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OLNEY'S MATHEMATICAL SERIES.
PLANE AND SPHERICAL.
BY EDWARD OLNEY,
MATHEMATICS IN THB UHIVKKSITY Off MICHIGAN
SHELDON & COMPANY,
No. 8 MURRAY STREET.
Entered according to Act of Congress in the year 1870. by
SHELDON & COMPANY,
In the Office of the Librarian of Congress at Washington.
PROF, OLNEY'S MATHEMATICAL COURSE,
INTRODUCTION TO ALGEBRA - - -
KEY TO COMPLETE ALGEBRA - . - -
KEY TO UNIVERSITY ALGEBRA -
A VOLUME OF TEST EXAMPLES IN ALGEBRA - - -
ELEMENTS OF GEOMETRY AND TRIGONOMETRY
ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University
ELEMENTS OF GEOMETRY, separate -
ELEMENTS OF TRIGONOMETRY, separate -
GENERAL GEOMETRY AND CALCULUS -
BELLOWS' TRIGONOMETRY -
PROF, OLNEY'S SERIES OF ARITHMETICS.
PRIMARY ARITHMETIC -
ELEMENTS OF ARITHMETIC - - - -
SCIENCE OF ARITHMETIC - - - - : - -
E. &. Trig.
PART IV. TRIGONOMETRY.
DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRI-
CAL FUNCTIONS OP AN ANGLE (OR ARC). PAGE.
Fundamental Relations of the Trigonometrical Functions of an
Signs of the Functions 7-10
Limiting Values of the Functions 10-14
Functions of Negative Arcs 15-16
Circular Functions 10
RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF DIFFERENT AN-
GLES (OR ARCS).
Functions of the Sum or Difference 20-25
Functions of Double and Half Angles 26
FORMULAE for rendering Calculable by Logarithms the Algebraic Sum
of Functions 29-30
CONSTRUCTION AND USE OF TABLES.
To Compute a Table of Natural Functions 32-33
To Compute a Table of Logarithmic Functions 33-34
Exercises in the Use of the Tables 34-39
Functions of Angles near the Limits of the Quadrant 39-42
Exercises . . 43
SOLUTION OF PLANE TRIANGLES. PAGE.
Of Right Angled Triangles 44-45
Exercises and Examples 45-48
Practical Applications 48-49
Of Oblique Angled Plane Triangles 49-51
Oblique Triangles Solved by means of Right Angled Triangles . . . 55-56
functions of the Angles in Terms of the Sides 58-59
Area of Plane Triangles 60-61
Practical Applications 61-64
PROJECTION OP SPHERICAL TRIANGLES.
Definitions and Fundamental Propositions 65-66
Projection of Right Angled Spherical Triangles 66-71
Projection of Oblique Angled Spherical Triangles 71-73
SOLUTION OP RIGHT ANGLED SPHERICAL TRIANGLES.
Definitions , 73-75
Napier's Rules 76-78
Determination of Species 79-81
Exercises in Solution of Right Angled Spherical Triangles 81-84
Quadrantal Triangles 84-85
OP OBLIQUE ANGLED SPHERICAL TRIANGLES.
To find the Segments of a Side made by a Perpendicular let fall
from the opposite angle 85-86
The Relation of the Sides and Opposite Angles 87
Solution of Oblique Spherical Triangles by Napier's Rules, in
Three Problems 87-90
Exercises .... 90-95
GENERAL FORMULAE. PAGE.
Angles as Functions of Sides, and Sides as Functions of Angles. 96-100
Gauss's Equations 100-101
Napier's Analogies 102
Exercises in the Use of these Formulae 102-107_
Aiea of Spherical Triangles 108-110
Practical Applications of Spherical Trigonometry 110-113
Introduction to the Table of Logarithms 1-10
Table of Logarithms of Numbers 11-28
Table of Natural Sines and Cosines, and of Logarithmic Sines, Cosines,
Tangents, and Cotangents 29-74
Table for Precise Calculation of Functions near their Limits 75-78
Table of Tangents and Cotangents 79-88
DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE
TRIGONOMETRICAL FUNCTIONS OF AN ANGLE (OR ARC).
1. Trigonometry is a part of Geometry which has for its sub-
ject-matter, Angles. It is chiefly occupied in presenting a scheme
for measuring and comparing angles, by means of certain auxiliary
lines called Trigonometrical Functions^ investigating the relations
between these functions, and in the solution of triangles by means
of the relations between their sides and the trigonometrical functions
of their angles.
2. Plane Trigonometry treats of plane angles and triangles,
in distinction from Spherical Trigonometry, which treats of spherical
angles and triangles.
