201
?7
8.79990
8.8c
) 076
. 19 924
9.99913
23
38
8.80 189
8.8(
) 277
201
. 19 ?23
9.99913
22
3 9
8.80 388
199
8.8c
' 4?6
I 99
. 19 524
9.99912
21
40
8.80 585
'97
8.80 6 7 4
I.19 326
9.99 911
20
4i
8.80 782
'97
8.8<
5872
1 06
I . I
9 128
9-99 9 10
19
42
8.8
3978
8.81 068
I. l8 932
9.99909
18
43
8.81 i 7 3
8.81 264
'9
i . 18 ?36
9.99909
17
194
'95
44
8.81 367
8.81 409
1. 18 54 1
9.99908
16
45
8.81 56o
8.81 653
1.18 34?
9.99907
i5
46
8.81 752
192
8.81 846
'93
1. 1 8 1 54
9.99906
i4
192
192
4 7
8. 8 1 944
8.82 o38
1.17 962
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
9.99 904
I I
50
8.82 5i3
8.82 610
1.17 3go
9.99 903
10
5i
8.82 701
187
8.8
2 799
1.17 201
9.99902
9
52
8.8
2888
8.82 987
1.17 oi3
9.99901
8
53
8.83 075
1 86
8.83 175
186
1.16825
9.99900
7
54
8.83 261
18
8.83 36i
i . 16 63g
9.99899
6
55
8.83 446
8.8354?
ifl
i.i6453
9.99898
5
56
8.83 63o
8.83 7 32
1.16 268
9.99898
4
183
-
57
8.83 8i3
8.83 916
1.16084
9.99 897
3
58
8.83 996
8.84 ioo
182
i . 1 5 900
9.99896
2
5 9
8.84 177
181
8.8
4 282
182
i .
5 718
9.99 895
I
60
8.84 358
8.84464
i.i5 536
9.99 894
L.
Cos. d.
L. Cotg.
d
L. Tang.
L.
Sin.
'
86.
PP
2OI
198
195
192
189
187
185
183
181
.1
2O. I
19.8
'9-5
.1
19.2
18.0
18.7
.1
i8. S
.8.3
18.1
.2
40.2
39- 6
39.0
.2
38-4
37-8
37-4
.2
37-o
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
5
100.5
99.0
97-5
5
96.0
Q4-.S
93-5
5
9 2 -5
9'-5
9-5
.6
120.6
118.8
117.0
.6
115.2
"3-4
1 1 2. 2
.6
III.O
109.8
108.6
7
140.7
138.6
136-5
7
134-4
I32-3
130.9
7
129.5
128.1
126.7
.8
160.8
158.4
150.0
.8
151.2
149.6
.8
148.0
146.4
144.8
.9 iSo.q
172.8 170.1
168.3
.9 166.5
164.7 162.9
4.
/
L.
Sin.
d.
L. Tang. d.
L.
Cotg.
L.
Cos.
8.84358
8.84464
182
I . i5 536
9.99894
60
I
8. 8453 9
179
8.84646
180
i .
5 354
9.99893
5y
2
8.84 718
8.84826
i . i 5 174
9.99 892
58
3
8.84897
179
8. 85 006
J -i4 994
9.99891
57
4
8.85 075
178
8.85 185
179
i.i48i5
9.99891
56
5
8.85 252
8.85 363
i .
4 637
9.99890
55
6
8.85 429
177
8.8554o
177
i . i4 46o
9.99889
54
176
'7i
7
8.85 605
8.85 717
176
i .
4283
9.99 888
53
8
8.85 780
8.858 9 3
i . i4 107
9.99887
52
9
8.85 955
'75
8.86 069
176
i .
3 g3i
9.99 886
01
10
8.86 128
173
8.86243
'74
i.i3 757
9.99 885
50
1 1
8.86 3oi
8.86417
'74
I .
3 583
9.99884
49
12
8.864?4
8.86 5 9 i
i . 1 3 4og
9.99883
48
i3
8.86645
171
8.86763
172
i .
3 237
9.99 882
47
171
"7
!
i4
8.86816
8.86935
i . 1 3 065
9.99 881
46
i5
8.86 987
169
8.87 106
i . 12 894
9.99 880
45
16
8.87 i56
169
8.87 277
171
170
1.12 723
9.99879
44
17
8.87 325
169
8. 87 447
160
i. 12 553
9.99879
43
18
8.87494
8.87 616
i. 12 384
9.99878
42
'9
8.87661
107
1 68
8.87785
169
1 . 12 2l5
9.99 877
4i
20
8.87 829
166
8.87 953
I . 12 047
9.99 876
40
21
8.87995
166
8.88 120
167
I .11 880
9.99875
3 9
22
8.88 161
16
8.88 287
i . 1 1 7i3
9.99874
38
23
8.8
8 326
164
8.8
8453
16s
i .
ii 547
9.99873
^7
24
8.88 490
164
8.8
8618
1. 1 1 382
9.99872
36
25
8.8
8654
8.88 7 83
i . 1 1 217
9.99871
35
26
8.8
8817
103
8.8
8 9 48
105
I . I I 052
9.99870
34
'63
16
*
27
O t
8 980
8.8
9111
I. 10 889
9.99869
33
28
8.8
9 142
8.8
9 274
i . 10 726
9.99868
32
29
8.8
9 3o4
8.8
9437
163
I .
10 563
9.99 867
3i
30
8. 89 464
8.89 598
I .
10 4oa
9.99 866
30
L.
Cos. i d.
L. Cotg.
d.
L.
Tang.
L.
Sin.
t
85 3O .
PP
181
179
177
175
173
.7>
168
1 66
164
.1
18.1
17.9
17-7
.1
'7-5
17-3
17.1
.1
16.8
16.6
16.4
.2
36-2
35-8
35-4
.2
35-o
34-6
34-2
.2
33-6
33-2
32.8
3
54-3
53-7
53-
3
52.5
5'-9
5' 3
3
50.4
49.8
49.2
4
724
71-6
70.8
4
70.0
69.2
68.4
4
67.2
66.4
6 5 6
5
9-5
88.5
5
8 7 -5
86. s
85.5
5
84.0
83.0
82.0
.6
108.6
107.4
106.3
.6
105.0
103.8
1 02. 6
.6
100.8
99.6
98.4
.7
126.7
125-3
123-0
7
122.5
121. 1
119.7
7
117.6
116. 2
114.8
.8
144.8
"43-2
41.6
.8
140.0
138.4
136.8
.8
'34-4
132.8
131.2
9
162.9
161.1 159-3
9
"57-5 '55-7
'53-9
151.2
141.4 147.6
38
4 3D .
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L
Cos.
30
8. 89 464
8.89 598
I . 10 402
9.99 866
30
3i
8.6
9 625
8.8
9 760
160
I . IO 240
9.99 865
29
32
8.
9?84
159
8.8
9 920
i . 10 080
9.99 864
28
33
M
9 943
'59
8.90 080
I .09 920
9.99 863
27
'59
j
34
8.90 1 02
TC8
8.90 24o
I .09 760
9.99 862
26
35
8.90 260
8 .90 399
i .09 601
9.99 861
25
36
8.90 417
157
8.90 557
158
I .09 443
9.99 860
24
157
15
-,
37
8.90 574
8.9
o? 1 !
i .09 285
9.99859
23
38
8.90 730
8.90 872
i .09 128
9.99 858
22
39
8. 90 885
155
8.91 029
i .08 971
9.99 857
21
40
8.91 o4o
155
8.91 185
150
i. 08 8i5
9.99 856
20
4i
8.91 195
155
8.91 34o
!55
i .08 660
9.99855
i
42
8.91 34g
8.91 4g5
i .08 Sog
9.99 854
18
43
8.91 5o2
153
8.91 650
D
i. 08 35o
9.99 853
i?
153
15
J
44
8.91 655
8.91 8o3
i .08 197
9.99 852
16
45
8.91 807
8.91 957
i. 08 o43
9.99 85i
i5
46
8.91 959
152
8.92 no
"S3
i .07 890
9.99850
i4
I e j
15
1
47
8.92 no
8.92 262
1.07 738
9.99848
i3
48
8.92 261
8.92 4i4
1.07 586
9.99847
12
49
8.92 4i i
ISO
8.92 565
15
1.07 435
9.99 846
I I
50
8.92 56i
ISO
8.92 716
i .07 284
9.99 845
10
5i
8.92 710
149
8.92 866
i .07 1 34
9.99 844
9
52
8.92 85g
8.93 016
i .06 984
9.99843
8
53
8.93 007
8.93 165
i.o6835
9 . 99 842
7
147
'4
54
8.93 i54
8.93 3i3
1.06687
9.99 84i
6
55
8.93 3oi
8.93462
i. 06 538
9.99 84o
5
56
8.93448
J 47
146
8.93 609
'47
i .06 391
9 . 99 83o
4
5?
8 . 93 5g4
ii6
8.93 756
i .06 244
9.99 838
3
58
8.93 740
8 .93 go3
i .06 097
9.99837
2
5 9
8.93 885
J 45
8 . 94 049
I4&
**6
i ,o5 g5i
9.99 836
I
60
8.94 o3o
8.94 igS
i .o5 805
9.99 834
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
<
85.
PP
162
160
159
157
155
'S3
151
149
M7
.1
16.2
16.0
15-9
.1
'5-7
15-5
IS-3 -I
IS-I
14.9
14.7
.2
32 4
32.0
3 i.8
.2
31-4
31.0
30.6 .2
30.2
29.8
29-4
3
48.6
48.0
47-7
3
47-'
40-5
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
73-5
.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.
"1-3
. 7
109.9
08.5
107.1 .7
ioS-7
104.3
102.9
.8
129.6
128.0
127.2
.8
125.6
24 o
122.4 .8
120.8
119.2
117.6
.9 145.8
144.0
'43- 1
9
J4i-3 '39-5
'37-7 -9 '35-9
134.1
132.
3 9
5.
'
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
L
Cos.
8.94 o3o
8.94 195
i .o5 805
9.99 834
60
I
8.94 i?4
144
8.94 34o
MS
i ,o5 660
9.99 833
5 9
2
8.94317
8. 9 4485
i .o5 515
9.99 832
58
3
8.94461
4 63o
MS
i .o5 370
9.99 83i
57
142
*4
i
4
8.g46o3
8.94773
i .o5 227
9.99 83o
56
5
8-94746
8.94 917
i.o5 o83
9.99 829
55
6
8.94887
141
8.95 060
M3
i .04 940
9.99 828
54
142
i j
2
7
8.g5 029
8.g5 202
i .04 798
9.99827
53
8
8.g5 170
8.95 344
i.o4656
9.99 825
52
9
8.95 3io
140
8. 9 5486
142
i.o4 5i4
9.99 824
5i
10
8.95 450
8.g5 627
141
i.o4 373
9.99 823
50
ii
8.95 58 9
139
8.95 767
140
i.o4 233
9.99 822
49
12
8.g5 728
8.95 908
i .04 092
9.99 821
48
i3
8.95 867
139
.38
8.96 047
139
140
i.o3 953
9.99 820
47
i4
8.c
)6 oo5
138
8.96 187
|
i.o3 8i3
9.99819
46
i5
8.96 i43
8.96 325
i .o3 675
9.99817
45
16
8.96 280
8.96464
'3V
i.o3 536
9.99 816
44
'37
i-
'7
8.96 417
116
8.96 602
i.o3 398
9.99 8i5
43
18
8.96 553
8.96 739
i .o3 261
9-99 8i4
42
[9
8.96689
136
8.96 877
i.o3 123
9.99 8i3
4i
20
8.96 825
8.97 oi3
I .02 987
9.99812
40
21
8.96 960
135
8.97 150
135
i .02 85o
9.99 810
39
22
8.97 095
8.97285
1.02 715
9.99809
38
23
8.97 229
8.97 421
1 .02 579
9.99 808
3?
'34
3
s
24
8.97 363
133
8.97 556
135
I .02 444
9.99807
36
25
8 .97 496
8.97 691
i .02 3og
9.99 806
35
26
8.97629
8.97 825
13
4
i .02 175
9.99 8o4
34
'33
M
^
27
8.97 762
8.97959
I .02 O4l
9.99 8o3
33
28
8.97894
8.9
8 092
I .OI 908
9.99 802
32
2 9
8.c
>8 026
8.9
8 225
i.oi 775
9.99 801
3i
30
8.98 157
8.98 358
I .Ol 642
9.99 800
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
84 3O
.
PP
MS
M3
141
139 '38
136
135
133
z
.1
.2
M-5
29.0
M-3
28.6
14.1
28.2
.1
.2
13.9 13.8
27.8 27.6
13.6 .1
27.2 .2
'3-5
27.0
III
'I' 1
26.2
3
43-5
42.9
42.3
3
4'-7 4'-4
40.8 . 3
4-5
39-9
39-3
4
58.0
57-2
56.4
4
55-6 55-2
54-4 -4
54-0
532
52-4
5
72-5
70-5
5
69.5 69.0
68.0 .5
67- S
66.5
65-5
.6
87.0
85.8
84.6
.6
83.4 82.8
81.6 .6
81.0
79.8
78.6
is
101.5
116.0
100. 1
114.4
98.7
112. 8
:l
97.3 96.6
III. 2 IIO.4
95.2 .7
108.8 .8
94-5
08.0
93-'
106.4
91.7
104.8
9 <3"-5
128.
126.9
I25.I 124.2
i??.4 .0 121.5
119.7
"7-9
4o
5 3O .
/
L. Sin.
d.
L, Tang. d.
L. Cotg.
L.
Cos.
30
8.98 167
8.98 358
I .01 642
9.99 800
30
3i
8.9
3 288
131
8.9
3 490
132
i .01 5io
9.99798
29
32
8.98 419
8.9
3 622
i .01 378
9-99797
28
33
8.9
3549
130
8.9
3 7 53
131
i .01 247
9-99 79 6
27
130
131
34
8.9
3 670
8.9
3884
i .01 1 16
9-99795
26
35
8.98 808
8.99015
i .00 985
9.99793
25
36
8.9
S 9 3 7
8.99 i45
130
i .00 855
9-99 79 2
24
129
130
3?
8.9
9 066
128
8.99 275
i .00 725
9.99791
23
38
8.99 194
8.99 405
J
i .00 5g5
9-99 79
22
3 9
8.99 322
8.99 534
i .00 466
9.99 788
21
40
8.99 450
8.99 662
i. oo 338
9.99 787
20
4i
8.99 577
127
8.99791
128
i .00 209
9.99 786
*9
42
8.99 704
8.99919
I .00 08 I
9-99 7 8 5
18
43
8.99 83o
126
9.00 046
128
0.99 954
9.99788
17
44
8.99 956
126
9.00 174
0.99 826
9.99 782
16
45
9.00 082
9.00 3oi
0.99699
9-99 7 Sl
i5
46
9.00 207
125
125
9.00 427
126
0.99 573
9.99 780
i4
4?
9.00 332
9.00 553
0.99 447
9.99 778
i3
48
9.00 456
9.00 679
0.99 32i
9-99777
12
49
9.00 58 1
125
9.00 805
0.99 195
9.99 776
I I
50
9 .00 704
9.00 930
0.99 070
9-99 775
10
5i
9.00 828
124
9.01 o55
124
O.(
)8 9 45
9.99 773
9
52
9.00 g5i
9.01 179
O.(
)8 821
9.99 772
8
53
9.01 074
9.01 3o3
O.(
>86 97
9-99 77 1
7
.
54
9.01 196
9.01 427
O.(
)8 573
9.99769
6
55
9.01 3i8
9.01 55o
o.c
18 450
9-99 7 68
5
56
9.01 44o
9.01 673
0.98 327
9-99 7 6 7
4
121
<
5?
9.01 56i
9.01 796
0.(
j8 204
9.9
9 765
3
58
9.01 682
9.01 918
O.(
j8 082
9-99 7 6 4
2
5 9
9.01 8o3
9.02 o4o
0.97 960
9-99 7 63
I
60
9.01 923
9.02 162
0.97838
9.99 761
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
.1
12.2
12. 1
12.0
.2
26.0
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- S
36.9
3
36.6
36.3
36.0
4
52.0
51-6
51-2
4
50-4
50.0
49.2
4
48.8
48.4
48.0
5
65.0
64-5
64.0
5
63.0
62.5
61.5
5
61.0
60.5
60. a
.6
78.0
77-4
76.8
.6
75-6
75-o
73-8
.6
73-2
72.6
72.0
7
91.0
90.3
89.6
. 7
88.2
87- S
86.1
. 7
8<v4
84.7
84.0
.8
104.0
103.2
102.4
.8
100.8
IOO.O
98-4
.8
97.6
96.8
96.0
9
117.0
116. i
115.2
9
113.4 112.5
110.7
9
109.8
108.9
toS.o
4r
6.
/
L.
Sin.
d.
L. Tang. d.
L.
Cotg.
L.
Cos.
9.01 923
9.02 162
0.97 838
9.99 761
60
I
9.02 o43
9.02 283
0.97 717
9-99 7 6
5 9
2
9.02 i63
9.02 4o4
0.97 596
9.99759
58
3
9.02 283
9.02 525
-97475
9.99757
S?
119
121
j
4
9 .02 4o2
118
9.02 645
0.97355
9. 997 56
56
5
9.02 52O
9.02 766
0.97 234
9-99 7 5 5
55
6
9.02 63g
119
9.02 885
119
0.97 115
9.99753
54
118
12'
'
7
9.02 757
9.0
3 005
0.96 995
9.99752
53
8
9.02 874
9.o3 124
0.96 876
9.99751
52
9
9.02 992
9.0
3 242
0.96 758
9-99 749
5i
10
g.o3 109
117
g.o3 36 1
119
TtP
o . 96 63g
9.99 748
50
1 1
9.o3 226
n6
9.03479
II
1
0.96 521
9.9974?
49
12
9.o3 342
9.o3 597
0.96 4o3
9-99 745
48
i3
9-03458
9-o3 714
117
0.96 286
9-99 744
47
116
II
i4
9.o3 5?4
116
9-o3 832
116
0.96 168
9.99 742
46
i5
9-o3 690
9.o3 948
0.96 o52
9-99 74i
45
16
g.o3 805
9.04 o65
"7
0.95 935
9-99 74o
44
"5
ii
17
g.oS 920
114
9.04 181
116
0.(
? 58i 9
9.99738
43
18
9.04 o34
9.04 297
O.I
>5 7 o3
9.9973?
42
'9
9.04 1 49
9 . o4 4 1 3
0.
?5 58 7
9.99 7 36
4i
20
9.04 262
9.04 528
"5
o . g5 4?2
9-99 7 3 4
40
21
g.o4 3?6
114
9.04 643
o
j5 35?
9.99 ?33
3 9
22
9.04 4go
9.04 758
O
}5 242
9.99731
38
23
9.04 6o3
9.04873
"5
o.
? 5 127
9.99730
3?
ii
i
24
9 . o4 7 1 5
"3
9-o4 987
o .
j5 oi3
9.99728
36
25
9.04828
9.o5 101
0.94 899
9.99727
3b
26
9.04 94o
9-o5 2i4
0.94 786
9-99 726
34
ii
L
27
9.0
>5 o52
9-o5 328
0.94 672
9-99724
33
28
9-o5 1 64
9.o5 44i
0.94 559
9-99723
32
2 9
9.o5 275
9.o5 553
0.94 44?
9.99 721
3i
30
g.o5 386
9.o5 666
"3
0.94 334
9.99 720
30
L.
Cos.
d.
L. Cotg.
d.
L.
Tang.
L.
Sin.
'
83 3O
.
PP
131
120
119
118
117
116
"5
114
"3
.1
12. 1
I2.O
II.9
.1
ii. 8
11.7
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
30-3
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
45 6
45
-5
60. S
OO.O
59- 5
5
59-
58.5
58.0
5
57-5
57-o
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
80.5
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. 3
104.4
IO2.6
101.7
6 3O .
/
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
30
9-o5 36
9 .o5 666
o. 9 4 334
9-99 7 20
30
3i
9.0
5497
9.0
5 778
0.94 222
9-99 7 l8
29
32
9.0
5 607
9.o5 890
0.94 no
9.99 717
28
33
g.oS 717
9.06 002
O.C
;3 998
9.99 716
27
no
in
34
g.o5 827
9.06 1 13
O.C
)3 887
9.99 714
26
35
9.06 937
9 .06 224
O.g3 776
9.99 713
25
36
9 .06 046
9.06 335
0.93 665
9.99 711
24
109
no
3?
9.06 1 55
9.06 445
O.C
>3 55,5
9.99710
23
38
9.06 264
9.06 556
0.93 444
9.99 708
22
3 9
9.06 372
9.06 666
0.93 334
9-99 77
21
40
9.06 48i
9.06 775
109
0.93 225
9-99 7 5
20
4i
9.06 589
9.06 885
o.g3 n5
9.99 704
'9
42
9.06 696
9.06 994
0.93 006
9.99702
18
43
9.06 8o4
9.07 io3
109
0.92 897
9.99 701
17
107
1 08
44
9.06 91 1
9.07 21 I
0.92 789
9.99699
16
45
9.07 018
9.07 820
109
0.92 680
9.99698
i5
46
9.07 124
9.07 428
0.92 572
9.99696
i4
107
4 7
9.07 281
9.07 536
0.92 464
9.99695
i3
48
9.07 337
9.07 643
0.92 357
9.99693
12
49
9 07 442
i5
9.07 751
0.92 249
9.99692
I I
50
9.07 548
9.07 858
107
0.92 142
9.99690
10
5i
9.07 653
9.07 964
0.92 o36
9.99689
9
62
9.07 758
9.0
8071
0.91 929
9.99687
8
53
9.07863
105
9.08 177
106
0.91 828
9.99 686
7
54
9.07 968
9.08 283
106
0.91 717
9.99 684
6
55
9.08 072
9.0
8 389
0.91 611
9.99 683
5
56
9.08 176
9.0
84 9 5
0.91 5o5
9.99 681
4
104
105
5?
9.0
8 280
9.08 600
0.91 4oo
9.99 680
3
58
9.0
8 383
9.08 705
0.91 295
9.99678
2
5 9
9.08 486
103
9.08 810
105
0.91 190
9.99 677
I
60
9.08 58g
9.0
8 914
0.91 086
9.99675
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
83.
PP
112
in
no
109
108 107
1 06
105
104
.1
II. 2
ii. i
II. O
.1
10.9
10.8 10.7
,!
10.6
10.5
10.4
.2
22-4
22.2
22.
.2
21.8
21.6 21.4
.2
21.2
21.
20.8
3
33-6
33-3
33-
3
32-7
32-4 3 2 -i
3
31.8
3'-5
31.2
4
44.8
44-4
44.0
-4
436
43.2 42.8
4
42.4
42.0
41.6
5
56.0
55-5
SS-
5
54-5
54- 53-5
5
53-o
52-5
52.0
.6
67.2
66.6
66.0
.6
65.4
64.8 64.2
.6
63.6
63.0
62.4
7
78.4
77-7
77.0
. 7
7 6 -3
75-6 74-9
7
74-2
73-5
72.8
8
89.6
oo. 8
88.0
.8
87.2
86 4 85.6
.8
84.8
84.0
83-2
9 TOO. 8
99.9 99.0
98.1 97.2 96.3
9 95-4
94-5
93.6
43
r
L.
Sin.
d.
L. Tang.
d.
L.
Cotg.
L.
Cos.
9.08 589
9.0
8 9 i4
0.91 086
9 . 99 675
60
1
9 .c
>8 692
103
9.09 019
105
0.90 981
9 . 99 674
5 9
2
9 .c
>8 795
9.09 123
0.90 877
9-99 6 7 2
58
3
9.08 897
9.09 227
104
0.90 773
9.99670
57
IO
1
4
9.08 999
9.09 33o
0.90 670
9.99669
56
5
9 .09 101
9.09 434
0.90 566
9.99 667
55
6
9.09 202
9.09 537
10
i
0.90 463
9.99 666
54
IO
1
7
9.09 3o4
9.09 64o
0.90 36o
9.99 664
53
8
9.09405
9.09 742
0.90 258
9.99 663
52
9
9.09 5o6
9.09 845
103
0.90 i55
9.99 661
1
10
9.09 606
9.09947
0.90 o53
9.99659
50
1 1
9.09 707
9.10 049
o.f
3 9 9 5i
9.99 658
49
12
9 .09 807
9.10 i5o
o.i
39 850
9.99 656
48
i3
9.09907
9. IO 252
o.i
39748
9.99655
47
99
i
i4
9. to 006
9.10 353
o.f
3 9 647
9.99 653
46
ID
9.10 106
9. 10 454
0.89 546
9.99 65i
45
16
9.10 205
99
9. 10 555
0.89445
9.99650
44
99
IO
:
17
9.10 3o4
08
9.10 656
o.f
39344
9.99 648
43
18
9. i
O 4O2
9.10 756
o.f
3g 244
9 . 99 647
42
'9
9.10 5oi
99
nR
9.10 856
o.i
3 9 1 44
9.99 645
4i
20
9 . 10 599
9.10 956
o.i
3 9 o44
9 . 99 643
40
21
9.10 697
98
9.11 o56
99
o.f
38 9 44
9.99 642
3 9
22
9. 10 795
08
9.11 i55
o.f
38 845
9.99 64o
38
23
9. 10 8 9 3
97
9.11 254
99
o.f
38 746
9.99 638
37
24
9. 10 990
97
9.11 353
99
o.f
38647
9.99637
36
25
9.11 087
9.11 452
o.f
38 548
9.99 635
35
26
9 . 1 1 184
97
9.11 55i
98
o.f
3844 9
9.99 633
34
27
9 . 1 1 281
06
9.11 64 9
|
o.f
3835i
9.99 632
33
28
9.11 3 7 7
9. ii 74?
1
0.88 253
9.99 63o
32
29
9- 1 1 4?4
06
9.11 845
n
1
o.f
58 155
9.99629
3i
30
9.11 570
9.11 943
o.f
38o5 7
9.99627
30
L.
Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
'
82 30
.
PP
105
104
103
103
101
1 00
99
98 97
.1
10.5
10.4
10.3
.i
IO. 2
10. I
10.
.1
9. 9
9.8 9.7
.2
21.
20.8
20 6
.2
20.4
20.2
20. o
.2
19 8
19.6 19.4
3
3i 5
31-2
3-9
3
30.6
30.3
30.0
3
29.7
29.4 29.1
4
42.0
41.6
41.2
4
40.8
40.4
40.0
4
19-6
392 3 8.8
5
52.5
52.0
S'-5
5
51.0
5-5
50.0
5
49-5
49.0 48.5
.6
63.0
62.4
61.8
.6
6l.2
60.6
60.0
.6
594
58.8 58.2
7
73-5
72.8
72.1
7
71.4
70.7
70.0
-7
69.3
68.6 67.9
8
84.0
8,. 2
82.4
.8
8l.6
80.8
80.0
.8
79.2
78.4 77.6
9 94- >
91.8 OO.g
90.0
89.1
88.2 87.3
44
7 3O .
,
L.
Sin.
d.
L. Tang.
d.
L. Cotg.
Cos.
30
9.11 570
9 6
95
96
95
95
95
94
95
94
94
93
94
93
93
93
93
93
92
92
92
92
9 1
92
9 1
9 1
90
9 1
go
9 1
90
9.11 943
97
98
97
97
96
97
96
96
96
96
95
95
95
95
95
94
95
94
94
93
94
93
93
93
93
92
92
93
9 1
92
d.
0.88 067
9-99 62 7
30
3i
32
33
34
35
36
3?
38
3 9
9.11 666
9.11 761
9.11 85y
9.11 g52
9.12 o4?
9.12 142
9.12 236
9.12 33i
9.12 425
9.12 o4o
9.12 i38
9.12 235
9.12 332
9.12 428
9.12 525
9.12 621
9.12 717
9.12 8i3
0.87 960
0.87 862
0.87 765
0.87 668
0.87 572
0.87475
0.87 379
0.87 283
0.87 187
9.99 625
9.99 624
9 .99 622
9.99 620
9.99 618
9.99617
9.99 6i5
9.99 6i3
9 . 99 612
29
28
2 7
26
25
24
23
22
2 I
40
9.12 5ig
9.12 909
0.87 091
9 .99 610
20
4i
42
43
44
45
46
4?
48
49
9.12 612
9.12 706
9.12 799
9.12 892
9.12 985
9. 1 3 078
9. i3 171
g.i3 263
9. i3 355
9. 1 3 oo4
9. 1 3 099
9 . 1 3 1 94
9. i3 289
9. 1 3 384
9.13478
g.i3 573
9-i3 667
g.iS 761
0.86 996
0.86 901
0.86 806
0.86 711
0.86 616
0.86 522
0.86 427
0.86 333
0.86 239
9 .99 608
9.99607
9.99605
9.99 6o3
9 .99 601
9.99 600
9-99 5 9 8
9.99 5 9 6
9.99595
'9
1 8
I?
16
i5
i4
i3
12
I I
50
9. i3 447
9.i3 854
0.86 i46
9.99 5 9 3
10
5i
52
53
54
55
56
5?
58
5 9
9. i3 53g
9. i3 63o
9. i3 722
9.i3 8i3
9. 1 3 904
9-i3 994
9. i4 o85
9.14 i?5
9. i4 266
9. 1 3 948
9 . 1 4 o4 1
9 .i4 1 34
9. i4 227
9. l4 32O
9.14412
9. 14 5o4
9. i4 597
9.14 688
0.86 o52
o.85 g5g
o.85 866
o.85 773
o.85 680
o.85 588
o.85 496
o.854o3
o.85 3i2
9.99 5 9 i
9.99589
9.99 588
9.99 586
9.99 584
9.99 582
9.99 58i
9.99579
9-99 5 77
9
8
7
6
5
4
3
2
I
60
9. i4 356
9 . i4 780
o.85 220
9.99 5 7 5
L.
Cos.
d.
L. Cotg.
L. Tang.
L.
t
82.
PP.
.1
.2
3
4
.6
.8
9
97
96
95
.1
2
3
4
.6
8
9
94 93
92
.1
.2
3
4
.6
',8
9
91
90
9-7
19.4
29.1
38.8
48.5
58.2
67.9
77.6
87.3
9 .6
19.2
28.8
38.4
48.0
57-6
67.2
76.8
86.4
9-5
19.0
28.5
38.0
47-5
57-o
66.5
76.0
85.5
9-4 9-3
18.8 18.6
28.2 27.9
37-6 37-2
47.0 46.5
56.4 55-8
63.8 65.1
75-2 74-4
84.6 81.7
9.2
i8. 4
27.6
3 6.8
46.0
55-2
64.4
73- 6
9 .i
18.2
- 2f -3..
36-4
45-5
54- 6
63-7
72.8
81.9
9.0
18.0
..27-0
36.0
45-o
54-o
63.0
72.0
81.0
45
8.
/
L. Sin.
d.
L. Tang. d. L. Cotg.
L. Cos.
9.14 356
9. i4 780
o.85 220
9
99 5 7 5
60
I
9.14445
9.14 872
92
o.85 128
9
99 574
5o
2
9. i4 535
9.14 963
o.85 037
9
.99 572
5*
3
9.14 624
Cy
9. 1 5 o54
9 1
0.84946
9
.99 570
3 7
4
9-i4 7'
4
90
80
9. i5 i45
9'
0.8485s
9
99 568
56
5
9. i4 8o3
9. 1 5 236
9 1
o.84 764
9
.99 566
55
6
9. i4 891
9. i5 327
9'
0.84673
9
99 56 5
54
7
9. i4 98
f(n
9. i
54i7
9
0.84583
9
.99 563
53
8
9. 1 5 069
9. 1 5 5o8
o.84 492
9
.99 56i
52
9
9. i5 i5j
9. i5 598
90
o.84 i
[O2
9
.99 55g
5i
10
9. 1 5 245
88
9.i5 688
90
o.84 3i2
9
.99 557
50
1 1
g.i5 333
88
9.1
5 777
9
0.84 223
9
.99 556
49
12
9. 1 5 421
9-i5 867
o.84 i33
9
.99 554
48
i3
9-i5 5o8
87
9. i5 956
9
o.84o44
9
.99 552
4?
i4
9. i5 596
88
8?
9.16 o46
90
0.83954
9
.99 55o
46
i5
9.i5 683
87
9.16135
0.83865
9
. 99 548
45
16
9 .i5 77
o
87
9. 16 224
89
88
o.83 776
9
.99 546
44
17
9. i5 857
87
9. 16 3i2
80
0.83688
9
.99 545
43
18
9. 1 5 94
4
86
9.16 4oi
o.83 599
9
.99 543
42
'9
9.16 o3o
86
9.16 48g
o.83 5n
9
.99 54r
4i
20
9.16 1 16
87
9. 16 577
0.83423
9
.99 SSg
40
21
9. 16 2o3
86
9.16665
88
o.83 335
9
.99537
3 9
22
9. 16 289
Be
9.16 753
o.83 247
9
.99 535
38
23
9.16 37
4-
86
9.16841
87
o.83
[5 9
9
.99 533
3?
24
9.16 46o
85
9. 16 928
88
o.83 072
9
.99 532
36
25
9.16 545
86
9.17 016
0.82 984
9
.99 53o
35
26
9.16 63i
9.17 io3
7
0.82 897
9
.99 528
34
85
87
27
9.16 716
85
9.17 190
87
0.82 810
9
.99 526
33
28
9.16 801
9.17277
0.82 723
9
.99 524
32
2 9
9.16 886
85
9.17 363
0.82 637
9
.99 522
3i
30
9. 16 970
84
9.17 450
87
0.82 55o
9
.99 52O
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
| L. Sin.
'
813O.
PP 9*
9*
90
89
88
87 86
.1 9.2
9.1
9.0
.1
8-9
8.8
.1
8.7 8.6
.2 18.4
18.2
1 8.0
.2
17.8
.7.6
.2
17.4 17.2
3 =7-6
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-o
S
43-5 43-o
.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
.9 828
-i. i
81.0
.9 80. i
9
78.3 77.4
46
8 3O .
'
L. Sin.
d.
L. Tang.
d.
L. Cotg. B L. Cos.
30
9. 16 970
9.17 450
0.82 55o
9
99 52O
30
3i
9.17 o5
=
85
8i
9.17 536
86
0.82 464
9
99 5i8
29
32
9.17 i3 9
9.17 622
0.82 378
9
99 5l 7
28
33
9.17 223
84
9.17 708
86
0.82 292
9
99515
27
34
9.17 307
9 . 1 7 794
86
0.82 206
9
99 5i3
26
35
9.17 3 9 i
4
9.17 880
O.82 120
9
99 5i i
25
36
9.174?'
i
8 3
9.17 965
85
86
0.82 c
3 5
9
99 Sog
24
37
9.17 558
9. 18 o5i
0.81 9
49
9
99 So?
23
38
9.17 64i
9.18 i36
0.81 864
9
99 5o5
22
3 9
9.17 724
83
9.18 221
85
0.81 779
9
99 5o3
21
40
9.17 807
83
9.18 3o6
85
0.81 694
9
.99 5oi
20
4i
9.17 89
3
83
83
9.18 391
85
84
0.81 609
9
99499
'9
42
9.17973
9.i84?5
0.81 525
9
99497
18
43
9.18 o55
9.18 56o
0.81 44o
9
99 4 9 5
17
84
44
9.18 i3
1
9.18 644
0.81 356
9
99494
16
45
9.18 220
3
9.18 728
0.81 272
9
.99 492
i5
46
9.18 3o2
9.18 812
84
0.81 188
9
99 490
i4
8l
84
4 7
9.18 383
9.18 896
R-3
0.81 io4
9
.99488
i3
48
9-i8465
9.18979
0.81 021
9
.99486
12
4 9
9.18 547
9.19 o63
84
0.80 987
9
99484
I I
50
9.18 628
9. 19 i46
0.80 854
9
.99 482
10
5i
9.18 709
81
9. 19 229
03
83
0.80 771
9
.99 48o
9
52
9.18 790
9.19 3i2
81
0.80 e
88
9
.99478
8
53
9.18871
9.19 3g5
83
0.80 605
9
99 476
7
54
9. 18 952
81
9.19 4?8
83
O.8o 522
9
.99 4?4
6
55
9.19 o33
9. 19 56i
O.8o 439
9
. 99 472
5
56
9.19 1 13
9.19 643
O.8o 35?
9
.99470
4
5?
9.19 193
80
9.19 725
82
O.8o 275
9
.99 468
3
58
9. 19 273
9.19 807
0.80 1
93
9
.99 466
2
5 9
9.19 353
g o
9. 19 889
82
O.8o III
9
99 464
I
60
9.19433
9.19 971
O.8o 029
9
.99 462
L. Cos.
d.
L. Cotg.
d.
L. Tang. \ L. Sin.
>
81.
PP 86
85 84
83
82
81
80
.1 8.6
8.5 8.4
.1
8.3
8.2
.1
8.1
8.0
.2 17.2
17.0 16.8
.2
16.6
16.4
.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-o 33- 6
4
33-2
32.8
4
32-4
32.0
5 43-o
42-5 42-0
5
4'-5
41.0
5
40.5
40.0
.6 S1 .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
567
56 o
.8 68.8
68.0 67.2
.8
66.4
656
.8
64.8
64.0
9 74-4
76.5 75.6
9 74-7 73- 8
9
72 9 72.0
47
9.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
9.19 433
9.19971
0.80 029
9
.99 462
60
I
9.19 5i3
9.20 o53
81
0.79947
9
.99 460
5 9
2
9.19 5g2
80
9.20 1 34
0.79 866
9
. 99 458
58
3
9. 19 672
9.20 216
0.79 784
9
.99 456
57
79
4
9.19 761
9.20 297
81
0.79 703
9
99 454
56
5
9. 19 83o
9.20 378
0.79 622
9
.99 452
b5
6
9.19909
79
9.20 45g
0.79 54i
.99450
54
7S
81
7
9.19 988
9.20 54o
81
0.79 46o
9
99448
53
8
9.20 067
9.20 621
0.79 379
9
.99 446
52
9
9.20 i45
78
9.20 701
0.79 299
9
99 444
5 i
10
9 . 2O 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
0.79 o58
9
.99 438
48
i3
9.20458
9.21 022
0.78 978
9
.99 436
47
77
i4
9.20 535
78
9.2110-2
80
0.78 f
$98
9
.99434
46
i5
9 .20 6i3
9.21 182
0.78 818
9
.99 432
45
16
9 .20 691
77
9.21 261
79
80
0.78 739
9
99 429
44
17
9.20 768
9.21 34i
o. 78 65g
9
99 427
43
18
9.20845
9.21 420
0.78 58o
9
.99 425
42
'9
9.20 922
77
9.21 499
79
0.78 5oi
9
.99 423
4i
20
9.20 999
9.21 578
79
0.78 422
9
.99 421
40
21
9.21 076
77
9.21 657
79
7O
0.78 343
9
.99419
3 9
22
9.21 i53
76
9.21 736
0.78 264
9
.99417
38
23
9.21 229
9.21 8i4
7
0.78 186
9
.99415
37
77
79
24
9.21 3o6
76
9.21 893
78
0.78 107
<;
. 99 4i3
36
25
9.21 382
-7 ft
9.21 971
o. 78 029
9
99 4n
35
26
9.21 458
7 6
9.22 049
78
o. 779 5i
9
99 409
34
2 7
9.21 534
76
9.22 127
78
0.77 873
9
.99407
33
28
9,21 610
9.22 2o5
78
0-77795
9
994o^
32
2 9
9.21 685
?6
9.22 283
78
0.77 717
9
.99 4O2
3 1
30
9.21 761
9 .22 36i
0.77 63g
9
.99 4oo
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
'
8O 3O .
PP 83
81
so
79
78
77
76
.1 8.2
8.1
8.0
.,
7-9
7 .8
.1
7-7
7.6
.2 16.4
16.2
16.0
.2
15.8
.5.6
.2
'5-4
15-2
3 24.6
2+- 3
24.0
3
23-7
23.4
3
23-'
22.8
.4 32.8
32.4
32.0
4
31.6
31-2
4
30 8
3-4
5 4'-o
40.5
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
53-2
.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 /''
7" 2
9
69.3
68.4
48
9 3O .
1
L. Sin.
d.
L. Tang. d. L. Cotg.
L. Cos.
30
9.21 761
75
76
75
75
75
74
75
75
74
74
74
74
74
74
73
74
73
73
73
73
73
73
72
73
72
72
73
7i
72
72
9 .22 36i
77
78
77
77
77
77
77
76
77
76
76
77
76
76
75
76
75
76
75
75
75
75
75
74
75
74
75
74
74
74
0.77 689
9
99 4oo
30
3i
32
33
34
35
36
3?
38
3 9
9.21 836
9.21 912
9.21 987
9 .22 062
9.22 187
9.22 211
g.22 286
9.22 36i
9.22 435
9.22 438
9.22 5i6
9.22 5g3
9 .22 670
9.22 747
9.22 824
9.22 901
9.22 977
9.28 o54
o. 77 562
0.77 484
0.77 407
0.77 33o
0.77 253
0.77 176
0.77 099
0.77 028
0.76 946
9-
9-
9-
9-
9-
9-
9
9
9
99 898
99 896
99 894
99 892
99 890
99 388
99 385
99 383
99 38i
29
28
2 7
26
25
24
23
22
21
40
9 .22 5og
9.23 i3o
o . 76 870
9
99 3 79
20
4i
42
43
44
45
46
4 7
48
49
9.22 583
9.22 657
9.22 781
9.22 805
9.22 878
9.22 g52
9-23 025
9. 23 098
9.23 171
9.23 206
9.23 283
9.23 SSg
9.28 435
9.23 5io
9.23 586
9.28 661
9.23 787
9.28 812
0.76 794
0.76 717
0.76 64i
0.76 565
0.76 490
0.76 4i4
0.76 889
0.76 268
0.76 188
9
9
9
9
9
9
9
9
9
99 3 77
99 3 75
99 872
99370
99 368
99 366
99 364
99 862
99 35 9
'9
18
17
16
i5
i4
i3
12
I I
50
g .23 244
9.28 887
0.76 1 13
9
99 35 7
10
5i
52
53
54
55
56
5y
58
5 9
9.23 317
9 .23 890
9.28 462
9.28 535
9.28 607
9.28 679
9-23 752
9.23 823
g.23 895
9.28 962
9.24 087
9 .24 112
9.24 186
9.24 261
9.24 335
9.24 4io
9.24484
9.24 558
0.76 o38
0.75 968
0.75888
0.75 8i4
0.75 789
0.75 665
0.75 Sgo
0.75 5i6
0.75 442
9.99 355
9.99 353
9.99 35i
9.99 348
9.99 346
9.99 344
9". 9 9 342
9.99 34o
9-99 33 7
9
8
7
6
5
4
3
2
I
60
9.28 967
9.24 682
0.75 368
9
.99 335
L. Cos. cl.
L. Cotg. d.
L. Tang.
L. Sin.
/
80.
PP 77
7 6
75
.1
.2
3
4
.6
.8
74
73
.1
.2
3
4
J
9
72
71
.1 7.7
.2 15.4
3 23.1
4 3-8
5 38.S
.6 46.2
7 53-9
,8 61.6
9 6 9-3
7 .6
15-2
22.8
30.4
38.0
45-6
53-2
60.8
68.4
7-5
15.0
22.5
30.0
37-5
45.0
52-5
60.0
ii
32.2
29.6
37-
44-4
51.8
59.2
7-3
14.6
21.9
29.2
36.5
43-8
5'-i
58.4
6s-7
7.2
14.4
21.6
28.8
36.0
43.2
50.4
57- 6
6 4 .3
7-i
14.2
21-3
28.4
35-5
42.6
497
568
63.9^
4 9
1O C
'
L. Sin.
d.
L.
Tang. d. L. Cotg.
L. Cos. d.
g .28
967
72
71
71
72
71
71
71
70
71
7
7'
70
70
70
70
70
70
69
70
69
69
69
69
69
69
69
68
69
68
68
9
24 632
74
73
74
73
74
73
73
73
73
73
72
73
72
73
72
72
72
72
72
7i
72
7
72
7'
7'
7'
7 1
70
7'
7'
0.75 368
9-99
335
2
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
60
I
2
3
4
5
6
7
8
9
9.24 oSg
9.24 no
9.24 181
9.24 253
9.24 324
9.24 395
9.24 466
9.24 536
9.24 607
9
9
9
9
9
9
9
9
9
24 706
24 779
24853
24 926
25 ooo
25 073
25 i46
25 219
25 292
o. 75 294
O.75 221
0.75 147
O.75 O74
0.75 ooo
0.74 927
0.74854
0.74 781
0.74 708
9-99
9.99
9-99
9-99
9.99
9.99
9-99
9.99
9.99
333
33i
328
326
324
322
319
3i 7
3,5
5 9
58
57
56
55
54
53
52
5i
10
9.24
6 77
9
25365
o. 7 4635
9-99
3i3
50
1 1
12
i3
i4
i5
16
i?
18
'9
9.24 748
9.24 818
9.24 888
9.24 g58
9.25 028
9.25 098
9-25 168
9.25 237
9.25 307
9
9
9
9
9
9
9
9
9
25437
25 5io
25 582
25 655
25 727
25 799
25 871
25 943
26 015
0.74 563
0.74 4go
0.74418
0.74345
0.74 273
O.74 2OI
O.74 129
0.74 057
o. 7 3 985
9.99 3 to
9.99 3o8
9.99 3o6
9.99 3o4
9 .99 3oi
9.99299
.9.99297
9.99294
9.99 292
49
48
4?
46
45
44
43
42
4i
20
9.25 376
9
26086
0.73 914
9-99
290
40
21
22
24
25
26
27
28
2 9
9.25 445
9.25 5i4
9-25 583
9.25 652
9.25 721
9 .25 790
9.25 858
9.25 927
9.25 995
9
9
9
9
9
9-
9
9
9
26 i58
26 229
26 3oi
26 372
26443
26 5i4
26 585
26655
26 726
0.73 842
0.73 771
0.73 699
0.73 628
0.73 557
0.73486
0.73415
0.73345
0.73 274
9.99 288
9.99 285
9.99 283
9.99 281
9.99278
9.99276
9.99274
9.99271
9.99269
39
38
37
36
35
34
33
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 3O'.
PP
.1
.2
3
4
.6
!s
'?
74
73 7*
.1
.2
3
4
.6
:l
71
70
69
6.9
13.8
20.7
27.6
34-5
41.4
48-3
55-2
62.1
.1
.2
3
4
.6
.8
68
3
7-4
14.8
22.2
29.6
37-o
44-4
5i.8
59- 2
7-3 7-2
14.6 14-4
21.9 21.6
29.2 28.8
36.5 36.0
43.8 43.2
51-1 504
58.4 57.6
6=1.7 64.8
7-'
14.2
21.3
28.4
35- 5
42.6
4 2i
56.8
7.0
14.0
2I.O
28.0
35-o
42.0
49.0
56.0
63.0
6.8
13.6
20.4
27.2
34-o
40.8
47.6
54-4
61.2
0.3
0.6
0.9
1.2
i-5
1.8
2.1
2.4
2 -7
5o
1O 30 .
,
L. Sin.
d.
L. Tang-, d. L.
Cotg.
L. Cos.
d.
30
9
.26 o63
9.26
797
0.73 2o3
9.99 267
30
3i
9
.26 i3
68
9.26 867
0.73 1 33
9.99 264
2
29
32
9
.26 199
9.26
9 3 7
0.73 o63
9.99
262
28
33
9
.26 267
9.27
008
7 1
0.72 992
9.99 260
27
68
7
3
34
9
.26335
68
9.27 078
0.72 922
9.99257
2
26
35
9
.26 4o3
9.27
i48
0.72 852
9.99255
25
36
9
.26 470
67
9.27
218
7
0.72 782
9.99 252
24
68
7
3 7
9
.26 538
f,-,
9.27
288
60
0.72 712
9.99
25o
2
23
38
9
.26 6o5 i
9.27
35 7
o. 72 643
9.99 248
22
3 9
9
. 26 672 7
9.27 427
70
0.72 573
9.99
245
21
40
9
.26 7 3g
9.27 4 9 6
9
0.72 5o4
9.99
243
20
4i
9
.26 806
VJ
67
9.27
566
7
60
0.72 434
9.99
241
'9
42
9
.26873
9.27
635
O. 7 2 365
9.99
238
18
43
9
.26 940
67
9.27
704
69
O. 7 2 296
9.99 236
'7
67
69
3
44
9
.27 007
66
9.27
77 3
fin
0.72 227
9.99
233
2
16
45
9
.27073
9.27 842
0.72 i58
9.99
23l
i5
46
9
.27 i4<
>
67
66
9.27 911
69
69
0^72 089
9.99229
3
i4
47
9.27 206
67
9.27 980
fin
0.72 020
9.99
226
2
i3
48
9.27 273
9.28 049
0.71 g5i
9.99 224
12
49
9
.27 33 9
9.28
117
0.71 883
9-99
221
I I
50
g .27 405
9.28
186
o.
71 8i4
9.99
219
10
5i
9
.27471
66
9.28
254
60
0.71 746
9.99
2I 7
9
52
9.27537
9.28
323
0.71 677
9.99
2l4
8
53
9.27 602
65
66
9.28
391
68
0.71 609
9.99
212
3
7
54
9.27 668
66
9.28 459
68
o.
71 54i
9.99
209
2
6
55
9.27 734
9.28
527
o.
71 4?3
9-99 2 7
5
56
9.27 799
65
9.28
^95
o.
7 i 4o5
9.99 204
4
65
67
57
9.27 864
66
9.28 662
68
0.
71 338
9.99
2O2
2
3
58
9. 27 g3o
9.28
73o
o.
7 I 2 7 O
9.99
2OO
2
5 9
9.27 995
65
9.28
798
fi-T
0.
7 I 202
9-99
'97
I
60
9.28 060
9.28
865
0.
7 I iSg
9.99
195
"
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-S
0.3
.2
140
13-8
13-6
.2 13.4
13.2
.2
13.0
0.6
3
21 O
20.7
20.4
. 3 20. i
19.8
3
J9-5
0.9
4
28 o
27.6
27.2
.4 26.8
26.4
4
26.0
1.2
5
350
34-5
34-o
5 33-5
33-o
5
32-5
"5
.6
42 o
41.4
40.8
.6 40.2
39- 6
.6
39-
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
_
63.0
62.1
61.2
.9 60.3
59-4
9
58.5 2.7
5i
11.
'
L. Sin.
d.
L
Tang.
d.
L. Cotg.
L. Cos.
d.
9.28 060
65
6 4
6s
65
64
6 4
65
6 4
6 4
6 4
63
6 4
6 4 -
63
6 3
6 3
9
28 865
68
67
67
67
67
67
66
66
67
66
66
66
66
66
66
66
65
66
65
66
65
65
65
64
0.71 1 35
9-99
'95
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
2
3
2
3
2
3
2
3
2
3
3
2
60
I
2
3
4
5
6
7
8
9
9.28 125
9.28 190
9.28 254
9.28 3ig
9.28 384
9.28448
9.28 5i2
9.28 577
9.28 64 1
9
y
9
9
9
9-
9
9
9
28 933
29 ooo
29 067
29 i34
29 201
29 268
29 335
29 4O2
29 468
0.71 067
0.71 ooo
0.70 933
0.70 866
0.70 799
0.70 732
0.70 665
0.70 5g8
0.70 532
9-99
9.99
9-99
9-99
9-99
9-99
9-99
9-99
9.99
192
190
187
185
182
1 80
177
175
172
5 9
58
57
56
55
54
53
52
5i
10
9.28 705
9
29 535
o. 70 465
9.99
170
50
1 1
12
i3
i4
i5
16
17
18
1 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
9-
9-
9
i ;
9
9
9
9
29 601
29 668
29 734
29 800
29 866
29 932
29 998
3oo64
3o i3o
0.70 399
0.70 332
0.70 266
0.70 200
o. 70 i34
0.70 068
0.70 002
0.69 g36
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
i5 7
155
I 52
150
i4 7
49
48
47
46
45
44
43
42
4i
20
9 .29 34o
3
9
3o 195
0.69 805
9.99
145
40
21
22
23
24
26
27
28
2 9
9.29 4o3
9.29 466
9.29 529
9 . 29 5gi
9.29 654
9.29 716
9.29 779
9.29 84i
9.29 903
6 3
63
62
63
62
63
62
62
9
9
9
9
9-
9-
9-
9-
9
3o 261
3o 326
3o 391
3o45?