3. A Function is a quantity, or a mathematical expression,
conceived as depending upon some other quantity or quantities for
ILL'S. A man's wages for a given time is a function of the amount received
per day ; or, in general, his wages is a function of both the time of service and
the amount received per day. Again, in the expressions y = 2ax*, y = x* -
2bx + 5,y = 2 log ax, y = a x ,y is a function of a: ; since, the numbers 2, 5, a and b
being considered fixed or constant, the value of y depends upon the value we
assign to x. For a like reason such expressions as \/a a x* , and 3oa; 3 2v/a"j
may be spoken of as functions of a:. Once more, the area of a triangle is a func-
tion of its base and altitude.
4. Angles as Functions of Arcs. Wo have learned in
Geometry (PART II., SEC. VI.), that angles and arcs may be treated
as functions of each other; and that, if the angles be taken at the
2 PLANE TRIGONOMETRY.
centre of the same or equal circles, the arcs intercepted nave the
game ratio as the angles themselves, and hence may be taken as then
measures or representatives. For trigonometrical purposes, an angle
is considered as measured by an arc struck with a radius 1, from the
angular point as a centre.
5. A Degree being the -g-J-^ part of the circumference of a circle,
becomes the measure of -fa of a right angle; and, for convenience, it
is customary to speak of such an angle as an angle of one degree, of
four times as large an angle as an angle of four degrees, etc., apply-
ing the term directly to the angle. A small circle written at the
right and a little above a number indicates degrees ().
6. A Minute is fa part of a degree. Minutes are designated by
an accent ('). A Second is -fa part of a minute. Seconds are
indicated by a double accent ("). Smaller divisions of angles (or
arcs) are most conveniently represented as decimals of a second
thougn the designations thirds, fourths, etc., are sometimes met with,
and signify further subdivisions into 60ths. 5 12' 16" 13'" is read,
" 5 degrees, 12 minutes, 16 seconds, and 13 thirds."
ILL'S. In Fig. I AOB is an angle of 35, because the measuring arc ab
contains 35 of the 360 equal parts into which
the circumference whose radius is 0a, could be
divided. In like manner BOC is an angle of 7.
BOC = iAOB = iAOC. Hence, it becomes evi-
dent that we may use the numbers 35, 7, and 42
to represent the respective angles AOB BOC
and AOC, or the corresponding arcs ab, be, and
7. A Quadrant is an arc of 90,
and is the measure of a right angle;
hence, a right angle is called an angle of
**> * 90. Thus arc ad, Fig. 1, = 90. or angle
AOD = 90.
8. The Complement of an angle or arc is what remains aftei
subtracting the angle or arc from 90. The Supplement of an angle
or arc is what remains after subtracting the angle or arc from 180
ILL'S. In Fig. 1, the angle BOD is the complement of AOB, and the arc 3d
is the complement of arc ab. The complement of 35 is 90 35- 55. Thf
supplement of 35 is 180- 35= 145.
DEFINITIONS AND FUNDAMENTAL RELATIONS. 3
,9. A Quadrant is often represented by ^T, since <n is the semi-
cirou inference when the radius is unity. When this notation is used,
the Unit Arc becomes - = 57.29578 nearly, or 57 17' 44 ".8 +,
which is an arc equal in length to the radius.
10. For trigonometrical purposes, an angle is conceived as ger
erated by the revolution of a line about the angular point, ana
hence may have any value whatever, not only from to 180, but
from to 360, and even to any number of degrees greater than
360, as 1280, etc. An angle of 45 is generated by of a revolu-
tion, 90 by J of a revolution, 180 by i a revolution, 270 by } of a
revolution, 360 by one revolution, 450 by 1J revolutions, 1280 by
3$ revolutions, etc., etc.
11. In accordance with the conception of an angle as generated
by a revolving line, the measuring arc is considered as originating
at the first position of the revolving line (i. e., with one side of the
angle), and terminating in the line after it has generated the angle
under consideration (i. e., with the other side of the angle), The
first extremity is called the Origin of the arc, and the other the
ILL'S. In Fig. 1, let the angle AOB be considered as generated by a line
starting from the position OA, and revolving around the point O, from right to
left,* till it reaches the position OB. Oa being taken as unity, the arc cub is the
measuring arc of the angle AOB ; a is its origin, and b its termination.
12. In the generation of angles by means of a revolving line, the
normal motion is considered to be from right to left, and the quad-
rants are numbered 1st, 2d, 3d, and 4th, in the order in which they
13. The Trigonometrical Functions are eight in num-
ber; viz., sine, cosine, tangent, cotangent, secant, cosecant, versed-
sine, and cover sed- sine. These lines are functions of angles, or.
what amounts to the same thing, of arcs considered as measures of
angles, and are the characteristic quantities of trigonometry.
14. The Sine of an angle (or arc) is a perpendicular let fall
from the termination of the measuring arc upon the diameter passing
through the origin of the arc. Thus in Fig. 2, bd is in each case
the sine of the angle AOB, or of the arc axb.