3o 522
3o58 7
3o 652
3o 717
3o 782
0.69 739
0.69 674
0.69 609
0.69 543
0.69 4?8
0.69 4i3
0.69 348
0.69 283
0.69 218
9.99
9-99
9-99
9-99
9.99
9.99
9-99
9.99
9.99
i4o
i3 7
132
i3o
127
124
122
39
38
3?
36
35
34
33
32
3i
30
9. 29 966
6 3
9
3o846
0.69 1 54
9-99 "9
30
L. Cos. d. L
Cotg. d.
L. Tang 1 .
L. Sin. d.
78 3O .
PP
.2
3
4
5
.6
.1
9
63
67 66
.i
.2
3
4
'.6
'.B
9
65
6 4
63
.1
.2
3
4
^6
'.S
9
62 3
6.8
,3.6
20.4
27.2
34-o
40.8
47.6
54 4
6.7 6.6
'3-4 13-;
20. i 19.8
26.8 26.4
33-5 330
40.2 39.6
46.9 46.2
53.6 52.8
60. 3 ' 59.4
6.5
13.0
'9-5
26.0
32-5
39.0
45 5
52.0
6. 4
12.8
19 2
2 5 6
32.0
38-4
44.8
5' 2
57-6
6 3
12.6
18.9
25.2
3I -
37-8
44.1
50.4
5*7
6.2 .3
12.4 .6
18.6 .9
24.8 1.2
31.0 I.S
37-2 1.8
43-4 2.1
496 2.4
55 8 2-7
52
'
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
9. 3o 028
62
9
3o 91 1
^
0.69 089
9.99
117
29
32
9-3o 090
9-
3o 975
4
o.6 9 025
9-99
u4
28
33
9. 3o 1 5j
9-
3 1 o4o
65
0.68 960
9.99
I 12
27
62
6 4
3
34
9-3o 21 3
62
9-
3i io4
g.
0.68 896
9.99
109
26
35
9.30275
9-
3i 1 68
0.6
3 832
9-99
I 06
25
36
g.3o 336
9-
3i 233
65
0.68 767
9.99
104
24
62
6 4
3
3 7
g.3o 398
61
9-
3i 297
0.68 703
9.99
101
23
38
9-3o 459
9-
3i 36i
4
0.6
3 63 9
9-99
099
22
3 9
g.3o
521
3 1 4^5
6 4
0.68 676
9.99
096
3
2 I
40
9. 3o 5b2
61
9-
3r 489
64
0.68 5 1 1
9.99
093
3
20
4i
g.3o 643
61
9-
3: 552
64
o.6i
3448
9-99
091
'9
42
9-3o 704
61
9-
3i 616
o.6i
3 384
9.99 088
18
43
9. 3o 765
61
9-
3i 679
6 4
0.68 32i
9.99 086
3
i?
44
9. 3o 826
61
9-
3i 743
6,
0.68 257
9.99 o83
16
45
g.3o 887
9-
3i 806
0.68 194
9.99 080
i5
46
9-3o 947
9-
3i 870
64
0.68 i3o
9.99078
i4
47
9. 3 1 008
60
9-
3i g3'
1
63
0.68 067
9.99
075
3
i3
48
9 . 3i 068
9-
3 1 996
3
0.68 oo4
9-99
072
12
49
9 .3i
[29
9-
32 o5c
)
63
0.67 941
9.99
070
I I
50
9 .3,
189
61
9
32 I 22
63
0.67 878
9.99
067
10
5i
9 .3i 250
60
9
32 i85
63
0.67 815
9.99
o64
9
52
9 . 3i 3io
9
32 248
J
0.67 752
9.99
062
8
53
9 . 3 1 370
9
32 3i i
63
0.67 689
9-99
oSg
7
60
62
3
54
9-3i 43o
60
9
32 3?3
6
0.67 627
9-99
066
6
55
g.3i
49
9
3a436
3
0.67 564
9.99
o54
5
56
9 .3i 549
59
9
32498
0.67 5o2
9.99 o5i
3
4
60
63
3
5 7
9. 3i 609
60
9
32 56i
62
0.67 43g
9.99 o48
2
3
58
9. 3 1 669
9
32 623
0.67 377
9-99
o46-
2
5 9
9 .3i
728
59
9
32 685
0.67 315
9.99
o43
I
60
9 .3,
788
9
32 747
0.67 253
9.99
o4o
L. Cos. d.
L
Cotg. d. L. Tang.
L. Sin. d.
'
78.
PP
65
64 63
62
61
60
59 3
.1
6.5
6.4 6.3
.1
6.2
6.1
6.0
.1
5-9 -3
2
12.8 12.6
.2
12.4
12.2
I2.O
.2
11.8 0.6
3
19-5
19.2 18.9
3
18.6
.8.3
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
5
32.5
32- 3'-5
5
31.0
3-S
30.0
5
29-5 i-5
.6
39-
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 5-4
.8
49.6
48.8
48.0
.8
47-2 2-4
9 58.5
57-6 56.7
9
55-8 54.9
54.Q
53-' 2.7
53
12.
,
L. Sin.
d.
L
Tang.
d.
L. Cotg.
L. Cos.
d.
9-3i
7 ss
9
.32 7 4 7
0.67 253
9.99 o4o
60
I
9-3i 84?
60
9
.32 8lO
62
0.67 190
9.99 o38
3
5 9
2
9-3i 907
9
32 872
0.67 128
9.99 o35
58
3
g.3i 966
59
9
32 933
62
0.67 067
9.99 o32
2
57
4
9. 32 025
59
9
32 995
62
0.67 005
9.99 o3o
56
5
g.32 o84
9
33o57
0.66 943
9.99027
55
6
g.32
i43
59
9
33 119
61
0.66881
9.99 024
54
7
g.32 202
59
9
33 180
62
0.66 820
9.99
O22
3
53
8
9. 32 261
rg
9
33 242
0.66 758
9.99019
52
9
9. 32
ii9
9
33 3o3
0.66 697
9.99
016
5i
10
g.32 378
9
33 365
0.66635
9.99
Ol3
50
1 1
9.32 437
58
9
33426
61
0.66 574
9.99011
49
12
9.32
495
9
33487
0.66 5i3
9-99
008
48
i3
9.32 553
5
9-
33 548
0.66 452
9.99 oo5
3
47
59
61
3
i4
9.32 612
58
9-
33 609
61
0.66 3gi
9.99 002
46
i5
9.32 670
9-
33 670
0.66 33o
9.99 ooo
45
16
9.32 728
5
9-
33 731
61
0.66 269
9.98
997
3
44
58
61
3
17
9.32 786
5 8
9
33 792
61
0.66 208
9.98
994
43
18
9.32 844
9
33853
0.66 147
9.98
991
42
'9
9.32 902
5
9
33 913
60
0.66 087
9.98
989
4i
20
9 . 32 960
9
33 974
0.66 026
9.98 986
3
40
21
9. 33 018
9-
34o34
61
o.65 966
9.98
9 83
J
3 9
22
9.33 075
9
34095
o.65 goS
9.98
980
38
23
9.33 i33
5
57
9
34 1 55
60
o.65 845
9.98
978
3
37
24
9-33 190
9
342i
)
61
0.65785
9.98
975
36
25
9-33 248
5
9-
34276
o.65 724
9.98
972
35
26
9.33 3o5
57
57
9-
34336
60
o.65 664
9.98
969
3
34
27
9-33 362
eg
9-
343 9 6
60
o.65 6o4
9.98
967
33
28
9.33 420
9-
34456
o.65 544
9.98
964
32
2 9
9-33477
57
9-
345i6
0.65484
9.98
961
3
3i
30
9-33 534
57
9-
34576
o.65 424
9.98
958
3
30
L. Cos.
d.
L
Cotg. d. L. Tang-.
L. Sin. d.
'
773O.
PP
63
62 61
60
59
58
57 3
.1
6.3
6. 2 6.1
.1
6.0
5-9
5.8
.1
5-7 0-3
.2
12.6
12-4 12.2
2
I2.O
n. 8
ii. 6
.2
11.4 0.6
3
18.9
18.6 18.3
3
18.0
'7-7
'7-4
3
17.1 0.9
4
25.2
24.8 244
4
24.0
23.6
23.2
4
22.8 1.2
5
3'-5
31.0 30.5
5
30.0
29.5
29.0
5
28.5 I.J
.6
37-8
37.2 36.6
.6
36.0
35-4
34-8
.6
34-z 1-8
7
44-'
43 4 42-7
7
42.0
4'-3
40.6
7
39-9 2-
.8
5-4
49.6 48.8
.8
48.0
47.2
46.4
.8
45-6 2.4
9 5 6 -7
55-8 . 54.9
9
S3- 1
S2.2
51.3 2.7
54
123O
/
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9.33 534
57
56
57
57
57
56
57
56
56
57
56
56
56
56
56
56
55
56
55
56
55
56
55
55
55
55
55
55
55
55
9-34
5 7 6
59
60
60
59
60
59
59
59
60
59
59
59
59
58
59
59
58
59
58
59
58
58
58
58
58
58
58
58
58
57
o.65 424
9.98
958
3
2
3
3
3
3
3
2
3
3
3
3
3
2
3
3
3
3
3
3
3
2
3
3
3
3
3
3
3
30
3i
32
33
34
35
36
3?
38
39
9 . 33 5gi
9.3364?
9.33 704
9-33 761
9.33 818
9.33874
9 . 33 g3i
9.33 987
9.34 o43
9.34635
9.34695
9-34 755
9.348i4
9. 348 7 4
9 .34 9 33
9.34 992
9.35 o5i
9-35 in
o.65 365
o.65 3o5
0.65245
o.65 186
o.65 126
o.65 067
o.65 008
o-64 949
o-64 889
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9 55
953
95
94?
944
94 1
9 38
936
933
29
28
2 7
26
25
24
23
22
21
40
9. 34 100
9 .36
170
o-64 83o
9. 98 g3o
20
4i
42
43
44
45
46
47
48
49
9.34 i56
9.34 212
9.34268
9.34324
9.34 38o
9.34436
9.34 491
9.34547
9.34 602
9. 35 229
9-35 288
9-35 347
9 .354o5
9-35464
9-35 523
9-35 58i
9-35 64o
9-35 698
o.64 771
o . 64 712
0.64653
o.64 595
0.64536
0.64477
o. 64 419
o.64 36o
o.64 3o2
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
927
924
921
919
916
910
907
9 o4
'9
18
16
i5
M
i3
12
I I
50
9-34 658
9.35
7 5 7
o.
64243
9.98
901
10
5i
52
53
54
55
56
5 7 .
58
5 9
9.34 71 3
9.34 769
9.34 824
9.34879
9.34989
9.35 o44
9-35 099
9 .35 1 54
9-35 815
9-35 873
9.35 g3i
9.35 989
9 . 36 o4?
9 .36 io5
9 .36 i63
9-36 221
9-36 279
o.64 i85
o.64 127
o . 64 069
o.64 on
o . 63 gSS
o.63 895
o.63 837
o.63 779
o.63 721
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
898
896
893
890
887
884
881
878
8 7 5
9
8
7
6
5
4
3
2
I
60
9/35 209
9-36
336
o.
63 664
9.98
8 7 2
L. Cos.
d.
L. Cotg.
d. L.
Tang.
L. Sin.
d.
1
77.
PP
.1
.2
3
4
5
.6
7
.8
60
59
58
57
56
.i
.2
3
4
'.6
9
55 3
6.0
12.
18.0
24.0
30.0
36.0
42.0
48.0
54-
5-9
'7-7
23.6
29 5
35-4
4'-3
47-2
53- *
5.8
ii. 6
17.4
23.2
29.0
34-8
40.6
46.4
i 5-7
.2 11.4
3 i/- 1
.4 22.8
5 28.5
.6 34.2
7 39-9
.8 45.6
9 5'-3
5 .6
II. 2
16.8
22.4
28.0
33-6
39- a
44-8
50.4
5-5 -3
ii. o 0.6
16.5 0.9
22. 1.2
27-5 1-5
33-o 1.8
38-5 2.1
44.0 2.4
49-5 2.7
55
13.
L. Sin
d.
L. Tang. d.
L.
Cotg.
L. Cos. d.
9. 35 209
54
55
55
54
54
55
54
54
54
54
54
54
54
54
54
53
54
53
54
53
53
53
54
53
53
53
52
53
53
53
9.36
336
58
58
57
57
58
57
57
57
57
57
57
57
57
57
56
57
56
57
56
57
56
56
56
56
56
56
56
56
56
55
0.63664
9.98
872
3
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
60
I
2
3
4
5
6
7
8
9
9 .35 263
9.35 3i8
9. 35 373
9.35 427
9.3548i
9-35 536
g.35 5go
9.35 644
9-35 698
9-36 3g4
9.36452
9. 36 Sog
9.36566
9.36624
9-3668i
9-36 738
9 . 36 795
9-36 852
o.
0.
o.
0.
o.
o.
0.
o.
o.
63 606
63 548
63 491
63434
63 376
63 3ig
63 262
63 205
63 1 48
9.98 869
9.98 867
9.98 864
9.98 861
9.98 858
9.98855
9.98 852
9.98849
9.98 846
5 9
58
5?
56
55
54
53
52
5i
10
9 . 35 7 52
9.36 909
0.
63 091
9.98 843
50
1 1
12
i3
i4
i5
16
17
18
'9
9-35 806
9.35 860
9-35 914
9-35 968
9. 36 022
9-36 076
g.36 129
g.36 182
9.36 236
9-36 966
9.37 O23
9.37 080
9 .3 7 i3 7
9.37 Ig3
9.37 250
9.37 3o6
9.37 363
9.37 419
0.
o.
o.
o.
o.
o.
o.
o.
o.
63o34
62977
62 920
62863
62 807
62 7 5o
62 6g4
62 637
62 58i
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
84o
83 7
834
83i
828
825
822
819
816
49
48
47
46
45
44
43
42
4i
20
9-36 289
9.37 4?6
0.
62 524
9.98
8i3
40
21
22
23
24
25
26
27
28
2 9
9-36 342
9 36 3g5
g.36 449
9-36 5o2
9.36555
g.36 608
9-35 660
9-36 713
9. 36 766
9.37 532
9 .3 7 588
9.37 700
9.37 7 56
9.37 812
9.37868
9.37 924
9.37 980
o.
o.
0.
o.
0.
0.
0.
o.
o.
62468
62 412
62 356
62 3oo
62 244
62 1 88
62 1 32
62 076
62 O2O
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
810
807
8o4
801
798
79 5
79 2
789
7 86
3 9
38
3?
36
35
34
33
32
3i
30
g.36 819
g.38o35
0.
61 965
9.98
7 83
30
L. Cos. | d.
L. Cotg.
d.
L.
Tang.
L. Sin. d.
'
76 3O .
PP 58
57
56
55
54
.1
.2
3
4
.6
53
3
i 5 8
.2 II. 6
3 '7-4
5-7
11.4
17.1
5-6
II. 2
16.8
i 5-5
.2 II. O
3 l6 -5
.4 22.0
5 27.5
6 33.0
7 38-5
.8 44.0
9 49- 5
10.8
16.2
21.6
27.0
32.4
37-8
43.2
10.6
'59
21.2
26.5
37- i
42.4
.6
9
1.2
1.5
2.1
24
2-7
.5 29.0
.6 34.8
.7 40.6
.8 46.4
28.5
34-2
39-9
45-6
5t-3
28.0
33-6
39.2
44.8
5 4
56
13 SO.
'
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos. d.
30
9 . 36 8i 9
52
53
52
52
53
52
52
52
52
52
52
52
52
y
5'
52
5'
52
5'
52
5i
5i
5i
5i
5'
5'
5i
5'
5'
5'
9
38o35
56
56
55
55
56
55
55
56
55
55
55
55
55
54
55
55
54
55
55
54
54
55
54
54
54
54
54
54
54
54
0.61 965
9.98
783
3
3
3
3
3
3
3
3
3
3
3
4
3
3
3
3
3
3
3
3
3
4
3
3
3
3
3
3
3
4
30
3i
82
33
34
35
36
37
38
3 9
9 .36 871
9 . 36 9 a4
9 . 36 976
9. 37 028
9 .37 08 1
9.37 i33
9.37 i85
9 .3 7 2 3 7
9.37 289
9
9
9
9
9
9
9
9
9
38 091
38 147
38 202
38 25 7
38 3i3
38 368
38423
38479
38 534
0.61 909
0.61 853
0.61 798
0.61 743
0.61 687
0.61 632
0.61 577
0.61 52i
0.61 466
9.98
9.98
9 . 9 8
9.98
9.98
9 . 9 8
9 . 9 8
9.98
9.98
780
777
774
771
768
7 6 5
762
7 5 9
756
2 9
28
27
26
25
24
23
22
21
40
9. 37 34i
9
38 589
0.61 4i i
9.98
7 53
20
4i
42
43
44
45
46
47
48
4 9
9. 37 3g3
9.37445
9.37497
9.37 54 9
9.37 600
9.37 652
9 . 37 703
9.37 755
9. 37 806
9
9
9
9
9
9
9
9
9
38 644
38 6 99
38 7 54
38 808
38863
38 918
38 972
3g 027
3g 082
0.61 356
o. 61 3oi
0.61 246
0.61 192
0.61 137
0.61 082
0.61 028
0.60 973
0.60 918
9.98 750
9.98 746
9-98 ?43
9-98 740
9.98 737
9.98 734
9.98 731
9.98728
9.98 725
'9
18
17
16
i5
i4
i3
12
II
50
9.37 858
9
39 1 36
0.60 864
9.98
722
10
5i
52
53
54
55
56
5?
58
5 9
9.37909
9.37 960
9 . 38 01 i
9.38 062
9-38 n3
9.38 1 64
g.38 2i5
g.38 266
9.38 3 1 7
9
9
9
9
9
9
9
9
9
3g 190
3 9 245
3 9 299
3 9 353
39 407
3 9 46i
3 9 5i5
3g 56g
3g 623
0.60 810
0.60 755
0.60 701
0.60 64?
0.60 5g3
0.60 SSg
0.60 485
0.60 43i
0.60 377
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
9.98
719
7 i5
712
709
706
703
700
697
6 9 4
9
8
7
6
5
4
3
2
I
60
9.38
368
9
39677
0.60 323
9.98
690
L. Cos.
d.
L
. Cotg.
d.
L, Tang.
L. Sin.
d.
'
76.
PP
.1
.2
3
4
.6
:i
56
55 54
.1
.2
3
4
.6
.8
53
52
51
4 3
5.6
II. 2
16.8
22.4
28.0
33-6
39-2
44-8
5-4
5-5 5-4
ii. o 10.8
16.5 16.2
22. 21.6
27.5 27.0
33.0 32.4
38.5 37.8
44.0 43.2
49.5 48.6
5-3
10.6
15.9
21.2
26.5
31-8
37-1
42.4
47-7
5-2
10.4
15.6
20.8
26.0
31.2
36.4
41.6
5-1
10.2
'5-3
20.4
25.5
30.6
35-7
40.8
45. s)
.1
.2
3
4
ii
'.8
9
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
57
14 C
,
L. Sin.-
d.
L.
Tang.
d.
L. Cotg.
L. Cos.
d.
9. 38 368
9
3 9 677
0.60 323
g.g8 6go
60
1
g.384i8
51
9
3g 73t
54
0.60 26g
9.98
687
3
5g
2
g.3846g
9
3g 785
0.60 2i5
9.98
684
58
3
g.38
5ig
9
3g 838
53
0.60 162
9.98
681
57
5'
54
3
4
9. 38 570
5
9
3g 8g2
0.60 1 08
9.98
678
3
56
5
g.38 620
9
3g g45
0.60 o5
9.98
675
55
6
9. 38 670
9
Sgggg
54
0.60 oo i
9.98
671
54
5'
53
3
7
9. 38
721
5
9
4o o52
o.5g g48
9.98668
53
8
g. 38 771
9
4o 1 06
o.5g 8g4
g.g8 665
52
9
g.38 821
9
4o i5g
53
o.5g 84i
9.98
662
5i
10
g.38 871
9
4o 212
53
o.5g 788
9.98
65g
50
1 1
g.38 g2i
9
4o 266
54
o.5g 734
9.98
656
4 9
12
9. 38 971
9
4o 3ig
o.5g 681
9.98
652
48
i3
g.3g 02 1
9
4o 372
53
o.5g 628
g. 9 8
64g
3
47
i4
g.3g 071
5
9
4o 425
53
o.5g 575
9.98
646
3
46
i5
g.3g
121
9-
4o 4?
3
o.5g 522
9.98
643
45
16
9 .3 9
170
49
9
4o 53i
53
o.5g 46g
g.g8
64o
3
44
, 7
g.3g 220
So
9
4o584
53
o.5g 4i6
9.98
636
4
43
18
g.3g 270
9
4o636
o.5g364
9.98
633
42
r 9
g. 3g 3ig
49
9
4o68g
53
o.5g 3n
9.98
63o
3
4i
20
g.3g 36g
5
9
4o 742
53
o.5g258
9.98
627
40
21
g.3g 4i8
49
9
4o 795
53
o.5g 2o5
9.98
6 2 3
4
3 9
22
g.3g 467
9-
4o84?
o.5g i53
9.98
620
38
23
g.Sg 5i7
5
9-
4o 900
53
o.5g 100
9.98
617
i
3?
24
g.Sg 566
49
9
4o g52
5*
o.5g o48
9.98
6i4
3
36
25
g.Sg 615
9-
4i 005
o.58 gg5
9.98
610
35
26
g.Sg 664
49
9
4i o5j
52
o.58 g43
9.98
607
J
34
27
9 .3 97 i3
49
9
4 1 log
52
o.58 891
9.98
6o4
3
33
28
g.3g 762
9-
4i 161
0.58 83g
9.98
60 1
32
29
g.Sg 8n
49
9
4l 21;
4
53
o.58 786
g.g8
5 97
4
3i
30
g. 3g 860
49
9
4i 266
52
o.58 734
9.98
5 9 4
3
30
L. Cos.
d.
L
Cotg.
d.
L. Tang.
L. Sin. d.
'
75 3O .
PP
54
53 53
51
50
49
4 3
.1
5-4
53 52
.j
5-i
5.0
49
.1
4 -3
.2
10.8
10.6 10.4
2
10.2
IO.O
9.8
.2
.8 .6
3
16.2
15-9 '5-6
3
'5-3
15.0
14.7
3
1.2 .9
4
21.6
21.2 20.8
4
20.4
20. o
19.6
4
1.6 1.2
5
27.0
26. 5 26.0
5
255
25.0
24-5
5
2.O 1.5
.6
32-4
31.8 31.2
6
30.6
30.0
29.4
.6
2.4 1.8
7
37.8
37- * 36-4
7
35-7
35-o
34-3
.-
2.8 2.1
.8
43-2
42-4 4'-6
8
40.8
40.0
39-2
.8
3 .2 2.4
9
48.6
47-7 46.8
9
45 9
45.0
44.1
58
143O
/
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.39 860
9-
4i 266
o.58 734
9.98 594
30
3i
9.39909
49
9-
4i 3i8
52
o.58 682
9.98 591
3
29
32
9.399
58
Q
9-
4i 870
0.58 63o
9.98 588
28
33
9.40 006
9-
4i 422
o.58 578
9.98 584
27
t9
52
H
34
9 ,4o c
48
9-
4i 4?/
52
o.58 526
9.98 58i
3
26
35
9.40 io3
9-
4i 5 2 6
o.584?4
9.98 578
25
36
9.40 i
52
9-
4i 5 7 8
o.58 422
9.98 574
24
a
51
3
37
9 .40 200
9-
4i 629
52
o.58 3?i
9.98 571
3
23
38
9.40 249
9-
4i 681
o. 58 3ig
9.98 568
22
3 9
9.40 297
9-
4i 733
o.58 267
9.98
565
21
40
g.4o 346
9-
4i 784
o.58 216
9.98
56i
20
4i
9.40 v
94
1
9-
4i 836
o.58 1 64
9.98 558
3
'9
42
9 .4o 44 2
9-
4i 887
o.58 1 13
9.98
55.5
18
43
9.40490
9-
4i 939
o.58 061
9.98
55i
17
B
51
3
44
g.4o 538
s
9-
4i 990
o.58 oio
9. 9 8
548
3
16
45
9.40 586
9-
42 o4i
0.57 959
9.98
54.5
i5
46
g.4o 634
("
9-
42 093
0.57 907
9.98
54 1
i4
H
51
3
47
9-4o 682
,8
9-
42 i44
o.5 7 856
9.98
538
3
i3
48
9-4o 7
3o
9-
42 ig5
0.57 805
9.98
53,5
12
49
9-4o 778
|fl
9-
42 246
51
0.57 754
9.98
53i
I I
50
9.40 825
47
,Q
9-
42 297
5 1
0.67 703
9.98
528
10
5i
9.40 873
18
9-
42 348
0.57 652
9.98
525
4
9
52
9.40 921
9-
42 399
0.57 601
9.98
521
8
53
g.4o 968
9-
42 45o
5'
0.57 550
9.98
5i8
7
V
51
54
9.41 016
9-
42 5oi
0.57 499
9.98
515
4
6
55
9.41 o63
9-
42 552
0.57448
9.98
5n
5
56
9-4i i
ii
|B
9-
42 6o3
Si
0.57 397
9.98
5o8
4
5 7
9.41 i58
47
9-
42 653
5
0.57 347
9.98
5o,5
4
3
58
9-4l 2
o5
9-
42 704
0.57 296
9.98
5oi
2
5 9
g.4l 252
47
9-
42 755
5i
o. 57 245
9.98
498
I
60
9 . 4 1 3oo
* u
9-
42 8o5
0.57 195
9.98
4 9 4
L. Cos.
d.
L.
Cotg.
d. L. Tang.
L. Sin.
d.
/
75.
PF
52
51
50
49
48
47
4
3
.1
5 2
5 .i
S-o
.1
4.9
4 .8
4-7
.1
0.4
-3
.2
10.4
IO. 2
IO. O
.2
9.8
96
9.4
.2
0.8
0.6
3
15.6
15-3
150
3
14.7
14.4
14.1
3
12
0.9
4
20.8
20.4
20. o
4
' 19.6
19.2
18.8
4
1.6
1.2
5
26.0
25.5
25.0
5
24-5
24.0
23-5
5
2.O
i 5
.6
31 2
30.6
30.0
.6
29.4
28.8
28.2
.6
24
1.8
7
36.4
35-7
35.0
7
34-3
33 6
32-9
7
2.8
2.1
.8
4 ..6
40.8
40.0
.8
39-2
38.4
37- 6
.8
3-2
2-4
9
46.8
45_9__
45
9
44.1 43.2
42.3
___
'-' 7
5 9
15 C
'
L.Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
9.41
3oo
9.42 bo5
5'
5
5'
5
5
5'
5
So
5
5
5
5
5
5
50
49
5
5
49
5
49
5
49
5
49
49
49
So
49
49
0.57 195
9
.9*494
3
3
4
3
4
3
3
4
3
4
3
4
3
3
4
3
4
3
4
3
4
3
4
3
3
4
3
4
3
4
60
I
2
3
4
5
6
7
8
9
9.41
9-4i
9.41
9,41
9-4i
9-4i
g.4i
9-4i
9-4i
34 7
3 9 4
44 1
483
535
682
628
6 7 5
722
47
47
47
47
47
46
47
47
46
47
46
47
46
47
46
46
47
46
9.42 856
9.42 906
9.42 957
9.43 007
9.43o57
9.43 108
9.43 1 58
9.43 208
9.43 258
0.57 i 44
o. 57 094
0.57 o43
o.56 993
o.56 9 43
0.66892
o.56 842
o.56 792
o.56 742
9
9
9
9
9
9
9
9
9
.98 491
.98488
.98484
.98481
.98477
.98474
.98471
.98467
.98464
5 9
58
5 7
56
55
54
53
52
5i
10
9-4i
768
9-43 3o8
o.56 692
9
.98460
50
1 1
12
i3
i4
i5
16
17
18
'9
9-4i
9.41
9.41
9.41
9.42
9.42
9.42
9.42
9.42
815
861
908
954
OOI
047
098
i4o
186
9-43358
9.434o8
9.43458
9 .435o8
9. 43 558
9.43 607
9.43 657
9-43 707
9.43756
o.56 642
o. 56 5 9 2
o.56 542
o.56 492
o.56 442
o.56 3g3
0.56343
o.56 293
o.56 244
9
9
9
y
9
'<;
9
9
9
.98457
.98453
.98450
.98447
. 9 8443
.98440
.98 436
.98433
.98 429
49
48
4?
46
45
44
43
42
4i
20
9.42
232
46
46
46
46
46
45
46
46
46
45
46
9.43 806
o.56 194
9
.98426
40
21
22
23
24
25
26
27
28
2 9
9.42
9.42
9.42
9.42
9.42
9.42
9 .4a
9.42
9.42
278
324.
870
4i6
46i
607
553
5 99
644
9-43855
9-43 905
9-43 954
9 . 44 oo4
9.44 o53
9-44 102
9-44 i5i
9-44 201
9-44 250
o.56 i4s
o.56 og5
o.56 o46
o.55 996
o.55 947
0.55898
0.55849
o.55 799
o.55 750
9
9
9
9
9
9
9
9
9
.98 422
.98 419
.984x5
.98412
. 98 4og
.98 4o5
.98 4O2
.98 398
.98 395
3 9
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. Cotg.
d.
L. Tang.
L. Sin.
d.
74 SO.
PP
.1
.2
3
4
:!
51
50
49
48
47
4 6
.1
.2
3
4
.6
.8
9
45
4
3
5-i
IO. 2
'5-3
20.4
25-5
30.6
35-7
40.8
45 9
5.0
IO. O
15.0
20.0
250
30.0
35-0"
40.0
49
9.8
14.7
19.6
24- S
39.4
34-3
39-
44 I
.1
.
3
4
'.6
.8
4 .8
9 .6
14.4
19.2
24.0
28.8
33-6
38.4
4S- 2
4-7
9-4
14.1
1 8.8
23-5
28.2
32.9
37-6
42 3
4-6
9.2
13- <*
18.4
.3.0
27.6
32.2
36.8
41 4
4-5
9.0
13-5
18.0
22.5
27.0
3' -5
30.0
Joy
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3-|
3-6
0.3
0.6
0.9
1.2
IS
1.8
2.1
2.4
2 -7
60
15 30
L. Sin.
d.
. Tang. d.
L. Cotg.
L. Cos. d.
30
9.42
690
9.44 299
o . 55 701
9 . 9
3 9 i
30
3 1
9.42
7 35
9
.44348
49
o.55 652
9.98
388
3
29
32
9.42
781
9
44 397
o. 55 6o3
9.98
384
28
33
9.42
826
9
.44446
o.55 554
9.98
38i
3
27
4 6
44
^
34
9.42 872
45
9
44 495
49
o.55 5o5
9.98
377
26
35
9.42
917
9
.44544
,0
o.55 456
9.98
3 7 3
20
36
9.42
962
9
.44 592
o.554o8
9.98
370
3
24
46
44
37
9-43
008
9
.4464i
49
o.55 35 9
9.98
366
23
38
9 .43 o53
9
.44 690
o.55 3io
9.98
363
22
3 9
9.43098
45
9
.44738
4
o.55 262
9.98
35g
4
21
40
9-43
i43
45
9
.44 787
o.55 2i3
9.98
356
3
20
4i
9-43
1 88
45
9
.44836
o.55 1 64
9.98
352
4
'9
42
9.43233
9
.4488
i
o.55 1 16
9.98
349
18
43
9-43
278
45
9
.44933
49
o.55 067
9 . 9 8
345
4
7
44
9-43
323
45
9
.4498
I
40
48
o.55 019
9.98
342
3
16
45
9-43
36?
9
.45 029
o.54 971
9.98
338
i5
46
9.43 412
45
9
.45078
49
o.54 922
9.98
334
4
i4
4 7
9.43457
45
9
.45 126
48
.
0.54874
9.98
33i
3
i3
48
9 .43 5o2
9
45 174
0.54826
9.98
327
12
49
9-43
546
44
9
.45 222
40
o.54 778
9.98
324
3
I I
50
9 .43 5gi
45
9-45 271
4y
0.54 729
9.98
32O
4
10
5i
9-43635
44
9
.45 319
,a
o.5468i
9.98
3i 7
3
9
52
9.4368o
9
.45 367
0.54633
9.98
3i3
8
53
9-43
724
44
9
.454i5
48
0.54585
9.98
3og
4
7
54
9-43
769
45
9
.45463
4*5
,a
0.54537
9.98
3o6
3
6
55
9-43
8i3
9
-455n
0.54489
9.98
302
56
9.4385?
44
9
.45559
4
o.5444i
9.98
299
3
4
57
9 .43 901
44
9
.45 606
47
48
o.54 394
9.98
295
4
3
58
9-43
9 46
9
.45654
0.54346
9.98
291
2
5 9
9.43 990
44
9
.45 702
48
o.54 298
9.98
288
3
I
60
9-44 o34
44
9
.45 750
48
o. 54 25o
9.98
284
4
L. Cos.
d.
L
. Cotg.
d.
L. Tang.
L. Sin.
d.
'
74
.
PP
49
4 8
47
4 6
45
44
4
3
.!
4.9
- 4 .8
4-7
.1
4 .6
4-5
4-4
.1
0.4
3
.2
9.8
Q.b
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
ig.2
18.8
4
z8. 4
18.0
,7.6
4
1.6
I 2
5
24 5
24.0
23-5
5
23.0
22.5
22.
5
2.O
i 5
.6
29.4
28.8
28.2
.6
27.6
27.0
26.4
.6
2-4
i 8
7
34-3
33 6
32.9
7
32.2
3'-5
30.8
7
2.8
2 I
.8
39 2
3-4
37-6
.8
36.8
36.0
35-2
.8
3-2
2 4
9 44-i
4?. 2 42. -^
9
41.4
40. 5
9
3.6
2 7
16.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
9-44 o34
9.45 750
o.54 z5o
S
.98 284
60
1
9.44 078
9-45 797
47
*.&
O.54 203
9
.98 281
5 9
2
9.44 122
9-45845
0.54 i55
9
.98 277
58
3
9-44 l66
9.45 892
47
o.54 1 08
9
.98 273
57
44
48
9
4
9.44 2IO
43
9.45 940
o.54 060
9
.98 270
4
56
5
9-44253
9.45987
o.54oi3
9
.98 266
55
6
9-44 297
9.46035
4
0.53965
9
.98 262
54
44
47
3
7
9-44 34 1
9-46 082
o.53 918
9
.98 259
53
8
9-44385
9-46 i3o
4
o.53 870
9
.98255
52
9
9.44428
43
9-46 177
47
o.53 823
9
.98 25 1
5i
10
9-44 472
9.46 224
47
o.53 776
9
.98 248
50
1 1
9.44 5i6
9.46 271
47
o.53 729
9
,98244
49
12
9 .44559
9.46 319
4
o.53 681
9
-98 240
48
i3
9-44 602
43
9.46 366
47
0.53634
9
.98 237
47
44
47
4
i4
9.44646
9.46 i
ii3
o.53 58?
g
.98 233
46
i5
9.44689
9-46 46o
47
o.53 54o
9
.98 229
45
16
9.44733
44
9-46 607
47
o.53 493
9
.98 226
3
44
'7
9-44 776
43
9. 46 554
47
0.53446
9
.98 222
4
43
18
9-44 819
9.46 601
o.53 399
.98 218
42
'9
9-44 862
43 __
9-46648
47
o.53 352
9
.98 215
4i
20
9-44 go5
43
9-46 694
46
o.53 3o6
9
.98 21 I
4
40
21
9.44948
43
9-46 74 1
47
o.53 2
5 9
9
.98 207
3
3o
22
9-44 992
9.46 788
O.53 212
.98 2O4
38
23
9-45 035
43
9.46 835
47
o.53 i65
9
.98 200
37
24
9.45077
42
9-46 88 1
46
o.53 119
9
.98 196
4
36
25
9.45 I2O
9 .46(
>28
o.53 072
9
.98 192
35
26
9.45 i63
43
9.46975
47
o.53 O25
9
.98 189
34
27
9.45 206
43
9.47 021
46
o.52 979
9
. 9 8 185
4
33
28
9.45 249
9.4? 068
47
o.52 9
32
9
.98 181
32
29
9.45 292
43
9-47
i4
4 b
o.52 886
9
.98 177
3i
30
9.45 334
42
9.47 160
46
o.52 84o
9
.98 174
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.Sin.
d.
'
73 30 .
PP
48
47
46
45
44
43
'
43
4 3
.1
4 .8
4-7
4-6
.,
4-5
4-4
4-3
.1
4-2
0.4 0.3
.2
9.6
9.4
9.2
.2
9.0
8.8
8.6
.2
8.4
0.8 0.6
3
M-4
. M i
13-8
3
'3'S
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
5
24.0
23-5
23.0
5
22.5
22.
21.5
5
21.
2.0 1.5
.6
28.8
28.2
27.6
.6
27.0
26. 4
25-8
.6
25.2
2.4 1.8
7
33-6
32-9
32.2
.7
31. s
30.8
30.1
7
29.4
2.8 2.1
.8
38.4
37-6
36.8
.8
36.0
352
V 4
.8
33-6
3.2 2.4
9 43-2
42.3 41.4
9
40. s 30.6 38.7
3.6 2.7
62
16 3O .
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
9-45
377
43
9.47 207
47
46
o. 52 793
9
.98 170
4
29
32
9-45
419
9.4? 253
o.52 747
9
.98 166
28
33
9.45 462
43
9.4? 299
46
o.52 701
9
.98 162
4
27
34
9-45
5o4
42
9-47 346
47
0.62 654
9
.98 i
59
3
26
35
9-45
54?
9.47 392
4
O.52 608
9
.98 165
25
36
9-45
58 9
9 . 47 438
46
o. 52 562
9
.98 i
)i
i
24
3?
9-45
632
43
9. 4? 484
4 b
-6
o.52 5i6
9
.98 147
4
23
38
9-45
6 7 4
9.47 53o
o.52 470
9
.98 1 44
22
39
9-45
716
42
9.47 576
46
O.52 424
9
.98 i4o
4
21
40
9-45
758
9.4? 622
46
o.52 378
9
.98 i36
4
20
4i
9-45
801
42
9.47 668
46
o.52 332
9
.98 i32
4
19
42
9-45
843
9-4? 1
i4
O.52 286
9
.98 129
18
43
9-45 885
9.4? 760
4
0.62 240
9
.98 125
4
17
42
46
4
44
9-45
927
42
9.47 806
46
o.52 194
9
.98 121
16
45
9.45
969
9.4? 852
o.52 i48
9
.98 117
i5
46
9-46
OI I
9.47897
45
o.52 io3
9
.98 n3
4
i4
42
46
3
47
9-46
o53
42
9.47 943
A.6
o.52 057
9
.98 no
i3
48
9.46 095
9.47 9 8 9
O.52
i
9
.98 1 06
12
49
9-46
i 36
9.48 035
46
o. 5i 965
9
.98 102
4
I I
50
9-46
178
9.48 080
45
o . 5 1 920
9
.98 o
?8
4
10
5i
9-46
220
42
9.48 126
o.5i 874
9
.98 094
9
52
9-46
262
9.48 171
o.5i 829
9
.98 090
8
53
9-46
3o3
9.48 217
o.5i 783
9
.98 087
3
7
42
45
4
54
9-46
34.5
9.48 262
o.5i 738
9
.98 o83
6
55
9-46
386
9. 48 3
07
o.5i 693
9
.98 079
5
56
9-46
428
9.48 353
46
o.5i 64?
9
.98 075
4
4
4 1
45
4
5?
g.46 469
9.48 398
o.5i 602
9
.98 071
3
58
9-46
5n
9-48443
o.5i 55 7
9
.98 067
2
5 9
9-46
552
4i
9.48489
46
o.5i 5i i
9
.98 o63
4
I
60
9-46
5 9 4
42
9.48 534
45
o.5i 466
9
.98 060
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
73.
PP 47
46
45
44 43 43
41
4 3
.1
4.7
4.6
4-5
.1
4.4 4.3 4.2
.1
4.1
0.4 o 3
.2
9-4
9.2
9.0
.2
8.8 8.6 8.4
2
8.2
0.8 06
3
14.1
13.8
>3-5
3
13.2 12.9 12.6
3
12.3
1.2 9
4
18.8
18.4
18.0
4
17.6 17.2 16.8
4
16.4
1.6 I 2
5
23-5
23.0
22.5
5
22. 21.5 21.
5
20.5
2.0 1 $
.6
28.2
27.6
27.0
.6
26.4 25.8 25.2
.6
24.6
2.4 i 8
7
32.9
32.2
31.5
7
30.8 30.1 29.4
7
28.7
2.8 2 I
.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
9
36.9
63
17 C
>
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos. d.
9.46 5g4
41
41
41
41
42
41
41
4
41
41
40
41
41
41
41
40
4'
40
41
40
4'
40
4i
40
40
4i
40
40
40
40
9-
4-3 534
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
7
8
9
9-46635
9.46 676
9-46 717
9-46 758
g.46 800
9.46 84 1
9.46 882
9:46 923
9.46 964
9-
9-
9-
9-
9-
9-
9-
9-
9-
48 579
48624
48669
48 714
48 7 5 9
488o4
48849
48894
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 061
9.98 o56
9.98 o52
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
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
5 9
58
57
56
55
54
53
52
5i
10
9.47 005
9-
48 9 84
o.5i 016
9.98
021
50
1 1
12
13
i4
i5
16
17
18
'9
9.47 o45
9.4? 086
9-47 127
9.47 1 63
9-4? 209
9.47 249
9.47 290
9.47 33o
9-47 371
9-
9-
9-
9-
9-
9-
9-
9-
9-
4g 029
49 073
49118
49 1 63
4g 207
49 252
4g 296
49 34 1
4g 385
o.5o 971
o.5o 927
o.5o882
o.5o83 7
o.5o 793
o.5o 748
o.5o 704
o.5o 65g
o.5o 615
9.98 017
9.98 oi3
9.98 009
9.98 oo5
9.98 ooi
9.97997
9.97993
9.97989
9.97986
49
48
47
46
45
44
43
42
4i
20
9-4? 4i i
9-
49 43o
o.5o 570
9-97
982
40
21
22
23
24
25
26
27
28
2 9
9.47 452
9.47 492
9.47533
9.47 5?3
9.47 6 1 3.
9.4? 654
9.47 6g4
9.47 7 3 4
9-47 774
9-
9-
9-
9-
9-
9-
9-
9-
9-
4g 474
4g 5ig
49 563
4g 607
49 65a
4g 696
49 740
4 9 7 84
49 828
o.5o 526
o.5o48i
0.60437
o.5o 3g3
o.5o348
o.5o 3o4
o.5o 260
o.5o 216
o.5o 172
9.97978
9.97 974
9.97970
9.97966
9.97 962
9.97958
9.97954
9.97950
9.97946
3 9
38
3?
36
35
34
33
32
3i
30
9-47 8i4
9-
49 872
o.5o 128
9.97942
30
L. Cos.
d.
L.
Cotg. | d. L. Tang.
L. Sin. d.
72 3O.
PP
.1
.2
3
4
.6
.8
45
44 43
.1
4*
41
40
.1
4 3
4-5
9.0
'3-5
18.0
22.5
27.0
3'-5
36.0
40. s
8.8 8.6
13.2 12.9
17.6 17.2
22.O 2I.S
26.4 25.8
30.8 30.1
35-2 34-4
r
4.1
4.0
o. 4 o. 3
o. 8 o. 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
3
4
'.6
:l
12.6
16.8
21.0
25.2
29.4
33-6
37-8
12.3
16.4
20.5
24.6
28.7
32.8
36.9
I2.O
16.0
20. o
24.0
28.0
32.0
3
4
5
.6
.8
64
17 30.
>
L. Sin.
d.
L
. Tang.
d.
L. Cotg.
L. Cos.
d.
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
4
4
4
4
4
4
4
4
4
4
30
9-4?
8i4
40
40
40
39
40
40
39
40
40
39
40
39
40
39
39
39
40
39
39
39
39
39
39
39
39
39
9
.49 872
44
44
44
44
44
44
44
43
44
44
44
43
44
43
44
43
44
43
44
43
43
44
43
43
43
43
43
44
43
43
o.5o 128
9-97
942
30
3i
32
33
34
35
36
37
38
3 9
9.47
9.47
9.47
9-4?
9.48
9-48
9-48
9-48
9-48
854
8 9 4
934
974
oi4
o54
094
i33
I 7 3
9
9
9
9
9
9
9
9
9
.49 916
.49 960
. 5o oo4
.5o o48
. 5o 092
.5o i36
.5o 180
,5o 223
. 5o 267
o.5o o84
o.5o o4o
0.49 996
0.49 952
o .49 908
0.49 864
0.49 820
0.49 777
0.49 733
9-97 9 38
9.97 934
9.97 9 3o
9.97926
9.97922
9.97 918
9-97 9'4
9.97910
9-97 9 6
29
28
27
26
25
24
23
22
21
40
9 .48
2l3
9
.5o 3n
0.49 689
9-97
902
20
4i
42
43
44
45
46
4 7
48
49
9-48 252
9-48 292
9-48 332
9-48 371
9-484U
9.48 45o
9 .48 490
9.48 52 9
9 .48 568
9
9
9
9
9
9
9
9
9
5o 355
5o 3 9 8
5o 442
5o485
5o 529
5o 572
5o 616
5o 65g
5o 703
0.49 645
0.49 602
0.49 558
0.49 5i5
o. 49 4?i
0.49 428
0.49 384
0.49 34i
0.49 297
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
898
8 9 4
890
886
882
878
8 7 4
870
866
'9
18
'7
16
i5
i4
i3
12
I I
50
9-48
607
9
5o 746
0.49 254
9-97
861
10
5i
52
53
54
55
56
57
58
5 9
9 .48
9-48
9.48
9-48
9-48
9-48
9-48
9-48
9-48
64?
686
725
8o3
842
881
920
9 5 9
9
9
9
9
9
9
9
9
9
5o 789
5o833
.50876
.5o 919
,5o 962
. 5 1 oo5
.5i o48
. 5 1 092
.5i 135
O.49 211
0.49 167
0.49 124
0.49 08 1
0.49 o38
0.48 995
o.48 g52
o.48 908
0.48865
9.97 857
9.97 853
9.97849
9.97 845
9-97 84i
9.97837
9.97 833
9.97 829
9-97 825
9
8
7
6
5
4
3
2
I
60
9-48
998
9
.5i 178
o.48 822
9-97
821
L. Cos. d.
L
. Cotg.
d.
L. Tang.
L. Sin. d.
'
72.
PP
.2
3
4
.6
.8
9
44
43
45
2
3
4
6
4
40
39
.1
.2
3
4
'.6
.8
9
5 4
4-4
8.8
13.2
,7.6
22.0
26.4
30.8
35 2
4-3
8.6
12.9
17.2
21.5
25.8
30.1
34-4
4.2
8.4
12.6
16.8
21.
25.2
29.4
33-6
37.8
4.1
8.2
12.3
16.4
20.5
24.0
28.7
32.8
4.0
8.0
I2.O
16.0
20. o
24.0
28.0
32.0
36.0
3-9
7.8
11.7
15.6
19.5
23-4
27-3
31.2
35- '
0.5 0.4
i.o 0.8
1-5 1.2
2.0 1.6
2-5 2.O
3.0 2.4
3-5 2.8
4-0 3.2
4-5 3-6
65
18.
L.Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
9.4^ 998
9-5i 178
43
43
42
43
43
43
43
42
43
43
42
43
43
42
43
42
42
43
42
43
42
42
42
43
42
42
42
42
42
42
o.48 822
9
97 821
4
5
4
4
4
4
4
4
4
5
4
4
4
4
4
5
4
4
4
4
60
I
2
3
4
5
6
7
8
9
9.49 oSy
9.49 076
9.49 us
9.49 i53
9.49 192
9.49 z3i
9.49 269
9.49 3o8
9.49 347
39
39
38
39
39
38
39
39
9.61 221
9. 5 1 264
9. 5 1 3o6
9. 5 1 349
9.5i 392
9.61435
9-5i 4?8
g.5i 520
g.5i 563
o.48 779
o.48 736
o.48 694
o.4865i
o.48 608
o.48 565
0.48522
o.4848o
0.48437
9
9
9
9
9
9
9
9
9
97817
97 812
97808
97 8o4
97 800
97796
97792
97788
97 7 84
59
58
57
56
55
54
53
52
5i
10
9.49 385
3
39
38
38
39
38
38
39
38
38
g.5i 606
o.48 3g4
9
97779
50
ii
12
i3
i4
i5
16
17
18
'9
9.49 424
9.49 462
9.49 5oo
9.49 53g
9.49 5 77
9.49 6 1 5
9.49654
9.49692
9.49 ?3o
9-5i 648
9-5i 691
g.5i 734
g.5i 776
9-5i 819
9-5i 861
9. 5 1 903
9. 5 1 946
9.5i 988
0.48352
0.48 3og
0.48266
0.48 224
o.48 181
0.48 i3g
0.48 097
o.48 o54
0.48 OI 2
9
9
9
9
9
9
9
9
9
97775
97771
97 767
977 63
97759
97754
97750
97 746
97 742
49
48
47
46
45
44
43
42
4i
20
9.49 768
38
9.02 o3i
0.47 969
9
97 7 3 ^
40
21
22
23
24
25
26
27
28
2 9
9.49 806
9.49844
9.49882
9.49920
9.49958
9.49996
9.5oo34
g.So 072
g.So 1 10
38
38
38
38
38
38
38
38
38
38
9.52 073
9.52 n5
9.52 157
9. 52 200
9.52 242
9.52 284
9.62 326
9.52 368
9.52 4io
0.47 927
0.47885
0.47843
0.47 800
0.47753
0.47 716
0.47 674
0.47632
0.47 590
9
9
9
9
9
9
9
9
9
97734
97729
97725
97721
97717
97713
97708
97 704
97 700
5
4
4
4
4
5
4
4
4
3 9
38
3 7
36
35
34
33
32
3i
30
g.5o i48
9.52 452
0.47543
9
97 696
30
L. Cos.
d.
L. Cot K.
d.
L. Tang.
L.Sin.
d.
713O.
PP
i
2
3
4
I
I
9
43
43
39
38
I
.2
3
4
'.6
j|
5
4
4-3
8.6
12.9
17.2
21.5
5-8
30-1
34-4
38.7
4-2
8 -*
12. 6
16.8
21.
25.2
29.4
33-6
37-8
i
2
3
4
6
I
3 2
7.8
11.7
.5.6
19.5
23-4
27.3
3i-a
***
7 .6
11.4
IS-2
19.0
22.8
26.6
3-4
3^2
o-S
I.O
'5
2.0
2-5
3-o
3-5
4.0
4-5
o-4
0.8
1.2
1.6
2.O
2-4
2.8
3-2
3-6
66
18 3O
L. Sin.
d.
L. Tang.
d.
L
Cotg.
L. Cos.
d.
30
9. 5o i48
9.5-2 452
O
47548
9.97696
30
3i
y .5o i85
38
9 .5 2
494
42
4? 5o6
9-97
69!
29
32
9. 5o 223
9.52 536
0.
47464
9-97
087
28
33
9-5o 261
3
9.52 578
O.
47 422
9-97
683
4
27
37
42
4
34
g.So 298
38
9-52 620
O.
4?38o
9.97679
26
35
9. 5o 336
38
9.52 661
O.
47339
9-97
6 7 4
25
36
9-5o 3 7
4
9.52
7o3
O.
47297
9-97
670
4
24
37
42
4
37
9. 5o 4i i
38
9 .52
745
42
O.
47255
9.97 666
23
38
9. 5o 449
9.52
787
O.
4? 2i3
9.97 662
22
39
9-5o486
37
9 .52
829
O.
47 171
9-97
65 7
5
21
40
9.5o 523
9.52 870
O.
47 i3o
9-97
653
20
4i
g.5o 56i
3
9.62
912
O.
47088
9-97
64g
4
19
42
9. 5o 5g
8
9.52
953
O.
4? 047
9-97
645
18
43
9. 5o 635
37
9.52 995
42
O.
47005
9.97 64o
5
17
44
9-5o 673
3*
9.53 037
42
0.
46 9 63
9-97636
4
16
45
9-5o 710
9.53078
O
46 922
9.97 632
i5
46
9-5o 747
37
9.53
120
42
O.
4688o
9.97 628
i4
37
41
5
47
9.5o 784
9.53
161
O.
4683 9
9-97
623
i3
48
9-5o 821
9. 53 202
O.
46 79 8
9.97619
12
49
9 .5o858
37
9.53 244
4 2
O.
46 7 56
9-97
615
I I
50
g.5o 896
38
9 .53
285
4i
0.
46 715
9-97
610
10
5i
9. 5o 933
37
9.53
327
42
O.
46673
9-97
606
4
9
52
9-5o 970
9.53368
O.
46 632
9.97 602
8
53
9. 5 1 007
37
9.53
409
4'
O.
465 9 i
9.97597
5
7
54
9-5i o43
30
9.5345o
4 1
O.
46 550
9-97
5 9 3
4
6
55
9.5i 080
37
9-53 492
O.
465o8
9.97 58g
5
56
9-5i 117
37
9.53
533
4 1
O.
46467
9.97 584
5
4
57
9. 5 1 1 54
37
9-53
5 7 4
4'
O.
46426
9.97 58o
4
3
58
g.Si 191
37
9-53 6i5
O.
46385
9.97576
2
5 9
9.61 227
36
9-53
656
4'
O.
46 344
9-97
5 7 i
5
I
60
9. 5 1 264
37
9.53 697
4'
O.
46 3o3
9.97 567
L. Cos.
d.
L. Cotg. d. L.
Tang.
L. Sin. d.
'
71.
PP
42
41
38
37
36
5 4
.1
4-2
4.1
3 .8
i 3-7
3 .6
.1
0.5 04
.2
8.4
8.3
7 .6
.2 7.4
7-2
.2
t.o 0.8
3
12.6
12.3
11.4
3 "-i
10.8
3
I.S 1.2
4
16.8
16.4
15-2
.4 14.8
14.4
4
2.O 1.6
5
21.
20.5
19.0
5 18.5
18.0
5
2.5 2.0
.6
25 2
24.6
22.8
.6 22.2
21.6
.6
3.0 2.4
7
29.4
28.7
26.6
7 25.9
25.2
7
3-5 2.8
.8
336
32.8
3<M
.8 29.6
28.8
.8
4.0 3-2
9 37-8
36.0 44.2
3 2 -4
4.5 3.6
67
19.
/
L. Sin.
d.
L. Tang.
d. L
. Cotg.
L. Cos.
d.
9.5i 264
9 .53 697
o
.46 3o3
9.97 567
60
I
9-5i 3oi
37
9.53 738
o
.46 262
9.97 563
5
5 9
2
9 .5i 338
,6
9.53779
o
.46 221
9-97
558
58
3
g.5i 3 r t
r4
9.53820
41
o
.46 180
9-97
554
57
37
41
4
4
9 .5i 4i
i
36
9.53 861
o
.46 1 39
9-97
550
5
56
5
9-5i 44?
9. 53 902
o
46 098
9-97
545
55
6
9. 5 1 484
9-53 943
o
46 057
9-97
54i
54
7
9-5i 620
37
9-53 984
41
o
46 016
9-97
536
53
8
9-5i 557
9.54 025
o
45 97 5
9-97
532
52
9
9. 5 1 5g3
3
9.54 o65
40
o
45 935
9-97
528
5i
10
9-5i 629
9.54 1 06
4 1
o
45 894
9-97
523
50
1 1
g.5i 666
36
9.54 i4?
4i
o
45 853
9-97
5i 9
49
12
9.5i 702
9.54 187
o
458i3
9-97
5ij
48
i3
9-5i 738
9.54 228
4i
45 772
9-97
5io
b
47
36
41
4
i4
9-5i 774
9.54 269
45 7 3i
9-97
5o6
46
i5
g.Si 81
i
9.54 3og
o
45 691
9-97
5oi
45
16
9.5i 84?
9.54 350
4'
0.
45 65o
9.97497
4
44
'7
g.5i 883
36
9.54 3go
4
o.
45 610
9.97492
b
43
18
y.5i 919
9-54431
o.
45 56 9
9-97
488
42
'9
9.61 9 55
30
9.54 471
40
45 529
9.97 484
4
4i
20
9-5i 991
9-54
5l2
4 1
0.
45488
9-97 479
40
21
9.52 027
3
9-54
55 2
o.
45448
9-97475
4
3 9
22
9-52 o63
9-54
5 9 3
o.
45 407
9.97470
38
23
9.52 099
3
9-54633
4
0.
4536 7
9.97 466
4
3 7
36
4
5
24
9-52 i3
5
,6
9.54673
o.
45 327
9-97
46i
36
25
9.52 171
9-54
714
0.
45286
9.97457
35
26
9.52 207
36
9.54
7 54
4
o.
45 246
9-97453
4
34
27
9.52 242
35
16
9.54 794
40
o.
45 206
9 . 9 7 448
5
33
28
9.52 278
g.54835
o.
45 i65
9-97
444
32
2 9
9.52 3i4
36
9.54875
40
o.
45 125
9.97439
b
3i
30
9 . 52 350
3 6
9.54915
40
o.
45o85
.9.97
435
4
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin. d.
'
7O 3O .
PF
4 j
40
37
36
35
5
4
.1
4 .i
4.0
3-7
.1 3.6
3-5
.1
0.5
0.4
.2
8.2
8.0
74
.2 7.2
7.0
.2
I.O
0.8
3
12.3
12.0
n. i
3 10.8
10.5
3
i-5
1.2
4
16.4
16.0
.48
4 M-4
14.0
4
2.0
1.6
5
20.5
20. o
18.5
.5 18.0
'7-5
5
2-5
2.0
.6
24.6
24.0
22.2
.6 21.6
31.
.6
3
2-4
7
28.7
28.0
5-9
7 25.2
24 5
7
3-5
2.8
.8
32.8
32.0
29.6
.8 28.8
28.0
.8
3-2
.g 36.9
36.0
9 ^ 2 -4
i i 3 'i 5
9
4-5
3-6
68
193O
/
L. Sin.
d.
L. Tang.
d. L
. Cotg.
L. Cos. d.
30
9.62 350
9.54915
O.
45 o5
9-97435
30
3i
9. 52 385
3 6
9.54 g55
40
o.
45 045
9.gy43o
4
2g
32
g.52 42
i
g.54
gg5
0.
45 005
9.97 426
28
33
9. 52 456
36
9 .55
o35
40
o.
44 965
9.97421
4
27
34
9.52 492
35
g.55
075
40
0.
44 g25
9-974i7
5
26
35
9. 52 527
16
g.55
1 15
o.
44885
9-97
4l2
25
36
9. 52 563
g.55
i55
o.
44845
9.97408
24
35
40
S
37
9. 52 5g8
36
9 .55
ig5
o.
44805
9.97 4o3
4
23
38
9. 52 634
g.55
235
o.
44765
9-97
3gg
22
3 9
9. 52 669
,6
9 .55
275
o.
44 725
9-97
3g4
21
40
g.5 705
9.55
3,5
0.
44 685
9-97
3go
20
4r
9. 52 740
35
g.55
355
o.
44645
9-97
385
4
'9
42
g.52 775
9 .55
3gS
0.
44 6o5
9-97
38i
18
43
g.52 81
i
g.55434
39
o.
44566
9-97
376
17
35
40
4
44
g.52 846
35
g.55 4?4
o.
44526
9-97
372
5
16
45
g.52 881
g.55
5i4
o.
44486
9-97
367
i5
46
9. 52 916
35
g.55 554
4
0.
44446
9-97
363
i4
35
39
4 7
g.52 g5 i
g.55
5g3
o.
44 407
9-97
358
5
i3
4*
9. 52 986
g.55
633
o.
44 36 7
9-97
353
12
49
g.53 02 1
35
g.55 6 7 3
40
0.
44 3 27
9-97
34g
I I
50
g. 53 o56
35
g.55
712
39
o.
44288
9-97
344
10
Si
g.53 092
34
g.55
752
39
o.
44248
9-97
34o
S
9
32
g. 53 1 26
g.55
79 1
o.
44 2og
9-97
335
8
53
g.53 161
35
g.55
83i
o.
44 i6g
9-97
33i
7
35
39
54
g.53 196
9.55870
4
o.
44 i3o
9-97
326
4
6
55
g . 5 3 2 3 1
g.55
gio
o.
44 ogo
9-97
322
5
56
g.53 266
35
g.55
g4g
39
o.
44o5i
9-97
3i 7
4
57
g.53 3oi
35
g.55 g8g
40
0.
44 on
9-97
3l2
4
3
58
g.53 336
g.56
028
o.
43 972
9-97
3o8
2
5 9
g. 53 370
34
g. 56 067
39
0.
43 9 33
9-97
3o3
I
60
g.53 4o5
35
g.56
107
40
0.
438g3
9-97
2 99
L. Cos.
d.
L. Cotg. d.
L.
Tang.
L. Sin.
d.
7O.
PP 40
39
36
35
34
5 4
i 4.0
3-9
3-6
i 3-5
3-4
.1
0.5 0.4
2 8.0
7.8
7.2
.2 7.0
6.8
.2
i.o 0.8
3 12.
" 7
10. S
3 io-5
10.2
3
1.5 i-a
4 16.0
15-6
14.4
.4 140
13.6
4
a.o 1.6
5 20.0
'9-5
18.0
5 i7-5
17.0
S
a. 5 2.0
6 24.0
23-4
21.6
.6 21.0
20.4
.6
3.0 2.4
7 28.0
27-3
25.2
7 24.5
23.8
7
3-5 2.8
8 32.0
28.8
.8 28.0
27.2
.8
4.0 3.2
9 36-0
3 2 -4
9 3'-5
^0.6
4-5 3-6
69
2O.
L. Sin
d.
L. Tang. d.
L.
Cotg.
L. Cos. d.
9.534o5
35
35
34
35
34
35
34
35
34
35
34
34
35
34
34
35
34
34
34
34
34
34
34
34
34
34
34
34
34
34
9-5b 107
39
39
39
4
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
38
39
39
39
38
39
39
38
39
38
39
o.
43 893
9-97
299
60
I
2
3
4
5
6
7
8
9
9.5344o
9.53475
9. 53 509
9-53544
9.53 578
9-536i3
9-53647
9.53682
9.53 716
9-56 146
9-56 i85
9-56 224
9-56 264
9-563o3
9-56342
9-5638i
9-56 420
9-56 45 9
o.
o.
0.
o.
o.
o.
o.
o.
o.
43854
438j5
43 776
43 7 36
43 697
43658
43 619
4358o
4354i
9.97 294
9.97 289
9.97285
9.97 280
9.97276
9.97271
9.97 266
9.97 262
9.97257
5
5
4
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
5
4
5 9
58
57
56
55
54
53
52
5i
10
9-53 751
9-56498
o.
43 5o 2
9-97
252
50
1 1
12
i3
i4
i5
16
18
'9
9.53 7 85
9-53 819
9-53854
9.53888
9.53 922
9 .53 9 5 7
9.53991
9.54 025
9.54 o5g
9.5653 7
9.565 7 6
9.566i5
9.56654
9.56693
9-56 732
9.56771
9-56 810
9.56 849
o.
o.
o.
o.
o.
0.
o.
o.
o.
43463
43424
43385
43 346
433o 7
43268
43 229
43 190
43 i5i
9.97 248
9.97 243
9.97 238
9-97 234
9.97 229
9-97 224
9.97 220
9.97 2i5
9-97 2IO
49
48
47
46
45
44
43
42
4i
20
9-54 093
9.56 887
o.
43 n3
9-97
206
40
21
22
23
24
25
26
2 7
28
2 9
9-54 127
9.54 161
9-54 195
9.54 229
9.54263
9-54 297
9-54 33i
9-54 365
9-54 399
9.56926
9.56965
9.57 oo4
9.57 042
9.57 081
9.57 120
9.57 i58
9-5 7 197
9.57 235
o.
0.
0.
o.
o.
o.
o.
o.
o.
43 074
43o35
42 996
42958
42 919
42 880
42842
42803
42 765
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
9-97
2OI
196
192
I8 7
182
I 7 8
I 7 3
168
i63
3 9
38
36
35
34
33
32
3i
30
9.54433
9-5 7
274
o.
42 726
9-97
1 5 9
30
L. Cos.
d.
L. Cotg-.
d.
L. Tang.
L. Sin.
d.
'
69 3O .
PP 40
39
38
35
34
,i
.2
3
4
'.6
'.&
9
5
4
0.4
0.8
1.2
1.6
2.0
2-4
2.8
3-2
3-6
.1 4.0
.2 8.0
.3 I2.O
.4 1 6.0
5 2O.O
.6 24.0
.7 28.0
.8 32.0
.9 36.0
3-9
7.8
11.7
15.6
19.5
23-4
27-3
31.2
35-1
3-8
7.6
11.4
15-2
19.0
22.8
26.6
30-4
34.2
i 3-5
.2 7-0
3 IO -5
4 M-o
5 >7-5
.6 21.
7 24.5
.8 28.0
9 3'-5
3-4
6.8
10.2
13-6
17.0
20.4
23 8
27.2
30.6
o-S
I.O
15
2.0
2-5
3-5
4.0
45
70
2O 3O
L. Sin.
d.
L. Tang. d. j L.
Cotg.
L. Cos. d.
30
9.54433
33
34
34
33
34
34
33
34
33
34
33
34
33
34
33
33
34
33
33
33
34
33
33
33
33
33
33
33
33
33
9 .5 7
2?4
38
39
38
39
38
38
39
38
38
39
38
38
38
38
39
38
38
38
38
38
38
38
38
38
38
38
37
38
38
38
O.42 726
9-97 l5 9
5
5
4
5
5
5
4
5
S
S
4
5
5
5
5
4
S
5
5
5
4
5
5
5
5
4
5
5
5
5
30
3i
32
33
34
35
36
3?
38
39
9.54466
9. 54 5oo
9.54534
9.54567
9.54 601
9.54635
9.54668
9.54 702
9.54735
9.57 3i2
9.57 35i
9.57 389
9.57428
9.57 466
9.57 5o4
9.57 543
9.57 58i
9.57 619
0.42 688
0.42 649
0.42 6 1 1
0.42 572
0.42 534
0.42 496
0.42 457
0.42 4>9
0.42 38 1
9-97
9-97
8.97
9-97
9-97
9-97
9-97
9-97
9-97
1 54
149
i45
i35
i3o
126
121
116
29
28
27
26
25
24
23
22
21
40
9.54769
9.5 7 658
o.
42 342
9-97
in
20
42
43
44
45
46
4 7
48
49
9.54 802
9.54836
9.54869
9.54903
9.54936
9.54969
9. 55 oo3
9.55o36
9.55 069
9.57 696
9.57734
9.57772
9.57 810
9.57849
9.5 7 88 7
9.57925
9.57 963
9-58 ooi
0.42 3o4
0.42 266
0.42 228
0.42 190
0.42 i5i
0.42 1 1 3
0.42 075
0.42 037
0.4 1 999
9.97 107
9.97 1 02
9.97097
9.97092
9.97087
9.97 o83
9.97078
9.97073
9.97 068
'9
18
7
16
i5
i4
i3
12
I I
50
9-55 102
9.58 039
o.
4i 961
9.97 o63
10
5i
52
53
54
55
56
5?
58
59
9.55 i36
9-55 169
9-55 202
9.55 235
9.55268
9-55 3oi
9.55334
9.55 367
9-55 4oo
9. 58 077
9.58 115
9.58 i53
9-58 191
9-58 229
9-58 267
9.583o4
9.58 342
9.58 38o
o.
o.
o.
0.
o.
o.
0.
o.
o.
4i 923
4i 885
41847
4i 809
4i 77'
4i ?33
4 1 696
4i 658
4i 620
9.97059
9.97054
9.97049
9.97044
9.97039
9.97035
9.97 o3o
9.97025
9.97 020
9
8
7
6
5
4
3
2
I
60
9.55433
9.584i8
o.
4i 582
9.97 01 5
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
69.
PP 39
38
37
34
33
9
5
4
i 3-9
.2 7.8
3 "?
4 '5-6
S '9-5
.6 23.4
7 27.3
.8 31.2
9 35-<
11.4
'5-2
19.0
22.8
26.6
3-4
34-2
3-7
7-4
n. i
14.8
18.5
22.2
259
29.6
33-3
I 3-4
.2 6.8
3 10.2
4 '3-6
.6 20.4
7 23.8
.8- 27.2
.9 30. 6
u
9-9
16.5
19.8
23-'
26.4
29.7
0.5
I.O
2.0
2-5
3-5
4.0
4-5
0.4
0.8
1.2
1.6
2.0
2.4
2.8
32
3 .6
21.
'
L. Sin.
d.
L. Tang. d. L. Cotg.
L. Cos.
d.
9.55 433
g.58 4i8
o.4i 582
9
g7 oi5
60
I
g.55466
33
g.58 455
37
18
o.4i 545
9
g7 oio
5
5 9
2
g.55 4gg
9. 584
9 3
o.4i 507
9-
g7 oo5
58
3
g.55 532
33
9. 58 53i
3 8
o.4i 46g
9-
g7 ooi
57
32
38
S
4
g.55 564
33
9. 58 56g
o.4i 43i
9-
g6 gg6
5
56
5
g.55 5g7
g.58 606
o.4i 3
94
9-
g6ggi
bb
6
g.55 63o
33
g.58 644
3 8
o.4i 356
9-
g6g86
54
33
37
S
7
g.55 663
g.5868i
o.4i 3
'9
9-
g6 g8i
5
53
8
g.55 6g5
g. 58 719
3
o.4i 281
9
g6 9 76
52
9
g.55 728
33
g.58 7 5 7
3 8
o.4i 243
9
9 6 97i
5i
10
g.55 761
g.58 79 4
37
o.4i 206
9
g6 g66
50
1 1
g . 55 7g3
3 2
g.58 832
3 8
o.4i 168
9
g6 9 62
5
49
12
g.55 826
g.5886g
o.4i i
3i
9
9 6g57
48
i3
g.55 858
32
g.58 907
3 8
o.4i og3
9-
g6 g52
47
i4
g.55 8g
I
33
g.58 9 44
37
o.4i o56
9-
9 6g47
b
5
46
i5
g.55 g23
g.58 9
81
o.4i 019
9
g6 g42
45
16
g.55 g56
33
g. 5g oig
38
o.4o g
81
9
9 6 9 3 7
44
17
g.55 g8
8
32
g.Sg o56
37
o.4o g44
9
g6 g32
b
43
18
g. 56 021
g.5gog4
3
o.4o 906
9
g6 g27
42
'9
9. 56 o53
32
g.Sg i
3i
37
o.4o 869
9
g6 g22
4i
20
g.56o85
32
g. 5g 168
37
o.4o 832
g.g6 gi7
S
40
21
9. 56 118
33
g . 5g 2o5
37
o.4o 7g5
9
g6 gi2
5
3 9
22
g. 56 i5o
g. 5g 243
3
o.4o. 757
9
g6 go7
38
23
g.56 182
32
g.5g 280
37
o.4o 720
9
g6 go3
37
24
g. 56 215
33
g.5g 317
37
o.4o683
9
96 898
b
5
36
25
g.56 247
32
g.Sg 354
37
o.4o 646
9
96 8g3
35
26
g.56 27g
32
g . 5g 3gi
37
o.4o 609
9
96888
34
2 7
g.56 3i
t
32
g.Sg 42g
3 8
o.4o 571
9
g6883
b
s
33
28
g.56 343
g.Sg 466
37
o.4o 534
9
96 878
32
2 9
g. 56 375
32
g . 5g 5o3
37
o.4o 4g?
9
96873
3i
30
9. 56 4o8
33
g. 5g 54o
37
o.4o 46o
9
96 868
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
68 3O .
PP
38
37
33
33
5
4
.1
3 .8
3-7
.,
3-3
3-2
.1
0.5
0.4
.2
7 .6
7-4
.2
6.6
6.4
.2
I.O
0.8
3
11.4
n. i
3
9-9
9.6
3
'5
1.2
4
IS .3
14.8
4
'3-2
12.8
4
3.0
1.6
5
19 o
18.5
5
16.5
16.0
S
2-5
2.O
.6
22.8
22.2
.6
19.8
19.2
.6
3-
2-4
7
26.6
25-9
7
23-'
22.4
7
3-5
2-8
.8
30.4
29.6
.8
26.4
25.6
.8
4.0
32
9 34- a
9
4-? 3-6
72
21 3O
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9-56 4oS
32
32
32
32
32
3"
32
S 2
32
9. 5g 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 460
9
.96 S68
5
5
5
5
5
S
5
S
5
5
5
5
5
5
5
5
5
5
6
5
5
S
S
5
5
5
5
5
5
S
30
3i
32
33
34
35
36
3?
38
3 9
9.56 44o
9-56 472
9-56 5o4
9.56536
g.56 568
9-56 599
g.56 63i
9.56663
9-56 6g5
9 .5 9 5 77
9.59 6i4
9.59 65i
9.59688
9 .5 9 725
9.59 762
9.59799
9.59 835
9.59 872
o.4o 423
o.4o 386
o.4o 349
o.4o 3i2
o.4o 275
o.4o 238
O.4O 2OI
o.4o 165
o.4o 128
9
9
9
9
9
9
9
9
9
. 9 6863
96858
.96853
.96848
.96843
96838
96833
96 828
96823
29
28
27
26
25
24
23
22
2 I
40
9-56 727
9- 5 9 99
o.4o 091
9
96 818
20
4i
42
43
44
45
46
4?
48
4 9
g.56 759
9. 56 790
9. 56 822
g.56854
9.56886
9-56 917
9-56 949
g.56 980
9.57 012
31
32
32
32
3i
32
31
3 2
9.59 946
9. 5g 983
9.60 019
9.60 o56
9.60 og3
9.60 i3o
9.60 166
9.60 2o3
9 . 60 24o
o.4o o54
o.4o 017
0.39 981
0.39 944
0.39 907
o. 3g 870
0.39 834
0.39 797
0.39 760
9
9
9
9
9
9
9
9
9
96 8i3
96 808
96 8o3
96 798
96 793
96 788
96 783
96 778
96 772
'9
18
'7
16
i5
i4
i3
12
I I
50
9.57 044
9 .60 276
0.39 724
9
96 767
10
5i
52
53
54
55
56
5?
58
5 9
9.57 075
9.57 107
9.57 1 38
9.57 169
9.57 2OI
9.57 232
9,57 264
9.57295
9.57 826
32
31
3'
32
3'
32
3i
3 1
9.60 3i3
9.60 34g
9.60 386
9.60 422
9.60 459
9.60 495
9.60 532
9.60 568
9. 60 605
0.39 687
0.39651
0.39 6i4
o.3g 578
o.3g 54i
o.3g 505
0.39468
o. 3g 432
o. 3g 3g5
9
9-
9-
9
9
9
9
9
9
96 762
96757
96 752
96 747
96 742
96 737
96 732
96 727
96 722
9
8
7
6
5
4
3
2
I
60
9.5 7 358
3 2
9.60 64 1
o.3g 35g
9
96 717
L. Cos.
d.
L. Cotg.
d. L, Tang.
L. Sin. d.
'
08.
PP
.1
.2
3
4
'.6
'.8
'?
37
36
.1
.2
3
4
'.6
.8
9
3*
31
.1
.2
3
4
'.6
9
6
5
3-7
7-4
n. I
.4.8
18.5
22.2
25.9
29.6
33-3
3.6
7-2
10.8
14.4
18.0
21.6
25.2
28.8
3 2 -4
3-2
6-4
9.6
12.8
16.0
19.2
22.4
25.6
11
9-3
12 .4
'5-5
18.6
21.7
24.8
27.9
0.6
1.2
1.8
2.4
3-
3-6
4-2
4.8
5-4
o-5
I.O
1.5
2.0
2-5
3.0
3-5
4.0
4-5
73
22.
L. Sin.
d.
L.
Tang.
d.
L. Cotg.
L. Cos.
d.
9- 5 7
358
9-
60 64
1
,6
o. 39 359
9.96
717
60
I
9.57
389
31
9-
60 677
37
0.39 323
9.96
711
5
5 9
2
9.5 7
420
9-
60 714
16
o. 39 286
9.96
706
58
3
9.57
45 1
9-
60 750
0.39 25o
9.96
701
57
31
3
S
4
9.57
482
32
9-
60 786
37
0.39 214
9.96
696
56
5
9.57
5i4
9-
60823
16
O.39 177
9.96
691
55
b
9< 5 7
545
9-
60859
0.39 i4i
9.96686
54
3'
30
5
7
9- 5 7
576
31
9-
60 895
36
0.39 105
9.96
681
5
53
8
9.57
607
9-
60 g3i
16
0.39 069
9.96 676
52
9
9.07
638
9-
60 967
o.3g o33
9.96
670
Dl
10
9.57 669
9-
6 1 oo4
36
o.38 996
9.96 665
50
1 1
9.57
700
31
9-
6 1 o4o
o.38
960
9.96
660
49
12
9.57
?3i
9-
61 076
36
0.38
924
9.96
655
48
i3
9.57
762
3'
9-
6l 1 12
36
0.38888
9.96
650
5
5
4 7
i4
9.57
79 3
31
9-
61 1 48
36
0.38852
9.96 645
46
t5
9' 5 7
824
9-
61 1 84
16
o.38 816
9.96 64o
45
16
9.57 855
3
9-
6l 220
36
o.38
780
9.96 634
5
44
17
9.57
885
31
9-
61 256
36
o.38
744
9.96 629
43
18
9.57
916
9-
61 292
76
o.38
708
9.96
624
42
! 9
9.57
947
9-
61 328
,6
o.38 672
9.96 619
4i
20
9.57
978
9-
61 364
76
0.38636
9.96 6i4
5
40
21
9.58 008
31
9-
6 1 4oo
36
o.38 600
9.96 608
3 9
22
9. 58
o3g
9-
61 436
o.38
564
9.96 6o3
38
23
9-58 070
3 1
9-
61 472
3
o.38
528
9.96
5 9 8
5
37
31
36
5
24
9.58
101
9-
6 1 5o8
36
0.38492
9.96
5g3
36
2!)
9-58
i3i
9.61 544
0.38456
9.96
588
35
26
9.58
162
3'
9.61 579
35
0.38421
9.96
582
34
3
36
5
27
9. 58
192
9.61 6i5
o.38
385
9.96
5 77
33
28
9. 58
223
9.61 65i
o.38
349
9.96
572
32
29
9 .58
253
3
9.61 687
3"
o.38
3i3
9.96
56 7
*
3i
30
g.58 284
3 1
9.61 722
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
39
31
30
6 5
i
3-7
3-6 35
I
3-2
3-'
3
,1
0.6 0.5
.2
74
7.2 7.0
.2
64
6.3
6.0
a
I 2 I.O
3
n. i
10 8 10.5
3
9.6
9-3
9.0
3
1.8 1.5
4
14.8
14.4 14
4
12.8
12.4
12.O
4
24 20
5
18.5
18.0 17.5
5
16.0
'5-5
15-0
5
3.0 2.5
.6
22.2
21.6 31.
6
19.2
18.6
18.0
.6
3.6 3.0
7
25.9
25-2 4 5
7
22 4
21 7
21.
. 7
42 3-5
.8
29.6
28. 8 28 o
.8
25.6
248
24.0
.8
48 4
9 31 3
3 2 4 3'5
9
27.0
9
54 4-5
22 3O .
,
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9
.58 284
3
3
3
3'
30
30
3
30
3
30
3
3
30
3
3
3
3
3
30
3
3
3"
29
30
3
3
9.61
722
3 6
36
36
35
36
35
36
36
35
36
35
36
35
36
35
36
35
35
36
35
35
36
35
35
35
36
35
35
35
35
o.38 278
9.96
562
6
S.
s
5
6
5
5
5
6
5
5
6
5
5
5
6
S
5
6
5
5
6
'5
5
6
's
5
6
5
5
30
3i
32
33
34
35
36
37
38
39
9
9
9
9
9
9
9
9
9
.58 3i4
.58 345
.58 3 7 5
.584o6
.58 436
.5846?
.58497
.58 52 7
.58 55 7
9.61 758
9.61 794
9.61 83o
g.6r 865
g ,61 go i
g .61 g36
g ,61 972
9.62 008
g .62 o43
o.38 242
o.38 206
o.38 170
o.38 i35
o. 38 ogg
o.38 o64
0.38 028
0.37 gg2
0.37 967
g.g6 556
g.g6 55i
g.g6 546
g .g6 54i
9.96 535
g.g6 53o
g.g6 525
g.g6 620
g.g6 5i4
29
28
27
26
25
24
23
22
2 I
40
9
.58 588
g.62 079
0.37 921
9-96
5og
20
4i
42
43
44
45
46
4 7
48
49
9
9
9
9
9
9
9
9
9
.586i8
.58 648
.586 7 8
.58 7og
.58 739
.58 769
.58 799
.58 829
.58 85g
9.62 u 4
g.62 150
g.62 i85
g.62 221
g.62 256
9.62 292
9.62 327
g.62 362
g.62 3g8
0.37 886
0.37 85o
0.37 815
0.37 779
0.37 744
0.37 708
0.37673
0.37 638
0.37 602
9.96 5o4
g.g6 4g8
g.g6 4g3
9 .g6488
g. 96 483
9.96477
g.g6 4?2
g.g6 467
9.96 46i
'9
18
16
i5
i4
i3
12
1 1
50
9
.58 88c
)
g.62 433
0.37 56 7
g.g6 456
10
5i
52
53
54
55
56
57
58
5g
9
9
9
9
9
9
9
9
.58 gig
.58 g4g
.58 979
.5g oog
.5g o6g
-5g og8
.5g 128
.5 9 i58
g.62 468
g.62 5o4
g.62 53g
9.62 674
9.62 6og
g.62 645
9.62 680
g.62 715
g.62 760
0.37 532
o.3 7 4g6
0.37 46i
0.37 426
0.37 3gi
0.37 355
0.37 32O
0.37 285
0.37 250
g.g6 45i
g.g6 445
9.96440
9.96 435
9.96 429
9.96 424
g.g6 4ig
9-96 4i3
g.g6 4o8
9
8
7
6
5
4
3
2
1
60
9
.5g 1 88
9.62
785
o.
37 215
g.g6 4o3
L. Cos. d.
L. Cotg-. d. L.
Tang.
L. Sin. d.
'
67.
PP
36
35
31
30
.1 3.0
2 D.O
3 9-
.4 12.
5 '5-0
6 18.0
.7 21.0
.8 24.0
29
.1
.2
3
4
'.6
.8
6
5
0.5
I.O
2.0
25
3-5
4
3 .6
3-5
2.9
5.8
8.7
ii 6
14.5
'7-4
20.3
23.2
26. i
0.6
1.2
1.8
2.4
3-
3-6
4.2
4.8
3
4
6
.8
10.8
'44
18.0
21.6
28.8
10.5
14.0
'7-5
21 O
24.5
28.0
Sj-3
9-3
12 .4
15-5
18.6
21.7
24.8
27.9
23.
L. Sin.
d.
L. Tang 1 , d.
L
. Cotg.
L. Cos. d.
9 .5 9 188
9.62 75
37215
9.90
4o3
60
I
9.59 218
9.62 820
35
o
37 180
9.96
397
5 9
2
9.59 247
9.62 855
o
37 145
9.96
392
58
3
9.59277
9.62 890
37 1 10
9.96
387
D 7
3
36
4
9.59 307
9.62 926
35
O.
3 7 o 7 4
9.96 38i
56
b
9.59 336
9.62 961
O.
3 7 o3g
9.96
376
bb
6
9-5 9 366
3
9.62 996
o.
37 oo4
9.96 370
54
3
35
s
7
9. 5g 3g6
9-63 o3i
35
o.
36 969
9.96365
53
8
g.Sg 425
9. 63 066
o.
36 934
9.96
36o
52
9
9. 5g 455
3
9-63 101
o
36 899
9.96
354
5i
10
9.5 9 484
29
9.63 i35
o.
36 865
9.96
349
50
1 1
9. 5g 5i4
3
9-63 170
o.
36 83o
9.96
343
49
12
9. 5g 543
9-63 2o5
Oi
36 795
9.96
338
48
i3
g.Sg 573
3
g.63 240
35
o.
36 760
9.96 333
47
29
35
6
i4
9. 5g 602
9.63
2 7 5
o.
36725
9.96 327
46
i5
9. 5g 632
9-63
3io
.0.
36 690
9.96
322
45
16
9. 59 661
29
9.63
345
35
o.
36655
9.96 3i6
44
i7
9.59 690
29
9.63
379
34
o.
3662!
9.96 3i i
5
6
43
18
9. 59 720
9-634i4
o.
36586
9.96
3o5
42
'9
9.59749
29
9-63 44g
35
o.
36 55i
9.96
3oo
5
4i
20
9.59778
29
9.63484
35
o.
36 5i6
9.96 294
40
21
9.59 808
9.63
5. 9
34
o.
3648i
9.96 289
5
39
22
9. 59 837
9-63
553
o.
36447
9.96 284
38
23
9.59866
29
9. 63 588
o.
364i2
9.96 278
37
29
35
5
24
9.59 895
g.63 623
34
o.
363 77
9.96
2 7 3
6
36
25
9.59924
9-63
65 7
0.
36343
9.96
267
35
26
9.59954
3
9-63 692
o.
363o8
9.96 262
5
34
29
34
6
27
g.Sg 983
9 .63
726
35
X).
36 274
9.96
256
33
28
9.60 OI2
9.63
761
o.
36 23g
9.96
25l
32
29
9.60 o4i
29
9 .63
79 6
35
o.
36 204
9.96 245
3i
30
9 .60 070
29
9. 63
83o
o.
36 170
9.96240
s
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin. d.
66 3O .
PP 36
35
34
30
29
6 5
3-6
3-5
3-4
.1 3.0
2-9
.1
0.6 0.5
.2 7.2
7.0
6.8
.2 6.0
5.8
.2
1.2 I.O
.3 10.8
10.5
10.2
3 9-
8-7
3
1.8 1.5
4 '4-4
14.0
,3.6
.4 12.
ii. 6
4
2.4 2.0
.5 18.0
J7-5
17.0
5 '5-o
14-5
5
3 2.5
.6 21.6
21. O
20.4
.6 18.0
6
3-6 3.0
7 2 5- 2
24-5
23.8
.7 21.
20.3
7
4-2 35
.8 28.8
28.0
27.2
,8 24.0
23.2
.8
4.8 4.0
9 324
31.5 30.6
.9 27.0
26 i
9
5-4 4-5
76
23 3O
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
.60 070
29
29
29
29
29
29
29
29
29
9 .63 83o
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
"dT
o.36 1 70
9.96 240
6
5
6
5
6
5
6
5
6
5
6
5
6
6
5
6
5
6
5
6
6
5
6
5
6
6
5
6
5
6
30
3i
32
33
34
35
36
3?
38
3 9
9
9
9
9
9
9
9
9
9
. 60 099
.60 128
.60 i5y
.60 186
.60 215
.60 244
.60 273
.60 3o2
.60 33i
9 . 63 863
9-63 899
9-63 934
9-63 968
9.64 oo3
9 . 64 037
9.64 072
9.64 1 06
9 .64 i4o
o.36 i35
o.36 101
o.36 066
o. 36 o32
o. 35 997
o.35 963
o.35 928
o.35 894
0.35 860
9 .96 234
9.96 229
9.96 223
9.96 218
9 .96 212
9.96 207
9.96 201
9.96 196
9.96 190
29
28
27
26
25
24
23
22
21
40
9
.60 35g
9 .64 175
0.35 825
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 4i?
.60 446
.60 4?4
.6o5o3
.60 532
.60 56i
.6o58 9
.60618
29
29
29
28
29
29
29
28
29
28
29
29
28
29
28
29
28
29
28
28
9.64 209
9.64 243
9.64 278
9 .64 3i2
9-64346
9-6438i
9.64 415
9.64 449
9-64483
o.35 791
o.35 757
o. 35 722
0.35688
0.35654
o.35 619
0.35585
o.3555i
o.35 5i 7
9.96 I 79
9.96 I74
9.96 l68
9.96 162
9.96 157
9 .96 i5i
9.96 i46
9 .96 i4o
9.96 185
'9
18
7
16
i5
i4
i3
12
I I
50
9
.60 646
9.64 5 1 7
0.35483
9.96 129
10
5i
52
53
54
55
56
5 7
58
5 9
9
9
9
9
9
9
9
9
9
.60 675
.60 704
.60 732
.60 761
.60 789
.60818
.60846
.60875
.60 903
9.64552
9-64586
9.64 620
9.64654
9.64688
9 .64 722
9-64 ?56
9-64 790
9.64824
0.35448
o.35 4i4
o.35 38o
o.35 346
o.35 3i2
o.35 278
o.35 244
o.35 210
o.35 176
9.96 123
9.96 1 18
9.96 112
9.96 1O7
9.96 ioi
9 .96 095
9.96 090
9.96 084
9.96079
9
8
7
6
5
4
3
2
I
60
.60 9 3r
9.64858
o.35 142
9.96 073
L. Cos.
d.
L. Cotg.
L. Tang.
L.
Sin.
d.
'
o.
PP
.1
.2
3
4
!e
'.8
9
35
34
29
28
.1
.2
3
4
^6
7
.8
9
6
5
3-5
7.0
J 5
14.0
'75
21.
24-5
Z8.0
3J-S
3-4
6.8
IO.2
13-6
I 7 .0
20.4
2 3 .8
27.2
30.6
.1
.2
3
4
.6
!s
9
2.9
5-8
8.7
ii.6
14.5
17.4
20 3
23.2
26.1
2.8
5.6
8.4
II. 2
14.0
16.8
19.6
22.4
25.2
0.6
1.2
1.8
2.4
3-
3-6
4.2
4.8
5-4
5
l.O
i-5
2.O
25
3-
3-5
4.0
4-5
77
24 C
'
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
9 . 60 93 1
9-64858
o.
35 142
9.96 073
60
I
9.60 960
28
9-64
892
34
o.
35 108
9.96 067
5
5 9
2
9.60 9 8
8
28
9.64 926
0.
35o 7 4
9.96 062
g
68
3
9.61 016
9.64 960
34
0.
35 o4o
9.96
o56
5 7
29
34
4
9.61 o4
5
28
9-64
994
o.
35 006
9.96 o5o
5
bb
5
9.61 073
28
9 .65
028
o.
34 972
9.96045
g
bb
6
9.61 101
9. 65
062
34
0.
34938
9.96
o3 9
b4
28
34
S
7
9.61 129
9-65 096
0.
34 904
9.96 o34
6
53
8
9.61 i58
9. 65
i3o
0.
34870
9.96 028
b2
9
9.61 186
28
9.65
1 64
34
o.
34836
9.96
022
5i
10
9.61 21 4
g.65
197
33
o.
348o3
9.96
017
g
50
1 1
9.61 242
28
9-65
23l
34
o.
34 769
9.96
01 I
ft
49
12
9.61 270
9-65
265
o.
34735
9.96 oo5
48
i3
9.61 298
9 .65
299
34
o.
34 701
9.96 ooo
47
i4
9.61 326
28
28
9 .65
333
34
o.
34667
9 . 9 5
994
6
46
i5
9.61 354
9. 65
366
o.
34634
9 . 9 5
988
45
16
9.61 382
g.65 4oo
34
o.
34 600
9-9 5
982
44
17
9.61 4i
i
29
9". 6 5 434
34
o.
34566
9.95
977
t
f,
43
18
9.61 438
9. 65
467
o.
34533
9.95
971
g
42
'9
9.61 466
9.65
5oi
34
o.
34 499
9.95
9 65
4i
20
9.61 4g4
9.65
535
34
0.
34465
9 . 9 5
960
g
40
21
9.61 522
28
9-65
568
33
o.
34432
9 . 9 5
9 54
6
39
22
9.61 55o
9. 65
602
34
o.
343 9 8
9 . 9 5 948
g
38
23
9.61 578
9 .65
636
34
o.
34364
9 . 9 5
942
37
24
9.61 606
28
28
9 .65
669
33
o.
3433i
9 . 9 5
9 3 7
S
6
36
25
9.61 634
9-65
7o3
34
o.
34 297
9.95
g3i
g
35
26
9.61 662
28
9. 65
736
33
o.
34264
9 . 9 5
925
34
27
9.61 68
9
27
28
9 .65
77
34
o.
34 23o
9 . 9 5
920
b
6
33
28
9.61 717
9. 65
8o3
33
o.
34 197
9 . 9 5
914
g
32
29
9.61 745
28
9 .6583 7
34
o.
34 1 63
9 . 9 5
908
g
Si
30
9.61 773
9.65
870
33
o.
34 i3o
9 -9 s
902
30
L. Cos. i d.
L. Cotg. d. L.
Tancr.
L. Sin. I d.
'
65 3O'.
PP 34
33
29
38
97
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 I.O
3 10.2
9-9
8.7
3 8.4
8.1
3
1.8 1.5
4 '3-6
13.2
n.6
4 II. 2
10.8
4
2.4 2.0
5 17.0
16.5
M 5
5 14.0
'3-5
5
3.0 2.5
.6 20.4
19.8
J7-4
6 16.8
16.2
.6
3.6 3.0
7 238
23 i
20.3
7 9-6
18.9
7
4-2 3-5
.8 27.2
26.4
23.2
8 22.4
21.6
.8
4.8 4.0
.0 70.6
24.7
5-4 4-5
78
24 3O
/
L. Sin.
d.
L. Tang.
d.
L. Cotg. | L. Cos.
d.
30
9
bi 773
27
28
28
27
28
28
27
28
27
28
9 .b5 870
34
33
34
33
34
33
33
34
33
33
34
33
33
33
34
33
33
33
33
34
33
33
33
33
33
33
33
33
33
33
o. 34 i 3o
9.96 902
5
6
6
6
6
5
6
6
6
6
5
6
6
6
6
S
6
6
6
6
6
5
6
6
6
6
6
6
6
5
30
3i
32
33
34
35
36
3?
38
39
9
9
9
9
9
9
9
9
9
bi 800
61 828
61 856
61 883
61911
61 g3g
61 966
61 994
62 02 1
9-65 904
9-65 937
9-65 971
9.66 oo4
9.66 o38
9.66 071
9.66 io4
9.66 i38
9.66 171
0.34 096
o.34o63
o.34 029
o.33 996
o.33 962
o.33 929
0.33 896
o.33 862
o.33 829
9.95897
9. gS 891
9 . 9 5 885
9.95 879'
9.95 873
9.95 868
9.96 862
9 <9 5 856
9.95 85o
29
28
27
26
25
24
23
22
2 I
40
9
62 o4g
27
28
27
28
27
28
27
27
28
9.66 204
o. 33 796
9. 9 5 844
20
4i
42
43
44
45
46
4?
48
49
9
9
9
9
9
9
9
9
9
62 076
62 io4
62 i3i
62 i5g
62 186
62 214
62 241
62 268
62 296
9.66 238
9.66 271
9.66 3o4
9.66 337
9.66 371
9.66 4o4
9 .66 437
9.66 4?o
9.66 5o3
o.33 762
0.33 729
o.33 696
0.33663
0.33 629
o.33 696
0.33563
o.33 53o
o.33 497
9.95 83g
9.95 833
9.95 827
9.95 821
9.95 8i5
9.95 810
9 .95 8o4
9.95 798
9 . 9 5 792
] 9
IS
'7
16
i5
i4
i3
12
I I
50
9
62 323
9.66 537
0.33463
9.95 786
10
5i
52
53
54
55
56
5?
58
5 9
9
9
9
9
9
9
9
9
9
62 35o
62 877
62 405
62 432
62 45g
62486
62 5i3
62 54 1
62 568
27
28
27
27
27
27
28
27
27
9.66 570
9.66 6o3
9. 66 636
9.66 669
9.66 702
9.66 735
9.66 768
9.66 801
9.66834
o.3343o
o.33 397
o.33 364
o.33 33i
o.33 298
o.33 265
o.33 232
o.33 199
o.33 166
9.95 780
9-9 5 775
9 . 9 5 769
9.95 763
9 . 9 5 757
9.95 75i
9.96 745
9 >9 5 739
9.95 733
9
8
7
6
5
4
3
2
I
60
9
62 595
9.66867
o.33 i33
9.95 728
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
d.
/
65.
PP
.1
.2
3
4
.'e
',8
9
34
33
.1
.2
3
4
.6
.8
9
28
37
.1
.2
3
4
6
6 5
3-4
6.8
10.2
13-6
17.0
20.4
23.8
27.2
30.6
6.6
9.9
13.2
16.5
19.8
23-1
26.4
29.7
2.8
5.6
8.4
11.2
14.0
1 6. 8
19.6
22.4
2.7
5-4
8.x
10.8
'3-5
1 6. 2
18.9
21.6
24-3
0.6 o. 5
1.2 I.O
1.8 1.5
2-4 2.0
3.0 2.5
3.6 3.0
4-2 3-5
4.8 4.0
5-4 4-5
79
25.
/
L. Sin.
d.
L. Tang.
d. L. Cotg.
L.
Cos. d.
9
.62 595
27
27
27
27
27
27
2 7
27
27
9 .t6 867
33
33
33
33
33
33
33
33
32
33
33
33
33
32
33
33
33
32
33
33
32
33
33
32
33
32
33
33
32
33
o.33 i33
9.95 728
6
6
6
6
6
6
6
6
6
6
5
6
6
6
6
6
6
6
6
60
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
.62 622
.62 64g
.62 676
.62 703
.62 730
.62 757
.62 784
.62 811
.62838
9.66 900
9.66 gSS
9.66 966
9.66999
9 .67 o32
9.67065
9.67 098
9.67 i3i
9.67 i63
o.33 100
o.33 067
o.33 o34
o.33 ooi
0.32 968
0.32 935
o.32 902
O.32 869
o.32 837
9 .95 722
9.95 716
9.95 710
9.95 704
9.95 698
9 .gS 692
9.95 686
9.95 680
9.95 674
5 9
58
5?
56
55
54
53
52
5i
10
9
.62 865
9.67 196
o.32 8o4
9.95 668
50
1 1
I 2
[3
i4
i5
16
'7
18
'9
9
9
9
9
9
9
9
9
9
.62 892
.62 918
.62 945
.62 972
.62 999
.63026
,63o52
.63 079
.63 1 06
26
27
27
27
27
26
27
27
2 7
26
27
27
26
7
26
27
26
27
9.67 229
9.67 262
9.67295
9.67 327
9.67 36o
9.67 393
9.67 426
9.67 458
9.67 491
o.32 771
o.32 738
o.32 705
o.32 673
O.32 640
o.32 607
0.32 5?4
0.32 542
o.32 5og
9.95 663
9.95 657
9.95 65i
9 . 9 5645
9.95 63g
9.95 633
9.95 627
9.95 621
9.95 615
49
48
47
46
45
44
43
42
4i
20
9
.63 i33
9.67 524
o.32 476
9.95 609
40
21
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.63 i5 9
.63 186
.63213
.63239
.63266
.63 292
.63 3ig
.63 345
.63 3 7 2
9.67 556
9.67 589
9.67 622
9.67 654
9.67 687
9.67 719
9.67 752
9.67 785
9.67 817
o.32 444
o.32 4i i
o.32 378
o.32 346
o.32 3i3
O.32 28l
0.32 248
0.32 2l5
o.32 i83
9.95 6o3
9.95 597
9.95 5gi
9.95585
9 . 9 5 579
9 .g5 573
9.95 567
9.95 56i
9.95 555
6
6
6
6
6
6
6
6
6
3 9
38
3?
36
35
34
33
32
3i
30
9
.63 3 9 8
9.67 850
o. 32 i5o
9.95 54g
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
64 3O .
PP
.i
.2
3
4
.6
!i
9
33
32
27
26
.1
.2
3
4
.6
.S
6
5
33
6.6
9.9
13.2
16.5
19.8
33.1
26.4
29.7
3-2
6-4
9.6
12.8
16.0
19.2
22. J
25.6
.1
.2
3
4
6
!s
2-7
5-4
8.1
10.8
13-5
16.2
18.9
21. D
24.3
2.6
52
7.8
10.4
13.0
15.6
18.2
20.8
2^-4
0.6
1.2
1.8
2-4
3-o
3-6
42
4.8
S
I.O
'5
2.O
2-5
3-
3-S
4.0
4' 5
80
25 3O .
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
63 3 9 S
9.67 850
o.32 i5o
9.95 549
30
3i
9
63 425
27
26
9.67 882
32
O. 32 I l8
9.95 543
g
29
32
9
6345i
9.67 915
o.32 o85
9.95 537
28
33
9
63 4?8
27
9.67 947
32
o.32 o53
9.95 53i
27
26
33
6
34
9
63 5o4
9.67 980
O.32 O2O
9.95 525
26
35
9
63 53i
9.68 012
o.3i 98
8
9.95 5ig
25
36
9
63 55?
9.68044
32
o.3i 96
6
9.95 5i3
24
37
9
63 583
26
9.68 077
33
o.3i 923
9.95 507
b
23
38
9
63 610
9.68 109
o.3i 891
9.95 5oo
22
3 9
9
63 636
26
9.68 142
33
o.3i 858
9 . 95 494
21
40
9
63 662
9.68 174
o.3i 826
9.95 488
20
4i
9
63 689
26
9.68 206
33
o.3i 794
9.95 482
6
19
42
9
63 715
26
9.68 239
i
o.3i 761
9.95 476
18
43
9
63 74 1
26
9.68 271
3 2
o.3i 729
9.95 470
6
i?
44
9
63 767
27
9.68 3o3
o.3i 697
9.95 464
6
16
45
9
63 794
26
9.68 336
o.3 1 664
9.95 458
i5
46
9
63 820
26
9.68 368
3 2
3 2
o.3i 632
9.95 452
6
i4
47
9
63846
26
9.68 4oo
o.3i 600
9.95 446
fi
i3
48
9
63872
9.68432
o.3i 568
9.95 44o
12
4 9
9
63 898
26
9.68 465
33
o.3i 535
9 . 9 5434
I t
50
9
63 924
26
9.68 497
S 2
o. 3i 5o3
9.95 427
10
5i
9
63 o5o
26
9.68 529
3 2
o.3i 471
9.95 421
6
9
52
9
.63 976
26
9.68 56i
o.3i 439
9.95 4i5
8
53
9
.64 002
9.68 59;
1
32
o.3i 407
9.95 409
7
26
33
54
9
.64 028
26
9.68 626
o.3i 374
9.95 4o3
6
6
55
9
.64o54
9.68658
o.3i 342
9 . 9 5 397
5
56
9
.64 080
9.68 690
3 2
o.3i 3io
9.95 391
4
26
32
7
5?
9
.64 1 06
26
9.68 722
o.3i 278
9.95 384
6
3
58
9
.64 1 32
9.68 754
o.3 1 246
9-9
5 3 7 8
2
5 9
9
.64 1 58
9.68 786
32
o.3i 214
9.95 372
I
60
9
.64 1 84
9.68 818
32
o.3i 182
9.95 366
L. Cos. d.
L. Cotg.
d.
L, Tang.
L.
Sin. d.
'
64.
PP
33
32
27
26
7
6
. x
3-3
3-2
.1
2-7
2.6
.,
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
104
4
2.8
2.4
5
16.5
16.0
5
13-5
13.0
5
3-5
3.0
.6
19.8
19.2
.6
16.2
15.6
.6
4-2
3-6
7
23- i
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
24.3 23.4
^3 5-4
81
26.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
9
.64 184
26
9.68 818
o.3i 182
9.95 366
g
CO
I
9
.64 210
26
9.68 85o
32
o.3i 150
9.95 366
6
5 9
2
V
.64236
26
9.68 882
o.3i 118
9 . 9 5 354
68
3
9
.64 262
26
9.68 914
32
32
o.3i 086
9.95 348
7
57
4
9
.64288
25
9.68 946
o.3i o54
9.95 34i
6
56
5
9
.643i3
26
9.68978
o.3i 022
9.95 335
55
6
9
.64339
26
9.69 oio
3 3
32
o. 3o 990
9.95 329
6
54
7
9
.64365
26
9.69 O42
o.3o 9
5S
9.95 323
6
53
8
9
.64 391
26
9.69 074
o.3o 926
9-9 5 3l 7
62
9
9
.64 4i7
9.69 I 06
3 2
o.3o 894
9.95 3io
5i
10
9
.64442
26
9.69 1 38
32
o.3o 862
9.95 3o4
g
50
ii
9
.64468
26
9.69 170
32
o.3o83o
9.95 298
fi
4 9
12
9
.64494
9.69 202
o.3o 798
9.95 292
48
i3
9
.64 5 1 9
25
9.69 234
32
o.3o 766
9.95 286
4?
i4
9
.64 545
26
9.69 266
32
o.3o 734
9.95279
7
6
46
i5
9
.64571
9.69 29^
I |
o.3o 702
9.95 273
45
16
9
.64596
25
26
9.69 329
3'
o.3o 671
9.95 267
6
44
17
9
.64 622
9.69 36i
o.3o 6
39
9.95 261
43
18
9
.64647
9.69 3g'
J
o.3o 607
9.95 254
42
'9
9
.64673
9.69425
32
o.3o 5
J^
9.95 248
4i
20
9
.646 9 8
25
9.69 457
32
o.3o 543
. 9 . 9 5242>
40
21
9
.64 724
9.6948*
J
o.3o 5i2
9.95 236
3 9
22
9
.64 749
9.69 52(
>
o.3o48o
9.95 229
38
23
9
.64 775
9.69 552
32
o.3o448
9.95 223
37
25
32
h
24
9
.64 800
26
9.69 584
o.3o 4
iG
9.95 217
36
25
9
.64826
9.69 6i5
o.3o 385
9.95 211
35
26
9
.6485i
25
9.69 647
S 2
o.3o 353
9.95 204
7
34
26
3 2
h
27
9
.64877
9.69679
o.3o 32i
g.gS 198
33
28
.64 902
9.69 710
o.3o 290
g.gS 192
32
2 9
9
.64 927
25
9.69 742
32
o.3o258
9.95 i85
7
3i
30
9
.64 g53
9.69 774
32
o.3o 226
9.95 179
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
63 3O .
PP
38
31
26
9
7
6
.i
3.2
3-'
.,
2.6
2-5
I
07
0.6
.2
6.4
6.2
2
5-2
5.0
2
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
IO.O
4
2.8
2-4
5
16.0
'5-5
5
13.0
12.5
5
3-5
.6
19.2
1 8.6
6
15.6
150
6
4-2
3-6
7
22.4
21.7
7
18.2
17-5
7
49
4-2
.8
25.6
24.8
8
20.8
20. o
8
5-6
4.8
9
21-4
9
82
26 30 .
/
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9.64953
9.69
774
o.
3o 226
9. 9 5
179
6
30
3i
9.64 978
9.69
8o5
3 1
o.
3o 195
9- 9 5
I 7 3
6
29
32
9-65 oo3
9.69
837
o.
3o i63
9. 9 5
167
28
33
9.65 029
9.69
868
3i
o.
3o i3z
9 >9 5
1 60
27
34
9-65 o54
25
9.69 900
32
o.
3o 100
9 . 9 5
1 54
fi
26
35
9-65 079
9.69
932
o.
3oo68
9 . 9 5
i48
25
36
9-65 io4
25
26
9.69 963
3 1
3 2
o.
3o 037
9 . 9 5
i4i
6
24
37
9
.65 i3o
9.69995
o.
3o oo5
9 . 9 5
i3g
fi
23
38
g.65 i5g
9.70 026
o.
29974
9 . 9 5
129
22
3 9
g.65 180
9.70 o58
3 2
o.
29 942
9 . 9 5
122
6
21
40
g.65 2o5
9.70 089
0.
29 91 1
9 . 9 5
116
6
20
4i
9-65 23o
9.70
121
o.
29879-
9 . 9 5
no
7
19
42
9-65 255
9.70
ID2
0.
29 848
9. 9 5
io3
18
43
g.65 281
9.70
1 84
32
o.
29 816
9 . 9 5
097
17
44
9.65 3o6
25
9.70 2i5
o.
29785
9 . 9 5
090
7
6
16
45
9.65 33i
9.70
247
o.
29 7 53
9 . 9 5
o84
i5
46
9.65 356
25
9.70 278
3 1
o.
29 722
9 . 9 5
078
i4
25
31
7
4 7
9-65 38i
9.70
309
o.
29 691
9-9 5
071
fi
i3
48
9
.65 4o6
9.70
34i
o.
29 65g
9 . 9 5
065
6
12
49
9
.6543i
25
9.70
3 7 2
3 1
o.
29 628
9-9 5
oSg
I I
50
9
.65456
9.70 4o4
o.
29 5g6
9.95 o52
6
10
5i
9
.6548i
25
9.70435
3 1
0.
29 565
9.95
o46
7
9
52
9
.65 5o6
9. 70 466
o.
29 534
9 . 9 5
o3g
g
8
53
9
.65 53i
25
9.70498
3 2
o.
29 5o2
9.95 o33
7
25
3 1
54
9
.65 556
9.70 529
o.
29 4?i
9 . 9 5
027
7
6
55
9
.65 58o
9.70 56o
o.
29 44o
9 . 9 5
020
5
56
9
.656o5
25
9.70 592
32
o.
29 408
9 . 9 5
oi4
4
57
9
,6563o
25
9. 70 623
o.
29377
g.gS 007
y
fi
3
58
9
.65 655
9. 70 654
o.
29 346
9 . 9 5
OOI
6
2
5 9
9. 65 680
23
9.70 685
3 1
o.
29 315
9-94995
I
60
9. 65 705
2 s
9.70
717
o.
29 283
9.94988
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
r
63.
PP
32
31
26
*5
34
7 6
.1
3-2
3-i
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
S
16.0
13.0
5 12.5
I2.O
5
3-5 3-
.6
19.2
18.6
15.6
.6 15.0
14.4
.6
4.2 3.6
7
22.4
21.7
18.2
7 17-5
16.8
7
49 4-2
.8
25.6
24.8
20.8
8 20.0
19.2
.8
5-6 4-8
9
23-4
.9 22.5
21.6
9
6-3 5-4
83
27.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
9-65 705
24
25
25
25
24
25
25
24
25
25
24
25
24
25
25
24
25
24
25
24
2 4
25
24
25
24
24
25
24
2 4
9.70717
31
31
31
3>
32
3'
3i
3'
3'
3i
3i
3'
3'
32
3'
31
3i
3'
3>
3'
3'
3'
3
3'
3'
3i
3i
3i
3'
3'
o
.29283
9.949*8
6
7
6
7
6
7
6
7
6
7
6
6
7
6
7
6
7
7
6
60
I
2
3
4
5
6
7
8
9
9. 65 729
9.65 7 54
9. 65 779
9-65 8o4
9-65 828
9-65853
9.63 878
9-65 902
9.65 927
9.70 748
9.70779
9.70 810
9.70 84i
9.70873
9.70904
9.70935
9.70 966
9.70997
o
o
o
o
o
o
o
o
. 29 25a
.29 221
.29 190
.29 1 59
.29 127
.29 096
.29 o65
.29034
.29 oo3
9.94 982
9.94975
9.94969
9.94962
9.94 956
9.94949
9.94943
9.94936
9.94 93o
5 9
58
57
56
55
54
53
52
5i
10
9-65 962
9.71 028
o
.28 972
9 . 9 4
923
50
i 1
12
i3
i4
i5
16
17
18
'9
9.65 976
9.66 ooi
9.66 O25
9.66 o5o
9.66075
9.66 099
9.66 124
9.66 i48
9.66 178
9.71 059
9.71 090
9.71 121
9.71 i53
9.71 184
9.71 215
9.71 246
9.71 277
9.71 3o8
o
o
o
o
o
o
o
o
.28941
,28 910
28879
28847
28 816
28 785
28 754
28 723
28 692
9.94917
9.94911
9.94904
9.94898
9.94 891
9.94885
9.94878
9.94871
9.94865
4 9
48
47
46
45
44
43
42
4i
20
9.66 197
9.71
339
o.
28661
9-94858
1
40
21
22
23
24
25
26
27
28
29
9.66 221
9.66246
9-66 270
9-66 295
9.66 319
9.66 343
9-66368
9.66 392
9-664i6
9.71
9.71
9.71
9.71
9.71
9.71
9.71
9.71
9.71
370
4i
43 1
462
493
524
555
586
617
o.
o.
o.
o.
o.
o.
o.
o.
o.
28 63o
28 599
28569
28 538
28 507
28476
28445
28414
28383
9.94 852
9-94845
9.94839
9.94832
9.94826
9.94819
9-948i3
9.94 806
9.94799
7
6
7
6
7
6
7
7
6
39
38
3?
36
35
34
33
32
3i
30
9.66 44i
9.71
648
o.
28 352
9 . 9 4
793
30
L. Cos.
d.
L. Cotg. d.
L.
Tang.
L. Sin. d.
62 3O .
PP
.1
.2
3
4
'e
.'s
9
3
31
30
.1
.2
3
4
'.6
.8
5
4
.1
.2
3
4
.6
'.8
9
7
6
3-2
6.4
9.6
12.8
16.0
19.2
22.4
256
11
93
12 .4
'5-5
18.6
21.7
24.8
27.9
3-
6.0
9.0
12.0
IS.O
18.0
21.
24.0
27.0
2-5
5-o
7-5
IO.O
12.5
15.0
'7-5
20. o
2.4
4 .8
7-2
9.6
I2.O
14.4
168
19.2
21.6
0.7
'4
2.1
2.8
3-5
42
4-9
5-6
1.2
1.8
2.4
3-
3.6
42
4.8
^4
84
27 3O .
'
L. Sin.
d.
L. Tang. d.
L.
Cotg.
L. Cos.
d.
30
9
,6644i
9.71
648
0.28 352
9.9479 3
30
3i
9.66465
9.71
679
3
0.28 321
9.94 786
6
29
32
9
.66489
9.71
709
0.28 291
9.94 780
28
33
9
.665i3
9.71
?4
31
0.28 260
9.94773
I
27
24
31
b
34
9
.6653 7
25
9.71
771
31
0.28 229
9.94767
7
26
35
9
.66 562
9.71
802
o.
28 198
9.94 760
25
36
9.66 586
9.71
833
31
0.28 167
9.94753
24
24
3
3?
9
.66 610
9.71
863
0.28 137
9.94747
23
38
9.66634
9.71
894
0.28 106
9.94 740
22
3 9
9.66658
24
9.71
9 2 5
3i
0.28 076
9 . 9 4
?34
21
40
9
.66682
9.71
9 55
0.28 045
9.94727
20
4i
9. 66 706
24
9.71
986
31
0.28 oi4
9.94 720
fi
i9
42
9.66 731
9.72 017
0.27983
9 . 94 7 1 4
18
43
9.66 755
24
9.72 o48
3i
0.27 952
9.94707
1
'7
24
3
7
44
9.66 779
9.72 078
0.27 922
9.94 700
6
16
45
9.66 8o3
9.72
109
0.27 891
9.94694
is
46
9.66 827
24
9.72
i4o
31
0.27 860
9.94 687
i4
24
3
7
47
9.6685i
9.72
170
0.27 83o
9.94 680
f,
i3
48
9.66875
9.72
201
0.27 799
9.94 674
12
49
9.66 899
24
9.72
23l
3
o.
27 769
9-94 667
I I
50
9.66 922
23
9.72
262
3 1
o.
27 788
9.94 660
6
10
5.1
9.66 946
24
9.72
293
3 1
o.
27 707
9-94654
9
52
9.66 970
9.72
323
0.
27 677
9.94,647
8
53
9.66994
24
9.72
354
3 1
o.
27 646
9.94 64o
1
7
54
55
9.67 018
9.67 O42
24
24
9.72
9.72
384
4i5
3
3 1
o.
0.
27616
27 585
9. 9 4634
9. 94 627
7
6
5
56
9.67 066
24
9.72
445
3
o.
27 555
9.94 620
4
5 7
9.67 090
2 4
9.72
4 7 6
3i
o.
27 524
9.94 6i4
3
58
9.67 1 13
9.72
5o6
o.
27 494
9.94607
2
59
9.67 187
24
9.72
53 7
3i
o.
27 463
9.94 600
J
I
60
9.67 161
24
9.72
56 7
3
0.
2 7 433
9.94 5g3
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
'
62.
PP
31
30
25
24
*3
7
6
.1
3- 1
3-o
2-5
.1 2.4
2-3
.1
0.7
0.6
.2
6.2
6.0
.2 4 .8
4.6
.2
1.4
1.2
3
9-3
9.0
7-5
3 7-2
6.9
3
2.1
1.8
4
12.4
12.0
IO.O
.4 9.6
9.2
4
2.8
2-4
5
iS-5
I 5 .0
12.5
5 12. o
11.5
5
3-5
3-
.6
18.6
18.0
15.0
.6 14.4
13-8
.6
4.2
3.6
.7
21.7
21.0
J 7-5
.7 16.8
16.1
7
4.9
4-2
.8
24.8
24.0
20. o
.8 19.2
18.4
.8
5-6
4.8
9
27.9
27.0 22.1;
.9 2r.6
20.7
9
6-3 5-4
85
28.
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
9
.67 161
9.72 567
31
3
3i
3
3
30
3
3i
3
3i
3
3
3'
30
30
3i
30
3
30
3i
30
3
30
3
3'
30
3
3
30
3
0.27 433
9.94 SgS
6
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
60
I
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
.67 185
.67 208
.67 232
.67 256
.67 280
.67 3o3
.6 7 3a 7
.67 35o
6 7 3 7 4
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 81 1
9.72 84i
0.27 402
0.27 372
0.27 34i
0.27 3i i
0.27 280
0.27 250
0.27 220
0.27 189
0.27 169
9.94587
9.94 58o
9.94 573
9.94 567
9.94 56o
9.94 553
9.94 546
9.94 54o
9.94 533
5 9
58
5?
56
55
54
53
52
Si
10
9
.67 3 9 8
9.72 872
0.27 128
9.94 526
50
1 1
12
i3
i4
i5
16
i?
18
'9
9
9
9
9
9
9
9
9
9
.67 421,
.67445
.67468
.67 492
.67 5i5
.67 53g
.6 7 562
.67 586
.67 609
24
23
24
23
24
23
24
23
9.72 902
9.72 932
9.72 963
9.72993
9.73 023
9.73 o54
9.73 084
9.73 n4
9.73 144
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 Sig
9.94 5i3
9.94 5o6
9.94499
9.94 492
9 . 9 4485
9.94479
9.94 472
9-94465
49
48
47
46
45
44
43
42,
4i
20
9
.67 633
9 . 7 3 175
0.26 8a5
9. 9 4458
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
.67 656
.67 680
.67 703
.67 726
.67 750
.6 777 3
.67 796
.67 820
.67 843
24
23
23
24
23
23
24
23
9.73 205
9.73 235
9.73 266
9.73 295
9.73 326
9.73 356
9.73 386
9.73416
9.73 446
0.26 795
0.26 765
0.26 735
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
9.94 43i
9.94 424
9.94417
9-94410
9.94 4o4
9.94 3 9 7
3 9
38
3?
36
35
34
33
32
3i
30
9
.67 866
9.73 476
0.26 524
9.94 390
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
61 3O.
PP
.1
.2
3
4
.6
!s
9
3
30
*4
23
.1
.2
3
4
.6
i
9
7
6
11
93
12.4
J5-5
18.6
21.7
24.8
27.9
3-
6.0
9.0
I2.O
15.0
18.0
21.
24.0
27.0
.1
.2
3
4
'.6
:l
2.4
4-8
7-2
9.6
I2.O
14.4
16.8
I 9 .2
2.3
4.6
6. 9
9.2
"5
.3-8
16.1
18.4
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
3
3-6
4.2
4.8
54
86
28 3O .
L. Sin.
d.
L. Tang. d. L. Cotg.
L. Cos.
d.
30
9.67866
9.73 476
3i
3
3
3
3
3
3
3
30
3
3
30
3
3
3
3
3<>
3
3
3
3
3
29
30
30
3
3
30
29
3
0.26 624
9.94 3go
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
3 9
9.67 890
9.67 gi3
9.67 936
9.67959
9 .67 982
9.68 006
9 .68 029
9.68062
9.68 075
23
23
23
23
24
23
23
23
9. 73 507
9.73537
9.73 567
9.73597
9.73 627
9.73 657
9.78 687
9.73 717
9 . 7 3 7 4?
0.26 493
0.26463
0.26433
0.26 4o3
0.26 3?3
0.26343
0.26 3i3
0.26 283
0.26253
9-94383
9.94 376
9.94 369
9.94 362
9.94355
9.94349
9 . 94 342
9.94335
9.94 328
29
28
27
26
25
24
23
22
21
40
9.68 098
9 . 7 3 777
0.26
223
9.94 32i
20
4i
42
43
44
45
46
4?
48
49
9.68 121
9.68 i44
9.68 167
9.68 190
9.68 2i3
9.68 237
9 .68 260
9.68 283
9.68 3o5
23
23
23
23
23
24
23
23
22
23
23
23
23
23
23
23
23
23
22
2 3
9.73 807
9.73837
9.73 867
9.73897
9.73927
9.73957
9.78987
9.74017
9.74047
0.26 193
0.26 i63
0.26 i33
0.26 io3
0.26 073
0.26 o43
0.26 oi3
o.25 983
O.25 953
9.94 3i4
9.94 3o7
9.94 3oo
9.94 293
9.94 286
9.94279
9.94 273
9.94 266
9.94 25g
'9
1 8
17
16
i5
i4
i3
12
II
50
9.68 328
9.74077
0.26
923
9.94
252
10
5i
52
53
54
55
56
5?
58
5 9
9.68 35i
9.68 374
9.68 397
9.68 420
9-68443
9.68466
9.68489
9.68 5i2
9.68 534
9.74 107
9-?4 i3?
9.74 166
9.74 196
9.74 226
9.74 256
9.74 286
9.74 3i6
9.74345
o.25 893
0.25863
0.25834
o.258o4
0.26 774
O.25 744
o.25 714
0.26 684
o.25 65
9.94245
9.94 238
9.94 23i
9.94224
9.94 217
9.94 2IO
9.94 2o3
9.94 196
9-94 189
9
8
7
6
5
4
3
2
I
60
9.68 557
9-
?43 7 5
0.25
625
9.94
l82
L. Cos.
d.
L.
Cotg.
d. L. Tang.
L. Sin.
d.
'
61.
PP
.1
.2
3
4
:i
7
.8
9
3
30 29
.1
.2
3
4
'.6
.8
*4
23
22
7
6
06
1.2
1.8
2.4
3-o
3-6
4.2
4.8
5-4
3-i
6.2
9-3
124
in
21.7
2 4 .8
27.
3-0 2.9
6.0 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
I2.O
14.4
16.8
19.2
21.6
2 '3
4.6
6.9
9.2
"5
13-8
16.1
18.4
2.2
4.4
6.6
8.8
II.
13.2
'5-4
17.6
19.8
i
2
3
4
6
8
0.7
>-4
2.1
2.8
3-5
4.2
4-9
5-6
6.3
87
2 c
'
L. Sin.
d.
L. Tang.
d. L. Cotg.
L.
Cos. d.
9
.68 557
9.74 375
3
3
3
29
3
3
29
3
3
30
29
3
3
29
3
3
29
3
29
3
29
3
3
29
3
29
3
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
7
60
I
2
3
4
5
6
7
8
9
y
y
y
y
y
y
y
9
y
.6858o
.686o3
.68625
.68648
.68 671
.68694
.68 716
.68 739
.68 762
23
22
23
23
23
22
23
23
9.74405
9.74435
9.74465
9.74494
9.74 524
9-74554
9.74 583
9.74 6i3
9.74 643
o.25 5g5
o.25 565
o.25 535
o.25 5o6
o.25 476
o.25 446
o.25 417
0.25 387
0.25 357
9 . 9 4 175
9.94 168
9.94 161
9.94 i54
9.94 i4?
9.94 i4o
9.94 1 33
9.94 126
9-94 119
5 9
58
56
55
54
53
52
Si
10
y
.68 784
9.74 673
O.25 327
9.94 H2
50
1 1
12
i3
i4
i5
16
17
18
'9
y
y
9
9
9
9
9
y
y
.68 807
.68 829
.68 852
.68875
.68 897
.68 920
.68 942
.68 965
.68 987
22
23
23
22
23
22
23
22
9.74 702
9.74 732
9.74 762
9-74 79 1
9. 74 821
9 . 7 485i
9.74 880
9.74 910
9.74 939
o.25 298
0.25 268
o.25 238
o. 25 209
o.25 179
o.25 149
O.25 120
o.25 090
o.25 061
9.94 105
9.94 098
9.94 090
9.94 o83
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
40
21
22
23
24
25
26
27
28
2 9
y
9
y
y
y
y
y
y
9
.69 o32
.69 o55
.69077
.69 too
.69 122
.69 144
.69 167
.69 189
.69 212
23
22
23
22
22
23
22
9.74998
9. 75 028
9.75 o58
9.75 087
9 . 7 5 117
9.75 i46
9.75 176
9.75 2o5
9 . 7 5 235
O.25 OO2
0.24 972
0.24 942
0.24 gi3
0.24883
0.24854
0.24 824
0.24 795
0.24 765
9.94 o34
9.94 027
9.94 020
9.94 OI2
9.94 oo5
9.93 998
9.93991
9.93 984
9.93 977
7
7
7
8
7
7
7
7
7
7
39
38
37
36
35
34
33
32
3i
30
9 . 69 234
9.75 264
0.24 736
9.93 970
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
-
6O 3O .
PP
.1
.2
3
4
.6
9
30
39
33
33
.1
.2
3
4
'.6
9
8
7
3-o
6.0
9.0
12.0
15.0
1 8.0
21.0
24.0
27.0
2-9
5-8
8.7
n.6
14.5
'7-4
20.3
23.2
.1
.2
3
4
'.6
2.3
46
6.9
9-2
ii-S
13-8
16.1
.8.4
20.7
2.3
4-4
6.6
8.8
II.
13.2
15.4
,7.6
10.8
0.8
1.6
2-4
3-2
4.0
4.8
I' 6
6.4
0.7
M
2.1
2.8
3-5
42
5^
6.3
29 3O .
'
L. Sin.
d.
L. Tang
. d. L. Cotg.
L.
Cos. d.
30
9.69 a34
22
2 3
22
22
22
23
22
22
22
9. 75 264
3
29
3<>
29
29
3
29
3
29
29
3
29
30
29
29
3
29
29
29
3
29
29
29
3
29
29
29
30
29
29
0.24 ?36
9.93 970
7
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
30
3i
32
33
34
35
36
3?
38
3 9
9 .69 256
9.69279
9.69 3oi
9. 69 323
9.69 345
9.69 368
9.69 Sgo
9.69 412
9.69434
9.75 294
9 . 75 323
9.75 353
9.75 382
9.75 4i i
9.75 44i
9.75 470
9 . 7 5 500
9.75 529
0.24 706
0.24 677
0.24 64?
0.24 618
0.24 58g
0.24 55 9
0.24 53o
0.24 5oo
0.24 471
9.93 963
9.93 955
9.93 948
9.93 941
9.93 934
9.93927
9.93 920
9.93 912
9.93 905
29
28
27
26
25
24
23
22
21
40
9.69 456
23
22
22
22
22
22
22
22
22
9 . 7 5 558
0.24 44a
9.93 898
20
4i
42
43
44
45
46
47
48
49
9.69479
9.69 5oi
9 . 69 523
9.69 545
9.69 56y
9.69 58g
9.69 61 1
9.69 633
9.69 655
9 . 7 5 588
9.75 617
9.75 647
9.75 676
9 . 75 705
9 . 7 5 735
9.75 764
9.75 7 9 3
9. 75 822
0.24 4i2
0.24383
0.24 353
O.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 . 9 3855
9-93847
9.93 84o
9 . 9 3 833
'9
18
7
16
i5
i4
i3
12
II
50
9.69 677
9.75 852
0.24 1 48
9 .93 826
10
5i
52
53
54
55
56
5?
58
5 9
9.69699
9.69 721
9.69 743
9.69 765
9.69 787
9.69 809
9.69831
9.69 853
9.69 875
22
22
22
22
22
22
22
22
9.75 881
9.75 910
9 . 7 5 939
9.75969
9.75998
9. 76 027
9. 76 o56
9. 76 086
9.76 115
0.24 119
0.24 090
0.24 061
0.24 o3i
o .24 002
o.23 973
o.23 944
o.23 914
o.23 885
9.93 819
9.98 81 1
9.93 8o4
9.93 797
9 . 9 3 789
9.93 782
9-9 3 775
9.93 768
9.93 760
9
8
7
6
5
4
3
2
I
60
9.69 897
9.76 144
o.23 856
9.93 753
L. Cos. ! d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
6O.
PP 30
29
23
22
.1
.2
3
4
'.6
:l
9
8
7
.1 3.0
2 6.0
3 9-
.4 12.0
5 '5-o
.6 iS.o
7 21.0
.8 24.0
.9 27.0
2.g
5-8
8.7
ii. 6
M-5
'7-4
20.3
23.2
26.1
.1
.2
3
4
.6
'.8
2 -3
4-6
6-9
9.2
11. 5
"3-8
16.1
18.4
2.2
4-4
6.6
88
II.
13.2
I5 1
17.6
19.8
0.8
1.6
2-4
3- 2
4.0
4.8
5.6
6-4
7.2
0.7
1.4
2.1
2.8
3-5
4.2
4-f
5-6
6.3
30.
'
L. Sin.
d.
L. Tang. d.
L
.Cotg.
L. Cos. d.
9.69 897
22
22
22
21
22
22
22
22
21
22
22
22
21
22
22
21
22
21
22
22
21
22
21
22
21
22
21
22
21
22
9.76 i44
29
29
29
3
29
29
29
29
29
29
29
29
29
29
29
29
3
29
29
28
29
29
29
29
29
29
29
29
29
29
o
2 3 856
9 . 9 3
753
7
8
7
7
7
8
7
7
8
7
7
8
7
8
7
7
8
7
7
60
I
2
3
4
5
6
7
8
9
9.69919
9.69 94 1
9.69 963
9.69 984
9 .70 006
9.70 028
9.70 050
9.70 072
9.70 093
9.76 173
9.76 202
9.76 23i
9.76 261
9.76 290
9.76 319
9.76 348
9.76377
9 .76 4o6
o
o.
o.
o.
o.
o.
o.
o.
e.
23 827
23 798
23 769
23 739
23 710
2368i
23652
23623
23 5g4
9.93 746
9.93 7 38
9.93 7 3i
9.93 724
9 . 9 3 717
9.93709
9.93 702
9.93 695
9.93 687
5 9
58
5?
56
55
54
53
D2
5i
10
9. 70 1 1 5
9.76435
o.
2356s
9 . 9 3
680
50
1 1
12
i3
i4
i5
16
i?
18
'9
9.70 i3 7
9.70 169
9.70 180
9.70 202
9.70 224
9.70 245
9.70 267
9.70 288
9.70 3 10
9.76 464
9.76493
9.76 522
9.76 55i
9.76 58o
9.76 609
9.76 639
9.76 668
9.76697
o.
o.
o.
'o,
o.
o.
o.
o.
23 536
23 507
23478
23 449
23 42O
23 391
23 36 1
23 332
23 3o3
9.93 673
9.93 665
9-93658
9.93 65o
9.93 643
9-93636
9.93 628
9.93 621
9.93 6i4
49
48
47
46
45
44
43
42
4i
20
9. 70 332
9.76 725
o.
23 275
9.93 606
40
21
22
23
24
25
26
27
28
29
9.70 353
9. 70 375
9. 70 396
9.70 4 1 8
9.70439
9.70 46i
9.70 482
9. 70 5o4
9.70 525
9.76 754
9.76 783
9.76 812
9v06_84i
9.76 870
9.76899
9.76 928
9.76957
9.76 986
o.
o.
o.
o.
o.
o.
o.
o.
o.
23246
23 217
23 188
23 i5g
23 i3o
23 IOI
23 072
23o43
23 Ol4
9.93599
9.93 591
9.93 584
9.935^7
9. 93 56g
9.93 562
9.93 554
9.93 547
9.93 53g
7
8
7
7
8
7
8
7
8
3 9
38
37
36
35
34
33
32
3i
30
9.70 547
9-77
015
o.
22 gSS
9 . 9 3
532
f
30
L. Cos. d.
L. Cotg.
d. ; L.
Tang.
L.Sin. d.
59 3O .
PP 30
39
38
.1
.2
3
4
:!
.s
9
33
31
i
2
3
4
6
i
9
8 7
.1 3.0
.2 6.O
3 9-
4 12.0
5 '5-
.6 18.0
7 21.0
8 24.0
.9 27.0
2.0
K
it.6
14.5
17.4
20.3
2J.2
2.8
5.6
8.4
II. 2
14.0
16.8
19.6
22.4
2.2
4-4
6.6
8.8
II.
13.2
'5-4
17.6
19.8
2.1
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 35
4.8 4.2
S- 6 4-9
6.4 5.6
90
3O 30 .
1
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9-
70 547
9-77 015
29
O.22 985
9.93 532
30
3i
9-
70 568
22
9.77 o44
29
O.22 gS
6
9.93 525
8
29
32
9-
70 5go
9.77073
28
O.22 927
9.93 517
28
33
9-
70 61 1
9.77 101
O.22 89
9
9.93 5io
27
29
34
9-
70 633
21
9. 77 i3o
29
0.22 870
9.93 5o2
26
35
9-
70 654
9-77 '$9
O.22 8^
i
9.93495
g
25
36
9-
70 675
9.77188
O.22 8l2
9.93487
24
29
7
37
9-
70 697
21
9.77217
29
0.22 783
9.93480
8
23
38
9-
70 718
9.77 246
O.22 754
9.93 472
22
3 9
9-
70 739
9. 77 27^
O.22 726
9.93 465
g
21
40
9
70 761
9.77 3o3
O.22 697
9.93 457
20
4i
9
70 782
21
9.77 332
O.22 668
9.93450
8
'9
42
9-
70 8o3
9.77 36i
O.22 6^
9
o . o3 442
18
43
9-
70 824
9.77 3 9 o
29
O.22 6lO
9.93435
i?
28
44
9-
70 846
21
9.77 4i8
O.22 582
9.93 427
7
16
45
9
70867
9.77447
0.22 553
9 .93 420
g
i5
46
9
70888
9.77476
29
O.22 524
9.93412
i4
29
7
4?
9
70 909
22
9.77505
28
O.22 49
5
9.93 405
8
i3
48
9
70 931
9.77 533
O.22 467
9.93 397
12
49
9
70 g52
9.77 562
29
0.22438
9 .93 390
g
I I
50
9
70973
9.77591
29
28
O.22 409
9.93 382
10
5i
9
70 994
9.77 619
29
0.22 38l
9.93 375
8
9
52
9
71 oi5
9-77 648
O.22 352
9.93 367
8
53
9
71 o36
9.77677
29
O.22 323
9.93 36o
7
22
29
54
9
71 o58
9.77706
28
O.22 294
9.93 352
8
6
55
9
71 079
9-77 7 3 4
O.22 266
9.93 344
5
56
9
71 100
9-77763
29
O.22 237
9.93 337
4
21
28
57
9
71 121
9.77791
O.22 209
9.93 329
7
3
58
9
71 142
9.77 820
O.22 l8o
9-9
3 322
g
2
5 9
9
71 i63
9-77 8 4g
29
O.22 l
I
9.93 3i4
I
60
9
71 i84
9-77 8 77
O.22 123
9.93 307
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
59.
PP
29
28
22
31
8 7
.1
2.9
2.8 I
2.2
2.1
,i
0.8 0.7
.2
5-8
5.6 2
4.4
4.2
,2
1.6 1.4
3
8.7
8-4 3
6.6
6-3
3
24 2.1
4
n.6
II. 2 4
8.8
8.4
4
3.2 2.8
5
14.5
14.0 5
II.
c-5
5
4.0 3.5
.6
17.4
16.8 6
13.2
2.6
.6
4.8 4.2
7
20.3
19.6 7
iS-4
4-7
7
5.6 4.9
.8
23.2
22.4 8
17.6
6.8
.8
6.4 5-6
26.1 25.2 9
19.8
8.9
9
7.2 6.3
9 1
31.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
9.71 1 84
9-7777
O.22 123
9.93 307
60
I
9.71 205
21
9.77906
29
0.22 094
9.93 299
8
5 9
2
9.71 226
9.7793
r )
28
0.22 o65
9.93 291
58
3
9.71 247
9.77963
0.22 o3 7
9.93 284
;
57
29
H
4
9.71 268
21
9.77992
28
O.22 OO8
9.93 276
56
5
9.71 289
9.78 020
O.2I 980
9.93 269
55
6
9.71 3io
9.78 049
O.2I 95 I
9.93 261
54
8
7
9.71 33i
21
9.78077
29
0.21 923
9.93 253
53
8
9.71 352
9.78 1 06
O.2I 894
9.93 246
52
9
9.71 3 7 3
9.78135
29
0.21 865
9.93 238
5i
10
9.71 393
9.78 i63
0.21 837
9.93 23o
50
1 1
9.71 4i4
21
9.78 192
28
0.21 808
9.93 223
7
8
49
12
9.71 435
9. 78 220
O.2I 780
9 . 93 2 1 5
48
i3
9.71 456
21
9.78 24
9
28
O.2I 761
9.93 207
47
i4
9.71 4?7
21
9.78277
29
0.21 723
9.93 2OO
46
i5
9.71 498
9.78 3o6
28
O.2I 694
9.93 192
45
16
9.71 519
9.78 334
0.21 666
9.93 1 84
44
29
17
9.71 539
21
9-78363
28
0.21 637
9.93 177
g
43
18
9.71 56o
9.78 391
28
0.21 609
9.93 169
42
9
9
.71 5Si
9.78 419
0.21 58i
9.93 161
4i
20
9
.71 602
9.78448
28
0.21 552
9.93 i54
7
40
21
9
.71 622
21
9.78476
29
0.21 524
9.93 1 46
8
3 9
22
9
.71 643
9.78 505
28
O.2I 495
9.93 i38
38
23
9
. 71 664
9.78 533
O.2I 467
9.93 i3i
7
3?
21
29
8
24
9
.71685
9.78 562
28
0.21 438
9.93 123
8
36
25
9
.71 705
9.78 590
28
0.21 4
[O
9.93 1 1 5
35
26
9
.71 726
9.78618
0.21 382
9.93 108
1
34
21
29
8
27
9
7 1 747
9.78647
28
0.21 353
9.93 too
g
33
28
9
.71 767
9. 7 86 7 5
0.21 325
9.93 092
32
29
9
.71 788
9.78 704
28
O.2I 296
9.93084
3i
30
9
.71 809
9.78 732
0.21 268
9.93 077
7
30
L. Cos. d.
L. Cotg
. d. L. Tang.
L.
Sin. d.
'
58 3O .
PP
29
28
21
20
8
7
.i
2.9
2.8
.1
2.1
2.0
.1
0.8
0.7
.2
5-8
S.6
.2
4.2
4.0
.2
1.6
1.4
3
8.7
84
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
14-5
14.0
5
10.5
IO.O
5
4.0
3-5
.6
17.4
16.8
.6
12.6
12.0
.6
4.8
42
7
20.3
19.6
7
14.7
14.0
7
56
49
.8
23.2
22.4
.8
16.8
16.0
.8
6.4
5.5
9
26. i
18.9 18.0
9
6.3
92
31 3O .
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
.71 809
9 .78 732
0.21 268
9.93 077
g
30
3i
9
.71 829
9. 78 760
O.2I 240
9 .93 069
8
29
32
9
.71 850
9.78 789
0.21 211
9.93 061
g
28
33
9
. 71 870
9.78 817
0.21 183
9-93o53
27
21
28
7
34
9
.71 891
9.78 845
O.2I I
55
9.98 o46
8
26
35
9
.71 911
9-78874
O.2I 126
9.93 o38
g
25
36
9
.71 982
9. 78 902
O.2I O
98
9.93 o3o
24
2O
37
9
.71 g52
9.78 g3o
O.2I 070
9.93 022
8
23
38
9
.71 973
9.78 9 5 9
0.21 o4i
9.93 oi4
22
3 9
9
.71 994
9.78987
28
0.21 Ol3
9.93 007
8
21
40
9
.72 oi4
9.79 oi5
O.2O 985
9.92 999
g
20
4i
9
.72 o34
9.79 o43
0.2O 957
9.92991
8
'9
42
9
.72 055
9.79072
O.2O 928
9.92 983
18
43
9
.72 075
21
9.79 100
28
O.2O 900
9.92976
8
i?
44
9
.72 096
9.79 128
28
O.2O 872
9.92 968
8
16
45
9
.72 1 16
9.79 i56
O.2O 844
9.92 960
8
i5
46
9
.72 i3 7
9.79 185
29
O.2O 8l 5
9.92 952
i4
20
28
4?
9
.72 167
9.79 2i3
28
O.2O 787
9.9
2 944
R
i3
48
9
.72 177
9-79 241
O.2O 759
9.92 986
12
49
9
.72 198
9.79269
0.20 781
9.92 929
8
I I
50
9
.72 218
9-79 2 97
0.20 703
9.92 921
g
10
5i
9
.72 238
9.79 326
29
28
O.2O 674
9.92 918
8
9
52
9
.72 25g
9.79 354
0.20 646
9 .92 go5
8
8
53
9
.72 279
9.79 382
0.20 618
9.92897
7
2O
54
9
.72 299
9.79410
28
O.2O 5(
)
9.92 889
8
6
55
9
.72 32O
9-79438
O.2O 562
9.92 881
5
56
9
.72 34o
9.79 466
0.2O 534
9.92 874
4
2O
29
57
9
.72 36o
9.79495
28
0.20 5o5
9.92 866
8
3
58
9
.72 38i
9. 79 523
O.2O 4 r
11
9.92 858
g
2
5 9
9
.72401
9.79 55i
28
O.2O 449
9.92 850
8
I
60
9
.72 4ai
9-79 5 79
O.2O 421
9.92 842
L. Cos.
d.
L. Cotg
. d.
L. Tang.
L.
Sin. d.
'
58.
PP
29
28
21
20
8
7
.!
2.9
2-8
.1
2.1
2.O
.1
0.8
0.7
.2
5-8
S.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
14.5
I4.O
.5
10.5
0.0
.5
4.0
3-5
.6
17.4
16.8
.6
12.6
2.O
.6
4.8
4.2
7
20.3
19.6
7
14.7
4.0
. 7
5-6
4.9
.8
23.2
22.4
.8
16.8
6.0
.8
6.4
5-6
9
26.1
9
18.9
8.0
9
7-2
6-3
93
32 C
'
L. Sin. d.
L.
Tang. d. L. Cotg.
L. Cos. d.
9. 72 421
9-79 5 79
O.2O 4^1
9.92
842
i
60
I
9.72 44i
20
9.79607
28
O.2O 393
9.92
834
f
5 9
2
9.72 46 1
9-
79 635
O.2O
365
9.92
826
c
58
3
9.72 482
20
9-
79663
28
O.2O 337
9.92
818
*7
4
9.72
5O2
20
9.79691
28
0.20 Sog
9.92
810
7
56
5
9.72
522
9-
79 7'9
0.20 28l
9.92
8o3
i
55
6
9.72
542
9-
79747
0.20
253
9.92
795
54
29
7
9.72
562
20
9-
79 77 6
28
O.2O 224
9.92
787
f
53
8
9.72
582
9-
79 8o4
O.2O 196
9.92
779
t
52
9
9.72
602
9-
79 832
O.2O
168
9.92
771
i
5i
10
9.72 622
9-
79 860
O.2O l4o
9.92
7 63
I
50
1 1
9.72
643
2O
9-
7988
8
28
O.2O 112
9.92
755
f
4 9
12
9.72 663
9-
79916
0.20 084
9.92 747
48
i3
9.72 683
9-
79 944
0.20 o56
9.92
7 3 9
47
28
i4
9.72
7o3
9-
79972
28
O.2O O28
9.92
7 3i
f
46
i5
9.72
723
9.80 ooo
O.2O OOO
9.92
723
45
16
9.72
743
9.80 028
o. 19 972
9.92
7 .5
44
20
28
i
17
9.72
763
9.80 o56
28
0.19
944
9.92
707
f
43
18
9.72
783
9.80084
o. 19
916
9.92
609
42
'9
9.72 8o3
9-
80 I 12
28
0.19
888
9.92 691
4i
20
9.72 823
9-
80 i4
o
o. 19 860
9.92
683
40
21
9.72 843
9.80 168
0.19
832
9.92
6 7 5
f
3 9
22
9.72 863
9.80 igS
o. 19
805
9.92
667
38
23
9.72 883
9.80 223
o. 19
777
9.92
65 9
37
24
9.72 902
'9
9.80 25l
28
0.19 749
9.92
65i
J
36
25
9.72
922
9.80 279
o. 19 721
9.92 643
35
26
9.72
942
9 . 80 3o,7
o. 19
6 9 3
9.92 635
34
20
27
9.72
962
9. 80 335
28
0.19 665
9.92
627
f
33
28
9.72 982
9-
80 363
0.19 637
9.92
619
32
29
9.73
002
9-
3o 3 9
i-
0.19 609
9.92
611
3i
30
9.73 022
9 .8o4i Q
o. 19 58i
9.92
6o3
30
L. Cos.
d.
L.
Cotg.
d.
L. Tang.
L. Sin.
d.
'
57 3O .
PP
29
28 27
21
20
19
8
7
i
2.9
2.8 3.7
i
2.1
2.0
1.9
.1
08
0.7
2
5-8
5-6 5-4
a
4.2
4.0
3-8
.2
1.6
1.4
3
8.7
8.4 8.1
3
6-3
6.0
5-7
3
2.4
2.1
4
ti.6
1 1. 2 IO 8
4
8.4
8.0
7-6
4
3-2
2.8
5
'4-5
14.0 13.5
5
0.5
o.o
95
5
4.0
3-5
6
'7-4
16.8 1 6. 2
6
2.6
3.O
11.4
.6
4.8
4-2
7
20.3
19.6 18.9
7
4-7
4-0
13-3
7
5-6
4-9
8
23.2
22.4 21.6
8
6.8
6.0
15-2
.8
6.4
5-6
9 26.1
2S-2 24.3
8.9
17.1
6.3
32 3O
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9-
73 022
9.80 419
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
o. 19 58i
9.92 6o3
8
8
8
8
8
8
9
8
8
g
30
3i
32
33
34
35
36
3?
38
39
9-
9-
9-
9-
9-
9-
9-
9-
9-
73 061
73o8i
73 101
73 121
73 i4o
73 160
73 180
73 200
20
20
2O
2O
'9
20
20
20
'9
20
20
19
2O
20
19
2O
2O
9
20
'9
20
'9
20
'9
2O
2O
'9
20
9.80447
9.8o4?4
9.80 5o2
9.8o53o
9. 80 558
9. 80 586
9.80 6i4
9.80 642
9.80 669
0.19 553
0.19 526
0.19 498
o. 19 470
0.19 442
o. 19 4i4
0.19386
0.19358
0.19331
9.92 595
9.92 587
9.92579
9.92 571
9.92 563
9.92 555
9.92 546
9-92538
9.92 53o
29
28
27
26
25
23
22
21
40
9-
73 219
9.80 697
o.i 9 3o3
9.92 522
20
4i
42
43
44
45
46
47
48
4 9
9
9
9
9
9
9
9
9
9
"732 3o
7 3 2 5O
73 278
73 298
73 3i8
7 333 7
73 357
73 377
73 396
9.80 725
9.80 753
9.80781
9.80808
9. 80 836
9.80864
9.80 892
9.80 919
9.80947
o. 19 275
0.19 247
o. 19 219
0.19 192
0.19 i64
0.19 i36
0.19 108
o. 19 08 1
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
8
8
8
8
9
8
8
8
8
8
8
9
8
8
8
8
8
9
8
'9
18
17
16
i5
i4
i3
12
II
50
9 . 7 34i6
9.80975
o. 19025
9.92 44i
10
5i
52
53
54
55
56
5 7
58
5 9
9
9
9
9
9
9
9
9
9
73435
73455
73494
735i3
73533
73552
73 572
73 591
9.81 oo3
9.81 o3o
9.81 o58
9.81 086
9.81 n3
9.81 i4i
9.81 169
9.81 i 96
9.81 224
0.18 997
0.18 970
0.18 942
0.18914
0.18887
0.18859
o.i883i
o.i88o4
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
73 61 1
9.81 252
0.18 748
9.92 359
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
57.
PP
.1
.2
3
4
.6
.3
9
38
27
20
19
.1
.2
3
4
7
9
9 8
2.8
5-6
8.4
II 2
14.0
16.8
ig.6
22.4
2.7 .1
5-4 -2
8.1 .3
10.8 .4
'3-5 -5
16.2 .6
18.9 .7
21.6 .8
24-3 -9
2.O
4-0
6.0
8.0
0.0
2.0
4.0
6.0
8.0
1.9
3-8
5-7
7.6
9-5
11.4
'3-3
'5-2
17.1
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
33.
'
L. Sin. d.
L. Tang. d.
L. Cotg.
L
Cos. d.
9.73 61 1
9.81 252
0.18 748
9.92 35g
60
I
9.73 63o
20
9.81 279
28
o. 1 8 721
9.92 35i
8
5 9
2
9.73 650
9.81 307
28
0.18 693
9.92 343
58
3
9.73 669
9.81 335
o.i8665
9.92335
57
27
9
4
9.73 689
9.81 362
28
0.18 638
9.92 326
8
56
5
9.73 708
9.81 3go
28
0.18 610
9.92 3i8
55
6
9.73 727
9.81 4i8
0.18 582
9.92 3io
54
20
27
8
7
9 . 7 3 747
9.81 445
28
0.18 555
9.92 3o2
53
8
9.73 766
9 81 473
o. 18 527
9.92 293
52
9
9 . 7 3 7 85
'9
9.81 5oo
28
o. 18 500
9.92 285
g
5i
10
9.73 805
9.81 528
28
0.18 472
9.92 277
50
1 1
9.73 824
19
9.81 556
27
0.18 444
9.92 269
49
12
9 . 7 3 843
9.81 583
28
0.18417
9.92 260
48
i3
9-73863
9.81 61 1
0.18 389
9.92 252
47
'9
2 7
i4
9.73 882
19
9.81 638
28
0.18 362
9.92 244
46
i5
9.73 901
9.81 666
0.18 334
9.92 235
45
16
9.73 921
9.81 693
0.18 307
9.92 227
44
'9
8
17
9.73 940
19
9.81 721
o. 18 279
9.92 219
fi
43
18
9
. 7 3 9 5 9
9.81 748
O.l8 252
9.92 211
42
'9
9.73978
9.81 776
o. 18 224
9.92 2O2
y
4i
20
9
.73 997
9.81 8o3
o. 18 197
9.92 194
g
40
21
9
.74017
9.81 83i
0.18 169
9.92 186
9
3 9
22
9
.74o36
9.81 858
o. 18 142
9.92 177
g
38
23
9
.74 o55
'9
9.81 886
0.18 1 14
9.92 169
37
19
27
24
9
.74074
9.81 gi3
28
0.18 087
9.92 161
9
36
25
9
.74 093
9.81 94
I
0.180
5 9
9.92 i52
35
26
9
. 7 4n3
9.81 968
27
o. 18 o32
9.92 1 44
34
28
27
9
74 i32
9.81 996
o. 1 8 oo4
9.92 i36
9
33
28
9
. 7 4 i5i
9.82 023
0.17977
9.92 127
g
32
2 9
9
?4 170
9.82 o5i
0.17 949
9.92 119
g
3i
30
9
74 189
9.82 078
2 7
0.17 922
9.92 in
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
'
56 3O .
PP
38
*7
ao
19
9
8
.1
2.8
2.7
i
2.O
I.O
i
0.9
0.8
.2
5.6
5-4
2
4.0
3.8
2
1.8
1.6
3
8-4
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
I4.O
13-5
5
o.o
9-5
5
4-5
4.0
.6
1 6. 8
16.3
6
2.0
"4
6
5-4
4.8
7
19.6
18.9
7
4-0
13-3
7
6.3
5-6
.8
22.4
21.6
8
6.0
15.2
8
7.2
6.4
9
25.2 24.3
8.0 17.1
8.1 7.2
96
33 30 .
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9.74 189
9.82 078
28
0.17 922
9.92 i ii
30
3i
9.74 208
19
9.82 106
27
0.178
94
9.92 IO2
9
8
29
32
9.74 227
9.82 i33
28
o. 17 867
9.92094
28
33
9
.74 246
9.82 161
0.17 8
*9
9. 9 2 086
27
19
27
9
34
9
.74 265
19
9.82 188
27
o. 17 812
9.92077
g
26
35
9
.74284
9.82 2i5
28
0.17785
9.92 069
25
36
9
.74 3o3
9.82 243
o.i 77 5 7
9.92 060
9
24
'9
27
K
3?
9
.74 322
19
9.82 270
28
o. 17 730
9.92 o52
8
23
38
9
74341
9.82298
0.17 702
9.92 o44
22
3 9
9
.74 36o
9.82 325
0.17675
9.92 o35
9
21
40
9
.74379
9.82 352
28
0.17 648
9 . 9 2 027
20
4i
9
. 7 43 9 8
9.82 38o
0.17 620
9 . 9 2 018
9
8
i 9
42
9
.74417
9.82 407
0.17 5(
;3
9.92 oio
18
43
9
. 7 4436
'9
9.82435
0.17 565
9.92 002
17
19
27
9
44
9
.74455
9.82 462
0.17 538
9.91 99 3
8
16
45
9
.74 4?4
9.82489
0.17 5u
9.91 985
i5
46
9
.74493
19
9.82 617
0.17483
9.91 976
9
i4
19
27
K
47
9
.74 5 12
9.82 544
o. 17 456
9.91 968
i3
48
9
.74 53i
9.82 571
0.17 429
9.91 959
12
49
9
74 54g
9.82 599
o. 17 4oi
9 . 9 i 9 5i
I I
50
9
74568
J 9
9.82 626
0.17 374
9.91 942
9
10
5i
9
7458 7
'9
9.82 653
27
28
0.17 347
9.91 934
9
52
9
.74 606
9.82681
0.17 3j
9
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
.74 662
9.82 762
0.17238
9.91 900
5
56
9
. 7 468i
'9
9.82 790
0.17 210
9.91 891
9
4
57
9
.74 700
'9
9.82817
27
0.17 i83
9.91 883
3
58
9
.74719
19
9.82 844
0.17 i56
9.91 874
2
5 9
9
18
9.82 871
27
0.17 129
9 . 9 i 866
I
60
9
.747^6
9.82 899
0.17 loi
9 . 9 i 857
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
i. 9
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
5
14.0
'3-5
5
9-5
9.0
5
45
4.0
.6
16.8
1 6. 2
.6
11.4
10.8
6
5-4
4.8
7
19.6
18.9
. 7
'3-3
12.6
7
6-3
5-6
.8
22.4
21.6
.8
14.4
8
7-2
6.4
9
25.2
24-3
17.1
16.2
9
S.j
7.2
97
34.
.
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
9
74 ?56
9.82 899
27
27
27
28
27
27
27
28
27
27
27
27
27
28
27
27
27
27
27
27
28
27
27
27
7
2 7
27
27
27
27
0.17 101
9.91 857
8
9
8
9
8
9
8
9
8
9
9
8
9
8
9
9
8
9
8
9
9
8
9
9
8
9
9
8
9
9
60
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
74 775
?4 794
.74812
. 7 483i
.74850
.74868
.74887
.74 906
.74 924
'9
18
'9
'9
18
'9
'9
18
9.82 926
9.82 953
9.82 980
9-83 008
9-83 oSg
9. 83 062
9.83089
9-83 117
9 .83 1 44
o. 17 074
0.17 047
0.17 O2O
o. 16 992
o. 16 965
o. 16 938
o. 16 911
0.16 883
0.16 856
9.91 84g
9.91 84o
9.91 832
9.91 823
9.91 815
9.91 806
9.91 798
9.91 789
9.91 781
5 9
58
57
56
55
54
53
52
Si
10
9
.74943
18
9-83 171
o. 16 829
9.91 772
50
1 1
12
i3
i4
i5
16
'7
18
'9
9
9
9
9
9
9
9
9
9
?4 961
.74 980
74999
.75 017
.75 o36
.75 o54
.75 073
.75 091
.75 no
19
'9
18
19
18
'9
18
'9
18
9-83 198
9-83 225
9.83252
9.83 280
9. 83 307
9.83334
9-83 36i
9.83 388
9-834(5
o. 1 6 802
o. 16 775
0.16 748
o. 16 720
0.16 693
0.16 666
o. 16 63g
o. 16 612
0.16 585
9.91 763
9.91 7 55
9.91 746
9.91 738
9.91 729
9.91 720
9.91 712
9.91 703
9.91 695
49
48
47
46
45
44
43
42
4i
20
9
.75 128
9.83442
0.16 558
9.91 686
40
21
22
23
24
25
26
27
28
2 9
9
9
9
9
9
9
9
9
9
.75 147
.75 i65
. 7 5 1 84
.75 202
.75 221
.75 239
.75258
.75 276
.75 294
18
'9
18
'9
18
'9
18
18
9. 83 4?o
9.83497
9.83 524
9.83 55i
9 .835 7 8
9.836o5
9-83 632
9.83 65g
9.83686
o. 16 53o
o. 16 5o3
o. 16 476
o. 16 449
o. 16 422
o. 16 395
0.16 368
o.i634i
0.16 3i4
9.91 677
9.91 669
9.91 660
9.91 65i
9.91 643
9.91 634
9.91 626
9.91 617
9.91 608
3 9
38
37
36
35
34
33
32
3i
30
9
. 7 5 3i3
9-83 713
o. 16 287
9.91 599
30
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
t
55 3O .
PP
.1
.2
3
4
.6
.8
9
23
37
9
18
.1
.2
3
4
.6
.S
9
9
8
2.8
5.6
8.4
II. 2
14.0
1 6.8
19.6
22.4
2.7
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
24-"!
i
a
3
4
6
I- 2
3-8
5-7
7.6
9-5
11.4
'3-3
15.2
17. i
1.8
36
5-4
7-2
9.0
10.8
12.6
J4-4
16. 2
0.0
i.S
2.7
3-6
4-5
5-4
6-3
7.2
8.1
0.8
1.6
2-4
32
4 2
4.8
5.6
6.4
7.2
34 3O'
f
L.Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
30
9
.75 3i3
18
9 .83
7 i3
o.
16 287
9.91
5 99
8
30
3i
9
.76 33i
19
9-83
?4o
28
0.
16 260
9.91
5 9 i
9
29
32
9
.75 350
18
9.83
768
o.
l6 232
9.91
582
28
33
9
.75 368
18
9.83
795
27
o.
16 2o5
9.91
5 7 3
8
27
34
9
.75 386
19
9-83
822
27
o.
16 178
9.91
56^
9
26
35
V
7 5 405
9-83
84 9
o.
16 i5i
9.91
556
25
36
9
. 75 4s
3
9-83 876
o.
1 6 124
9.91
547
24
27
9
^7
9
75 44
i
it
9.83
903
27
o.
1 6 097
9.91
538
8
23
38
9
.75459
9-83
93o
o.
16 070
9.91
53o
22
3 9
9
.75478
r 9
18
9-83 957
27
o.
16 o43
9.91
521
21
40
9
.75 4y6
it
9.83 984
o.
16 o 1 6
9.91
5l2
8
20
4i
9
.75 5i4
9 .84
01 I
o.
i5 989
9.91
5o4
9
'9
42
9
.76 533
18
9 .84o38
o.
i5 962
9.91
495
18
43
9
. 7 5 55i
9-84
065
o.
i5 935
9.91
486
J?
27
9
44
9
.75 56 9
18
9-84
092
o.
i5 908
9.91
477
8
16
45
9
.75 58?
18
9-84
119
o.
i588i
9.91
46 9
i5
46
9
.75 6o5
9-84
1 46
27
o.
i5854
9.91
46o
i4
4 7
9
.75 624
18
9-84
I 7 3
27
o.
i5 827
9.91
45i
9
i3
48
9
. 75 642
9-84
200
o.
1 5 800
9.91
442
12
49
9
.75 660
18
9-84
227
27
o.
i5 77 3
9.91
433
8
I I
50
9
.75 678
18
9-84
254
27
26
o.
i5 746
9.91
425
10
5i
9
.75 696
18
9-84
280
27
o.
1 5 720
9.91
4i6
Q
9
52
9
.75 714
9-84
307
o.
1 5 693
9.91
407
8
53
9.75 733
'9
9-84
334
o.
i5 666
9.91
3 9 8
7
18
27
9
54
9
. 7 5 7 5i
18
9-84
36i
o.
i5 63g
9.91
38 9
8
6
55
9.75 769
9-84
388
o.
i5 612
9.91
38i
5
56
9.75 787
9.844i5
27
o.
i5 585
9.91
372
4
ll
27
9
57
9.75 805
18
9-84
442
o.
1 5 558
9.91
363
9
3
58
9.75 823
9.84469
o.
i5 53i
9.91
354
2
5 9
9 . 7 584i
9.84496
27
o.
1 5 5o4
9.91
345
I
60
9 . 7 585 9
9-84
523
27
o.
i5 477
9.91
336
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin.
d.
1
55.
PP
a
97
26
19
18
9 8
.1
2.8
2-7
2.6
.1 I.O
1.8
.1
0.9 0.8
.2
5.6
5-4
5.2
.2 3.8
3-6
.2
1.8 1.6
3
8.4
8.1
7.8
3 5-7
5-4
3
2.7 2.4
4
II. 2
10.8
10.4
4 7-6
7.2
4
3.6 3.2
5
14.0
13-5
13.0
5 9-5
9.0
5
4-5 4-
.6
16.8
16. 2
156
.6 11.4
10.8
.6
5-4 4-8
7
19.6
18.9
18.2
7 13.3
12.6
. 7
6-3 5.6
.8
22.4
21.6
20.8
.8 15.2
14.4
.8
7.2 6.4
9
25.2
24.3 23.4
9 '7-i
16.2
9
8.1 7.2
99
35.
t
L. Sin. d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
9 . 7 55 9
it
18
18
18
18
18
18
18
18
9.84 523
27
26
27
27
27
27
27
27
26
27
27
27
27
27
26
27
27
27
27
26
27
27
27
26
27
27
27
26
27
27
o.i5 477
9.91 336
8
9
9
9
9
9
9
8
9
9
9
9
9
9
9
9
9
9
9
9
9
8
9
9
9
9
9
9
9
9
60
1
2
3
4
5
6
7
8
9
9 . 7 58 77
9 . 7 58 9 5
9.75913
9.75931
9.75949
9.75967
9 . 7 5 9 S5
9.76 oo3
9. 76 O2I
9.84550
9.84576
9-846o3
9-8463o
9.84 657
9-84684
9.84 711
9.84?38
9-84 ?64
o. i5 45o
o. i5 424
o. 1 5 397
o. i5 370
o.i 5 343
o.i5 3i6
o. i5 289
o. i5 262
o.i5 236
9.91 328
9.91 319
9.91 3io
9.91 3oi
9.91 292
9.91 2*3
9.91 274
9.91 266
9.91 25 7
5 9
58
57
56
55
54
53
52
5i
10
9
.76 o3g
18
18
18
18
18
'7
18
18
18
18
18
9.84 791
o. 1 5 209
9.91 248
50
1 1
12
i3
i4
i5
16
17
18
'9
9
9
9
9
9
9
9
9
9
. 76 057
.76075
.76 093
.76 in
.76 129
.76 i46
.76 i64
.76 182
.76 200
9.84818
9-84845
9.84872
9.84899
9.84 925
9.84 952
9.84979
9.85 006
9.85o33
o. i5 182
o. i5 1 55
o.i5 128
o. i5 101
o.i5o75
o. 1 5 o48
0. l5 021
0.14994
o. i4 967
9.91 239
9.91 23o
9.91 221
9.91 212
9.91 2o3
9.91 194
9.91 i85
9.91 176
9.91 167
49
48
4?
46
45
44
43
42
4i
20
9
.76 218
9.85 059
o. 1 4 941
9.91 i58
40
21
22
23
24
25
26
2 7
28
2 9
9
9
9
9
9
9
9
9
9
.76236
.76253
.76 271
.76 289
.76 307
.76 324
.76 342
.76 36o
.76378
'7
18
18
18
'7
18
18
18
9. 85 086
9.85 ii3
9-85 i4o
g.85 166
9-85 193
9-85 220
9-85 247
9.85 273
9.85 3oo
o. i4 9'4
0.14887
o. 14 860
o.i4834
o. i4 807
o.i4 780
0.14753
o.i4 727
o. 1 4 700
9.91 i4g
9.91 i4i
9.91 i32
9.91 123
9.91 n4
9.91 105
9.91 096
9.91 087
9.91 078
3 9
38
3?
36
35
34
33
32
3i
30
9
.76395
9.85 327
0.14673
9.91 069
30
L. Cos.
d.
L. Cot?.
d.
L. Tang.
L.
Sin.
d.
'
54 3O .
P?
. i
.1
3
4
5
.6
.8
'?
37
26
18
17
.1
.2
3
4
.6
.8
9
9
8
27
5-4
8 i
10.8
'3-5
1 6. 2
iS.q
21.6
MJ
2.6
5-2
7.8
10.4
13.0
15.6
18.2
20.8
21 (
.1
.2
3
4
'.6
:l
1.8
3-6
5-4
7- a
9.0
10.8
12.6
'44
16.2
1 7
3-4
5-i
6.8
8-S
10.2
"9
13.6
'S-3
0.9
1.8
2-7
3-6
45
5-4
6.3
7.2
0.8
1.6
2.4
3-2
4.0
4.8
5-6
6.4
7-2
35 30 '.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9
. 76 3y5
9.85 327
27
26
2 ?
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. 1 4 673
9.91 069
9
9
9
9
10
9
9
9
9
9
9
9
9
9
9
9
9
9
10
9
9
9
9
9
9
IO
9
9
9
9
30
3i
32
33
34
35
36
3 7
38
3 9
9
9
9
9
9
9
9
9
9
764i3
.76431
. 7 6448
.76466
.76484
.76 5oi
.76 5ig
. 7 653 7
.76 554
18
'7
18
18
'7
18
18
'7
18
9 .85 354
9.85 38o
9-85 407
9.85434
9-85 46o
9.85487
9.855i4
9-85 54o
9-85 567
o. i4 646
o. :4 620
o. i4 5 9 3
o.i4566
o. 1 4 54o
o. 14 5i3
o.i4486
o. i4 46o
o.i4433
9.91 060
9.91 o5i
9.91 O42
9.91 o33
9.91 O23
9.91 oi4
9.91 oo5
9.90996
9.90987
29
28
27
26
25
24
22
2 I
40
9
.76 572
9.85 5 9 4
o. i4 4o6
9.90978
20
4i
42
43
44
45
46
4 7
48
49
9
9
9
9
9
9
9
9
9
. 76 5go
. 76 607
.76 625
.76 642
.76 660
.76677
.76695
.76 712
.76 73o
17
18
'7
18
7
18
i?
18
9.85 620
9.85 647
9-85 674
g.85 700
9-85 727
9-85 7 54
9.85 780
9.85 807
9.85 834
o.i4 38o
o.i4353
o. i4 326
o. i4 3oo
o. i4 278
0.14 246
o. i4 220
o. i4 193
o. 1 4 166
9.90969
9.90 960
9.90 9 5i
9.90 942
9.90 933
9.90 924
9.90915
9.90 906
9.90 896
'9
18
'7
16
i5
i4
i3
12
I I
50
9
.76 747
9.85 860
o. 1 4 i4o
9.90 887
10
5i
52
53
54
55
56
57
58
5 9
9
9
9
9
9
9
9
9
9
.76 765
.76 782
.76 800
.76 817
.76 835
.76 852
.76 870
.76 887
.76 904
'7
18
17
18
'7
18
17
'7
iR
9.85 887
9.85 918
9-85 9 4o
g.85 967
9-85 993
9.86 020
9.86 o46
9.86 073
9.86 100
o.i4 u3
o. i4 087
o. 1 4 060
o.i4 o33
o. 1 4 007
o. i3 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
60
. 76 Q22
9.86 126
o.i3 874
9.90796
L. Cos.
d.
L. Cotg
. d.
L. Tang.
L.
Sin.
d.
f
54.
PP
.1
.2
3
4
5
.6
.8
q
7
26
18
17
.1
.2
3
4
'.6
9
IO
9
2.7
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
24-3
2.6
5-2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
.1
.2
3
4
:i
9
1.8
3-6
5-4
7.2
9.0
10.8
13.6
H.4
IO.2
'7
3-4
5-'
6.8
8-5
IO.2
,1.0
13.6
1^ .3
1.0
2.0
3-
4.0
5-o
6.0
7.0
8.0
0.0
0.9
1.8
2-7
3-6
4-5
5-4
6-3
7.2
8.1
101
36.
'
L.Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
9
.76 922
9.86 126
27
26
27
26
27
26
27
26
27
27
26
27
26
27
26
27
26
26
37
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
10
9
9
9
10
60
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
. 7 6 9 3 9
.76957
.76974
.76991
.77009
.77 026
77 43
.77 061
.77078
18
'7
i?
18
>7
'7
18
'7
9.86 i53
9.86 179
9.86 206
9.86282
9.86259
9 .86285
9.86 3i2
9.86338
9.86365
o.i3 847
o.i382i
o.i3 794
0.18 768
0.18 741
0.18 715
o.h3688
o.i 3 662
0.13635
9.90 787
9.90777
9.90 768
9.90759
9.90750
9.90 741
9.90 781
9.90 722
9.90 71 3
5 9
58
5?
56
55
54
53
52
5i
10
9
.77095
>7
'7
18
'7
17
'7
18
'7
'7
i?
9.86 392
'o.i3 608
9.90 704
50
1 1
12
i3
i4
i5
16
'7
18
'9
9
9
9
9
9
9
9
9
9
77 112
.77 i3o
77 '47
77 i64
.77 181
77 '99
.77 216
.77233
.77 25o
9.86418
9.86445
9.86471
9.86498
9.86 524
9 .8655i
9.86577
9.866o3
9.8663o
o.i3 682
o.i3555
0.18 529
o. 1 3 5o2
0.18476
o. 1 3 449
0.18428
0.18 397
o. i3 870
9.90694
9.90 685
9.90 676
9.90 667
9.90 657
9.90 648
9.90 689
9.90 63o
9.90 620
49
48
47
46
45
44
43
42
4i
20
9
.77268
'7
'7
i?
'7
'7
'7
17
18
>7
'7
9.86656
o.i 3 344
9.90 61 1
y
40
21
22
23
24
25
26
27
28
29
9
9
9
9
9
9
9
9
9
77 285
.77 3o2
.77819
.77 336
.77353
77 3 7
.77887
774o5
.77422
9.86688
9.86 709
9.86 736
9.86 762
9.86 789
9.86 8i5
9.86 842
9.86868
9.86894
o.i3 317
o.i 3 291
0.18 264
0.18288
0. l3 21 I
0.18 185
0.18 i58
0.18 182
o . 1 3 1 06
9.90 602
9.90 5g2
9-90583
9.90 574
9.90 665
9.90 555
9.90 546
9.90 537
9.90 527
y
10
9
9
9
10
9
9
10
9
3 9
38
37
36
35
34
33
32
3i
30
9
. 77 43 9
9.86 921
o. i3 079
9.90 5 1 8
30
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin. d.
53 3O .
PP
.1
.2
3
4
.6
:l
9
27
26
18
17
I
2
3
4
1
I
10
9
2.7
i;
10.8
'3-5
16.2
18.9
21.6
..i- ;
2.6
S 'o
7-8
10.4
13.0
15-6
18.2
20.8
2T..A
i
2
3
4
5
6
8
9
1.8
3-6
5-4
7.2
90
10.8
12.6
14-4
'7
3-4
5-t
6.8
8.5
IO.3
II.9
13-6
I.O
2.0
3
4.0
5-
6.0
7-
8.0
0.9
2.7
3.6
45
5-4
6-3
7.2
8 i
102
36 3O
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
77439
9 . S6 92 1
26
o. 1 3 079
9.90 5i8
30
3i
9
77456
17
9.86 947
27
o. i3 o53
9 . 90 5og
2 9
32
9
7?4?3
9.86974
26
o. i3 026
9.90499
28
33
9
77490
9.87 ooo
o. 1 3 ooo
9.90 490
9
27
17
27
IO
34
9
77607
17
9.87 027
26
o. 12 973
9.90 480
26
35
9
77 524
9.87053
26
o. 12 947
9.90471
2D
36
9
77 54i
9.87079
O. 12 921
9.90 462
y
24
17
27
10
37
9
77558
9.87 106
26
0.12 894
9.90 452
23
38
9
77 5 ? 5
9.87 i32
26
0,12 868
9.90443
22
3 9
9
77 592
9.87 1 58
0.12 842
9.90434
9
21
40
9
77 609
9.87 185
26
o. 12 8i5
9.90424
20
4i
9
77626
9.87 21 I
O. 12 789
9.90415
9
'9
42
9
77 643
9.87238
O. 12 762
9.90 4o5
18
43
9
77 660
'7
9.87 264
O. 12 736
9.90 396
y
17
17
26
10
44
9
77677
9.87 290
O. 12 710
9.90386
16
45
9
77 6g4
9 .8 7 3i 7
0.12 683
9.90377
i5
46
9
77711
'7
9.87 343
O. 12 667
9.90368
9
i4
17
26
IO
47
9
77728
16
9.87 369
o. 12 63i
9. 90 358
i3
48
9
77 744
9.87 396
o. 12 6o4
9.90 349
12
49
9
77 ?6t
'7
9 .87 422
O. 12 578
9.90 339
I I
50
9
77 77 8
'7
9. 87 448
O. 12 552
9.90 33o
9
10
5i
9
77 79 5
17
9.87475
27
26
O. 12 525
9.90 32o
9
52
9
77812
9.87 5oi
O. 12 499
9.90 3 1 1
8
53
9
77 829
17
9.87 527
O. 12 4?3
9.90 3oi
7
54
9
77846
'7
16
9.8 7 554
27
26
o. 12 446
9.90 292
9
6
55
9
77 862
9.87 58o
O. 12 42O
9.90 282
5
56
9
.77879
'7
9.87 606
o. 12 3g4
9.90 273
9
4
5?
9
. 77 896
'7
9.87633
27
26
O. 12 367
9.90 263
IO
3
58
9
.77913
'7
9.87 65$
0.12 34 I
9.90 254
2
5 9
9
. 7 7 9 3o
17
9.87 685
o. 12 315
9.90 244
I
60
9
.77 946
9.87711
O. 12 289
9.90235
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
53.
PP
37
26
17
16
10 g
.1
2.7
2.6
.1
'7
1.6
.1
i.o 0.9
2
5-4
5-2
.2
3-4
3.2
.2
2.O 1.8
3
8.1
M
3
4.8
3
3.0 2.7
4
10.8
10.4
.4
6.8
6.4
4
4-o 3-6
5
'3-5
13.0
5
8.5
8.0
5
5.0 4.5
.6
16.2
15.6
.6
IO. 2
96
.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
15-3 144
90 8.1
io3
37.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
9
77 946
9.87 711
0.12 289
9.90 235
60
I
9
77963
17
9.87 738
26
O. 12 262
9.90 225
9
5 9
2
9
77980
9.87 764
O. 12 236
9.90 21 6
58
3
9
77997
16
9.87790
27
0. 12 2IO
9.90 2O6
p
5?
4
9
.78013
17
9.87817
26
0.12 i83
9.90197
IO
56
5
9
.78 o3o
9.87843
26
0. 12 l5
7
9.90 187
55
6
9
.78 047
16
9.87 869
26
0. 12 l2
i
9.90 178
10
54
7
9
.78 o63
17
9.87895
27
0.12 105
9.90 1 68
9
53
8
9
.78080
9.87 922
0.12 078
9.90 i5g
52
9
9
.78097
16
9.87948
O. 12 o52
9.90 149
5i
10
9
.78 n3
9.87974
O. 12 O26
9.90 i3g
50
ii
9
.78 i3o
'7
9.88 ooo
O. 12 OOO
9.90 i3o
49
12
9
.78 147
16
9.88 027
o.ii 973
9.90 1 20
48
i3
9
.78 i63
9.88o53
o.ii 947
9.90 in
y
47
'7
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. ii 895
9.90 091
45
16
9
.78 2i3
9.88 i3i
o.ii 869
9.90 082
y
44
'7
27
10
17
9
.78 23o
16
9.88 i58
26
O.II 842
9.90 072
43
18
9
.78 246
9.88 i84
0.11 8l6
9.90 o63
42
'9
9
.78 263
9.88 2IO
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
9.88 262
o.ii 7!
53
9.90 o34
y
3 9
22
9
783i3
16
9.88 289
o.ii 711
9.90 024
38
23
9
.78 329
9.88 3i
i
>
o.ii 685
9.90 oi4
37
'7
26
9
24
9
.78346
16
9.8834i
26
o.ii 659
9.90005
36
25
9
. 7 836 2
9.88 367
o.ii 633
9.89995
35
26
9
.78 379
9.88 3 9 ;
i
o.ii 607
9.8
99 85
34
27
9
. 7 83 9 5
17
9.88420
27
26
o.ii 58o
9.89976
9
33
28
9
.78 412
16
9.88446
o.ii 554
9.8
9 966
32
2 9
9
.78428
9.88472
O.II 528
9.8
9 9 56
3i
30
9
.78445
9.88 498
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
.1
2.7
2.6
i
'7
1.6
.1
I.O
0.9
.2
54
5-2
2
3-4
3-2
.2
2.0
j.8
3
7.8
3
4.8
3
3-o
27
4
10.8
10.4
4
6.8
6-4
4
4.0
3.6
5
'3-5
13.0
8.5
8.0
5
5
4 5
.6
16.2
15.6
6
10.2
9.6
.6
6.0
5-4
.7
18.9
18.2
7
II-9
II. 2
7
7.0
63
.8
21.6
20.8
8
I 3 .6
12.8
.8
8.0
7.2
9
24-3 2 3 4
'4-4
9.0 8.1
io4
37 30
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
78445
9.88498
O. I I 5O2
9.89947
30
3i
9
78461
9.88 524
26
0. I I 4?6
9-8c
) 93?
29
32
9
78478
9.88 55o
o. 1 1 450
9-8(
3927
28
33
9-
78494
16
9.88 577
27
26
O. I I 423
9.89 918
y
IO
2 7
34
9-
78 5io
9.88 6o3
26
o. 1 1 397
9.8<
3908
26
35
9
78 527
16
9.88 629
o. 1 1 371
9.89898
25
36
9
78 543
9.88 655
O.I I 345
9.89888
24
17
26
9
37
9-
78 56o
16
9.88 681
26
o. 1 1 319
9.8
? 8 79
23
38
9-
78 576
9.88 707
o. 1 1 293
9.89869
22
3 9
9
78 592
9.88 733
26
o. 1 1 207
9.8
? 85 9
21
40
9
78 609
16
9.88 759
O. I I 24l
9.8
? 84 9
20
4i
9
78 625
9.88 786
O. I I 2l4
9.8
9 84o
y
'9
42
9
78 642
16
9.88 812
26
o.n 188
9.8
983o
18
43
9
78658
16
9.88838
26
o. 1 1 162
9.8
9 820
IO
17
44
9
78674
9.88864
26
o.n i36
9.8
9 810
16
45
9
78 691
16
9.88890
i
o.n no
9.8
9 801
i5
46
9
78 707
9.88 916
0. II 084
9.8
9791
i4
26
IO
47
9
78723
16
9.88 942
26
o.n o58
9.8
9 781
i3
48
9
7 8 7 3 9
9.88 966
1
O. I I O32
9.8
9 77 1
12
4 9
9
78 756
'7
9.88994
26
o. 1 1 006
9.89 761
I I
50
9
78 772
16
9.89 020
26
o.io 980
9.89 752
y
10
5i
9
78 788
17
9.89 o46
27
o. 10 954
9.8
9 742
9
52
9
.78 805
9.89 073
26
o. 10 927
9.8
9 732
8
53
9
78 821
16
9.89099
26
o.io 901
9.89 722
10
7
54
9
78837
16
9.89 125
26
o.io 875
9.8
9712
6
55
9
78853
16
9.89 161
26
o.io 849
9.8
9 702
5
56
9
78 869
9.89 177
O. IO 823
9.8
9 693
4
17
IO
57
9
78886
16
9.89 2o3
26
o. 10 797
9.8
9683
3
58
9
.78 902
9.89 229
26
o.io 771
9.8
9 6 7 3
2
5 9
9
.78 918
9.89 255
26
O.IO 745
9.8
9 663
I
60
9
.78934
9.89 281
o. 10 719
9.8
9 653
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
1.7
1.6
.1
I.O
0.9
.2
5-4
5-2
.2
3-4
3-2
.2
2.O
1.8
3
8.1
7.8
3
5-'
4-8
3
3-0
2.7
4
10.8
10.4
4
6.8
6.4
4
4.0
3-6
'3-5
1 6.2
13.0
iS.6
J
8-5
10.2
8.0
9.6
.6
5-0
6.0
4-5
5-4
7
18.9
18.2
7
II.9
II. 2
7
7.0
6.3
.8
21.6
20.8
.8
I 3 .6
12.8
.8
8.0
7-2
243 23-4
9
'5-3 '4-4
9.0 8.1
io5
38 C
'
L. Sin.
d.
L. Tang.
d.
L.
Cotg.
L. Cos.
d.
9.78 934
16
9.89
28l
26
o.
10 719
9.89
653
60
I
9.78 g5o
17
9.89
307
26
o.
10693
9.89 643
10
5 9
2
9.78 967
16
9.89
333
26
o.
10 667
9.89
633
58
3
9.78 983
16
9.89
35 9
26
o.
10 64 1
9.89
624
IO
*7
4
9.78999
16
9.89
385
26
o.
10615
9.89
6i4
IO
56
5
9.79015
16
9.89
4u
26
o.
10 589
9.89
6o4
55
6
9.79 o3i
16
9.89437
26
0.
10 563
9.89
5 9 4
10
54
7
9.79047
16
9.89
463
26
o.
10 S3?
9.89
584
10
53
8
9.79 o63
9.89
48 9
26
o.
10 5i i
9.89
5 7 4
52
9
9-79 79
9.89
5i5
26
o.
10 485
9.89
564
Si
10
9-79 9 5
16
9.89
54 1
26
o.
10 459
9.89
554
50
1 1
9.79 in
17
9.89
56y
26
o.
10433
9.89
544
IO
49
12
9.79 128
16
9.89
5 9 3
26
o.
10 407
9.89
534
48
i3
9.79 i44
16
9.89
619
26
o.
10 38 1
9.89
524
IO
47
i4
9.79 160
16
9 .8 9
645
26
o.
10 355
9.89
5i4
46
i5
9.79 176
16
9.89
671
26
o.
10 329
9.89
5o4
45
16
9.79 192
16
9 .8 9
697
26
o.
10 3o3
9.89
495
IO
44
'7
9.79 208
16
9.89
7 23
26
o.
10 277
9.89
485
IO
43
18
9.79 224
16
9.89
749
o.
IO 25l
9.89
475
42
[9
9.79 24
o
16
9.89
775
26
o.
10 225
9.89
465
4i
20
9.79 256
16
9.89
801
o.
10 199
9.89
455
40
21
9.79272
16
9.89 827
26
o.
10 173
9.89445
10
3 9
22
9.79 288
16
9.89
853
o.
10 i47
9.89435
38
23
9.79 3o4
9.89879
o.
IO 121
9.89425
37
15
26
24
9.79 3ig
16
9 .8 9
95
26
o.
10 og5
9.89
415
IO
36
25
9.79335
9.89
o.
10 069
9.89
405
35
26
9.79 35i
16
9 .8 9
9 5 7
26
0.
10 o43
9.89395
IO
34
27
9.79367
16
9.89
9 83
26
0.
10 017
9.89
385
IO
33
28
9.79 383
16
9.90 009
0.
09991
9.89
375
32
29
9-79 3 99
16
9.90035
o.
09 965
9.89
364
3i
30
9.79 415
9.90 061
o.
o 99 3 9
9.89
354
30
L. Cos.
d.
L. Cotg.
d.
L.
Tang.
L. Sin. d.
'
513O.
pp 36
17
16
15
n
10
9
.1 26
'7
1.6
.1 i-S
n
.1
I.O
0.9
.2 5.2
3-4
3.2
.2 3.0
2.2
.2
2.O
1.8
3 7-8
5-'
4-8
3 4-5
3-3
3
3.0
2-7
4 '0-4
6.8
6.4
.4 6.0
4-4
4
4.0
3-6
5 '3-0
8.5
8.0
5 75
5-5
5
5-
4-5
.6 15.6
IO. 2
9.6
.6 9.0
6.6
.6
6.0
5-4
.7 18.2
II-9
II. 2
7 '0-5
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
53 M4
9 !;
8.1
106
38 3O'.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.79415
16
9.90 061
o . 09 939
9.89 354
30
3i
9
7943i
16
9.90 086
26
0.09 914
9.8
9344
2 9
32
9
.79447
16
9.90 112
26
0.09 86
8
9.8
9 334
28
33
9
.79 463
IS
9.90 i38
26
0.09 862
9.8
9 324
IO
27
34
9
.79478
16
9.90 1 64
26
0.09836
9.8
9 3i4
26
35
9
.79 4g4
9.90 190
26
0.09 8 10
9.8
9 3o4
25
36
9
.79 5io
16
9.90 216
26
0.09 784
9.8
9 294
IO
24
3 7
9
.79 526
16
9.90 242
26
0.09 758
9.8
9 284
23
38
9
.79 542
9.90 268
26
0.09 782
9.8
9 274
22
3 9
9
.79 558
9.90 294
26
0.09 706
9.8
9 264
81
40
9
79 5 7 3
16
9.90 320
26
0.09 680
9. 89 254
20
4i
9
.79 58 9
16
9.90 346
25
0.09 654
9.8
9 244
'9
42
9
.79605
16
9.90 371
0.09 629
9.8
9233
18
43
9
.79 621
15
9.90397
26
0.09 6o3
9.8
9 223
IO
7
44
9
.79 636
16
9.90 423
26
0.09 577
9.8
9 2i3
16
45
9
. 79 652
16
9.90449
0.09 55i
9.8
9 2o3
i5
46
9
.79 668
16
9.90475
26
0.09 525
9.8
9 193
10
i4
47
9
.79 684
15
9.90 5or
26
0.09 499
9.8
9 i83
i3
48
9
.79699
9.90 527
0.09 4?3
9.8
9 !7 3
12
49
9
79 7 l5
16
9.90 553
0.09447
9.8
9 162
I I
50
9
.79 731
9.90 578
25
26
0.09 422
9.89 i52
10
5i
9
79 746
16
9.90 6o4
26
o . 09 396
9.8
9 i42
9
52
9
.79 762
9.90 63o
26
0.09 370
9.8
9 i3 2
8
53
9
79 778
15
9.90 656
26
0.09 344
9.8
9 122
IO
7
54
9
79 79 3
16
9.90 682
26
o . 09 3 1 8
9.8
9112
6
55
9
.79809
9.90 708
26
0.09 292
9.8
9 ioi
5
56
9
.79825
9.90 734
0.09 266
9.8
9091
4
15
25
IO
5?
9
.79 84o
16
9.90759
26
0.09 24 1
9.8
9 08 1
3
58
9
.79 856
9.90 785
o . 09 2 1 5
9.89 071
2
5 9
9
.79 872
9.90 811
0.09 i
^9
9.8
9 060
I
60
9
.79887
9.90 887
0.09 1 63
9.8
9 o5o
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
51.
PP
26
25
16
15
ii
IO
.1
2.6
2-5
.1
1.6
i. 5
.1
i.i
I.O
.2
5-2
5.0
.2
3-2
3-o
.2
2.2
2.0 .
3
7.8
7-5
3
4.8
4-5
3
3-3
3-
4
10.4
10.0
4
6.4
6.0
4
4-4
4.0
5
13.0
12.5
5
8.0
7-5
5
5-5
5-0
.6
15.6
15.0
.6
9.6
9.0
.6
6.6
6.0
. 7
18.2
17-5
7
II. 2
10.5
7
7-7
7.0
.8
20.8
20. o
.8
12.8
I2.O
.8
8.8
8.0
9
2^.4 '.!2. S
9
14.4 13.5
9
9-9 Q.O
107
!
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
9
.79887
16
9.90 837
26
0.09 i63
9.89 o5o
60
i
9
.79903
15
9.90 863
26
0.09 137
9.8
9 o4o
IO
5 9
2
9
.79918
16
9 . 90 88
;
0.09 in
9.89 o3o
58
3
9
.79934
16
9.90914
26
0.09 086
9.8
9 020
ii
3 7
4
9
.79950
15
9.90 940
26
0.09 060
9.89 009
TO
56
5
9
.79965
16
9.90 966
26
0.09 o34
9.88999
55
6
9
.79981
15
9.90992
26
o . 09 008
9.8
8 989
II
54
7
9
. 79 996
16
9.91 018
25
0.08 9
3 a
9.8
8978
10
53
8
9
.80 012
9.91 o43
26
0.08 957
9.8
8968
02
9
9
.80 O27
16
9.91 069
26
0.08 9
>i
9.8
8958
Si
10
9
.80043
15
9.91 og5
26
0.08 905
9.8
8948
50
1 1
9
.8oo58
16
9.91 121
26
0.08 879
9.88 937
49
12
9
.80 074
9.91 i4'
i
0.08 853
9.8
8927
48
i3
9
.80 089
16
9.91 172
26
0.08 828
9.8
8917
II
4?
t4
9
.80 105
15
9.91 19*
]
26
0.08 802
9.8
8906
46
i5
9
.80 120
16
9.91 224
26
0.08 776
9.8
8896
45
16
9
.80 i36
'5
9.91 250
26
0.08 750
9.8
8886
II
44
17
9
.80 i5i
15
9.91 276
2 5
0.08 724
9.8
88 7 5
IO
43
18
9
.80 166
16
9.91 3oi
26
0.08 699
9.8
8865
42
'9
9
.80 182
9.91 3a-
i
26
0.08 673
9.88 855
4i
20
9
.80 197
16
9.91 353
26
0.08 647
9.8
8844
40
21
9
.80 2i3
15
9.91 379
25
0.08 621
9-88834
3 9
22
9
.80 228
16
9.91 4o4
0.08 5(
fi
9.88*824
38
23
9
.80244
9.91 43o
0.08 570
9-888i3
3?
15
26
10
24
9
.80 25g
15
9.91 456
26
0.08 544
9.8
88o3
36
25
9
.80 274
mtf
9.91 482
0.08 5i8
9.8
8 79 3
35
26
9
.80 290
9.91 507
2 S
0.08 4
f>
9.8
8 782
34
IS
26
IO
2 7
9
.80 3o5
15
9.91 533
26
0.08 467
9.88772
33
28
9
.80 32o
mtt
9.91 55c
1
0.08 44i
9.8
8 761
32
2 9
9
.80 336
9.91 585
0.08 4i5
9.8
8 7 5i
3i
30
.80 35i
9.91 610
'5
o . 08 390
9.8
8 741
30
L. Cos.
d.
L. Cotg.
d. L. Tang.
L.
Sin. d.
'
50 30 .
PP
26
as
it
15
n
IO
.1
2.6
3-5
i
1.6
J-5
.1
i.i
I O
2
5-2
S-o
2
32
3-o
.2
2.2
20
3
7.8
7-5
3
4.8
4-5
3
3-3
3-
4
10.4
1 0.0
4
6.4
6.0
4
4-4
4.0
5
13.0
12 .5
5
8.0
7-5
-5
5 5
5.0
.6
15.6
IS.O
6
9,6
9.0
.6
6.6.
6.0
. 7
1 8.9
'7-5
7
II. 2
10.5
7
77
7.0
.8
20.8
20.
8
12.8
I2.O
.8
8.8
8.0
9
234 22.5
'4-4
'3-5
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
3 9
9
9
9
9
9
9
9
9
9
.80 366
80 382
.80397
. 80 412
80428
80443
80 458
8o4?3
80489
16
15
IS
16
i5
15
15
16
9.91 636
9.91 662
9.91 688
9.91 718
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 1 84
0.^8 1 58
9.88 730
9.88 720
9.88 709
9.88 699
9.88 688
9.88 678
9.88668
9.8865?
9.8864?
10
II
IO
II
10
IO
II
IO
29
28
27
26
25
24
23
22
2 I
40
9
80 5o4
9.91 868
0.08 i32
9.8
3636
20
4i
42
43
44
45
46
4?
48
49
9
9
9
9
9
9
9
9
9
80 5ig
80 534
80 Sgo
.80 565
.8o58o
.SoSgS
.80 610
.80625
.8o64i
IS
15
16
15
IS
iS
IS
15
16
9.91 893
9.91 919
9.91 945
9.91 971
9.91 996
9.92 022
9.92 o48
9.92 073
9.92099
0.08 107
0.08 081
0.08 o55
0.08 029
0.08 oo4
0.07 978
0.07 952
0.07 927
0.07 901
9.88 626
9-886i5
9^88605
9.88 594
9.88584
9.88573
9.88563
9.88552
9.88 542
II
IO
II
10
II
IO
11
IO
II
10
II
II
IO
II
10
II
10
II
II
19
18
i?
16
i5
i4
i3
12
I I
50
9
.80 656
15
'5
15
15
15
15
IS
16
IS
'S
9.92 125
0.07 875
9.8
8 53:
10
5i
52
53
54
55
56
5?
58
5 9
9
9
9
9
9
9
9
9
9
.80671
.80686
.80 701
.80 716
.80731
.80746
.80 762
.80 777
.80 792
9.92 i5o
9.92 176
9.92 202
9.92 227
9.92 253
9.92 279
9.92 3o4
9.92 33o
9.92 356
0.07 850
0.07 824
0.07 798
0.07 773
0.07 747
0.07 721
0.07 696
0.07 670
0.07 644
9.88 52i
9.88 5io
9.88499
9.88 489
9.88478
9.88468
9.88457
9-88447
9.88436
9
8
7
6
5
4
3
2
I
60
9
.80807
IS
9.92 38 1
0.07 619
9.88425
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
/
5O.
PP
.1
.2
3
4
.6
.8
9
26
5
16
15
.1
.2
3
4
:!
.B
9
ii
IO
2.6
5-2
7.8
10.4
13.0
156
18.2
20.8
23.4
2.5
5-o
7-5
IO.O
12.5
15-0
'7-S
20.0
22.5
.i
.2
3
4
.6
'.S
9
1.6
3- 2
4.8
6.4
8.0
9.6
1 1.2
12.8
14.4
'5
3-
45
6.0
7-5
9.0
10.5
12.
13-5
i.i
2.2
3-3
4-4
1.1
i!fi
9-9
1.0
2.0
3-0
4.0
5-o
6.0
7.0
8.0
9-
109
40.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
9.80 807
15
15
15
15
15
15
15
15
'5
'S
15
>5
% 15
15
'5
15
14
15
'5
'S
IS
. iS
IS
IS
14
15
15
IS
IS
14
9.92 38i
26
26
23
26
26
25
26
26
25
26
25
26
26
25
26
26
25
26
25
26
26
23
26
25
26
26
25
26
25
26
0.07 619
9.88425
IO
II
10
II
II
IO
II
II
10
II
II
10
II
II
IO
II
II
10
II
GO
I
2
3
4
5
6
7
8
9
9.80822
9.80837
9.80 852
9.80867
9.80882
9.80 897
9.80 912
9.80 927
9.80 942
9.92 407
9.92433
9.92 458
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
0.07 465
0.07 43g
0.07 4i3
0.07 388
9.88415
9-884o4
9.88 3 9 4
9.88383
9.88 372
9.88 362
9-8835i
9.88 34o
9.88 33o
5 9
58
57
56
55
54
53
52
5i
10
9
.80 957
9.92 638
0.07 362
9.8
8 3i 9
50
1 1
12
i3
i4
i5
16
17
18
1 9
9
9
9
9
9
9
9
9
9
.80 972
.80987
.81 002
.81 017
.81 o32
.81 047
.81 061
.81 076
.81 091
9.92 663
9.92 689
9.92 715
9.92 740
9.92 766
9.92 792
9.92 817
9.92 843
9.92 868
0.07 337
0.07 3n
0.07 285
0.07 260
0.07 234
0.07 208
0.07 i83
0.07 167
0.07 i32
9.88 3o8
9.88 298
9.88 287
9.88276
9.88 266
9.88 255
9.88244
9.88 234
9.88 223
49
48
4?
46
45
44
43
42
4i
20
9
.81 106
9.92 894
0.07 1 06
9.8
8 212
40
21
22
23
24
25
26
27
28
2 9
9
9
9
9
9
9
9
9
9
.8l 121
.81 i36
.81 i5i
.81 166
.81 180
.81 196
.8l 2IO
.8( 225
. 81 240
9.92 920
9.92 945
9.92 971
9.92996
9 .93 022
9-93o48
9.93 073
9.93099
9.93 124
0.07 080
0.07 oSg
0.07 029
0.07 oo4
0.06 978
0.06 952
0.06 927
0.06 901
0.06 876
9.88 2OI
9-88 igi
9.88 I 80
9.88 169
9.88 i58
9.88 i48
9.88 137
9.88 126
9.88 u5
10
II
II
II
10
II
II
II
3 9
38
3 7
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.
'
49 3O .
PP
.1
.2
3
4
!e
:2
9
36
25
15
14
.1
.2
3
4
.6
.8
9
ii
10
2.6
5-2
7-8
10.4
13.0
'5-6
18.2
20.8
23.4
2.5
5-0
7-5
IO.O
12-5
15-0
'7-5
2O.O
22.5
.1
.2
3
4
:I
.'s
i-S
3-o
4-5
6.0
7-5
9-
io-5
12.
'3-5
i-4
2.8
4-2
56
7.0
8.4
9.8
II. 2
12.6
i.i
2.2
3-3
4-4
5-5
6.6
H
0.9
1.0
2.0
3-
4-0
5-o
6.0
7.0
8.0
4O 3O .
/
L. Sin. d.
L. Tang. d. i L. Cotg.
Cos.
d.
ii
ii
ii
ii
IO
II
II
II
II
II
II
IO
II
II
II
II
II
II
II
30
9
.81 254
15
'5
'5
'5
14
15
15
M
15
15
9.93 150
25
26
26
25
26
25
26
25
26
26
25
26
25
26
25
26
25
26
26
25
26
25
26
25
26
25
26
25
26
25
0.06 85o
9.88 105
30
3i
33
34
35
36
3?
38
3 9
9
9
9
9
9
9
9
9
9
. 8 1 269
.81 284
.81 299
.81 3i4
.81 328
.81 343
.81 358
.81 3y2
.81 38 7
9.93 175
9.93 2OI
9.93 227
9.93 252
9.93 278
9.93 3o3
9.93 3ag
9.93 354
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 9 4
9.88 o83
9.88 072
9.88 06 1
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
.8l 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 46i
.81475
.81 490
.81 505
.81 5i 9
.81 534
M
15
15
M
15
15
M
15
9.93 43i
9.93 457
9.93482
9.93 5o8
9.93 533
9.93 55g
9 . 9 3 584
9.93 610
9.93 636
0.06 569
0.06 543
0.06 5i8
0.06 492
0.06 467
0.06 44i
0.06 4i6
0.06 390
0.06 364
9.87985
9- 8 7975
9.87 964
9.87953
9.87 942
9.87 9 3i
9.87 920
9.87 909
9.87 898
19
1 8
17
16
i5
i4
i3
12
I I
50
9
.81 549
9 .93 661
0.06 339
9.87 887
10
5i
52
53
54
55
56
5?
58
5 9
9
9
9
9
9
9
9
9
9
.81 563
.81 5 7 8
.81 592
.81 607
.81 622
.81 636
.81 65i
.81 665
.81 680
5
14
15
15
'4
15
'4
15
9.93 687
9.93 712
9.93 738
9.93 763
9 . 9 3 789
9.93 8i4
9.98 84o
9.93 865
9.93 891
0.06 3i3
0.06288
0.06 262
0.06 237
O.O6 21 I
O.O6 l86
O.O6 I 60
0.06 i35
0.06 109
9.87877
9.87 866
9.87 855
9-87844
9.87 833
9.87 822
9.87 811
9.87 800
9.87 789
II
II
II
II
II
II
II
II
9
8
7
6
5
4
3
2
I
60
9
.81 6g4
9.93 916
0.06 084
9.87 778
L. Cos. d.
L. Cotg. | d.
L. Tang.
L.
Sin.
d.
f
49.
PP
.1
.2
3
4
.6
.8
26
25
5
M
.1
.2
3
4
.6
9
ii
10
2.6
52
7.8
10.4
13.0
15-6
l8.2
20.8
23.4
2-5
5-0
75
IO.O
12 5
15.0
'7-5
20.0
.1
.2
3
4
.6
.8
-0
i-S
3.0
4-5
6.0
7-5
9.0
10.5
I2.O
'3-5
1.4
2.8
4.2
5-6
7.0
8.4
9.8
II. 2
i.i
2.2
3-3
44
5-5
6.6
11
9-9
1.0
2.0
3-
4.0
5-
6.0
7.0
8.0
0.0
41.
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
9.81 694
9.93 916
26
0.06 o84
9.87 77 8
60
I
9.81 709
14
9.93 942
25
0.06 o58
9.87 767
5 Q
2
9.81 723
9.93967
26
0.06 o33
9.87 756
58
3
9.81 738
9.93993
0.06 007
9.87745
57
14
25
ii
4
9.81 762
is
9.94 018
26
o.o5 982
9-87734
56
5
9.81 767
9.94044
0.05966
9.87 723
55
6
9.81 781
15
9.94069
26
o.o5 931
9.87 712
ii
54
7
9.81 796
14
9.94095
25
o.o5 905
9.87 701
53
8
9.81 810
9.94 120
26
o.o5 880
9.87 690
52
9
9.8! 825
9.94 1 46
o.o5854
9.87679
5i
10
9.81 83g
15
9.94 17'
26
o.o5 829
9.87 668
50
ii
9.81 854
14
9.94 197
25
o.o5 8o3
9 .8 7 65 7
ii
49
12
9.81 868
9.94 222
26
o.o5 778
9.87 646
48
i3
9.81 882
15
9.94248
25
o.o5 752
9.87635
ii
4?
i4
9.81 897
*4
9.94 273
26
o.o5 727
9.87624
ii
46
i5
9.81 911
9.94299
o.o5 701
9.87613
45
16
9.81 926
M
9.94 324
26
o.o5 676
9.87 601
ii
44
17
9.81 940
15
9.94 350
25
o.o5 65o
9.87 590
ii
43
18
9.81955
9 . 9 43 7 5
26
o.o5 625
9.87579
42
J 9
9.81 969
9.94401
o.o5 599
9. 87 568
4i
20
9.81 983
9.94426
26
o.o5 574
^.87.557
40
21
9
.81 998
14
9.944^2
25
o.o5 548
9.87 546
ii
39
22
9
.82 OI2
9.94477
26
o.o5 523
9.87535
38
23
9
.82 O26
9.94 5o3
o.o5 497
9.87 524
37
15
25
24
9
.82o4l
14
9.94 528
26
o.o5 472
9.87513
12
36
25
9
.82 055
9.94 554
o.o5 446
9.87 5oi
35
26
9
.82 069
9.94579
o.o5 421
9.87490
34
15
2 5
27
9
.82084
14
9.94604
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
9.94 655
26
o.o5 345
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 3O .
PP
36
*5
5
M
12
II
.x
2.6
25
.1
i-5
fc l
.1
1.2
I.I
.2
5-2
5-o
.3
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
10.
4
6.0
5.6
4
4.8
4-4
5
13.0
12-5
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
ill
'75
. 7
10. s
9.8
7
8.4
7-7
.8
20.8
20. o
/ -8
I2.O
1 1. 2
.8
9.6
8.8
'?
2^.4 22.5
9
13.5 12.6
9
10.8 9.9
41 3O .
'
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
30
9.82 126
9.94 68 1
o.o5 319
9.87446
30
3i
9.82 i4i
9.94 706
26
o.o5 294
9.87434
29
32
9.82 155
9.94 73
2
o.o5 268
9.87 423
28
33
9
.82 169
9.94757
o.o5 243
9.87 412
27
15
34
9
.82 1 84
14
9.94783
25
o.o5 217
9.87 4oi
26
35
9
.82 198
9 . 9 4 80
I
o.o5 192
9.87 390
25
36
9
.82 212
9. 9 4834
o.o5 166
9.87 378
24
14
25
II
37
9
.82 226
14
9.94 85g
25
o.o5 i
4i
9.87 367
23
38
9
.82 240
9. 9 4884
o.o5 116
9.87 356
22
3 9
9
.82 255
9.94 910
o.o5 090
9.87345
II
21
40
9
.82 26 9
9.94 935
26
o.o5 065
9.87 334
20
4i
9
.82283
'4
9.94 961
2 5
o.o5 o3g
9.87 322
i 9
42
9
.82 297
9.94 986
26
o.o5 oi4
9.87 3n
1 8
43
9
.82 3n
9.95 012
o.o4 9
38
9.87 3oo
7
'5
25
12
44
9
.82 326
14
9- 9 5 o3
7
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
4?
9
.82 368
9.95 1 13
26
o.o4 887
9.87 255
i3
48
9
.82 382
9 . 9 5 i3<
)
o.o486i
9.87 243
12
49
9
.82 396
14
9.95 i64
25
26
o.o4 836
9.87 232
I I
50
9.82 4io
J 4
9.95 190
o . o4 8 1 o
9.87 221
10
5i
9
.82424
15
9.95 2i5
2 5
o.o4 785
9.87 209
9
52
9
.82439
9.95 240
26
o.o4 760
9.87 I 9 8
8
53
9
.82453
9 .95 266
o.o4 734
9.87187
7
14
25
12
54
9
.82467
9.95 291
26
o.o4 709
9.87 175
6
55
q
.82481
9- 9 5 317
o.o4 683
9.87 i64
5
56
9
.82 495
'4
14
9.95 34s
i
26
o.o4658
9.87 i53
12
4
5 7
9
.82 .609
9.95 368
2 5
o.o4 632
9.87 i4i
3
58
9
.82 523
9.95 3g!
\
o.o4 607
9.87 i3o
2
5 9
9
.82 537
'4
9 . 9 5 4i
\
26
o.o4 582
9.87 119
I
60
9
.82 55i
14
9.95 444
o.o4 556
9.87 107
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
48.
PP
26
*5
15
14
12
ii
.1
2.6
2-5
x
i-5
1.4
.1
1.2
i.i
.2
5-2
2
3.0
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
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
'7-5
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
9
23-4
9
13.5 12.6
9
10 8
9-9
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos. d.
9
.82 55i
14
14
14
M
'4
14
H
'4
M
9.95 444
25
26
25
25
26
25
26
25
*5
26
25
25
26
25
26
25
25
26
25
26
25
25
26
25
25
26
25
26
25
25
o.o4556
9.87 107
ii
ii
12
II
12
II
II
12
II
60
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
.82 565
.82 579
.82 5g3
.82 607
.82 621
.82635
.82 64g
.82 663
.82 677
9.95 46y
9.95495
g.gS 52O
9.95 545
9.95 571
9.95 596
9.95 622
9.95 64?
9.95 672
o.o4 53i
o.o4 5o5
o.o4 48o
o.o4 455
o.o4 429
o.o4 4o4
0.04378
o.o4353
0.04 328
9.87 096
9.87 085
9.87073
9.87 062
9.87 o5o
9.87 o3g
9.87028
9.87 016
9.87005
5 9
58
5 7
56
55
54
53
52
Si
10
9
.82 691
J 4
M
'4
14
M
H
14
13
14
14
9.95 698
o.o4 3o2
9.86993
50
1 1
12
i3
i4
i5
16
i?
18
1 9
9
9
9
9
9
9
9
9
9
.82 705
.82 719
.82 733
.82 7 4?
.82 761
.82775
.82 788
.82 802
.82 816
9.95 723
9.95748
9-9 5 774
9.95799
9.95825
9.95 85o
9.96 8y5
9.95 901
9.95 926
0.04 277
O.o4 252
o.o4 226
o.o4 20 1
o.o4 i?5
o.o4 150
o.o4 125
o.o4 099
o.o4 074
9.86 982
9.86 970
9.86 959
9.86 947
9.86936
9.86 924
9.86 gi3
9.86 902
9.86890
12
II
13
II
12
II
II
12
49
48
4?
46
45
44
43
42
4i
20
9
.82 83o
14
9.95 952
o.o4 o48
9.86 879
40
21
22
23
24
25
26
2 7
28
2 9
9
9
9
9
9
9
9
9
9
.82 844
.82858
.82 872
.82886
.82899
.82 913
.82 927
.82 g4i
.82955
14
'4
*3
'4
14
14
'4
'4
9.95977
9.96 002
9.96 028
9.96 o53
9.96 078
9.96 io4
9.96 129
9.96 165
9.96 1 80
o.o4 O23
o.o3 998
o.o3 972
o.o3 947
o.o3 922
0.08896
o.o3 871
o.o3845
o.o3 820
9.86 867
9.86855
9-86844
9.86 832
9.86 821
9.86809
9.86798
9.86 786
9.86 775
12
II
12
II
13
II
12
II
3 9
38
37
36
35
34
33
32
3i
30
9
.82 968
J 3
9 .96 2o5
o.o3 795
9.86 763
30
L. Cos. d.
L. Cotg.
d.
L. Tang.
L.
Sin.
d.
'
47 30'.
PP
.1
.2
3
4
.6
: 8 7
9
36
as
M
13
.1
.2
3
4
.6
!s
9
13
ii
2.6
5 "o
7.8
10.4
13.0
15.6
18.2
20.8
-: 4
2-5
5
75
10.
12.5
15.0
'7-5
20. o
i
2
3
4
.6
'.B
9
;i
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
ii. 7
1.2
2-4
3-6
4-8
6.0
72
8-4
9.6
10.8
i.i
2.2
33
4-4
5-5
6.6
7-7
8.8
42 30
L. Sin.
d.
L. Tang.
d.
L. Cotg.
Cos.
d.
30
9
.82 968
9 .96 2o5
26
o.o3 795
9.86 763
30
3i
9
.82 982
J 4
9.96 23l
2 5
o.o3 769
9.86 762
29
32
9
.82 996
9.96 256
o.o3 744
9.86 740
28
33
9
.83 oio
13
9.96 281
26
o.o3 719
9.86.728
II
27
34
9
83023
14
9.96 307
25
o.o3 693
9.8
6 717
26
35
9
83 o3y
9.96 332
o.o3 668
9.8
6 705
2D
36
9
83 o5i
14
9.96 357
26
o.o3 643
9.86 694
12
24
3?
9
83o65
13
9.96 383
25
o.o3 617
9.86 682
23
38
9
83078
9 .96 4o8
o.o3 592
9.86 670
22
3 9
9
83 092
9.96433
26
o.o3 567
9.86 659
21
40
9
83 106
9.96 459
o.o3 54i
9.86647
20
*i
9
83 120
13
9. 96 484
26
o.o3 5i6
9.8
6635
J 9
42
9
83 i33
9.96 5io
o.o3 490
9.86624
18
43
9
83 i4y
9.96 535
o.o3 465
9.8
6612
17
14
25
12
44
9
83 161
13
9.96 56o
26
o.o3 44o
9.8
6 600
16
45
9
83 i 7 4
9.96 586
o.o34i4
9.86 58g
i5
46
9
83 188
9.96 6 1 1
25
o.o3 389
9.86577
i4
14
2 5
12
47
9
83 202
13
9.96 636
26
o.o3 364
g.86565
i3
48
9
83 2i5
9.96 662
o.o3 338
g.86554
12
49
9
83 229
9.96 687
25
o.o3 3i3
9.86 542
I I
50
9
83 242
9.96 712
o.o3 288
9.8
6 53o
10
5i
9
83 2 56
9.96 738
o.o3 262
9.8
65i8
9
52
9
832yo
9.96 763
o.o3 237
9.8
6 507
8
53
9
83283
13
9.96 788
25
o.o3 212
9.86495
7
14
26
12
54
9
83297
9.96 8i4
o.o3 186
9.86483
6
55
9
.833io
9.96 83g
o.o3 161
9.86 472
5
56
9
.83324
M
9.96864
25
o.o3 i36
9.86460
4
14
26
12
5?
9
.83338
9.96 8gc
ft
o.o3 1 10
9.86448
3
58
9
.83 35i
9.96915
o.o3o85
9-86436
2
5 9
9
.83 365
14
9.96 940
25
o.o3 060
9.86 425
I
60
9
.833-8
13
9.96 966
o.o3 o34
9.864i3
L. Cos.
d.
L. Cotg. d.
L. Tang.
L.
Sin.
d.
'
47.
PP
26
25
M
13
12 II
.1
2.6
2-5
.1
1.4
'3
.1
1.2 I.I
.2
5-2
5-0
.2
2.8
2.6
.2
2.4 2.2
3
7.8
7-5
3
4.2
3-9
3
3- 6 3-3
-4
10.4
10.
4
5.6
5-2
4
4.8 4.4
5
13.0
12.5
5
7.0
6-5
5
6.0 5.5
.6
15.6
15.0
.6
8.4
7.8
.6
7.2 6.6
7
18.2
'7-5
7
9.8
9.1
7
8.4 7.7
.8
20.8
20. o
8
II. 2
10.4
.8
9.6 8.8
9
23.4 22.5
9
12.6 11.7
9
jo. 8 9.9
n5
43.
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
9
.83 3 7 8
9.96 966
25
S
26
25
25
26
25
25
25
26
25
25
26
2 S
25
26
25
25
26
25
25
26
25
25
25
26
25
25
26
25
o.o3 o34
9.864i3
12
12
12
II
12
12
12
12
12
II
12
12
12
12
12
12
12
II
12
12
12
12
12
12
12
12
12
12
12
60
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
.83 392
.834o5
.83419
.83432
.83446'
.83459
.834?3
.83486
.83 500
>4
'3
M
13
M
13
'4
'3
'4
9.96991
9.97 016
9.97 o4a
9.97067
9.97092
9.97 118
9.97 i43
9.97 168
9-97 i9 3
o.o3 009
0.02 984
0.02 958
0.02 933
O.O2 908
0.02 882
0.02 857
O.O2 832
O.O2 807
9.86 4oi
9.86 389
9.86377
9.86366
9-86354
9.86 342
9.86 33o
9-863i8
9.86 3o6
5 9
58
57
56
55
54
53
52
5i
10
9
.83 5i3
'3
9.97 219
O.O2 781
9.86 295
50
1 1
12
i3
i4
i5
16
17
18
'9
9
9
9
9
9
9
9
9
9
.83 52 7
.8354o
.83554
.83 56 7
.83 58i
.83 5g4
.83 608
.83 621
.83634
13
'4
'3
M
'3
'4
'3
'3
9.97 244
9.97 269
9.97295
9.97 32o
9.97345
9-97371
9.97 3 9 6
9.97 421
9.97447
0.02 756
0.02 73i
0.02 705
O.O2 680
O.O2 65g
O.O2 629
0.02 6o4
O.O2 579
0.02 553
9.86283
9.86 271
9.86 259
9.86 247
9.86 235
9.86 223
9.86 211
9.86 2OO
9.86 188
4 9
48
47
46
45
44
43
42
4i
20
9
.83 648
'4
9.97472
0.02 528
9.86 176
40
21
22
23
24
25
26
27
28
2 9
9
9
9
9
9
9
9
9
9
.83 661
.83674
.83688
.83 701
.83715
.83 728
.83 7 4i
.83 755
.83 768
'3
'4
'3
14
'3
'3
14
3
9.97497
9.97 523
9.97 548
9.97573
9.97598
9.97 624
9.97649
9.97 674
9.97 700
0.02 5o3
O.O2 477
0.02 452
O.O2 427
O.O2 4O2
O.O2 376
0.02 35i
0.02 326
0.02 3oo
9.86 1 64
9.86 i52
9.86 i4o
9.86 128
9.86 116
9.86 104
9.86 092
9.86 080
9.86068
3 9
38
37
36
35
34
33
32
3r
30
9
.83 781
9-97 7 2 5
O.O2 275
9.86 o56
30
L. Cos.
d.
L. Cotg.
d. L. Tang.
L.
Sin.
d.
'
46 3O .
PP
.1
.2
3
4
!e
i
g
26
5
14
3
.1
.2
3
4
'.6
.8
13
II
2.6
5-2
7.8
10.4
13.0
,5.6
18.2
20.8
23.4
a- 5
S-o
7-5
IO.O
"5
15-0
'7-5
20. o
.1
.2
3
4
:I
.8
1.4
2.8
4.2
5.6
1
8.4
9.8
II. 2
.2.6
i-3
2.6
3-9
5-2
M
7.8
9.1
10.4
it. 7
1.2
2.4
3-6
4.8
6.0
7.2
8.4
*
I.I
2.2
3-3
4-4
55
6.6
7-7
8.8
116
4.3 3O
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9
83 781
9-97 7 2 5
O.O2 275
9-
86 o56
30
3 1
9
83 795
'4
9.97 750
25
26
O.O2 250
9-
86o44
29
32
9
83 808
9.97 776
O.O2 224
9-
86 o32
28
33
9
83 821
'3
9.97 801
25
O.O2 199
9-
86 020
27
13
25
12
34
9
83834
9.97 826
O.O2 I 74
9-
86008
26
35
9
83848
9.97 85i
0.02 149
9-
85 996
25
36
9
83 861
13
9.97877
O.O2 123
9-
85 984
24
J 3
2 5
12
^7
9
838 7 4
9.97902
O.O2 098
9
85 972
23
38
9
83 887
9.97 927
O.O2 O73
9
85 960
22
39
9
83 901
9.97 9 53
0.02 047
9
85 9 48
21
40
9
83 yi4
9-97 97 8
O.O2 O22
9
85 9 36
20
4i
9
S3 927
9.98 oo3
2 5
26
o.oi 997
9
85 924
'9
42
9
83 940
9.98 029
o.oi 971
9
85 912
18
43
9
83 954
14
9.98 o54
25
o.oi 946
9
85 900
17
13
25
12
44
9
83967
9.9
8079
o.oi 921
9-
85888
16
45
9
83 980
9.9
8 io4
o.oi 896
9-
85 876
i5
46
9
83 993
'3
9 .98 i3o
o.oi 870
9
85864
i4
47
9
84 006
'3
9.9
8 i55
25
o.oi 845
9-
8585i
i3
48
9
84 020
9.98 180
o.oi 820
9
85 839
12
49
9
84o33
'3
9.9
8 206
o.oi 794
9
85 827
1 I
50
9
84 o46
9.9
8 2 3i
o.oi 769
9
85 8i5
10
Si
9
84 o5g
13
9.98 256
o.oi 744
9
858o3
9
52
9
84 072
9.98 281
o.oi 719
9
85 791
8
53
9
84o85
13
9-9
8 307
o.oi 6g3
9
85 779
7
54
9
84 098
13
9.9
8 332
25
o.oi 668
9
85 766
13
6
55
9
.84 112
9 .98 357
o.oi 643
9
85 7 54
5
56
9
.84 125
'3
9.9
8 383
o.oi 617
9
85 742
4
13
25
12
57
9
.84 i38
9-9
84o8
o.oi 592
9
85 730
3
58
9
.84 i5i
9-9
8433
o.oi 567
9
85 718
2
5 9
9
.84 164
13
9.9
8458
2 S
26
O.OI 542
9
85 706
I
60
9
.84 177
9.98 484
o.oi 5i6
9
85 6 9 3
L. Cos. ; d.
L. Cotg. d.
L. Tang.
L. Sin. d.
"
46.
PP
26
55
M
13
12
.1
2.6
2-5
.1
1.4
'3
.1
1.2
.2
5-2
.2
2.8
2.6
.2
2-4
3
7.8
7-5
3
4-2
3-9
3
3-6
4
10.4
IO.O
4
5.6
5-2
4
4 .8
5
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. o
.8
II. 2
10.4
.8
9.6
23.4 22.5
.9 12.6
TI-7
9
10.8
117
44.
/
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos. d.
9
.84 177
'3
'3
'3
'3
'3
'3
M
'3
'3
13
'3
13
'3
'3
'3
12
'3
'3
'3
'3
'3
'3
'3
13
13
'3
'3
12
'3
'3
9. 98 484
25
25
26
25
25
25
26
25
25
26
25
25
23
26
25
25
25
26
25
25
26
25
25
25
26
25
25
25
26
25
o.oi 5i6
9
85 6y3
12
12
12
12
'3
12
12
12
13
12
12
12
'3
la
12
13
12
12
'3
12
12
'3
12
12
13
12
3
12
12
'3
GO
I
2
3
4
5
6
7
8
9
9
9
9
9
9
9
9
9
9
.84 190
.84 2o3
.84216
.84 229
.84242
.84255
.84 269
.84282
.84295
9.98 5og
9.98 534
9.98 56o
9.98 585
9.98 610
9.98 635
9.98 661
9.98 686
9.98 711
o.oi 491
o.oi 466
o.oi 44"
o.oi 4t5
o.oi 390
o.oi 365
o.oi 33g
o.oi 3i4
o.oi 289
9
9
9
9
9
9
9
9
9
8568i
85 669
8565 7
85 645
85 632
85 620
85 608
85 5 9 6
85 583
5y
58
5?
56
55
54
53
52
5i
10
9
.84 3o8
9.98 737
o.oi 263
9
85 5 7 i
50
1 1
12
i3
i4
i5
16
'7
18
9
9
9
9
9
9
9
9
9
9
.84 32i
.84334
.84347
.8436o
.843 7 3
.84 385
.84398
.844n
.84424
9.98 762
9.98 787
9.98 812
9.98 838
9.98 863
9.98888
9.98 gi3
9.98939
9.98 964
o.oi 238
O.OI 2l3
o.oi 188
o.oi 162
o.oi 1 37
O.OI 112
o.oi 087
o.oi 06 1
o.oi o36
9
9
9
9
9
9
9
9
9
85 SSg
85 547
85 534
.85 522
.85 5io
.85497
.85485
.854?3
.85 46o
4 9
48
4?
46
45
44
43
42
4i
20
9
.8443 7
9.9
8 989
O.OI OI I
9
.85448
40
21
22
23
24
25
26
27
28
2 9
9
9
9
9
9
9
9
9
9
.8445o
.84463
.84476
.8448 9
.84 5o2
.845i5
.84 528
.8454o
.84 553
9.99015
9.99 o4o
9.99 o65
9.99090
9.99 i i 6
9.99 i4i
9.99 166
9.99 191
9.99 217
o.oo 985
o.oo 960
o.oo 935
o . oo 9 1 o
o.oo 884
o.oo 85g
o.oo834
o.oo 809
o.oo 783
9
9
9
9
9
9
9
9
9
.85 436
.85423
.85 4n
.85 399
.85 386
.85 3 7 4
.85 36i
.85 349
.85 33 7
3 9
38
37
36
35
34
33
32
Q,
30
9
.84 566
9.99 242
o.oo 758
9
.85 324
30
L. Cos. d.
L. Cotg.
d. L. Tang.
L. Sin. d.
<
45 3O .
PP
.1
.2
3
4
!e
: 7 8
9
26
5
.1
.2
3
4
'.6
.8
9
M
13
.1
.2
3
4
5
.6
.8
9
13
2.6
5-2
7-8
10.4
"3-?
15.6
i8.z
20.8
23.4
2.5
S-o
7-5
IO.O
'2-5
15.0
17-5
20. o
22.5
'4
2.8
4-2
5.6
7.0
8.4
9.8
II. 2
.2.6
1:1
3-9
5-2
6.5
7.8
9-i
10.4
11.7
1.2
2-4
3-6
4.8
6.0
7-2
8 -*
9-6
10.8
118
44 3O
I
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30
9.84566
9.99 242
o.oo 758
9
.85 324
30
3i
9.84579
13
9.99267
26
o.oo 733
9
.853i2
29
32
9.84 592
9 99293
o.oo 707
9
.85 299
28
33
9.84605
9. 99 3i8
o.oo 682
9
.85 287
27
'3
13
34
9.84618
12
9.99843
25
o.oo 667
9
.85 274
26
35
9.8463o
9.99 368
26
o.oo 632
9
.85 262
25
36
g.84643
9-99 3 94
o.oo 606
9
.85 250
24
13
25
13
3?
g.84656
9.99419
25
o.oo 58 1
9
.85 237
23
38
9.84669
9.99 444
o.oo 556
9
.85 225
22
3 9
9-84 682
J 3
9.99469
26
o.oo 53i
9
.85 212
3
21
40
9 .846 9 4
9.99495
o.oo 5o5
9
.85 200
20
4i
9.84 707
'3
9.99 620
25
o.oo 48o
9
.85 187
$
42
9.84 720
9.99 545
o.oo 455
9
.85 175
18
43
9
.84 ?33
12
9.99 5 7 o
26
o.oo 43o
9
.85 162
'j
12
17
44
9.34745
'3
9.99696
o.oo 4o4
9
.85 150
16
45
9.84768
9.99 621
o.oo 379
9
.85 i3 7
i5
46
9.84 771
9.99 646
o.oo 354
9
.85i25
i4
'3
26
13
4?
9.84784
12
9.99672
O.OO 328
9
.85 112
i3
48
9
.84796
9.99697
o.oo 3o3
9
.85 100
12
49
9
.84809
9.99 722
25
o.oo 278
9
.85 087
'3
I I
50
9
.84 822
9.99 7 4?
25
26
o.oo 253
9
.85 074
10
5i
9
.84835
12
9.99773
25
o.oo 227
9
.85 062
13
9
52
9
.8484?
9-99 79 8
O.OO 2O2
9
.85 049
8
53
9
.8486o
9.99828
o.oo 177
9
.85 037
7
13
25
*3
54
9
.84873
9.99848
26
o.oo i5a
9
.85024
6
55
9
.84885
9.99874
o.oo 126
9
.85oi2
5
56
9
.84898
'3
9.99899
O.OO IOI
9
84 999
'3
4
J 3
25
J 3
5 7
9
.84911
9-999 2 4
o.oo 076
9
.84986
3
58
9
.84 923
9.99949
o.oo o5i
9
.84974
2
5 9
9
.84 9 36
'3
9-99975
o.oo 026
9
.84961
13
I
60
9
.84949
o.oo ooo
25
o . oo ooo
9
.84949
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
/
45.
PP
26
25
14
'3
12
.1
2.6
2-5
.1
1.4
'3
.1
1.2
.2
5.2
5-
.2
2.8
2.6
.2
2.4
3
7.8
7-5
3
4.2
3-9
3
3-6
4
10.4
IO.O
4
5.6
5-2
. -4
4.8
S
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
208
20. o
.8
II. 2
10.4
.8
9.6
9
23.4 22.5
9 12.6
11.7
.9 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 <JO TO 89
52' ; TO EVERY TEN SECONDS FROM 90 TO 88.
FUNCTIONS OF SMALL ANGLES.
LOGARITHMIC SINE AND TANGENT.
0"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
o
10
20
5-68557
98660
68557
72697
1.00779
98660
76476
4,02800
799b2
*0473
,28763 $38454 #46373
83170, 861671 88969
,06579 ,08351 4,10055
91602
,11694
,58866
94085
'3273
,63982
96433
*'4797
*68 5 57
98660
50
40
3
30
4
5
6. 16270
28763
38454
17694
29836
39315
19072
30882
40158
20409
31904
40985
21705
32903
4'797
22964
33879
42594
24188
34833
43376
25378
35767
44M5
26536
36682
44900
27664
37577
45643
28763
38454
46373
20
IO
o 59
1 o
IO
20
6.46373
6. 5 3067
8866
7090
9406
7797
4291
9939
8492
4890
4,0465
9'75
#0985
9849
6064
*'499
4,0512
6639
4,2007
7207
* 2 509
4,1808
7767
4,3006
8320
4,3496
*8866
5
40
30
3
40
50
6.63982
8557
6. 7 2697
4462
8090
3090
4936
9418
3479
5406
9841
3865
587
4,0261
4248
6330
4,0676
4627
6785
5003
7235
4,1496
5376
7680
4,1900
5746
8121
4,2300
6112
8557
4.2697
6476
20
IO
o 58
2 o
10
20
6476
9952
6.83170
6836
4,0285
3479
7'93
*o6ig
3786
7548
*943
4091
7900
4,1268
4394
8248
*'59'
4694
8595
4,1911
4993
8938
4,2230
5289
9278
*2545
5584
9616
4.2859
5876
* 9952
6^7
5
4
3
3
40
50
6167
8969
6.9 1602
6455
9240
'857
6742
9509
2IIO
7027
9776
2362
73 10
4,0042
2612
759'
4,0306
2861
7870
4,0568
3'9
8'47
4,0829
3355
8423
4,1088
3599
8697
*38
8969
4,1602
4085
20
10
57
3 o
10
20
4085
6433
8660
4325
6661
8877
4565
6888
9093
4803
7113
9307
5039
7338
9520
5275
7561
9733
5509
7783
9944
5742
8004
4.0155
5973
8224
6204
8443
*572
6433
8660
*0779
5
4
3
40
50
7.00779
2800
4730
0986
2997
49'9
1191
3'93
5106
'395
3388
5293
'599
3582
5479
1801
3776
5664
2003
3968
5849
2203
4160
6032
2403
435'
6215
2602
4541
6397
2800
473
6579
2O
IO
o 56
4 o
IO
20
6579
8351
7- 1 0055
6759
8525
0222
6939
8698
0388
7118
8870
553
7296
9041
0718
7474
9211
0882
765.
1046
7827
955'
1209
8003
9719
J37'
8177
9887
533
835 '_
1694
5
40
30
30
40
50
1694
3273
4797
1854
3428
4947
2014
3582
5096
2174
3736
5244
2333
3889
5392
2491
4042
554
' 2648
4194
5687
2805
4346
5833
2962
4497
5979
3"8
4647
6125
3273
4797
6270
20
IO
o 55
5 o
IO
20
7.1 6270
7694
9072
6414
7834
9208
6558
7973
9343
6702
8112
9478
6845
8250
9612
6987
8389
9746
7130
8526
9879
7271
8663
4,0012
74'3
8800
4,0145
7553
8937
*277
7694
9072
4,0409
50
40
3
30
40
5
7. 2 0409
1705
2964
0540
>833
3088
0671
1960
3212
0802
2087
3335
0932
2213
3458
1062
2339
358o
1191
2465
3702
1320
2590
3824
'449
2715
3946
'577
2840
4067
1705
2964
4188
20
10
o 54
6 o
10
20
4l88
5378
6536
4308
5495
6650
44*8
5612
6764
4548
5728
6877
4668
584!
6991
4787
59 6 '
7104
496
6076
7216
5024
6192
7329
5 '42
6307
744'
5260
6421
7552
5378
6536
7664
50
40
3
30
40
50
7664
8763
9836
7775
8872
9942
7886
8980
*47
7997
9088
8107
9196
4,0257
8217
933
4,0362
8327
9410
4,0467
8437
95'7
*57'
8546
9623
8655
9730
4,0779
8763
9836
4,0882
20
IO
o 53
7 o
IO
20
7.30882
1904
2903
0986
2005
3001
1089
2106
3100
1191
2206
3198
1294
2306
3296
1396
2406
3393
1498
2506
349'
1600
2606
3S88
1702
2705
3685
1803
2804
3782
1904
2903
3879
5
40
3
3
40
5"
3879
483?
576?
3975
4928
5860
4071
5022
5952
4167
56
6044
4263
5209
6i35
4359
5303
6227
4454
63^8
4549
5489
6409
4644
5582
6500
4739
5675
6591
4833
5767
6682
20
10
o 52
10"
9"
8"
7"
6"
5"
4"
3"
2"
1"
0"
"
LOGARITHMIC COSINE AND COTANGENT.
89 C
FUNCTIONS OP SMALL ANGLES.
L. Sin.
L. Tang.
,
L. Sin.
L. Tang.
o
IO
20
3o
4o
5o
5.68 55 7
5.98 660
6.16 270
6.28 7 63
6.38454
5.6855 7
5.98 660
6. 16 270
6.28763
6.38454
o 60
5o
4o
3o
20
IO
73o
4o
5o
7.33879
7-34833
7.35 767
7-33 879
7-34833
7.35767
3o
20
IO
8 o
IO
20
3o
4o
5o
7.36 682
7.37577
7.38454
7.39 3i4
7-4o i58
7.40985
7.36 b2
7 .3 7 5 77
7.38455
7-393I5
7-4o i5S
7.40985
o 52
5o
4o
3o
20
10
1 o
10
20
3o
4o
5o
6.46 373
6.53 067
6.58866
6.63 982
6.68 55?
6. 72 697
6.46 3 7 3
6.53 067
6.58866
6. 63 982
6.68557
6.72 697
o 59
5o
4o
3o
20
IO
9 o
IO
20
3o
4o
5o
7-4i 797
7.42 594
7.433 7 6
7.44 i4s
7.44900
7-45643
7.41 797
7.42 594
7.433 7 6
7-44 i45
7-44 900
7-45643
o 51
5o
4o
3o
20
IO
2 o
10
20
3o
4o
5o
6.76 4 7 6
6 79 g52
6.83 170
6.86 167
6.88 969
6.91 602
6.76476
6.79 952
6.83 170
6.86 167
6.88969
6.91 602
o 58
5o
4o
3o
20
IO
10 o
10
20
3o
4o
5o
7.46373
7.47 090
7.47797
7-48491
7-49 176
7.49 84g
7 .463 7 3
7 -4 7 091
7-47797
7.48492
7 .49 i 7 6
7.49 849
o 50
5o
4o
3o
20
IO
3 o
IO
20
3o
4o
5o
6.94 085
6.96433
6.98 660
7.00779
7.02 800
7.04 780
6.94 085
6.96433
6.98 660
7.00779
7.02 800
7.04 73o
o 57
5o
4o
3o
20
IO
11 o
IO
20
3o
4o
5o
7.5o 5i2
7-5i 165
7.5i 808
7.52 44a
7.53 067
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
IO
20
3o
4o
5o
7.06 379
7.08 35i
7.10 o5<5
7.11 6g4
7-i3 2 7 3
7.14 797
7.06 579
7.08 352
7. 10 oSg
7.11 694
7. i3 2 7 3
7.14 797
o 56
5o
4o
3o
20
IO
12 o
IO
20
3o
4o
5o
7.54 291
7. 54 890
7.5548i
7-56 o64
7.5663 9
7.57 206
7-54 291
7 .54 890
7.5548i
7-56 o64
7 .5663 9
7.57 207
o 48
5o
4o
3o
20
IO
5 o
IO
20
3o
4o
5o
7.16 270
7.17 6g4
7.19 072
7.20 409
7.21 705
7.22 964
7. 16 270
7.17 6g4
7.19073
7.20 409
7.21 70 r >
7.22 964
o 55
5o
4o
3o
20
10
13 o
IO
20
3o
4o
5o
7.57 767
7-58 320
7-58866
7.59 4o6
7.59939
7.60 465
7.57 767
7.58 320
7 .5886 7
7.59 4o6
7.59939
7.60 466
o 47
5o
4o
3o
20
IO
6 o
IO
20
3o
4o
5o
7.24 188
7.25 378
7.26 536
7.27 664
7.28 763
7.29 836
7.24 188
7.25 378
7.26536
7.87 664
7.28 764
7.29 836
o 54
5o
4o
3o
20
IO
14 o
IO
20
3o
4o
5o
7.60 985
7.61 499
7.62 007
7.62 5og
7.63 006
7-(3 4g6
7 .6o 986
7 .6i 500
7.62 008
7 .62 5io
7 .63 006
7 .63 497
o 46
5o
4o
3o
20
10
7 o
10
20
3o
7.3o 882
7. 3 1 904
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.
L. Cos.
L. Cotg.
" '
12 3
89.
FUNCTIONS OF SMALL ANGLES.
0.
, n
L. Sin.
L. Tang.
,
L. Sin.
L. Tang.
15 o
10
20
7. (53 982
7-6446i
7.64936
7.63 982
7.64 462
7.64 937
o 45
5o
4o
22 3<>
4o
5o
7.81 5gi
7.81 911
7.82 229
7.81 5yi
7.81 912
7.82 23o
3o
20
10
3o
4o
5o
7.654o6
7.65 870
7.66 33o
7-65 4o6
7.65 871
7.66 33o
3o
20
10
23 o
IO
20
7.82545
7.82 35g
7-83 170
7.82 546
7.82 860
7.83 171
o 37
5o
4o
16 o
10
20
7.66 784
7.67235
7.67680
7.66785
7.67 235
7.67 680
o 44
5o
4o
3o
4o
5o
7-83479
7.83786
7.84 091.
7.83480
7. 83 787
7 .84 092
3o
20
IO
3o
4o
5o
7.68 121
7.68557
7.68989
7.68 121
7.68558
7.68 990
3o
20
to
24 o
IO
20
7.84 3 9 3
7-846 9 4
7-84 992
7.843 9 4
7.84695
7.84 993
o 36
5o
4o
17 o
10
20
7.69 47
7.69841
7.70 26l
7.69 4i8
7.69 842
7.70 261
o 43
5o
4o
3o
4o
5o
7.85 289
7.85583
7.85876
7-85 290
7.85 584
7. 85 877
3o
20
IO
3o
4o
5o
7.70 676
7.71 088
7.71 4g6
7.70677
7.71 088
7.71 496
3o
20
10
25 o
10
20
7.86 166
7.86455
7 86 741
7.86 167
7.86456
7.86743
o 35
5o
4o
18 o
10
20
7.71 gOO
7.72 3oo
7.72697
7.71 900
7.72 3oi
7.72697
o 42
5o
4o
3o
4o
5o
7.87 026
7.87 309
7 .87 Sgo
7.87 027
7.87 3io
7.87591
3o
20
IO
3o
4o
5o
7. 73 090
7. 7 3479
7-73865
7. 73 090
7.73 48o
7.73866
3o
20
IO
26 o
10
20
7.87 870
7.88 i4?
7.88423
7.87871
7.88 1 48
7.88424
o 34
5o
4o
19 o
10
20
7.74 248
7.74 627
7.75 oo3
7.74 248
7.74 628
7.75 oo4
o 41
5o
4o
3o
4o
5o
7.88697
7.88969
7.89 240
7.88698
7.88 970
7.89 24 1
3o
20
10
3o
4o
5o
7 . 7 53 7 6
7.75745
7.76 112
7.75377
7.75 746
7.76 u3
3o
29
IO
27 o
IO
20
7.89 509
7.89776
7.90 o4 1
7.89 5io
7.89777
7.90 o43
o 33
5o
4o
20 o
10
20
7.76475
7-76836
7-77 i9 3
7.76 476
7.76837
7.77 i 9 4
o 40
5o
4o
3o
4o
5o
7.90 3o5
7.90 568
7.90 829
7.90 307
7 .90 569
7.90 83o
3o
20
IO
3o
4o
5o
7-77548
7.77899
7.78248
7-77 5 49
7.77900
7.78 249
3o
20
IO
28 o
10
20
7.91 088
7.91 346
7.91 602
7.91 089
7.91 34?
7.91 6o3
o 32
5o
4o
21 o
10
20
7.78 5 9 4
7.789^
7.79278
7.78 5g5
7.78938
7.79279
o 39
5o
4o
3o
4o
5o
7.91 857
7.92 1 10
7.92 362
7.91 858
7.92 iii
7.92 363
3o
20
IO
3o
4o
5o
7.79 6 1 6
7.79952
7.80284
7.79617
7.79952
7.80 285
3o
20
10
29 o
10
20
7.92 612
7.92 861
7.93 108
7.92 61 3
7.92 862
7.93 1 10
o 31
5o
4o
22 o
10
20
7.80615
7.80 g4a
7.81 268
7.8o6i5
7.80943
7.81 269
o 38
5o
4o
3o
4o
5o
7.93354
7.93 5gg
7.93 842
7.93 356
7.93 601
7-93844
3o
20
10
3o
7.81 5 9 i
7.81 5gt
3o37
30 o
7-<;4 o84
7.94 086
o 30
L; COS.
L. Cotg.
" '
L. Cos.
L. Cotg.
" '
124
FUNCTIONS OF SMALL ANGLES.
0.
, '
L. Sin.
L. Tang.
,
L. Sin.
L. Tang.
30 o
10
20
3o
4o
5o
7.94 o84
7.94325
7.94 564
7.94 802
7.95 o3g
7.96 274
7.94 086
7.94 326
7.94 566
7.94 8o4
7 .gS o4o
7.95 276
o 30
5o
4o
3o
20
10
37 3o
4o
5o
8.o3 775
8.o3 967
8.o4 i Sg
8 .o3 777
8 .o3 970
8.o4 162
3o
20
10
38 o
10
20
3o
4o
5o
8.o435o
8.o454o
8.o4 729
8.04918
8.o5 io5
8.o5 292
8.o4 353
8.o4543
8.o4 732
8.o4 921
8.o5 1 08
8.o5 295
o 22
5o
4o
3o
20
10
31 o
10
20
3o
4o
5o
7 .96 5o8
7-9 5 ?4i
7.95^73
7.96 2o3
7.96 432
7.96 660
7.95 5io
7.95 743
7.95974
7.96 205
7.96434
7 .96 662
o 29
5o
4o
3o
20
10
39 o
10
20
3o
4o
5o
8.o5 478
8.o5 663
8.o5848
8.o6o3i
8.06 214
8.06 396
8.o5 48 1
8.o5666
8.o585i
8.o6o34
8.06 217
3.o6 399
o 21
5o
4o
3o
20
IO
32 o
10
20
3o
4o
5o
7.96 887
7-97 n3
7.97 33 7
7.97 56o
7 .97 782
7.98 oo3
7.96 889
7.97 ii'4
7.97339
7.97 562
7.97784
7.98 oo5
o 28
5o
4o
3o
20
10
40 o
10
20
3o
4o
5o
8.06578
8.06 758
8.06938
8.07 117
8.07 295
8.07473
8.06 58i
8.06 761
8.06 94 1
8.07 120
8.07 298
8.07 476
o 20
5o
4o
3o
20
10
33 o
10
20
3o
4o
5o
7.98 ^23
7.98 442
7.98 660
7.98 876
7.99092
7 .99 3o6
7 .98 225
7.98444
7.98 662
7.98878
7.99094
7.99 3o8
o 27
5o
4o
3o
20
10
41 o
10
20
3o
4o
5o
8.07 650
8.07826
8.08 002
8.08 176
8.08 35o
8.08 524
8. 07 653
8.07 829
8.08 005
8.08 180
8.08 354
8.08 527
o 19
5o
4o
3o
20
10
34 o
10
20
3o
4o
5o
7 .99 620
7.99 732
7.99943
8.00 i54
8.00 363
8.00 671
7.99 522
7.99 7 34
7.99946
8.00 1 56
8.oo365
8.00 574
o 26
5o
4o
3o
20
10
42 o
10
20
3o
4o
5o
8.08696
8.08868
8 . 09 o4o
8.09 210
8.09 38o
8.09 550
8.08 700
8.08 872
8.09043
8.09 214
8.09 384
8.09553
o 18
5o
4o
3o
20
IO
35 o
10
20
3o
4o
5o
8.00 779
8.00 985
8.01 190
8.01 395
8.01 698
8.01 801
8.00 781
8.00 987
8.01 193
8.01 397
8.01 600
8.01 8o3
o 25
5o
4o
3o
20
JO
43 o
10
20
3o
4o
5o
8.09 718
8.09886
8.ioo54
8. 10 220
8.io386
8.10 552
8.09 722
8.09 890
8. 10 067
8. 10 224
8. 10 3go
8.io555
o 17
5o
4o
3o
20
10
36 o
10
20
3o
4o
5o
8 .02 002
8.02 203
8. 02 402
8. 02 601
8. 02 799
8. 02 996
8.02 oo4
8. 02 2o5
8. 02 405
8. 02 6o4
8.02801
8.02 998
o 24
5o
4o
3o
20
10
44 o
IO
20
3o
4o
5o
8.10717
8.10881
8.11 o44
8. ii 207
8. ii 370
8. ii 53i
8. 10 720
8.io884
8 . 1 1 o48
8. II 211
8. ii 373
8. ii 535
o 16
5o
4o
3o
20
IO
37 o
10
20
3o
8.o3 192
8.o3 387
8.o3 58i
8.o3 775
8.o3 194
8.o3 3go
8.o3584
8 .o3 777
o 23
5o
4o
3o22
45 o
8. 1 1 6g3
8. 1 1 696
3 15
L. Cos.
L. Cotg.
" '
L. Cos.
L. Cotg.
" '
125
89.
FUNCTIONS OF SMALL ANGLES.
0.
,
L. Sin.
L.Tang.
,
L. Sin.
L. Tang.
45 o
10
20
8. ii 693
8. ii 853
8. 12 oi3
8. 1 1 696
8. ii 85?
8.12 017
o 15
5o
4o
523o
4o
5o
8.18 ^87
8.i8524
8.18662
8.18 392
8.18 53o
8.18 667
3o
20
IO
3o
4o
5o
8.12 172
8.12 33i
8.12 489
8.12 176
8.12335
8.12493
3o
20
10
53 o
IO
20
8.18 798
8.18935
8.19 071
8.18 8o4
8.18940
8.19 076
o 7
5o
4o
46 o
IO
20
8.12647
8.12 8o4
8.12 961
8. 12 65i
8.12 808
8.12 965
o 14
5o
4o
3o
4o
5o
8.19 206
8.19 34i
8.19 4?6
8.19 212
8.19 347
8.I948J
3o
20
10
3o
4o
5o
8.i3 117
8.i3 272
8.13427
8.i3 121
8.i3 276
8.i343i
3o
20
IO
54 o
IO
20
8.19610
8.19 744
8.19877
8.19 bib
8.19 749
8.19 883
o 6
5o
4o
47 o
10
20
8.i3 58i
8.i3 7 35
8.i3888
8.i3 56
8.i3 739
8.13892
o 13
5o
4o
3o
4o
5o
8. 20 oio
8.20 i43
8. 20 275
8.20 016
8 . 20 i4c;
8.20 281
3o
20
10
3o
4o
5o
8.i4o4i
8.i4 193
8.i4344
8.i4o45
8.i4 197
8.i4348
3o
20
IO
55 o
IO
20
8. 20 407
8.20538
8.20 669
8.2o4i3
8.20 544
8.20675
o 5
5o
4o
48 o
10
20
8.14495
8.i4646
8. i4 796
8. i4 500
8.i465o
8.i48oo
o 12
5o
4o
3o
4o
5o
8. 20 800
8. 20 g3o
8.21 060
8.20806
8.20936
8.21 066
3o
20
IO
3o
4o
5o
8.14945
8. 1 5 094
8.i5 243
8.14950
8. i5 099
8.i5 247
3o
20
IO
56 o
10
20
8.21 189
8.21 319
8.21 447
8.21 195
8.21 324
8.21 453
o 4
5o
4o
49 o
IO
20
8.i5 391
8.i5 538
8.i5685
8.i5 3 9 5
8.i5 543
8. 1 5 690
o 11
5o
4o
3o
4o
5o
8.21 576
8.21 703
8.21 83i
8.21 58i
8.21 709
8.21 83 7
3o
20
IO
3o
4o
5o
8.15832
8.i5 978
8.l6 123
8.15836
8.15982
8.16 128
3o
20
IO
57 o
IO
20
8.21 958
8.22085
8.22 211
8.21 964
8.22 091
8.22 217
o 3
5o
4o
50 o
10
20
8.16 268
8.i64t3
8.t655 7
8.16273
8.16417
8.i656i
o 10
5o
4o
3o
4o
5o
8.22 33 7
8.22463
8.22 588
8.22 343
8.22 469
8.22 595
3o
20
IO
3o
4o
5o
8.16 700
8.i6843
8.16986
8.16 705
8.16848
8.16 991
3o
20
IO
58 o
10
20
8.22 713
8.22838
8.22 962
8.22 720
8.22 844
8.22968
o 2
5o
4o
51 o
IO
20
8.17 128
8.17 270
8.17411
8.17 i33
8.17275
8.17416
o 9
5o
4o
3o
4o
5o
8. 23 086
8.23 2IO
8.23333
8.23 092
8.23 216
8.23 33 9
3o
20
IO
3o
4o
5o
8.i 7 55 2
8.17 692
8.i 7 83 2
8.17 557
8.17 697
8.i 7 83 7
3o
20
10
59 o
IO
20
8. 23456
8.23578
8.23 700
8.23462
8.23 585
8.23 707
o 1
5o
4o
52 o
10
20
8.17971
8. 18 no
8.18 249
8.17 976
8.18 n5
8.18 254
o 8
5o
4o
3o
4o
5o
8.23 822
8.23 944
8.24065
8.23829
8.23950
8.24 071
3o
20
10
3o
8.18 887
8.18 392
3o 7
60 o
8.24 186
8.24 192
o
L. Cos.
L. Cot jf.
" '
L. Cos.
L. Cotg.
" '
126
89.
FUNCTIONS OP SMALL ANGLES.
1.
/ //
L. Sin.
L. Tang.
, n
L. Sin.
L.Tang.
o
IO
20
8.24 186
8.24 3o6
8.24426
8.24 192
8.243i3
8.24433
o 60
5o
4o
1 60
4o
5o
8 . 29 3oo
8.29 407
8.29 5i4
8.29 3og
8.29 4'6
8.29 523
So
20
IO
3o
4o
5o
8.24546
8.24665
8.24 785
8.24553
8.24 672
8.24 791
3o
20
IO
8 o
IO
20
8.29 621
8.29 727
8.29 833
8.29 629
8.29 736
8.29842
o 52
5o
4o
1 o
IO
20
8.24 go3
8.25 022
8.25 i4o
8.24 910
8.25 029
8.25 147
o 59
5o
4o
3o
4o
5o
8.29 939
8.3oo44
8 .3o 150
8.29 947
8.3oo53
8.3o i58
3o
20
IO
3o
4o
5o
8. 25 2 58
8.25 3y5
8.25 4g3
8.25 265
8.25 38 2
8. 25 500
3o
20
IO
9 o
IO
20
8.3o 255
8.3o359
8.3o464
8.30263
8.3o368
8.3o473
o 51
5o
4o
2 o
IO
20
8.25 609
8.25 726
8.25 842
8.25 616
8.25 7 33
8.25849
o 58
5o
4o
3o
4o
5o
8.3o568
8.30672
8.3o 776
8.3o577
8.3o68i
8.30785
3o
20
10
3o
4o
5o
8.25958
8.26 074
8.26 189
8.25965
8.26081
8.26 196
3o
20
IO
10 o
IO
20
8.30879
8.3o 983
8.3i 086
8.3o888
8.3o 992
8.3i 095
o 50
5o
4o
3 o
IO
20
8.263o4
8.26 419
8.26533
8.26 3l2
8.26426
8.2654i
o 57
So
4o
3o
4o
5o
8.3i 188
8.3i 291
8.3i 3g3
8.3i 198
8 . 3 1 3oo
8.3i 4o3
3o
20
IO
3o
4o
5o
8.26648
8.26761
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
8.3i 809
8.3i 911
8. 32 OI2
3o
20
IO
3o
4o
5o
8.27 326
8.27438
8.27 Sgo
8.27334
8.27446
8. 27 558
3o
20
IO
12 o
IO
20
8.3a io3
8.32 2o3
8.32 3o3
8.32 112
8.32 2i3
8.32 3i3
o 48
5o
4o
5 o
10
20
8.27 Obi
8.27773
8. 27 883
8.27 669
8.27 780
8.27891
o 55
5o
4o
3o
4o
5o
8.324o3
8.32 5o3
8.32602
8.324i3
8.325i3
8.32 612
3o
20
IO
3o
4o
5o
8.27 994
8.28 io4
8.28 215
8.28 002
8.28 112
8.28223
3o
20
IO
13 o
10
20
8.32 702
8.32801
8.32899
8.32 711
8.32 811
8.32 909
o 47
5o
4o
6 o
IO
20
8.28324
8.28434
8.28 543
8.28 332
8.28442
8.2855i
o 54
5o
4o
3o
4o
5o
8.32998
8.33 096
8.33 195
8. 33 008
8.33 106
8.33 205
3o
20
IO
3o
4o
5o
8.28652
8.28 761
8.28869
8.28660
8.28 769
8.28 877
3o
20
10
14 o
IO
20
8.33 292
8.33 3go
8.33488
8.33 3o2
8.334oo
8.33498
o 46
5o
4o
7 o
IO
20
8.28 977
8.29085
8.29 198
8.28986
8.29 094
8.29 2OI
o 53
5o
4o
3o
4o
5o
8.33585
8.33682
8.33779
8.335 9 5
8.33692
8.33789
3o
20
IO
3o
8.29 3oo
8.29 3og
3o52
15 o
L338-
8.33886
o 45
L. Cos.
L. Cotg.
" '
L. Cos.
L. Cotg.
" '
127
88.
FUNCTIONS OP SMALL ANGLES.
1.
,
L. Sin.
L. Tang.
,
L.Sin.
L.Tang.
15 i>
10
20
3o
4o
5o
8.33 876
8.33 972
8.34o68
8.34 i64
8.34260
8.34355
8.33 88b
8.33982
8.34078
8.34 174
8.34270
8.34366
o 45
5o
4o
3o
20
10
223
4o
5o
8.38 oi4
8.38 101
8.38 189
8.38 026
8.38 n4
8.38 202
3o
20
10
23 o
10
20
3o
4o
5o
8.38 276
8.38363
8.3845o
8.3853 7
8.38624
8.38710
8.38 289
8.383 7 6
8.38463
8.38650
8.38 636
8.38 723
o 37
5o
4o
3o
20
10
16 o
10
20
3o
4o
5o
8.3445o
8.34546
8.3464o
8.34735
8.3483o
8.34924
8.3446i
8.34556
8.3465i
8. 34 ?46
8.3484o
8.34935
o 44
5o
4o
3o
20
IO
24 o
10
20
3o
4o
5o
8.38 796
8.38882
8. 38 968
8.39054
8.39 1 39
8.3 9 225
8.38 809
8. 38 8 9 5
8.38 981
8.39 067
8.39 1 53
8.3 9 238
o 36
5o
4o
3o
20
10
17 o
IO
20
3o
4o
5o
8.35oi8
8.35 112
8.35 206
8. 35 299
8.35 392
8.3545
8.35 029
8.35 123
8.352i 7
8.35 3io
8.354o3
8.35497
o 43
5o
4o
3o
20
10
25 o
10
20
3o
4o
5o
8.3 9 3io
8.3g 395
8.39480
8.39 565
8. 3 9 649
8.3 97 34
8.39 323
8.39408
8.39493
8.39578
8.3 9 663
8.3 97 47
o 35
5o
4o
3o
20
IO
18 o
10
20
3o
4o
5o
8.35 578
8.35671
8.35764
8.35856
8.35 9 48
8.36o4o
8.35 590
8.35682
8.35 77 5
8.35867
8.35959
8.36o5i
o 42
5o
4o
3o
20
10
26 o
10
20
3o
4o
5o
8.39 818
8.3g 902
8.39986
8 .4o 070
8.4o i53
8.40237
8.39832
8.39916
8.4o ooo
8.4oo83
8.4o 167
8.4o25i
o 34
5o
4o
3o
20
IO
19 o
10
20
3o
4o
5o
8.36 i3i
8.36223
8.363i4
8.364o5
8.36496
8.36587
8.36 i43
8.36235
8.36326
8.36417
8.36 5o8
8.36 599
o 41
5o
4o
3o
20
IO
27 o
10
20
3o
4o
5o
8.4o 3ao
8.4o4o3
8.4o486
8.4o569
8.4o65i
8.4o 734
8.4o334
8.4o 417
8.4o 500
8.4o583
8.4o665
8.40748
o 33
5o
4o
3o
20
IO
20 o
10
20
3o
4o
5o
8. 36 678
8. 36 768
8.36858
8.36948
8.37038
8.37 128
8.36 689
8.36780
8.36870
8.36960
8.37050
8.37 i4o
o 40
5o
4o
3o
20
10
28 o
IO
20
3o
4o
5o
8.40816
8.40898
8.40980
8.4i 062
8.4i i44
8.4i 225
8.4o83o
8.40913
8.40995
8.4i 077
8.4i i58
8.4i 240
o 32
5o
4o
3o
20
10
21 o
10
20
3o
4o
5o
8.37 217
8.37306
8.37395
8.37484
8.37573
8.37662
8.37229
8.37 3i8
8.37408
8. 3 7 497
8.3 7 585
8.37674
o 39
5o
4o
3o
20
10
29 o
IO
20
3o
4o
5o
8.4i 307
8.4i 388
8.4i 469
8.4i 55o
8.4i 63i
8.4t 711
8.4i 32i
8.4i 4o3
8.4i484
8.4i 565
8.4i 646
8.4i 726
o 31
5o
4o
3o
20
IO
22 o
10
20
3o
8.37 750
8.37838
8.37 926
8.38oi4
8.37762
8.3 7 85o
8.37938
8.38026
o 38
5o
4o
3o37
30 o
8.4i 792
8.4i 807
o 30
. L. Cos.
L. Cotg.
" '
L. Cos.
L. Cotg.
' "
128
88.
FUNCTIONS OF SMALL ANGLES.
1.
, n
L. Sin.
L.Tang.
i n
L. Sin.
L.Tang.
30 o
10
20
8.4i 792
8.4i 872
8.4i 962
8.4i 807
8.4i 887
8.4i 967
o 30
5o
4o
373o
4o
5o
8.45 267
8.45 34i
8.454i5
8.45 2a5
8.45 35 9
8.45433
3o
20
10
3o
4o
5o
8.42032
8.42 112
8.42 192
8.42 o48
8.42 J2"
8.42 207
3o
20
10
38 o
JO
20
8.45489
8.45563
8.45637
8.45 So?
8.4558i
8.45655
o 22
5o
4o
31 o
10
20
8.42 272
8.4235i
8.4243o
a. 42 207
8.42 366
8.42446
o 29
5o
4o
3o
4o
5o
8.45 710
8.45 7 84
8.4585 7
8.45 728
8.45 802
8.45875
3o
20
10
3o
4o
5o
8.42 5io
8.42689
8.42667
8.42 525
8.42 6o4
8.42683
3o
20
10
39 o
JO
20
8.45930
8.46oo3
8.46 076
a. 45 948
8.46 O2I
8.46094
o 21
5o
4o
32 o
JO
20
8.42 746
8.42 825
8.42 903
8.42 762
8.42 84o
8 .42 919
o 28
5o
4o
3o
4o
5o
8.46 149
8.46 222
8.46294
8.46 167
8.4624o
8.46 3i2
3o
20
10
3o
4o
5o
8.42 982
8. 43 060
8.43 i38
8.42 997
8.43075
8.43 1 54
3o
20
10
40 o
JO
20
a. 46 366
8.4643 9
8.465ii
8.46385
8.4645 7
8.46629
o 20
5o
4o
33 o
10
20
a. 43 2iti
8.43 2 9 3
8.43 871
8.43232
8.43 309
8.43387
o 27
5o
4o
3o
4o
5o
8.46583
8.46655
8.46 727
8.46602
8.46674
8.46745
3o
20
JO
3o
4o
5o
8.43448
8.43 626
8.436o3
8.43464
8.43 542
8.436i 9
3o
20
JO
41 o
JO
20
a. 46 799
8.46870
8.46942
8.46817
8.46889
8.46960
o 19
5o
4o
34 o
JO
20
a. 4-1 6ao
8.43 7 5 7
8.43834
a. 43 6y6
8.43 773
8.4385o
o 26
5o
4o
3o
4o
5o
8.47 oi3
8.47084
8.47 1 55
8.4? o32
8.47 io3
8.4? i ?4
3o
20
JO
3o
4o
5o
8 .43 910
8.43987
8.44o63
8.43 927
8.44oo3
8. 44 080
3o
20
10
42 o
10
20
a.4 1 ; 226
8.47 297
8.47368
8.47245
8.473i6
8. 4? 387
o 18
5o
4o
35 o
10
20
8.44 i3g
8.442i6
8.44 292
8.44 i56
8.44232
8.443o8
o 25
5o
4o
3o
4o
5o
8.47439
8.47 5og
8.47 58o
8. 4? 458
8. 4? 528
8.47 599
3o
20
JO
3o
4o
5o
8.44367
8.44443
8.445iQ
8.44384
8.4446o
8.44536
3o
20
JO
43 o
10
20
8.4? 650
8.4? 720
8.4? 790
8.47669
8.4? ?4o
8.47810
o 17
5o
4o
36 o
10
20
a. 445 9 4
8.44669
8.44745
8.446ii
8.44686
8.44 762
o 24
5o
4o
3o
4o
5o
8.47860
8.4? 93o
8.48ooo
8.47880
8.4795
8.48020
3o
20
JO
3o
4o
5o
8.44820
8.44895
8.44969
8.44837
8.44 912
8.44987
3o
20
JO
44 o
JO
20
8.48069
8.48 1 39
8.48208
8.48090
8.48 i5 9
8.48 228
o 16
5o
4o
37 o
10
20
8.45o44
8.45 119
8.45 193
8.45o6i
8.45 i36
8.45 210
o 23
5o
4o
3o
4o
5o
8.48 278
8.4834?
8.484i6
8.48298
8.48 367
8.48436
3o
20
10
3o
8.45267
8.45 285
3o22
45 o
8.48485
8.48 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
IO
20
3o
4o
5o
S. 48 485
8.48554
8.48622
8.48691
8.48 760
8.48828
8.48 5o5
8.48 574
8.48643
8.48 711
8.48 780
8.48849
o 15
5o
4o
3o
20
IO
~o~14~
5o
4o
3o
20
10
52 : J ><>
4o
5o
8.5i 48o
8.5i 544
8. 5 1 609
8.5i 5o3
8.5i 568
8.5i 632
3o
20
IO
53 o
10
20
3o
4o
5o
8.5i 673
8.5i 737
8.5i 801
8.5i 864
8.5i 928
8.5i 992
8.5i 696
8.5i 760
8.5i 824
8.5i 888
8.5i 9 52
8.52 oi5
7
5o
4o
3o
20
IO
^~ir
5o
4o
3o
20
IO
46 o
10
20
3o
4o
5o
8.48896
8.48965
8.49033
8.4g 101
8.49 169
8.49236
8.48917
8.48985
8.49053
8.49 121
8.49 189
8.49 257
54 o
IO
20
3o
4o
5o
8.52 o55
8.52 119
8.52 182
8.52 245
8.52 3o8
8.52 371
8 . 52 079
8.52 i43
8.52 206
8.52 269
8.5 2 332
8.52 3 9 6
47 o
JO
20
3o
4o
5o
8.4g 3o4
8.49372
8.49439
8.4g 5o6
8.49574
8.49641
8.49 325
8.49 3g3
8.49460
8.49528
8.49595
8.49662
o 13
5o
4o
3o
20
IO
~^~i2
5o
4o
3o
20
IO
55 o
IO
20
3o
4o
5o
8.52434
8.52497
8.52 56o
8.52 623
8.52685
8.52 748
8.52459
8.52 522
8.52 584
8.52647
8.52 710
8.52 772
o 5
5o
4o
3o
20
10
48 o
10
20
3o
4o
5o
8.49 708
8 -49775
8.4g 84a
8.49908
8.4g 9?5
8.5o 042
8.49 729
8.4g 796
8.49863
8.49 g3o
8.49997
8.5oo63
56 o
10
20
3o
4o
5o
8.52810
8.52 872
8.52 985
8.52 997
8.53 oSg
8.53 121
8.5a 835
8.52897
8.52 960
8.53 022
8.53o84
8.53 i46
o 4
5o
4o
3o
20
10
49 o
10
20
3o
4o
5o
8.5o 108
8.5o 174
8.5o24i
8.5o3o7
8. 5o373
8. 5o439
8.5o i3o
8.5o 196
8.50263
8.5o 329
8.5o 395
8.5o46i
o 11
5o
4o
3o
20
IO
57 o
10
20
3o
4o
5o
8.53 i83
8.53245
8.533o6
8.53368
8.53429
8.53491
8.53 208
8.53 270
8.53 332
8.53 3g3
8.53455
8.535i6
o 3
5o
4o
3o
20
IO
50 o
IO
20
3o
4o
5o
8.5o5o4
8.5o57o
8.5o636
8.5o 701
8.5o 767
8.5o832
8.5o 527
8.5o593
8.5o658
8.5o 724
8.50789
8.5o855
o 10
5o
4o
3o
20
10
58 o
IO
20
3o
4o
5o
8.53552
8.536i4
8.53675
8.53736
8.53 797
8.53858
8.53 578
8.5363 9
8.53 700
8.53762
8.53823
8.53884
2
5o
4o
3o
20
IO
51 o
10
20
3o
4o
5o
8.5o 897
8.50963
8.5i 028
8. 5 1 092
8.5i 167
8.5l 222
8.5o 920
8.50985
8.5i o5o
8.5i n5
8. 5 1 1 80
8. 5 1 245
o 9
5o
4o
3o
20
10
59 o
IO
20
3o
4o'
5o
8.53919
8.53 979
8.54o4o
8.54 ioi
8.54i6i
8.54 222
8.53 945
8.54oo5
8. 54 066
8.54 127
8.54 187
8.54248
o 1
5o
4o
3o
20
10
52 o
10
20
3o
8.5i 287
8.5i 35i
8.5i4i6
8.5i 48o
8.5i 3io
8.5i 374
8.5i 439
8.5i 5o3
o 8
5o
4o
3o 7
60 o
8.54282
8.54 3o8
o
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
4.6062
.01
5.3 9 48
IOOO
6.9078
.001
7.0922
IOOOO
9.2108
.0001
10.7897
IOOOOO
1 1 . 6129
.00001
12.4871
IOOOOOO
i3.8i55
.00000 I
i4.i845
IOOOOOOO
16.1181
.000000 I
77.8819
IOOOOOOOO
18.4207
.00000001
19.5793
IOOOOOOOOO
20.7233
.00000000 I
21.9767
Num.
Log.
Num.
Log.
LOGARITHMS OF NUMBERS FROM i TO 10.
N
1
2
3
4
5
6
7
8
9
1.0
o.oooo
OIOO
0198
0296
o3g2
o488
o583
0677
0770
0862
i . i
.2
.3
0.0953
0.1823
0.2624
io44
1906
2700
n33
1989
2776
1222
2070
2852
i3io
2l5l
2927
i3g8
223l
3ooi
i484
23l I
3075
1670
2390
3i48
i655
2469
3221
i?4o
2546
3293
.4
.5
.6
0.3365
o.4o55
0.4700
3436
4l2I
4762
35o7
4i8 7
4824
35 77
4253
4886
3646
43i8
494?
3716
4383
5oo8
3 7 84
4447
5o68
3853
45n
5i28
3920
45 7 4
5i88
3 9 88
463 7
5247
7
.8
9
o.53o6
0.5878
0.6419
5365
5 9 33
64? i
5423
5 9 88
6523
548i
6o43
65 7 5
553 9
6098
6627
55 9 6
6i52
6678
5653
6206
6729
5710
6269
6780
5 7 66
63i3
683i
5822
6366
6881
2.0
o.6g3i
6981
7o3i
7080
7129
7178
7227
7 2 7 5
7324
7 3 7
N
1
2
3
4
5
6
7
8
9
132
NAPERIAN LOGARITHMS.
N
1
o
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. 7 885
0.8329
7467
7 9 3o
8372
7 5i4
7975
84i6
756i
8020
845 9
7608
8o65
85o2
7655
8109
8544
7701
8 1 54
858 7
774?
8198
8629
7793
8242
8671
73 9
8286
8 7 i3
2.4
2.5
2.6
0.8755
o.gi63
o. 9 555
8796
9203
g5 9 4
8838
9243
9632
8879
9282
9670
8920
9022
9708
8961
g36i
9746
9002
9400
973
9042
943 9
9821
9083
94 7 8
9858
9123
9 5 '7
9 8 9 5
2.7
2.8
2.9
3.0
3.i
3.2
3.3
o.ggSS
i .0296
1.0647
9969
o332
0682
6006
0367
0716
6o43
o4o3
0750
6080
o438
0784
61 16
o4?3
0818
6i52
o5o8
o852
0188
o543
0886
0225
0578
0919
6260
06 1 3
ogSS
i .0986
1019
io53
1086
1119
1 1 5i
ii84
1217
1249
1282
i.i3i4
i.i632
i .1939
1 346
i663
1969
i3 7 8
1694
2OOO
i4io
1725
2o3o
i442
1756
2060
i4?4
1787
2090
i5o6
1817
21 19
i53 7
1 848
2149
i56g
1878
2179
1600
1909
2208
3.4
3.5
3.6
1.2238
r.2528
i .2809
2267
2556
2837
2296
2585
2865
2326
26i3
2892
2 355
2641
2920
2384
2669
2947
24l3
2698
2975
2442
2726
3002
2470
2754
3029
2499
2782
3o56
3-7
3.8
3. 9
4.0
4.i
4.2
4.3
i.3o83
i.335o
i.36io
3no
33 7 6
3635
3i3 7
34o3
366i
3i64
3429
3686
3igi
3455
3712
32i8
348 1
3 7 3 7
3244
35o7
3762
3271
3533
3 7 88
32 97
3558
38i3
33 2 4
3584
3838
1.3863
3888
3gi3
3 9 38
3962
3 9 8 7
4012
4o36
4o6i
4o85
i.4no
i.435i
1.4586
4i34
43 7 5
4609
4i5g
43 9 8
4633
4i83
4422
4656
4207
4446
4679
423i
446 9
4702
4255
4493
4725
4279
45i6
4748
43o3
454o
4 77 o
432 7
4563
4793
4-4
4.5
4.6
i.48i6
i .5o4i
1.5261
483g
5o63
5282
486 1
5o85
53o4
4884
5 1 07
5326
4907
5129
534?
4929
5i5i
536 9
4g5i
5i?3
5390
4974
5i 9 5
54 12
4996
5217
5433
Soig
523g
5454
4-7
4.8
4-9
5.0
5.i
5.2
5.3
1.5476
1.5686
1.5892
54 9 7
5 7 o 7
5gi3
55i8
5728
5g33
553 9
5?48
5 9 53
556o
5769
5 974
558i
5790
5994
56o2
58io
60 1 4
5623
583i
6o34
5644
585i
6o54
5665
5872
6074
i .6og4
6n4
6i34
6i54
6174
6ig4
6214
6233
6 2 53
6273
i .6292
i.648 7
1.6677
63i2
65o6
6696
6332
6525
6 7 i5
635i
6544
6 7 34
63?!
6563
6752
63go
6582
6771
6409
6601
6790
6429
6620
6808
6448
663g
6827
646 7
6658
6845
5.4
5.5
5.6
1.6864
1.7047
i .7228
6882
7066
7246
6901
7 o84
7263
6919
7102
7281
6 9 38
7120
7299
6 9 56
7 i38
7 3i 7
6974
7 i56
?334
6993
7i74
7 352
701 1
7192
7 3 7 o
7029
7210
7 38 7
5-7
5.8
5-9
6.0
i.74o5
1.7579
r.775o
7422
7 5 9 6
7766
?44o
7 6i3
7783
7457
763o
7800
7 4 7 5
7647
7817
7492
7664
?834
7 5o 9
7681
7 85i
7 52 7
7699
7867
7 544
7716
7 884
7 56i
7733
79'
1.7918
79 34
79 5i
79 6 7
794
8001
8017
8o34
8o5o
8066
1
2
3
4
5
6
7
8
9
i33
NAPERIAN LOGARITHMS.
N
1
2
3
4
5
6
7
8
9
6.0
i .7918
7934
7951
7967
79^4
8001
8017
8o34
8o5o
8066
6.1
6.2
6.3
.8o83
.8245
.84o5
8099
8262
8421
8116
8278
843 7
8i32
8294
8453
8i48
83io
846 9
8i65
8326
8485
8181
8342
85oo
8197
8358
85i6
82i3
83 7 4
853 2
8229
83 9 o
854 7
6.4
6.5
6.6
.8563
.8718
.8871
85 79
8733
8886
85g4
8 7 4 9
8901
8610
8764
8916
8625
8779
8g3i
864i
8 79 5
8 9 46
8656
8810
8961
8672
8825
8976
8687
884o
8991
8 7 o3
8856
9006
6.7
6.8
6.9
.9021
.9169
. 9 3i5
9o36
9184
933o
905 1
9199
9344
9066
92 1 3
gSSg
9081
9228
9 3 7 3
9 o 9 5
9242
9 38 7
91 10
9257
g4o2
9125
9272
94 1 6
9140
9286
g43o
gi55
gSoi
9445
7.0
.9459
9 4?3
g488
9502
g5i6
953o
9544
9 55 9
9 5 7 3
9 58 7
7- 1
7-2
7 .3
.9601
9?4i
.9879
9615
9755
9892
9629
9769
9906
9643
9782
9920
9 65 7
9796
9933
9671
9810
9947
9 685
9824
9961
9699
9 838
9974
97 i3
985i
9988
97 2 7
9 865
OOOl
7-4
7 .5
7.6
2.OOJ5
2.0149
2.0281
0028
0162
0295
0042
0176
o3o8
oo55
0189
0321
0069
O2O2
o334
0082
02 1 5
o347
0096
0229
o36o
0109
0242
0373
OI22
O255
o386
oi36
0268
0399
7-7
7.8
7-9
2 . O4 I 2
2.o54l
2.0669
o425
o554
0681
o438
0567
0694
o45i
o58o
0707
o464
0592
0719
0477
o6o5
0732
0490
0618
0744
o5o3
o63i
7 5 7
o5i6
o643
0769
o528
o656
O 7 82
8.0
2 .0794
0807
0819
o832
o844
0857
0869
0882
0894
0906
8.1
8.2
8.3
2.0919
2 . I O4 I
2.II63
0931
io54
1 175
0943
1066
1187
og56
1078
1199
0968
1090
121 I
0980
I IO2
1223
0992
ii i4
1235
ioo5
1 126
1247
1017
1 1 38
1258
1029
n5o
I2 7
8.4
8.5
8.6
2. 1282
2. l4oi
2.i5i8
1294
l4l2
1529
i3o6
1424
i54i
i3i8
i436
i552
i33o
1 448
i564
I 342
i45 9
1576
i353
i4?i
i58 7
1 365
i483
1 599
i3 77
i4g4
1610
i38 9
i5o6
1622
8.7
8.8
8.9
2.i633
2.1748
2.1861
1 645
i 7 5 9
1872
i656
1770
i883
1668
1782
1894
1679
1793
igoS
1691
1804
1917
1702
i8i5
1928
I 7 i3
1827
i 9 3 9
I 7 25
1 838
igSo
i 7 36
i84 9
1961
9.0
2. 1972
1983
i 99 4
2006
2017
2028
2039
2060
2061
2O 7 2
9.1
9.2
9 .3
2.2083
2.2192
2.23OO
2094
22O3
23ll
2105
22l4
2^22
2116
2225
2332
2127
2235
2343
2i38
2246
2354
2148
225 7
2364
2i5g
2268
2375
2I 7 O
22 7 9
2386
2l8l
2289
2396
9 .4
9 .5
9.6
2.24O7
2.25l3
2.2618
2418
2523
2628
2428
2534
2638
2439
2 544
2649
245o
2 555
265g
2460
2565
2670
2471
2576
2680
2481
2586
2690
2492
25 97
2 7 OI
25O2
26o 7
2 7 II
9-7
9.8
9.9
2.2721
2.2824
2.2925
2732
2834
2935
2742
2844
2946
2752
2854
2g56
2762
2865
2966
2 77 3
2875
2976
2783
2885
2986
2 79 3
2895
2996
28o3
2 9 o5
3oo6
28l4
29l5
3oi6
10.0
2. 3O26
3i26
3224
3322
34i8
35i4
3609
3703
3 79 6
3888
N
1
2
3
4
5
6
7
8
9
1 34
TABLE V
FOUR-PLACE LOGARITHMS
OF NUMBERS
FOUR-PLACE LOGARITHMS.
N
1
2
3
4
5
6
7
8
9
10
oooo
o43
086
128
170
212
253
94
334
3 7 4
1 1
4i4
453
492
53i
56 9
607
645
682
7i9
755
12
792
828
864
899
934
969
ioo4
io38
I0 7 2
1106
i3
1 1 39
i 7 3
206
271
3o3
335
367
3 99
43o
i4
46 1
492
523
553
584
6i4
644
6 7 3
7 o3
732
i5
I 7 6i
79
818
847
8 7 5
90
3
9
3i
9 5 9
987
20l4
16
20
il
068
095
122
i48
i 7 5
2OI
227
253
2 79
17
3o4
33o
355
38o
4o5
43o
455
48o
5o4
529
18
553
577
60 1
625
648
6 7 2
6 9 5
718
7 42
7 65
'9
788
810
833
856
8 7 8
900
923
945
967
989
20
3oio
032
o54
075
096
lib
1 39
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
5 79
5 9 8
23
617
636
655
674
692
7 1I
729
747
766
7 84
24
802
820
838
856
874
892
909
927
945
962
25
3 979
997
4oi4
4o3i
4o48
4o65
4082
4099
4n6
4 1 33
26
4i5o
166
i83
200
216
232
249
265
281
298
27
3i4
33o
346
862
378
3 9 3
4<
>y
4
25
44o
456
28
472
48 7
502
5i8
533
548
564
5 79
5 9 4
609
29
624
639
654
669
683
698
7 i3
728
742
7 5 7
30
4771
786
800
8i4
829
843
85 7
87,
886
900
3i
9
4
928
942
9 55
969
9 83
997
5oi i
5o24
5o38
32
5o5i
o65
079
092
io5
"9
I 32
i45
i5g
I 7 2
33
i85
198
21 I
224
237
25o
263
276
289
302
34
3i5
328
34o
353
366
3 7 8
3 9 i
4o3
4i6
428
35
544 1
453
465
4 7 8
490
5O2
5i4
527
53 9
55i
36
563
5 7 5
58 7
5 99
611
623
635
647
658
6 7 o
37
682
6 9 4
7o5
717
729
74<
>
7 52
7 63
77 5
786
38
798
809
821
832
843
855
866
877
888
899
39
911
922
933
944
9 55
966
977
9
88
999
6010
40
6021
o3i
042
M^^M^M
o53
o64
075 o85
096
^^^^^i
107
117
N
1
2
3
4
5
6
7
8
9
PP
38
32 28
as
99
21
19
18
17
:6
.1
3-8
3.2 2.8
a-5
.,
2.2
2.1
1.9
.1
1.8
'7
1.6
.2
6.4 5.6
5.0
.2
4-4
4-2
3-8
.2
3.6
3-4
3-2
3
n-4
9.6 8.4
7-5
3
6.6
6. 3
5-7
3
5-4
5-'
4.8
4
15.2
12 8 II. 2
1 0.0
4
8.8
8.4
7.6
4
7-2
6.8
6.4
5
19.0
1 6.0 14.0
12.5
5
II. O
10.5
9-5
5
9.0
8-5
8.0
.6
22.8
19.2 16.8
15-0
.6
13-8
13.6
11.4
.6
10.8
IO. 2
9.6
7
26.6
22.4 19.6
I7-S
7
'5-4
14.7
'3-3
. 7
12.6
II-9
II. 2
.8
3-4
25.6 22.4
20. o
.8
17.6
16.8
15.2
.8
14.4
I 3 .6
12.8
9
34-2
28.8 25.2 22.5
.9 19.8 18.9
17.1
9
16.2 15.3 14.4
i36
FOUR-PLACE LOGARITHMS.
N
1
2
3
4
5
6
7
8
9
40
6021
o3i
042
o53
o64
075
o5
096
107
117
4i
128
[3
i4 9
1 60
170
180
191
201
212
222
42
232
243
253
263
274
284
294
3o4
3i4
325
43
335
345
355
365
3 7 5
385
3 9 5
4o5
4i5
425
44
435
444
454
464
4 7 4
484
493
5o3
5i3
522
45
653 2
542
55i
56i
5 7 r
58o
590
5 99
609
618
46
628
63 7
646
656
665
6 7 5
684
6 9 3
702
712
47
721
780
7 3 9
?4g
7 58
767
776
7 85
794
8o3
48
812
821
83o
83 9
848
85 7
866
8 7 5
884
8 9 3
49
902
911
9 2O
928
9 3 7
9 46
9 55
9 64
97 2
9 8i
50
6990
998
7007
7016
7 O24
7o33
7042
7o5o
7 o5 9
7067
5i
7076
084
o 9 3
101
I IO
118
126
i35
i43
l52
52
160
168
177
i85
I 9 3
202
210
218
226
235
53
243
s5i
267
2 7 5
284
292
3oo
3o8
3i6
54
324
332
34o
348
356
364
3 7 2
38o
388
3 9 6
55
?4<
54
4l2
4i 9
427
435
443
45i
45 9
466
4 7 4
56
482
490
4 9 7
5o5
5i3
52O
528
5
36
543
55i
57
55 9
566
5 7 4
58 2
58 9
5 97
6o4
612
6i 9
627
58
634
642
64g
65 7
664
6 7 2
679
686
6 9 4
701
5 9
709
716
723
7 3i
7 38
7 45
752
760
767
774
60
7782
789
796
8o3
810
818
825
832
83 9
846
61
853
860
868
8 7 5
882
88c
)
8^
6
. 93
9 IO
917
62
924
9 3i
g38
945
9 52
9 5 9
966
97 3
9 8o
987
63
99 3
8000
8007
8oi4
8021
8028
8o35
8o4i
8o48
8o55
64
8062
069
075
082
089
096
102
109
116
122
65
8129
i36
1 42
i4 9
i56
162
169
176
182
i8 9
66
i 9 5
202
209
2l5
222
228
235
24 1
248
254
67
261
26 7
274
280
287
2 9 :
\
299
3o6
3l2
3i 9
68
325
33i
338
344
35i
35 7
363
870
3 7 6
382
69
388
3 9 5
4oi
4o 7
4i4
420
426
432
43 9
445
70
45i
45 7
463
470
4?6
482
488
4
94
5oo
mmmmmmm
5o6
N
1
2
3
4
5
6
7
8
9
PP
15
14
'3
12
IX
10
9
8
7
6
0.8
0.6
.2
3-
2.8
2.6
2.4
.2
2.2
2.0
!s
.2
0.7
1.4
1.2
3
45
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
5-5
4-5
5
4.0
3-5
3.0
.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
12.
II. 2
10.4
9.6
.8
8m
.a
8.0
7.2
8
6-4
5-6
48
9 '3-5
12 6
ii 7 10.8
8.1
9
7.2
6.3
5-4
i3 7
FOUR-PLACE LOGARITHMS.
N
1
2
3
4
5
6
7
8
9
70
7 1
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
9 1
92
9 3
9 4
9 5
96
97
98
99
100
<^^MM
N
845i
45 7
463
470
476
482
488
494
5oo
5o6
5i3
573
633
692
8 7 5i
808
865
921
976
5i 9
579
63 9
698
7 56
8i4
871
927
982
525
585
645
704
762
820
876
9 32
987
53i
Soi
65i
710
768
8 2 5
882
9 38
99 3
53 7
5 9 7
657
716
774
83i
887
9 43
998
543
6o3
663
722
779
83 7
8 9 3
9 4 9
9 oo4
~o58~
549
6o 9
66 9
727
785
842
899
9 54
9009
555
6i5
6 7 5
7 33
791
848
904
960
9015
56i
621
681
739
797
854
910
9 65
9020
56 7
62 7
686
7 45
802
85 9
gi5
97i
9025
9 o3i
o36
o42
o47
o53
o63
069
o 7 4
079
o85
1 38
191
243
9 2 9 4
345
3 9 5
445
4 9 4
090
i43
196
248
299
35o
4oo
45o
499
096
i4 9
2OI
253
3o4
355
4o5
455
5o4
IOI
1 54
206
258
3o 9
36o
4 i <>
46o
5o 9
106
i5 9
212
263
3i5
365
4i5
465
5i3
I 12
i65
217
26 9
320
370
420
46 9
5i8
117
170
222
274
325
3 7 5
425
4 7 4
523
122
i 7 5
22 7
279
33o
38o
43o
479
528
128
1 80
232
284
335
385
435
484
533
i33
186
238
28 9
34o
3 9 o
44o
48 9
538
9 542
547
552
55 7
562
566
671
5 7 6
58 1
586
5 9 o
638
685
7 3i
9777
823
868
912
9 56
595
643
689
736
782
827
872
9'7
9 6i
600
647
6 9 4
74 1
786
832
877
921
9 65
6o5
652
699
745
7 9 i
836
881
9 26
9 6 9
609
65?
7 o3
75o
79 5
84 1
886
gSo
974
6i4
661
708
754
800
845
8 9 o
9 34
978
619
666
7-3
7 5 9
8o5
85o
8 9 4
939
9 83
624
671
7'7
7 63
809
854
899
9 43
98?
628
6 7 5
722
768
8i4
85 9
9 o3
9 48
99 i
633
680
7 2 7
773
818
863
9 o8
9 52
99 6
oooo
oo4
1
009
2
oi3
3
017
4
022
5
026
^
6
o3o
-..
7
o35
8
o4o
9
PP
.2
3
4
.6
.8
9
7
6
.1
.2
3
4
'.6
.8
5
4
0.7
'4
2.1
2.8
3-5
4.2
4.9
5-6
6.3
0.6
1.2
1.8
2.4
3'
3.6
4-2
4.8
5-4
0.5
I.O
'5
20
2-5
3.0
3-5
4.0
4-5
0.4
0.8
1.2
1.6
2.O
2.4
2.8
H
36
i38
TABLE VI
FOUR-PLACE LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
TO EVERY TEN MINUTES
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L.Tang.
d.
L. Cotg.
L. Cos.
d.
o
10
20
3o
4o
5o
7
7
7
E
3011
1760
1250
969
792
669
5 8o
5"
458
4'3
378
348
300
280
263
248
235
322
212
203
193
l8 S
177
I 7
I 5 8
152
'47
7 .463 7
7.7648
7 . 9409
8.o658
8.1627
3011
1761
1249
969
792
670
512
457
4'5
378
348
323
300
281
263
249
235
223
213
202
194
'85
,78
171
165
'58
'54
148
2.5363
2.2352
2.O59I
I .9342
1.8373
o.oooo
o . oooo
o.oooo
o.oooo
o.oooo
o.oooo
o
o
o
o 90
5o
4o
3o
20
1 O
.463 7
.7648
.9408
.o658
. 1627
1 o
10
20
3o
4o
5o
8
8
a
8
8
8
.2419
.3o88
.3668
.4i79
.463 7
.5o5o
8.2419
8.3089
8.3669
8.4i8i
8.4638
8.5o53
i. 7 58i
i .691 i
i. 633 i
1.5819
1.5362
9-9999
0.9999
9.9999
9.9999
9.9998
9.9998
I
o
o
I
o
I
o
I
o
I
o
I
I
I
I
I
o 89
5o
4o
3o
20
IO
2 o
10
20
3o
4o
5o
8.5428
8.5 77 6
8.6097
8.6397
8.6677
8.6940
8.543i
8.5 779
8.6101
8.64oi
8.6682
8.6945
.4569
.4221
.3899
.35 99
.33i8
.3o55
9.9997
9-9997
9.9996
9.9996
9.9995
9.9995
o 88
5o
4o
3o
20
10
3 o
10
20
3o
4o
5o
8
8
8
8
8
8
.7188
. 7 423
. 7 645
. 7 85 7
.825i
8 . 7 i 94
8. 7 652
8.7865
8.8067
8.8261
i .2806
I .25 7 I
1.2348
.2i35
.1933
.i 7 3 9
9.9994
9 . 999 3
9.9993
9.9992
9.9991
9.9990
o 87
5o
4o
3o
20
IO
4 o
10
20
3o
4o
5o
8
8
8
8
8
8
.8436
.86i3
.8 7 83
.8946
.9104
.9256
8.8446
8.8624
8.8795
8.8960
8.9118
8.9272
.i554
.1376
.1205
. io4o
.0882
.0728
9.9989
9.9989
9.9988
9.9987
9.9986
9.9985
o
I
I
I
I
2
o 86
5o
4o
3o
20
IO
5
8 . 94o3
8.9420
i.o58o
9 . 998 3
o 85
L
.Cos
d.
L. Cotg.
d.
L. Tang.
L.Sin.
d.
'
PP
.2
3
4
.6
.8
9
348
300
36 3
26.3
52.6
78.9
IOS-2
I3I-5
157.8
l8 4 .,
210.4
.1
.2
3
4
.1
: 8
9
*35
313
185
.1
.2
3
4
.6
9
171
158
147
34-8
69.6
104.4
139.2
174.0
208.8
243-6
278.4
30
60
90
120
ISO
1 80
210
240
47-C
94-c
117.5
141.0
164.5
i88.c
211.5
21.,
42. c
63.5
85.2
106.5
I27-J
149.1
170.4
191.7
.8.5
37 o
55-5
74.0
92.5
III.O
129.5
148.0
166.5
17.1
342
5'-3
68.4
85.5
IO2.D
II9.7
136.8
'53-9
15.
3i.<
47-'
63.
79- <
94-
I 10.
126.
142.
i 14.7
) 29.4
\ 44 '
i 588
> 73-5
! 88.2
i 102.9
I "7- 6
132.3
i4o
FOUR-PLACE LOGARITHMIC FUNCTIONS.
o
i
L. Sin.
d.
L. Tang. d.
L. Cotg.
L. Cos. d.
5 o
10
20
3o
4o
5o
8
8
8
8
8
9
,g4o3
. 9 545
.9682
.9816
.9945
.0070
142
137
'34
129
125
122
II 9
"5
"3
109
107
104
IO2
99
97
95
93
9'
89
87
85
84
82
80
79
78
76
75
73
73
8.9420
8. 9 563
8.9701
8. 9 836
8.9966
9.0093
143
138
'35
13
127
123
1 20
117
"4
in
108
105
104
101
98
97
94
93
9i
89
87
86
84
82
81
80
78
77
76
74
i. 0680
i .0437
i .0299
i .0164
i.oo34
o.ggo?
g.gg83
9.9982
g.ggSi
9.9980
9-9979
9.9977
i
i
i
i
2
I
I
2
I
I
2
I
2
2
I
2
2
I
2
2
2
2
2
2
2
2
2
2
2
2
o 85
5o
4o
3o
20
JO
6 o
10
20
3o
4o
5o
9.0192
9.o3i i
9.0426
g.o53g
9.0648
9.0755
9.0216
g.o336
g.o453
9.0667
9.0678
9.0786
o.g784
o.g664
o.g547
o.g433
o.g322
o.g2i4
9.9976
9-99?5
9.9973
9.9972
9-9971
9.9969
o 84
5o
4o
3o
20
IO
7 o
10
20
3o
4o
5o
g.o85g
9.0961
9. 1060
9.1157
g. 1252
9.1345
9.0891
9.0996
9. 1096
9.1194
9. 1291
g.i385
o.giog
0.9006
o.8go4
0.8806
o.87og
0.8616
9.9968
g.gg66
9-9964
g.gg63
9.9961
g.ggSg
o 83
5o
4o
3o
20
10
8 o
10
20
3o
4o
5o
9. i436
9. i525
9. 1612
9.1697
9.1781
9 .i863
g.i47^
g. 1669
g.i658
9.1745
g.i83i
g. 1916
0.8622
o.843i
0.8342
0.8266
o.8i6g
0.8086
g.gg58
g .9966
g.gg54
9.gg52
9.9960
g.gg48
o 82
5o
4o
3o
20
10
9 o
10
20
3o
4o
5o
9.1943
9.2022
9.2100
9.2176
9.2261
9.2324
9.1997
9.2078
9.2168
9. 2236
g.23i3
9.2389
o.8oo3
o.7g22
0.7842
0.7764
o. 7687
0.761 1
9.9946
9-gg44
g.gg42
g.gg4o
g.gg38
g.gg36
o 81
5o
4o
3o
20
10
10
O
9.2397
g.2463
0.7637
g.gg34
o 80
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' o
PP
.1
.2
3
4
5
.6
:!
9
138
"5
117
.1
.2
3
4
'.6
9
104
97
8 9
.1
.2
3
4
.6
.S
9
84
78 73
13.8
27.6
41.4
55-2
69.0
82.8
96.6
110.4
124.2
'2-5
25.0
37-5
50.0
62.5
75-0
87.5
1OO.O
112.5
11.7
23-4
35-1
46.8
58.5
70.2
Sr.g
93.6
-'5-3
10.4
20.8
31-2
4 ..6
52.0
62.4
72.8
83.2
93.6
9-7
19.4
29.1
38.8
48.5
58.2
67.9
77.6
87.3
8. 9
.i
26.7
35-6
44-5
53-4
62.3
71.2
80. i.
8.4
1 6. 8
25.2
33-6
42.0
50.4
58.8
67.2
75-6
7.8 7.3
15.6 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
FOUR-PLACE LOGARITHMIC FUNCTIONS.
'
L. Sin.
d.
L.Tang. d.
L. Cotg.
L. Cos. j d.
10
o
9
.23 97
9.2463
0.7637
9.9934
o 80
10
9
.2468
9.2636
0.7464
9.9931
3
5o
2
9
.2538
70
9 .26o 9
73
0.7391
9.9929
2
4o
68
7 1
3o
9
.2606
68
9.2680
0.7320
9.9927
3o
4o
9
.2674
9.2760
0.7260
9-9924
3
20
5o
9
.2740
9 .28i 9
09
0.7181
9.9922
2
IO
11
9
.2806
64
9 .2887
66
0.7113
9.9919
3
o 79
IO
9
.2870
9 .2 9 53
0.7047
9.9917
5o
20
9
.2 9 34
64
9.3020
t>7
0.6980
9.9914
3
4o
3o
9
.2997
63
61
9 .3o85
65
6*
0.6916
9.9912
2
3o
4o
9
.3o58
9. 3i
4 9
0.6861
9.9909
3
20
5o
9
.3u 9
9.3212
03
0.6788
9.9907
2
IO
12
9
.3i 79
9.3276
03
61
0.6726
9.9904
3
o 78
10
9
.3238
9 .3336
0.6664
9.9901
3
5o
20
9
.3296
50
9 .33 97
o.66o3
9.9899
2
4o
3o
9
.3353
57
9 .3458
61
0.6642
9.9896
3
3o
4o
9
.34io
9 .35i7
0.6483
9 .9893
3
20
5o
9
.3466
56
9 .3576
59
eft
0.6424
9.9890
3
IO
13
9
.3521
9. 3634
0.6366
9.9887
3
o 77
10
9
.3676
9.3691
o.63og
9.9
884
3
5o
20
9
.3629
54
9 .3 7 48
57
0.6262
9-9
881
3
4o
53
56
3o
9
.3682
9 .38o4
0.6196
9.9878
3o
4o.
.3 7 34
9 .385 9
o.6i4i
9.9876
3
20
5o
y
.3 7 86
52
9 .3 9 i4
55
0.6086
9.9872
3
10
14
9
.3837
5 1
9 .3 9 68
54
o.6o32
9.9869
3
o 76
10
9
.3887
9 .4o2i
0.6979
9.9866
3
5o
20
9
.3 9 3 7
5
9.4074
53
0.6926
9.9863
3
4o
3o
9
.3 9 86
49
9.4127
53
o.58 7 3
9.9869
4
3o
4o
9
.4o35
49
9.4178
0.6822
9.9866
3
20
5o
9
.4o83
48
9.42
3o
52
0.6770
9.9863
3
IO
15
O
9 .4i3o
47
9.4281
5'
0.6719
9.9849
4
o 75
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
PP
71
68
66
64
61
58
55
53 Si
.1
7-i
6.8
6.6
.1
6. 4
6.1
5-8
.1
5-5
5-3 5-i
.2
14.2
13.6
13.2
.2
12.8
12.2
ii. 6
.2
II.
IO.6 IO.2
3
21.3
20.4
19.8
3
19.2
l8. 3
17.4
3
16.5
'5-9 >5-3
4
28.4
27.2
26.4
4
25.6
24-4
23.2
4
22.
21.2 2O.4
5
35-5
34-o
33-o
5
^2.0
3-5
29.0
5
27.5
26.5 2S.5
.6
42.6
40.8
39- 6
.6
sM
36.6
34-8
.6
33-o
31.8 30.6
7
49-7
47.6
46.2
. 7
44.8
42.7
40.6
7
38.5
37- 1 357
.8
56.8
54-4
52.8
.8
51.2
48.8
46.4
.8
44.0
42.4 40.8
61.2
59-4
9
57-6 54-9 52-2
142
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L.Sin. d.
L. Tang.
d.
L. Cotg.
L.
Cos.
d.
15
O
g.4i3o
9.4281
0.6719
9.9849
o 75
IO
9.417
7
9 .433i
0.6669
9-
? 846
5o
20
9-4223
46
9.4
5-81
0.6619
9.9843
3
4o
3o
9.4269
4 6
9-443o
49
o. 6670
9.9839
4
3o
4o
g.43i4
9.4479
0.6621
9.9836
20
5o
9.4369
45
9.4627
4
0.6473
9 .9832
4
IO
16
g.44o3
44
9.4676
0.5426
9-
3828
4
o 74
10
9.4447
9.4622
0.5378
9.9826
5o
20
9.4491
44
9.4669
47
o.533i
9.9821
4
4o
3o
9.4533
42
9.4716
47
xfi
0.6284
9.9817
4
3o
4o
9.4676
9.4762
0.5238
9.9814
20
;
)0
9.4618
4 2
9.4808
46
0.6192
9.9810
4
IO
17
9.4669
9. 4853
45
0.6147
9 .9806
4
o 73
10
9.4700
9.4898
0.6102
9-
3802
5o
20
9.474
9.4943
45
0.6067
9.9798
4
4o
3o
9.4781
40
9.4987
44
o.5oi3
9 979 4
4
3o
4o
9.4821
9-5o3i
0.4969
9.9790
20
5o
9.4861
40
9.6076
44
0.4926
9.9786
4
IO
18
o
9.4900
39
9.6118
43
0.4882
9.9782
4
o 72
10
9.4939
9.6161
o.483g
9.9778
5o
20
9.4977
38
9.6203
42
0.4797
9.9774
4
4o
3o
9.6016
3
9.6245
42
0.4766
9.9770
4
3o
4o
9. 6062
9.6287
o.47i3
9.9766
20
5o
9. 6090
38
9.5329
42
0.4671
9.9761
4
IO
19
o
9.6126
36
9.5370
4 1
o.463o
9.9767
71
IO
9.5i63
9.641 i
0.4689
9.9762
5o
20
9.6199
36
9.545i
40
0.4549
9.9748
4
4o
3o
9.6235
30
9.5491
40
0.4609
9.9743
5
3o
4o
9.6270
9 . 553i
0.4469
9.9739
20
5o
9.53o6
36
9.6671
40
0.4429
9-<
J7^4
5
IO
20
9.534i
35
9.6611
40
0.4389
9.9730
4
o 70
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' O
PP
49 47
45
44
43
41
40 38
36
.1
4-9 4-7
4-5
.1
4-4
4-3
4.1
.1
40
3.8
3-6
.2
9.8 9.4
9.0
.2
8.8
8.6
8.2
.2
80
7 .6
7-2
3
14.7 14.1
13-5
3
13.2
12.9
12.3
3
12 O
11.4
10.8
4
19.6 18.8
18.0
4
17.6
17.2
16.4
4
16 o
1^.2
14.4
5
24.5 23.5
22.5
5
22.
21.5
20.5
5
20 o
ig.O
18.0
.6
29.4 28.2
27.0
.6
26.4
25.8
24.6
.6
340
22.8
21.6
7
34-3 32.9
3'-5
7
30.8
30.1
28.7
. 7
280
26.6
25.2
.8
39.2 37.6
36.0
.8
35-2
34-4
32.8
.8
32 o
30.4
28.8
9
44-1 42.3
40.5
39- 6
38.7 36.9
9
360 34.2
32-4
i43
FOUR-PLACE LOGARITHMIC FUNCTIONS.
>
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L.
Cos. d.
20 o
10
20
3o
4o
5o
9.5341
9.5376
9.5409
9.5443
9.5477
9.55io
34
34
34
34
33
33
33
33
32
32
3'
3 2
3i
3i
30
3'
30
3
29
3
29
29
2 9
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
0.4389
o.435o
o.43i i
0.4273
o.4a34
0.4196
9.9730
9.9725
9.9721
9.9716
9.9711
9.9706
5
4
5
5
o 70
5o
4o
3o
20
10
21 o
10
20
3o
4o
5o
9.5543
9 .55 7 6
9.5609
9.5fi4i
9 .56 7 3
9.5704
9.5842
9.0879
9.5917
9.5954
9.5991
9.6028
o-4i58
o.4i 21
o.4o83
o.4o46
0.4009
0.3972
9.9702
9.9697
9.9692
9.9687
9.9682
9.9677
5
5
5
5
5
5
5
6
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
9.585g
9.5889
9.6064
9.6100
9.6i36
9.6172
9.6208
9.6243
0.3936
0.3900
0.3864
0.3828
0.3792
o.3 7 5 7
9.9672
9.9667
9.9661
9.9656
9.9651
9 .9646
o 68
5o
4o
3o
20
I O
23 o
10
20
3o
4o
5o
9.5919
9 .5 9 48
9.5978
9.6007
9-6o36
9-6o65
9.6279
9-63i4
9-6348
g.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
10
20
3o
4o
5o
9.6093
9.6121
9.6149
9.6177
9.6205
9.6232
9.6486
9-652O
9-6553
9.6587
9.6620
g.6654
o.35i4
o.348o
0.3447
o. 34 i 3
o.338o
0.3346
9.9607
9.9602
9.9596
9.9590
9 . 9 584
9-9 5 79
5
6
6
6
5
o 66
5o
4o
3o
20
IO
25
9.6259
9.6687
o.33i3
9. 9 5 7 3
o 65
L. Cos.
d.
L.Cotg.
d.
L. Tang.
L. Sin.
d.
' O
PP
.i
.2
3
4
'.6
7
.8
9
39 37
35
.i
.2
3
4
'.6
.8
34
33
33
.1
.2
3
4
.6
.8
9
31
30
29
3-9 3-7
7-8 7-4
11.7 u. i
15.6 14.8
19-5 '8.5
23.4 22.2
27.3 25.9
3 1. 1 29.6
35- i 33- 3
3-5
7.0
10.5
14.0
'7-5
21.
24.5
28.0
3'-5
3-4
6.8
IO.2
13.6
I 7 .0
20.4
2 3 .8
27.2
30.6
II
9-9
13.2
16.5
19.8
23-'
26.4
32
6. 4
9 .6
12.8
16 o
19.2
22.4
5- 6
28.8
3-i
6.2
9-3
12.4
iS-5
18.6
21.7
24.8
27.0
3-
6.0
9.0
12.0
15.0
18.0
21.
24.0
27.0
2 'i
58
8.7
ii. 6
'45
17-4
20.3
23.2
26. i
1 44
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
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
24
25'
25
24
24
24
9.6687
9.6720
9.6762
9.6786
9.6817
9.6860
33
S 2
33
32
33
32
32
32
32
32
3'
3'
3
3
3
3
3
3
29
30
29
3
29
29
o.33i3
0.3280
0.3248
o.32i5
o.3i83
o.3i5o
9.9673
9.9667
9.9661
9.9655
9.9649
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.3ii8
o.3o86
o.3o54
o.3o23
0.2991
0.2960
9.9637
9.9630
9.9624
9.9618
9.9612
9.9606
7
6
6
6
7
6
7
6
7
6
7
o 64
5o
4o
3o
20
IO
27 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6670
.65 9 5
.6620
.6644
.6668
. 6692
9.7072
9.7103
9.7134
9. 7166
9.7196
9.7226
0.2928
0.2897
0.2866
0.2835
0.2804
0.2774
9.9499
9.9492
9.9486
9.9479
9-94?3
9.9466
o 63
5o
4o
3o
20
10
28 o
10
20
3o
4o
5o
9
9
9
9
9
9
.6716
.6740
.6763
.6787
.6810
.6833
24
24
23
24
23
23
23
22
23
22
23
22
22
9.7267
9.7287
9 . 7 3i 7
9 . 7 348
9.7408
0.2743
0.2713
0.2683
0.2662
0.2622
0.2692
9.9469
9.9453
9.9446
9.9439
9.9432
9.9426
7
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
.6 9 46
.6968
9. 7 438
9.7467
9.7497
9.7626
9.7666
9 . 7 585
0.2662
0.2533
o.25o3
0.2474'
O.2444
0.24 i 5
9.9418
9.9411
9.9404
9-9 3 97
9.9390
9-9383
7
7
7
7
7
o 61
5o
4o
3o
20
10
30
9
.6990
9.7614
0.2386
9.9376
o 60
L
. Cos.
d.
L. Cotg.
d.
L.Tang.
L. Sin.
d.
1 O
PP
28
27
26
25 24
23
22
7
6
o 6
.2
3
4
'.I
.9
5-6
8.4
II. 2
14.0
16.8
19.6
22.4
25.2
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
24.3
5-2
7-8
10.4
13.0
15-6
18.2
20.8
23.4
.2
3
4
9
5.0 4-8
7-5 7-2
10.0 9.6
I2.S I2.O
15.0 14.4
17.5 16.8
20.0 19.2
22.5 21.6
?
9-2
"5
13-8
16.1
18.4
20.7
.2
3
4
'.6
'.&
4-4
6.6
8.8
II.
13.2
19.8
1.4
2.1
2.8
3-5
4.2
4.9
5-6
6-3
1.2
1.8
2.4
3'
3-6
4-2
4.8
5-4
i.45
FOUR-PLACE LOGARITHMIC FUNCTIONS
O '
L. Sin.
d.
L. Tang.
d.
L. Cotg.
L. Cos.
d.
30 o
IO
20
3o
4o
5o
9.6990
9.7012
9.7033
9.7065
9.7076
9.7097
22
21
22
21
21
21
21
21
21
2O
21
3O
2O
2O
20
20
2O
19
19
2O
J 9
19
19
'9
18
19
ll
'9
18
18
9.7614
9.7644
9.7673
9.7701
9 . 77 3o
9.7769
30
29
28
29
29
29
28
29
28
29
28
28
28
28
28
28
27
28
28
27
28
27
28
27
27
27
27
27
27
27
0.2386
0.2356
0.2327
0.2299
0.2270
0.224l
9.9375
9.9868
9.9361
9.9353
9.9346
9.9338
7
7
8
7
8
o 60
5o
4o
3o
20
IO
31 o
IO
20
3o
4o
5o
9.7118
9.7189
9.7160
9.7181
9.7201
9. 7222
9.7788
9.7816
9.7845
9.7873
9.7902
9.7930
'O.22I2
0.2184
o.2i55
0.2127
0.2098
0.2070
9.9331
9.9323
9 . 9 3i5
9.9308
9.9300
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
9.7302
9.7322
9.7342
9.7968
9.7986
9.8014
9.8042
9.8070
9.8097
O.2O42
O.20l4
o. 1986
o. 1968
o. igSo
o. 1903
9.9284
9.9276
9.9268
9.9260
9.9262
9.9244
8
8
8
8
8
o 58
5o
4o
3o
20
10
33 o
IO
20
3o
4o
5o
9.7361
9.7380
9.7400
9.7419
9. 7 438
9.7457
9.8125
g.8i53
9.8180
9.8208
9.8235
9.8263
0.1875
0.1847
o. 1820
o. 1792
o. 1765
0.1737
9.9236
9.9228
9.9219
9.9211
9.9203
9.9194
8
9
8
8
9
o 57
5o
4o
3o
20
IO
'34 o
IO
20
3o
4o
5o
9.7476
9.7494
9.7613
9.7531
9.7550
9.7668
9.8290
9.8317
9 .8344
9.8371
9.8398
9.8426
0.1710
o.i683
o. i656
o. 1629
o. 1602
o. 1676
9.9186
9.9177
9.9169
9.9160
9.9161
9.9142
9
8
9
9
9
8
o 56
5o
4o
3o
20
IO
35
9.7586
9.845a
o.i548
9.9134
o 55
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
'
PP
.i
29 38
37
.1
33
21
30
.1
19
8
7
2. Q 2.8
t8 z 6
2.7
'2.2
2.1
2.0
1.9
3 8
0.8
1.6
0.7
3
4
'.6
'.8
9
.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
ft
10.8
'3-5
16.2
18.9
21.6
24.3
3
4
'.6
:l
9
6.6
8.8
II. O
13.2
15-4
17.6
19.8
6-3
8.4
10.5
12.
M-7
16.8
18.9
6.0
8.0
o.o
2.O
4.0
6.0
3
4
'.6
:l
9
5-7
7.6
9-5
11.4
J3-3
15.2
17.1
2-4
3-2
*%
4.8
5.6
6.4
7-2
2.1
2.8
3-5
4-2
4.9
56
6-3
1 46
FOUR PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L.Tang.
d.
L. Cotg
L. Cos.
9
9
9
9
9
9
10
9
9
IO
9
35 o
10
20
3o
4o
5o
9.7686
9. 7604
9.7622
9.7640
9.7667
9.7676
18
18
18
17
18
'7
18
'7
16
17
16
17
16
16
16
15
16
16
9.8462
9.8479
9.8606
9. 8533
9.8669
9.8686
27
27
27
26
27
27
26
27
26
26
27
26
26
27
26
26
26
26
26
26
26
26
26
26
26
25
26
26
25
26
o.i548
o. 1621
0.1494
o. 1467
o. i 44 i
O. I-i I 4
ON ON ON ON ON ON
9134
9126
91 16
9107
9098
90^9
o 55
5o
4o
3o
20
IO
36 o
10
20
3o
4o
5o
9. 7692
9.7710
9.7727
9.7744
9.7761
9.7778
9-86i3
9.863g
9.8666
9.8692
9.8718
9.8746
0.1387
o. i36i
o. i 334
o.i3o8
o. 1282
o. 1266
9.9080
9.9070
9.9061
9.9062
9.9042
9.9033
o 54
5o
4o
3o
20
10
37 o
10
20
3o
4o
5o
9.7796
9.781 i
9.7828
9-7844
9.7861
9.7877
9.8771
9.8797
9.8824
9.8860
9.8876
9.8902
o. 1229
O. I2O3
o. i 176
o. 1160
o. i 124
o. 1098
9.9023
9.9014
9 . 9004
9.8996
9.8986
9.8976
9
10
9
IO
10
o 53
5o
4o
3o
20
10
38 o
10
20
3o
4o
5o
9
9
9
9
9
9
.7893
.7910
. 7926
.7941
.7967
.7973
9.8928
9.8964
9.8980
9.9006
9.9032
9.9068
o. 1072
o. io46
o. 1020
0.0994
0.0968
0.0942
9.8966
9.8966
9.8945
9.8935
9.8926
9.8916
IO
IO
10
IO
10
o 52
5o
4o
3o
20
10
39 o
10
20
3o
4o
5o
9
9
9
9
9
9
.7989
.8004
.8020
.8o35
.8060
.8066
15
16
15
16
9.9084
9.9110
9.9135
9.9161
9.9187
9.9212
0.0916
0.0890
o.o865
0.0839
o.o8i3
0.0788
9.8906
9.8896
9.884
9.8874
9.8864
9.8853
IO
II
10
IO
II
o 51
5o
4o
3o
20
10
40
9
.8081
15
9.9238
0.0762
9.8843
o 50
L
.Cos.
d.
L. Cotg.
d.
L.Tang.
L. Sin.
d.
' O
PP
.1
.2
3
4
9
26
25
18
.1
.2
3
4
'.6
'.S
9
'7
16
15
.2
3
4
.6
9
ii
IO
9
2.6
5-2
7-8
10.4
13.0
15-6
18.2
20.8
23.4
2-5
5.0
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
16 2
'7
3-4
6.8
8-5
10.2
II.O
13.6
1.6
32
4.8
6.4
8.0
9.6
II. 2
12.8
14.4
i-5
4-5
6.0
7-5
9.0
10.5
12.0
1 3- 5
i.i
2.2
3-3
4-4
u
u
9.9
I.O
2.O
4.0
6!o
7.0
8.0
0.9
1.8
2.7
3-6
4-5
5-4
6.3
7.2
8.1
FOUR-PLACE LOGARITHMIC FUNCTIONS.
O '
L. Sin.
d.
L.Tang.
d.
L. Cotg.
L
.Cos.
d.
40 o
10
20
3o
4o
5o
9.8081
9.8096
9.8111
9.8125
9.8140
9.8r55
15
'5
M
15
'5
M
'5
M
'5
'4
'4
9.9238
9.9264
9.9289
9-93i5
9 . 9 34i
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
S
0.0762
0.0736
0.071 1
o.o685
0.0659
o.o634
9
9
9
9
9
9
.8843
.8832
.8821
.8810
.8800
.8789
ii
ii
ii
IO
II
II
II
II
II
12
II
o 50
5o
4o
3o
20
10
41 o
10
20
3o
4o
5o
9.8169
9.8184
9.8198
9.8213
9.8227
9.8241
9.9392
9.9417
9.9443
9.9468
9.9494
9 . 9 5i 9
0.0608
o.o583
o.o557
o.o532
o.o5o6
o.o48i
9
9
9
9
9
9
.8778
.8767
.8756
.8 7 45
.8733
.8722
o 49
5o
4o
3o
20
10
42 o
IO
20
3o
4o
5o
9.8255
9.8269
9.8283
9.8297
9-83u
9-8324
M
'4
'4
M
'3
M
'3
'4
'3
13
14
3
'3
'3
'3
12
13
13
9.9544
9.9370
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
10
20
3o
4o
5o
9.8338
9.835i
9.8365
9.8378
9.8391
9.84o5
9.9697
9.9722
9-974?
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
.85 9 4
.8582
12
II
12
12
12
'3
12
12
'3
12
'3
12
o 47
5o
4o
3o
20
10
44 o
10
20
3o
4o
5o
9.84(8
9.843i
9-8444
9.8457
9.8469
9.8482
9.9848
9.9874
9.9899
9.9924
9.9949
9.9975
O.Ol52
0.0126
O.OIOI
0.0076
o.ooSi
O.OO25
9
9
9
9
9
9
.8569
-855 7
.8545
.8532
.8520
-85o7
o 46
5o
4o
3o
20
10
45 o
9.8495
o.oooo
o . oooo
9
.8495
o 45
L. Cos.
d.
L. Cotg.
d.
L. Tang.
L. Sin.
d.
' O
PP
36
*5
15
14
13
13
.1
ii
10
.1
2.6
2-5
1.5 .1
1.4
"3
1.2
i.i
I.O
3
4
'.6
'.8
'?
78
10.4
'3
156
iS 2
20.8
23 4
7-5
10.0
12 5
150
'7 5
20. o
4-5 -3
6.0 .4
75 -5
9.0 .6
10.5 .7
12.0 .8
4-2
5-6
7.0
8.4
9.8
II. 2
12.6
3-9
5.2
6 -l
7.8
9.1
104
3.6
4.8
6.0
7-2
8.4
9 '2
10.8
3
4
.6
.S
3-3
4-4
5-5
6.6
11
9-9
3-
4.0
5-o
6.0
7
8.0
9.0
i48
TABLE VII
FOUR-PLACE
NATURAL TRIGONOMETRIC
FUNCTIONS
TO EVERY TEN MINUTES
FOUR-PLACE NATURAL FUNCTIONS.
'
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
o
10
2O
3o
4o
5o
, o.oooo
0.0029
o.oo58
0.0087
o.oi 16
o.oi45
29
29
29
29
29
o.oooo
0.0029
0.0068
0.0087
o.oi 16
o.oi45
29
29
29
29
29
30
29
29
29
29
29
29
29
29
3
29
29
29
29
29
3
29
29
29
3
29
29
29
3
29
infinit.
171.8864
114.6887
86.9398
68.7601
1st
6'.
477
J*
313
fc
22C.
188
,6;
143
at
112
IOL
9<:
Si
74
6h
62
5/
5=
45
4:
4-
3<
I
I
I
o
o
.0000
. oo'oo
.0000
.0000
9999
9999
o
o
I
o
I
o
I
I
I
I
I
2
I
I
2
I
2
2
I
2
2
2
3
2
2
3
2
o 90
5o
4o
3o
20
10
1 o
10
20
3o
4o
5o
0.0176
O.O2O4
O.O233
0.0262
0.0291
O.O32O
29
29
29
29
29
0.0176
O.O2O4
0.0233
0.0262
0.0291
0. O320
67.2900
49. 1039
42.9641
38.1885
34.3678
3 i .2416
61
tf
5''
01
'. 2
o
o
o
.9998
.9998
9997
9997
.9996
.9996
o 89
5o
4o
3o
20
IO
2 o
10
20
3o
4o
5o
0.0378
0.0407
o.o436
o.o465
0.0494
29
29
29
29
29
29
29
29
29
29
3
29
29
29
29
29
29
29
29
0.0378
0.0407
o.o466
0.0496
28.6363
26.43i6
22.9038
21 .4?o4
2O.2O56
33
47
98
So
34
4
o
o
o
o
o
o
.9994
. 999 3
.9992
.9990
.9989
.9988
o 88
5o
4o
3o
20
10
3 o
10
20
3o
5o
0.0623 _
0.0662
0.0681
0.0610
o.o64o
0.0669
0.0624
o.o553
0.0682
0.0612
o.o64 i
0.0670
19.081 I
18.0760
17. ID93
16.3499
i5.6o48
14.9244
6 1
57
'.'4
5i
04
37
4 u
y8
"7
57
43
6,
o
o
o
.9986
.9986
.9933
.9981
.9980
.9978
o 87
5o
4o
3o
20
IO
4 o
10
20
3o
4o
5o
0.0698
0.0727
0.0766
0.0786
0.08 i 4
o.o843
0.0699
0.0729
0.0768
0.0787
0.0816
o.o846
14.3007
13.7267
i 3 . i 969
12.7062
12.2606
i i .8262
o
o
o
I)
.9976
9974
.9971
.9969
.9967
.9964
o 86
5o
4o
3o
20
10
5 o
o.
0872
0.0876
i i .43oi
(1
.9962
o 85
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' O
PP
.1
.2
3
4
.8
9
26053
16380
"245
8194
623*
4907
.1
.2
3
4
5
.6
: 7 8
9
396:
30 29
2605
5211
7816
10421
13027
15632
18237
20842
23448
1638
3276
4914
6552
8190
9828
11466
13104
14742
1125 .1
2249 .2
3374 -3
4498 .4
5623 .5
6747 .6
7872 .7
8996 .8
IOI2I .9
819.4
1638.8
3277-6
4097.0
4916.4
5735-8
6555-2
'374 6
623.7
1247.4
1871.1
2494.8
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
35*49
3.0 2.0
6.0 5.8
9.0 8.7
12. II. 6
iS-o M-5
18.0 17.4
21. 2O.3
24.0 23.2
270 26.1
160
FOUR-PLACE NATURAL FUNCTIONS.
o /
Sin. d.
Tang.
d.
Cotg.
d.
Cos.
d.
3
2
3
3
3
5 o
IO
20
3o
4o
5o
0.0872
0.0901
0.0929
o.ogSS
0.0987
o. 1016
29
28
29
29
29
29
29
29
29
29
29
29
29
28
29
29
29
29
29
28
29
29
29
28
29
29
28
29
29
28
d.
o
o
.0875
.0904
.0934
.0963
.0992
. 1 022
29
30
29
29
30
29
29
30
29
3
29
3
29
30
3
29
3
29
3
30
3
29
3
3
30
3
29
30
3
30
d.
n.43oi
II .0594
10.7119
10.3854
10.0780
9.78*2
37
34
32
30
28
*7
as
a*
33
22
at
20
'9
it
*7
16
16
*5
14
*3
13
12
12
II
II
IO
to
so
9
^H
d
37
75
53
74
^8
38
,"
55
29
'4
->7
3
26
+6
7i
00
33
-2
'3
57
06
58
IO
68
26
B6
5
81
mmm
.
O
o
o
c
.9962
99 5 9
99 5 7
. 99 54
-99 51
9948
o 85
5o
4o
3o
20
IO
6 o
10
20
3o
4o
5o
o.
o.
0.
o.
o.
o.
io45
1074
I io3
Il32
1 161
1 190
o
. io5i
. 1080
. i no
. 1 1 3g
. 1 169
.1198
9.5i44
9.2553
9.0098
8.7769
8.5555
8.345o
o
o
9945
9942
99 3 9
.9936
.9932
.9929
3
3
3
3
4
3
o 84
5o
4o
3o
20
IO
7 o
10
20
3o
4o
5o
o.
o.
0.
o.
o.
o.
1219
1248
1276
i3o5
1 334
i363
.1228
. 1257
.1287
.1346
.i3 7 6
8.1443
7.9530
7.7704
7-5 9 58
7.4287
7.2687
0.9925
0.9922
0.9918
0.9914
0.991 1
0.9907
4
3
4
4
3
4
o 83
'5o
4o
3o
20
IO
8 o
10
20
3o
4o
5o
o.
o.
0.
0.
0.
o.
1392
i449
i4?8
i5o7
i536
o
o
o
o
o
.i4o5
.i435
.1465
. i495
. 1 524
.i554
.i584'
.1614
.1644
.1673
. 1703
.i 7 33
7. i i 54
6.9682
6.8269
6.6912
6.56o6
6.4348
0.9903
0.9899
0.9894
0.9890
0.9886
0.9881
4
4
5
4
4
5
4
5
4
5
5
5
o 82
5o
4o
3o
20
10
9 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
0.
1 564
i5g3
1622
i65o
1679
1708
6. 3 i 38
6. 1970
6.o844
5.9758
5.8708
5.7694
0.9877
0.9872
0.9868
0.9863
0.9858
0.9853
o 81
5o
4o
3o
20
10
10 o
i
o. 1736
Cos.
o. 1763
Cotg.
5.6 7 i3
Tang.
Ml
.9848
^cr
Sin.
HIM
d.
o 80
ommm^m*
f
PP 2738
1533
981
.1
.2
3
4
'.6
3
*9
28
.1
.2
3
5 4
3
.1 273.8
2 547- 6
3 821.4
-4 I09S-2
-5 1369-0
.6 1642.8
.7 1916.6
.8 2190.4
.9 2464.2
'53-3
306.6
459-9
613.2
766.5
919.8
1073.1
1226.4
1370.7
98.1
196.2
294-3
392-4
490.5
588.6
686.7
784.8
882.9
3-0
6.0
9.0
2.9
i: 7
2.8
5-6
8.4
0.5
I.O
0.4
0.8
1.2
0.6
-9
10.0
21.0
24.0
27.0
"7-4
20.3
23.2
26.1
14.0
16.8
19.6
22.4
25.2
.6
9
2.5
3-
3-5
4.0
4-5
2.O
2-4
2.8
3-2
3-6
'5
1.8
2.1
2-4
2-7
FOUR-PLACE NATURAL FUNCTIONS.
o
'
<
Jin.
(1.
Tang
d.
Cot
?.
C
L
Cos.
d.
10
to
10
3o
$0
5o
O
O
o.
o.
o.
o.
1736
1765
1794
1822
i85i
1880
29
29
28
29
29
o. 176
0.179
0.182
o.i85
0.188
0.191
3
3
3
3
3
4
3
3
3
30
3'
5.67
5.5?
5.48
5.3 9
5.3o
5.22
i3
64
45
55
9 3
57
9
9
a
&
8
49
9
< >
Sa
;'.
0.9848
0.9843
0.9838
0.9833
0.9827
0.9822
5
5
5
6
5
o 80
5o
4o
3o
20
IO
11
,
o
to
10
Jo
io
io
o.
o.
o.
o.
o.
0.
1908
1937
i 9 65
1994-
2O22
2o5i
29
28
29
28
29
0.194
0.197
O.2OO
O.2o3
O.2O6
0.209
4
4
4
5
5
5
3
3
3 1
3
3
5.i4
5.o6
4.98
4.91
4.84
4-77
46
58
94
52
3o
2 9
t
t
7
y
7
f.
3S
'4
22
11
3,
0.9816
0.9811
0.9805
0.9799
o. 979 3
0.9787
S
6
6
6
6
o 79
5o
4o
3o
20
IO
12
t
10
JO
Jo
fo
)0
0.
o.
o.
o.
o.
o.
2079
2108
2i36
2164
2193
2221
29
28
28
29
28
O.2I2
O.2l5
0.218
O.22I
O.224
O.227
6
6
6
7
7
s
3
3
3i
3
3i
4.70
4.63
4.5 7
4.5i
4.44
4.38
46
82
36
07
94
97
6
6
6
A
5
4
t*
20
3
'7
b
0.9781
0.9775
0.9769
0.9763
0.9757
0.9750
6
6
6
6
7
o 78
5o
4o t
3o
20
10
13
i
JO
io
fo
o.
o.
o.
o.
o.
o.
2260
2278
23o6
2334
2363
23 9 i
29
28
28
28
29
28
O.23o
0.233
0.237
O.24o
0.243
O.246
9
9
I
2
2
30
3'
3'
3'
30
4.33
4.27
4.21
4.16
4. 1 1
4.06
i5
47
9 3
53
26
1 1
5
5
5
5
S
'.8
M
to
IJ
5
0.9744
0.9737
0.9730
0.9724
0.9717
0.9710
7
7
6
7
7
o 77
5o
4o
3o
20
10
14
i
\
10
Jo
|o
io
o.
o.
o.
o.
0.
o.
2419
2447
2476
25o4
2.532-
256o
28
29
28
28
28
O.249
O.252
o.255
o.s58
0.261
0.264
i
4
5
6
7
S
31
3"
3'
3'
3>
4.01
3.96
3.91
3.86
3.82
3. 77
38
17
36
5?
38
5o
4
4
4<
4!
44
)'
b
Sj
9
s
0.9703
0.9696
0.9689
0.9681
0.9674
0.9667
7
7
7
8
7
7
o 76
5o
4o
3o
20
IO
15
o.
2588
0.267
9
3.73
21
0.9659
o 75
c
OS.
d.
Cotg
d.
Tan
S-
d
.
Sin.
d.
'
PP
7
43
448
31
30
29
a8
7
6
5
.i
.2
3
4
'.6
IB
g
3
14
K
~<
35
44
Ji
4.2
8.4
2.6
,6.8
I.O
5-2
9-4
3.6
7.8
44.8
89.6
134-4
179.2
224.0
268.8
313-6
3584
6.
9-
12.
15-
18.
21.
24-
i .1
2 .2
3 -3
4 -4
5 -5
5 .6
7 -7
8 .8
i
i
i
2
2
3-o
6.0
9.0
2.O
5-o
8.0
I.O
4-o
2.9
* 5-8
8-7
n.4
M-5
17-4
20.3
*3- 2
2.
s'.
II.
14-
16.
19.
22.
s
6
4
2
)
3
6
4
1
i 0.7
2 1.4
3 2.1
4 2.8
5 3-5
6 4.2
7 4-9
8 5.6
0.6
1.2
1.8
2-4
3-
3.6
4-2
4-8
-5
I.O
'5
2.O
2-5
3-o
3-5
4.0
152
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg,
d.
Cos.
d.
15 o
10
20
3o
4o
5o
o.
0.
o.
0.
o.
o.
2588
2616
2644
2672
2700
2728
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
27
28
28
27
28
28
27
28
=7
28
27
27
28
27
o. 2679
0.271 1
0.2742
0.2773
o.28o5
0.2836
32
31
32
31
3'
3 2
32
3'
32
3 1
32
3 2
32
32
32
3 2
S 2
33
32
S 2
33
32
33
S 2
33
33
33
33
3.7321
3.6891
3.64?o
3.6o59
3.5656
3.526i
4;
3*
3'
3
35
*
3!
3!
31
3:
3:
y
31
9
3<
3 r
2(
a
ai
2S
5
5
at
at
2
2_
a;
I
>3
B
7
9
i
5
7
o
3
3
B
5
9
3
>7
12
7
i
7
i
7
2
3
3
9
5
o
0.9659
0.9652
0.9644
0.9635
0.9628
0.9621
7
8
8
S
7
o 75
5o
4o
3o
20
IO
16 o
10
20
3o
4o
5o
o.
0.
o.
o.
o.
0.
2 7 56
2784
2812
2840
2868
2896
0.2867
o. 2899
0.2931
0.2962
0.2994
o. 3o26
3. 48 7 4
3.4495
3.3769
3. 34o2
3.3o52
0.9613
0.9605
0.9596
0.9688
0.9580
0.9572
8
9
8
8
8
o 74
5o
4o
3o
20
IO
17 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
2924
2952
2979
3007
3o35
3o62
o. 3o8g
0.3I2I
o.3i53
o.3i85
0.3217
3.2709
3.2371
3.2o4i
3.1716
3. 1 397
3.io84
0.9553
o. 9 555
0.9546
0.9537
0.9528
0.9520
9
8
9
9
9
8
9
9
IO
9
9
9
o 73
5o
4o
3o
20
10
18 o
10
20
3o
5o
o.
o.
o.
o.
0.
o.
3ogo
3n8
3i45
3i 7 3
32OI
3228
0.3249
o.328i
o.33i4
0.3346
0.3378
o.34n
3.0777
3.0178
2.9887
2.9600
2.9319
o.g5i i
0.9602
0.9492
0-9483
0.9474
o 72
5o
4o
3o
20
IO
19 o
10
20
3o
4o
5o
o.
o.
o.
o.
0.
o.
3256
3 2 83
33n
3338
3365
33 9 3
0.3443
0.3476
o.35o8
o.354i
0.3574
0.3607
2.9042
2.8770
2.85O2
2.8239
2.7980
2.7725
0.9455
0.9446
0.9436
0.9426
0.9417
0.9407
9
IO
IO
9
IO
o 71
5o
4o
3o
20
10
20
o.
3420
o.364o
2.7475
0.9397
o 70
Cos.
d.
Cotg.
d.
Tang.
d
Sin.
d.
'
PP
.1
,2
3
4
'.6
:l
9
255
33
3*
31
28
37
,i
.2
3
4
.6
!i
9
IO
9
8
25-5
5.o
76.5
102.0
127-5
153-0
178.5
204.0
6.6
9-9
%l
19.8
23-1
26.4
3.2 .1
6.4 .2
9.6 .3
12.8 .4
16.0 .5
19.2 .6
22.4 .7
25.6 .8
28.8 .9
6.' 2
9-3
12.4
'5-5
1 8.6
21.7
24.8
2.8
5-6
8-4
II. 2
14.0
1 6. 8
19.6
22.4
2-7
5-4
8.1
10.8
13 5
16.2
18.9
21.6
24-3
I.O
2.0
3.0
4.0
6.0
7.0
8.0
0.9
1.8
2-7
3-6
4-5
5-4
6-3
7.2
8.1
0.8
1.6
2-4
3-2
4.0
48
5-6
6.4
i53
FOUR-PLACE NATURAL FUNCTIONS.
O f
Sin.
d.
as
27
27
27
28
27
27
27
27
27
27
27
27
27
27
27
27
26
27
27
26
27
27
26
27
26
27
26
27
26
Tang
. d.
Cotg. d.
Cos. d.
20 o
10
20
3o
4o
5o
o.
o.
o.
o.
'o.
0.
3420
3448
34 7 5
35o2
3539
355 7
o. 364o
o.36 7 3
o.3 7 o6
o.3 7 39
o.3 77 2
o.38o5
33
33
33
33
33
34
33
34
33
34
33
34
34
34
34
34
34
2. 7 4 7 5
2. 7 228
2.6985
2.6 7 46
2. 65 i i
2.6279
-'.
-'-
2_
2;
2;
2:
th
2!
21
21
21
2C
2C
2C
2C
'9
19
iC
'9
if
18
if
18
'7
'7
'7
'7
16
16
16
7
3
9
5
2
S
t
I
9
4
2
9
3
o
7
5
i
6
5
i
11
7
4
3
n
S
i
o. 9 3 97
o.9~3 7
-9 3 77
o.936 7
0.9356
0.9346
10
n
IO
' 70
5o
4o
3o
20
10
21 o
10
20
3o
4o
5o
o.
o.
0.
o.
o.
o.
3584
36n
3638
3665
3692
3719
i
i
I
(
<
.383 9
.38 7 2
.3go6
.3 9 3 9
.3 97 3
.4oo6
2.6o5i
2.5826
2 .56o5
2.5386
2.5l 7 2
2 .4960
0.9336
o. 9 325
o.gSiS
0.9304
0.9293
0.9283
II
10
II
II
IO
o 69
5o
4o
3o
20
IO
22 o
10
20
3o
4o
5o
o.
o.
b.
o.
o.
o.
3 7 46
3 77 3
38oo
382 7
3854
388i
(
<
4o4o
4o 7 4
.4108
.4142
4i 7 6
.4210
2.4 7 5i .
2.4545
2.4342
2.4i4z
2.3945
2.3 7 5o
0^9272
0.9261
0.9250
0.9239
0.9228
0.9216
II
II
II
II
12
o 68
5o
4o
3o
20
10
23 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
3 97
3 9 34
3961
3 9 8 7
4oi4
4o4i
b
o
.4245
42 7 9
.43i4
.4348
.4383
44i 7
34
35
34
35
34
35
35
35
35
35
36
35
2.3559
2.3369
2.3i83
2.2998
2.28l 7
2.263 7
0.9205
0.9194
0.9182
o.9i 7 i
0.91 59
o.9i4 7
II
12
II
12
12
o 67
5o
4o
3o
20
IO
24 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
4o6 7
4094
4l2O
4i4 7
4'i?3
4200
o
.4452
448 7
.4522
.455 7
.4592
.4628
2.246O
2.2286
2.21 l3
2.1 9 43
2.1 77 5
2. 1609
o
o
o
o
o
.9135
.9124
.9112
.9100
.9088
. 9 o 7 5
II
12
12
12
'3
12
o 66
5o
4o
3o
20
IO
25
0.
4226
(1
.4663
2.1445
o
.9063
o 65
Cos.
d.
cotg.
d.
Tang.
d.
Sin.
d.
' O
PP
.1
.2
3
4
'.6
7
.8
9
177
35
34
3-4
6.8
IO.2
.3-6
17.0
20.4
2 3 .8
27.2
30.6
ii
.2
3
4
.(,
.8
33
97
26
,i
.2
3
4
!e
.8
12
ii
10
17.7
35-4
53-i
70.8
88.5
106.2
123.9
141.6
'59-3
3-5
7.0
10.5
14.0
'7-5
21.0
24-5
28.0
3-3
6.6
9-9
'3-2
.6.5
19.8
23.1
26.4
2.7
5-4
8.1
10.8
'3-5
16.2
18.9
21.6
24.3
2.6
5 "o
7 .8
10.4
I 3 .0
I 5 .6
18.2
208
23-4
1.2
2.4
3-6
4.8
6.0
7.2
8.4
9.6
10.8
i.i
2.2
3-3
4-4
u
7-7
8.8
9-9
I.O
2.0
3
4.0
5
6.0
7.0
8.0
9.0
1 54
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang. 1 d.
Cotg.
d.
Cos.
d.
25 o
10
20
3o
4o
5o
0.
o.
o.
o.
0.
o.
4226
4253
4279
43o5
433i
4358
27
26
26
26
27
26
26
26
26
26
26
26
26
26
25
26
26
26
25 .
26
26
25
26
25
26
25
25
26
25
25
0.4663
0.4699
o .4?34
0.4770
o.48o6
o.484i
36
35
36
36
35
36
36
37
36
36
37
36
37
37
37
37
37
37
37
38
38
37
38
38
38
38
39
38
39
39
2. I 445
2.1283
2. 1123
2.0965
2.0809
2 .o655
162
160
158
'56
i54
152
150
'49
'47
MS
144
142
140
139
137
136
'34
133 .
IS'
130
128
127
126
"5
123
121
121
119
116
o
o
t
.9063
.9038
.9026
.9013
.9001
12
'3
12
'3
12
o 65
5o
4o
3o
20
IO
26 o
10
20
3o
4o
5o
o.
o.
o.
o.
' O.
0.
4384
44 10
4436
4462
4488
45i4
0.4877
o.4gi3
o . 495o
0.4986
0.5o22
o.5o5g
2.o5o3
2.O2O4
2.0057
1.9912
1.9768
0.8988
0.8975
0.8962
0.8949
o.8 9 36
0.8923
13
'3
'3
13
13
'3
'3
'3
14
'3
M
14
'3
'4
M
M
'4
'4
'4
'5
'4
o 64
5o
4o
3o
20
IO
27 o
10
20
3o
4o
5o
o.
o.
o.
0.
o.
o.
454o
4566
4592
4617
4643
4669
o . SogS
o.5 i 32
o.5i6g
o.52o6
0.5243
0.5280
i .9626
i .9486
1.9347
i .9210
i .9074
0.8910
0.8897
0.8884
0.8870
0.8857
0.8843
o 63
5o
4o
3o
20
IO
28 o
10
20
3o
4o
5o
0.
o.
o.
o.
0.
o.
4720
4?46
4772
4797
4823
o.53i7
0.5354
0.5392
o. 543o
<?.546 7
o.55o5
1.8807
1.8676
1.8546
i.84i8'
i .8291
i.8i65
0.8829
0.8816
0.8802
0.8788
0.8774
0.8760
o 62
5o
4o
3o
20
10
29 o
10
20
3o
4o
5o
o.
o.
o.
0.
0.
o.
4848
48 7 4
4899
4924
4975
0.5543
o.558i
0.5619
0.5658
0.5696
o.5f35
i.8o4o
1.7917
1.7796
1.7675
1.7556
1.743?
0.8746
0.8732
0.8718
0.8704
0.8689
0.8675
o 61
5o
4o
3o
20
IO
30
o
o .
5ooo
0.5774
I .?32I
0.8660
o 60
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' O
PP
.1
149
131
39
38
37
36
,i
25
1.4
2.8
13
14.9
3-9 - 1
7.8 .2
$
3-7
3-6
2-5
'3
2.6
3
4
i
44-7
59- 6
74-5
89.4
104.3
119.2
134.1
39-3
52.4
65.5
78.6
91.7
104.8
117 9
"7 -3
15.6 .4
19-5 -5
23.4 .6
27-3 '7
31.2 .8
3S. i -9
11.4
15-2
19.0
22.8
26.6
3-4
34-2
II. I
14.8
18.5
22.2
25-9
29.6
10.8
14.4
18.0
21.6
25.2
28.8
32.4
3
4
.6
9
7-5
10.
12.5
15.0
17-5
20. o
22.5
4-2
5-6
7.0
8.4
9.8
II. 2
3-9
5-2
6.5
7.8
9.1
10.4
n.y
i55
FOUR-PLACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
30 o
10
20
3o
4o
5o
0.
o.
o.
0.
o.
o.
5ooo
5o25
5o5o
5o 7 5
5ioo
5i25,
25
25
25
25
25
25
25
25
25
25
25
24
25
24
25
25
24
24
25
24
34
25
24
o.5 77 4
o.58i2
o.585i
0.5890
o.SgSo
o. 5969
38
39
39
40
39
40
39
40
4
4
40
41
40
4'
4i
4'
4'
4'
42
4i
42
42
42
I . 7 32I
I . 7 2O5
i . 7 o9o
i.6 977
1.6864
i.6 7 53
116
"5
113
"3
in
no
109
108
107
107
105
0.8660
o.864a
o.863i
0.8616
0.8601
o.858 7
14
15
15
15
M
o 60
5o
4o
3o
20
10
31 o
10
20
3o
4o
5o
0.
o.
o.
o.
o.
o:
5i5o
5i 7 5
520O
5225
525o
5 2? 5
0.6009
o.6o48
0.6088
0.6128
0.6168
0.6208
1.6643
1.6534
i .6426
i.63i9
i .6212
i .6io 7
o.85 7 2
o.855 7
0.8542
0.8526
o.85 1 1-
0.8496
'5
15
15
16
'5
15
16
o 59
5o
4o
3o
20
10
32 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
5299
5324
5348
53 7 3
53 9 8
5422
0.6249
0.6289
o.633o
o.63 7 i
O.64l2
0.6453
i .6oo3
i . 5goo
i.5 7 98
i.569 7
i .559 7
i .549 7
104
103
102
101
100
100
o.848o
0.8465
o.845o
0.8434
o.84i8
o.84o3
'5
IS
16
16
'5
16
16
16
16
16
16
o 58
5o
4o .
3o
20
10
33 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
5446
54 7 i
54g5
55ig
5544
5568
o.64g4
0.6536
o.65 77
0.6619
0.6661
o.6 7 o3
i .5399
i.53oi
i .5204
i.5io8
i'.5oi3
i .4919
98
98
97
96
95
94
o.838 7
o.83 7 i
0.8.355
0.8339
0.8323
o.83o7
o 57
5o
4o
3o
20
10
34 o
10
20
3o
4o
5o
0.5592
o.56i6
o. 564o
0.5664
0.5688
0.5712
24
24
24
24
24
24
24
o
u
o
o
.6 7 45
.6 7 8 7
.683o
.68 7 3
.6916
.6959
42
43
43
43
43
43
1.4826
i.4 7 33
i.464i
i.455o
i .446o
i.43 7 o
93
92
9 1
90
90
0.8290
O.82 7 4
0.8258
0.8241
0.8225
0.8208
16
16
17
16
'7
16
o 56
5o
4o
3o
20
IO
35 o
o.
5 7 36
. 7 OO2
1.4281
0.8192
o 55
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' O
PP 43
42
4*
.1
.2
3
4
5
.6
40
as
*4
17
16
15
i 4-3
.2 8.6
3 '2-9
4 >7-2
5 21.5
.6 25.8
7 3-i
8 34-4
9 38.7
4.2
8. 4
12.6
16.8
21.0
25.2
29.4
33-6
I 1
8.2
12.3
,6.4
20.5
24.6
28.7
32.8
36.9
4.0
8.0
12.0
16.0
20. o
24.0
28.0
32.0
2.5
S.o
7-5
IO.O
12.5
15.0
'7-5
2O.O
2. 4 .1
4.8
7-2 -3
9.6 .4
12. .5
14.4 .6
16.8 .7
19.2 .8
21.6 .9
'7
3-4
5-1
6.8
8-5
IO.2
II.9
I 3 .6
1.6
11
&
9.6
II. 2
12.8
14-4
'5
3-
45
6.0
75
9.0
10.5
12.0
'3-5
1 56
FOUR PLACE NATURAL FUNCTIONS.
o /
Sin.
d.
Tang.
d.
Cotg.
d.
Cos.
d.
35 o
10
20
3o
4o
5o
o.
o.
0.
o.
0.
o.
5 7 36
6760
5 7 83
6807
583i
5854
24
23
24
24
23
24
23
24
23
24
23
23
23
24
23
23
23
o. 7002
0.7046
0.7089
0.7133
0.7177
0.7221
44
43
44
44
44
44
45
45
45
45
45
46
45
46
46
47
46
1.4281
1.4193
i .4io6
i .4019
i .3g34
1.3848
88
87
87
85
86
84
84
83
83
82
81
81
80
79
79
78
78
77
76
76
75
75
74
0.8192
0.8175
o.8i58
o.8i4i
0.8124
0.8107
17
J 7
i?
!7
'7
o 55
5o
4o
3o
20
IO
36 o
10
20
3o
. 4o
5o
o.
o.
o.
0.
o.
o.
58 7 8
5goi
5920
5 9 48
5 97 2
5995
0.7265
0.7310
'o. 7 355
0.7400
0.7445
0.7490
1.3764
i.368o
i .3597
i.35i4
1.3432
i.335i
0.8090
0.8073
o.8o56
0.8039
0.8021
o.8oo4
J 7
*7
'7
18
i?
o 54
5o
4o
3o
20
IO
37 o
10
20
3o
4o
5o
o.
o.
o.
0.
o.
o.
6018
6o4i
6o65
6088
61 1 1
6 1 34
0.7536
0.7581
0.7627
0.7673
0.7720
0.7766
i .3270
i .3190
i.3m
i.3o32
i .2954
1.2876
0.7986
o. 7969
0.7951
0.7934
0.7916
0.7898
17
18
'7
18
18
o 53
5o
4o
3o
20
IO
38 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
0.
6i5 7
6180
6202 '
6225
6248
6271
23
23
22
23
23
2 3
22
23
22
23
22
23
0.7813
0.7860
0.7907
0.7954
0.8002
o.8o5o
47
47
47
47
48
48
1.2799
i .2723
i .2647
i .2572
1.2497
1.2423
(.
i
o
.7880
7?62
.7844
.7826
.7808
779
18
18
18
18
18
o 52
5o
4o
3o
20
10
39 o
10
20
3o
4o
5o
o.
o.
o.
o.
o.
o.
6293
63i6
6338
636i
6383
64o6
0.8098
o.8i46
0.8195
0.8243
0.8292
0.8342
40
48
49
48
49
5
I .2349
I .2276
I .22O3
I.2l3l
i . 2o5g
1.1988
74
73
73
72
72
7i
o
.7771
. 77 53
. 77 35
.7716
.7698
.7679
*9
18
18
J 9
18
19
o 51
5o '
4o
3o
20
IO
40
O
o.
6428
0.8391
49
i . 1918
70
c
.7660
'9
o 50
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' O
PP
.1
.2
3
4
'.6
.8
9
48
47
4 6
45
44
*3
.1
.2
3
4
3
1
23
19
18
4-8
9.6
14.4
19.2
24.0
28.8
33-6
38.4
43.2
4-7
9-4
14.1
"18.8
23-5
28.2
32-9
37-6
4 .6
92 .2
I 3 .8 . 3
l8. 4 .4
23.0 .5
27.6 .6
32.2 .7
36.8 .8
41.4 .9
4-5
9.0
'3-5
18.0
22.5
27.0
3'-5
36.0
4 4
8.8
13.2
17.6
22.
26.4
30.8
35-2
2-3
4.6
6.9
9.2
"5
13.8
16.1
.8.4
2.2
4-4
6.6
8.8
II. O
'3-2
15.4
17.6
19.8
i. 9
3-8
5-7
7-6
9-5
11.4
13.3
'5.2
17.1
1.8
3-6
5-4
7-2
9.0
10.8
12.6
14.4
16.2
i5 7
FOUR PL ACE NATURAL FUNCTIONS.
O '
Sin.
d.
Tang.
d.
Cotg. d.
Cos.
d.
40 o
10
20
3o
4o
5o
o.
o.
o.
o .
o.
0.
6428
645o
64?2
64g4
65i 7
653 9
22
22
22
23
22
22
22
21
22
22
22
22
21
22
21
22
21
21
22
21
21
21
2O
21
21
21
2O
21
0.8391
o.844i
0.8491
o.854i
o.SSgi
0.8642
5
5
5
SO
Si
5'
5'
52
Si
52
53
52
53
53
53
54
54
54
55
55
55
55
56
56
56
57
57
57
58
58
i . 1918
I.I84?
1.1778
i . 1708
i . i64o
i . 1571
7i
69
70
68
69
67
68
67
66
66
66
65
65
64
64
63
64
62
63
62
61
61
61
61
60
60
59
59
59
58
0.7660
0.7642
0.7623
0.7604
0.7585
o. 7566
18
'9
'9
'9
19
o 50
5o
4o
3o
20
10
41 o
IO
20
3o
4o
5o
o.
o.
0.
0.
0.
o.
656i
6583
66o4
6626
6648
6670
0.8693
0.8744
0.8796
0.8847
0.8899
0.8952
i . i5o4
i.i436
1.1369
i.i3o3
i . 1237
i . 1171
0.754?
0.7528
0.7509
0.7490
0.7470
o.745i
'9
'9
'9
'9
20
'9
o 49
5o
4o
3o
20
IO
42 o
10
20
3o
4o
5o
0.
o.
o.
o.
o.
0.
6691
6713
6 7 34
6 7 56
6777
6799
0.9004
0.9057
0.9110
0.9163
0.9217
0.9271
i . i i 06
i . io4i
1.0977
i . 09 i 3
i .o85o
1.0786
o . 74,3 1
0.7412
0.7392
0.7373
0.7353
0.7333
'9
20
'9
20
20
o 48
5o
4o
3o
20
IO
43 o
10
20
3o
4o
5o
0.
o.
o.
o.
o.
o.
6820
684 1
6862
6884
6906
6926
0.9325
o.g38o
0.9435
0.9490
0.9545
0.9601
i .0724
i .0661
i .0699
i.o538
1.0477
i . o4 i 6
0.7314
0.7294
0.7274
0.7254
0.7234
0.7214
'9
20
20
20
20
20
o 47
5o
4o
3o
20
10
44 o
IO
20
3o
4o
5o
o.
o.
o.
o.
o.
0.
6g4 7
6967
6988
7009
7o3o
7060
0.9657
0.9713
0.9770
0.9827
0.9884
0.9942
i -o355
i .0295
i .O235
i .0176
i .01 17
i .oo58
0.7193
0.7173
o.7i53
0.7133
0.7112
0.7092
20
20
2O
21
20
o 46
5o
4o
3o
20
10
45
o.
7071
i .0000
i .0000
0.7071
o 45
Cos.
d.
Cotg.
d.
Tang.
d.
Sin.
d.
' o
PP
.1
.2
3
4
'.6
'.8
9
57
55
54
53
51
23 31
30
'9
5-7
11.4
17.1
22.8
28.5
34-2
39-9
45-6
gj
5-5
II.
16.5
22.
27-5
33-o
38.5
44-0
5-4 -i
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
10.6
15.9
21.2
26.5
3 1.8
37-i
42-4
5-'
IO. 2
'5-3
20.4
2 5-5
30.0
35-7
40.8
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
15.4 .7 14.7
17.6 .8 16.8
19.8 .9 18.9
2.O
4.0
6.0
8.0
IO.O
12.0
14.0
16.0
18.0
1.9
3-8
5-7
7.6
9-5
ii. 4
'3-3
'5-2
.7.1
i58
TABLE VIII.
SQUARES AND SQUARE ROOTS OP NUMBERS.
SQUARES OF INTEGERS FROM 10 TO 100.
N
1
2
3
4
5
6
7
8
9
10
IOO
121
1 44
169
196
226
256
289
324
36i
20
4oo
44 1
484
629
576
626
676
729
74
84i
3o
900
961
1024
1089
n56
1225
1296
i36g
1 444
l52I
4o
1600
1681
1764
1849
ig36
2O25
2116
2209
23o4
24OI
5o
2600
2601
2704
2809
2916
3o25
3i36
324g
3364
348 1
60
36oo
3 7 2I
3844
3969
4096
4226
4356
448 9
4624
4?6i
70
4900
5o4i
5i84
5329
54?6
5625
5 77 6
5 9 2 9
6o84
6241
80
64oo
656i
6724
6889
7066
7225
7 3 9 6
7 56 9
7744
7921
9
8100
8281
8464
8649
8836
9025
9216
9409
9604
9801
SQUARE ROOTS OF NUMBERS FROM TO 10; AT INTERVALS OF .1.
N
.0
.1
.2
.44?
.3
.4
.5
.6
.7
.837
.8
.9
o
.3i6
.548
.632
.707
775
.894
.949
i
2
3
1. 000
i.4i4
1.732
1/049
1.449
1.761
i. og5
1.483
1.789
i.i4o
1.517
1.817
i.i83
1.549
1.844
1.225
i. 58 1
1.871
1.265
i. 612
1.897
i.3o4
1.643
1.924
1.342
i.6 7 3
1.949
1.878
1.703
'975
4
5
6
2.OOO
2.236
2.449
2.025
2.258
2.470
2.o4g
2.280
2.490
2.074
2.3O2
2.5lO
2.098
2.324
2.53o
2. 121
2.345
2.55o
2.145
2.366
2.569
2.168
2.387
2.588
2.191
2.4o8
2.608
2.2l4
2.429
2.627
7
8
9
2.646
2.828
3.000
2.665
2.846
3.017
2.683
2.864
3.o33
2.702
2.881
3.o5o
2.720
2.898
3.o66
2. 7 3 9
2.915
3.082
2.757
2.933
3.098
2.775
2.950
3.n4
2.793
2.966
3.i3o
2.811
2.983
3.i46
SQUARE ROOTS OF INTEGERS FROM 10 TO 100.
N
1
2
3
4
5
6
7
8
9
10
3.162
3.3 1 7
3.464
3.6o6
3.742
3.8 7 3
4.ooo
4.123
4.243
4.35 9
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
5.65 7
5. 7 45
5.83i
5.916
6.000
6.o83
6.i64
6.245
4o
6.325
6.4o3
6.48i
6.55 7
6.633
6.708
6.782
6.856
6.928
7.000
5o
7.071
7-i4i
7.21 1
7.280
7.348
7.4i6
7-483
7.55o
7.616
7.681
60
7.746
7.810
7.874
7.937
8.000
8.062
8.124
8.i85
8.246
8.307
70
8.367
8.426
8.485
8.544
8.602
8.660
8.718
8-775
8.832
8.888
80
8. 9 44
9.000
g.o55
9.1 10
9.i65
9.220
9.274
9.327
9.38i
9-434
9
9-487
9.539
9.592
9-644
9.695
9-747
9.798
9-849
9.899
g.gSo
TABLE IX.
THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF
NUMBERS FROM TO 2.5, AT INTERVALS OF .1.
C
cosh ./
sinh ./
tanh oc
e*
e~'
. I
.2
.3
i .000
o
i .000
i .000
i .oo5
I .020
1. 045
. IOO
.201
.3o5
. IOO
.197
.291
i . io5
1 .221
i.35o
.905
.819
74i
.4
.5
.6
1.081
1.128
i.i85
.4u
.521
.63 7
.38o
.462
.53 7
i .492
i .649
i .822
.670
.607
.549
7
.8
9
1.0
i . i
I .2
1.3
1.255
i.33 7
1.433
7 5 9
.888
i .027
.6o4
.664
.716
2.0l4
2.226
2.460
497
.449
.407
1.543
i .175
.762
2.718
.368
i .669
i'.8n
1.971
j.336
i .5og
1.698
.801
.834
.862
3 ,oo4
3. 32o
3.669
.333
.3oi
.2 7 3
i.4
i.5
1.6
2. l5l
2.352
2.577
1.904
2. 129
2.376
.885
.goS
.922
4.o55
4.482
4. 9 53
.247
.223
.202
i-7
1.8
1.9
2.0
2. I
2.2
2.3
2.828
3. 107
3.4i8
2.646
2.942
3.268
.935
947
956
5.4?4
6.o5o
6.686
.i83
.i65
.i5o
3.762
3.627
.964
7 .38 9
-.i35
4.i44
4.568
5.o37
4.022
4.45 7
4.937
.970
.976
.980
8.]66
9.025
9-974
. 122
.III
. IOO
2.4
2.5
5.55 7
6.i32
5.466
6.o5o
.984
.987
I I .023
12. l82
.091
.082
1 60
TABLE X
CONSTANTS
MEASURES AND WEIGHTS
AND OTHER CONSTANTS
MEASURES AND WEIGHTS
English Measures
Metric M 'ensures
LENGTH
LENGTH
12 inches (in.) = i foot (ft.).
10 millimeters (mm.) = i centimeter (cm.).
3 feet = i yard (yd.).
10 centimeters = i decimeter (dcm.).
16^ feet = i rod (rd.).
10 decimeters = i meter (m.).
5280 feet = i mife (m.).
10 meters = i dekameter (dkm.).
6080.3 feet = i nautical mile.
10 dekameters = i hektometer (hkm.).
5.}$ yards = i rod.
10 hektometers = i kilometer (km.).
4 rods -i chain (ch.).
i foot =30.48 centimeters.
( = 39-37 inches,
i meter
\^ ( = 3.2808 feet.
i yard = .9144 meter.
i kilometer = 0.6214 mile.
i mile = 1.6093 kilometers.
SURFACE
SURFACE
144 sq. inches = - 1 sq. foot.
100 sq. millimeters = i sq. centimeter.
9 sq. feet = i sq. yard.
too sq. centimeters = i sq. decimeter.
3oJ sq. yards = i sq. rod.
160 sq. rods = i acre.
( = i sq. meter.
100 sq. decimeters <
(=i centare (ca.).
43560 sq. feet = i acre.
100 sq. meters = i are (a.).
640 acres = i sq. mile.
loo ares = t hektare (hka.).
i sq. inch =6.4516 sq. centimeters.
i sq. centimeter = 0.1550 sq. inch.
i sq. foot = 0.0929 sq. meter.
1= 1.196 sq. yards.
i sq. yard =0.8361 sq. meter.
, ,
= 10.764 sq. feet.
i acre =0.4047 hectare.
i are = 1076.48 sq. feet.
i hektare = 2.471 acres.
VOLUME
VOLUME
1728 cu. inches = i cu. foot.
looo cu. millimeters = i cu. centimeter.
27 cu. feet = i cu. yard.
1000 cu. centimeters = i cu. decimeter.
128 cu. feet = i cord (cd).
( = i cu. meter,
.ooocu. decimeters { = igtere(st)
i cu. inch = 16.387 cu. centimeters.
i cu. foot = 0.028 cu. meter.
i cu. cantimeter = 0.06 1 cu. inch.
i cu. yard = 0.7646 cu. meter.
(= 35-3M cu. feet.
i cord = 3.625 steres.
{= 1.308 cu. yards.
i stere = 0.2759 cord.
CAPACITY
CAPACITY
i liq. gal. = 3. 785 liters = 231 cu. in.
100 centiliters (cl.) = i liter (1.).
i dry gal. = 4-404 liters = 268.8 cu. in.
loo liters = i hektoliter (hkl).
i bushel =0.3524 hkl. =2150.42 cu. in.
i liter = 1.0567 liq. qts. = i cu. dcm.
AVOIRDUPOIS WEIGHT
METRIC WEIGHT
16 ounces (oz.) = i pound (lb.).
looo grams (gm.) = i kilogram (kilo.).
loo Ibs. = i hundredweight (cwt.).
looo kilograms = i tonneau (t.).
20 hundredweight = i ton (T.).
i gram = 15-432 grains.
i pound = .4536 kilo. = 7000 grains.
i kilogram = 2.2046 pounds.
i ton =.9071 tonneau (t).
i tonneau = 1.1023 tons.
TROY WEIGHT
i pound = 5760 grains = 12 ounces.
162
MEASURES AND WEIGHTS Continued
60 seconds (")= i minute (').
60 minutes = i degree ().
90 degrees = i right angle.
radians = i right angle.
It 3.141
27T = 6.2831853
41T =12. 5663706
= 1.0471976
- 4 "" = 4.1887902
-^ = 0.7853982
-r = 0.5235988
= 0.3183099
*= 9.8696044
I
= 0.1013212
Vr = 1.7724539
= 0.5641896
VTT
CONSTANTS
log it =0.4971499
log 2?r =0.7981799
log 47r =1.0992099
log =0.1961199
log = 0.0200286
log =0.6220886
3
log =9.8950899 10
log = 9.7189986 10
log =9. 5028501 10
log 7r = 0.9942997
log t = 9.0057003 10
logVw =0.2485749
log ' =9.7514251 10
Vit
= i. 1447299
e =2,718281828459
M =0.4342945
log* =0.4342945
log M = 9.6377843 10
^=2.3025851
log = 0.3622157
( = 57- 295779 5
Radian j =. 3437-747'
' = 206264.8"
log 57.2957795 = 1.7581226
log 3437-747 = 3-5362739
log 206264.8 = 5.3144251
i degree =0.0174533 radians
i minute = 0.0002909 radians
i second = 0.0000048 radians
log 0.0174533 = 8.2418774 10
log 0.0002909 = 6.4637261 10
log 0.0000048 = 4.6857749 10
i63
QA
531
Phillips-Elements of trigonometry;
plane and spherical
UNIVERSITY OF CALIFORNIA LIBRARY
Los Angeles
This book is DUE on the last date stamped below.
Form L9-116m-8,'62(D1237s8)444
Form L-9
23m-2,'43(5205)
^UNIVERSITY of CALTFORMTA
K
JOL72