* The pupil will understand that, if he imagines himself standing at the centre of moti .n.
a* the moving body or point passes before him, the distinctions " from right to left." and
14 from left to right," are easily made.
15. The Trigonometrical Tangent of an angle (or arc)
is a tangent drawn to the measuring arc at its origin, and limited
by the produced diameter passing through the termination of the
arc. Thus in Fig. %, ac is in each case the tangent of the angle AOB,
or of the arc axb.
16. The Secant of an angle (or arc) is the distance from the
angular point, or centre of the measuring circle, to the extremity of
the tangent of the same angle (or arc). Thus in Fig. 2, Oc is in
each case the secant of the angle AOB, or of the arc axb.
17. The Versed-Sine of an angle (or arc) is the distance
from the foot of the sine of the same angle (or arc) to the origin of
the measuring arc. Thus, in Fig. 2, da is in each case the versed-
sine of the angle AOB, or of the arc axb.
18. The prefix co, in the names of the four trigonometrical func-
tions in which it occurs, is an abbreviation for the word complement.
Thus cosine means complement-sine, i. e., the sine of the comple-
ment; cotangent means tangent of the complement; etc. The co-
sine of 40 is the sine of 90 40, or 50 ; the cosine of 110 is the
sine of 90 110, or 20; the cotangent of 30 is the tangent of
60 ; the cosecant of 200 is the secant of - 110.
j.9. Construction of the Complementary Functions.
Let us now see how the complementary functions are constructed with refer-
uce to their primitives, premising that all arcs in ffy. 3, reckoned from A, are
DEFINITIONS AND FUNDAMENTAL RELATIONS.
to foe reckoned around from right to left in this discussion. 1st. Let AP be
uny arc less than 90 ; then 90' AP = aP is its complement. Now considering
a as the origin and P the termination of
this complementary arc, Pd is its sine, at
its tangent, Ot its secant, and ad its versed -
sine. Hence, Pd, at, Ot, and ad are respect-
ively the cosine, cotangent, cosecant, and
coversed-sine of the arc AP, or the angle
AOP. 2d. Letting APP' be any arc between
90 and 180, its complement is 90 APP'
or aP', the sign signifying that the arc
is reckoned backward from P f to a. But as
the values of the functions will be the same
whether the origin be taken at P' or at a,
we may take a as the origin of this comple-
mentary arc, and P' as its termination,
whence P' d' becomes its sine, at' its tan-
gen t, Ot' its secant, and ad' its versed-sine.
Therefore P'd', at', Of, and ad', are respect-
ively the cosine, cotangent, cosecant, and Fl - 3.
coversed-sine of the arc APP', or the angle AOP'. 3d. In like manner, aP' is
shown to be the complement of arc APP'P" ; and as P"d", at, Ot, and ad!"
are respectively the sine, tangent, secant, and versed-sine of this complement,
they are the corresponding cofunctions of the arc APP'P", or the salient angle
AOP" 4th. In the same way, it appears that P r "d'", at', Of, and ad'" are the
cosine, cotangent, cosecant, and coversed-sine of the arc APP'P" P'", or the
salient angle AOP'". Observe that a, a point on the measuring arc 90 from the
primitive origin, is the origin of all the complementary functions.
SCH. It will readily appear from the figure that the cosine of an angle
(or arc) is always equal to the distance from the foot of tJie sine to the vertex of the
angle (or the centre of the measuring arc). This is the more convenient prac-
tical definition. Thus the cosine of AP is Pd = DO; the cosine of APP' is
P'd 1 = D'O, etc.
20. Notation. Letting x represent any angle (or arc), the
several trigonometrical functions of it are writteL sin a:, cos a;, tana;,
cots, sec a;, cosecz, versa;, and covers x. They are read "sinea,"
" cosine x" " tangent x" " cotangent <B," etc.
FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRICAL
FUNCTIONS OF AN ANGLE (OR ARC).
[Note. These fundamental relations rant he made perfectly familiar They must be
memorized, and be as familiar as the Multiplication Table. The student can do nothing in
trigonometry without them.]
Hf The discussions in this treatise all proceed upon one general
plan ; viz., First obtain the particular propwty of the
sine and cosine, and from this deduce all the other*
according to the dependencies shown in the follow-
21. Prop. The Fundamental Relations which the Trigono-
metrical Functiw* *ustain to each other are:
(1) sin 2 x + cos 9 x = 1 ;
. . sin a;
(2) tan x= ;
v ' '
/ox cos a;
(3) cot x = -r
tan x '
(5) seca; = :
. ' sin a;'
(7) sec 2 x = 1 '4- tan 8 x ;
(8) cosec'a; = 1 + cot 8 a;
(9) vers x = 1 cos x ;
(10) covers # = 1 sin a:.
(The forms sin a a;, sec 2 a;, etc., signify the square of the sine, the
square of the secant of x, etc., and are read " sine square x," " secant
square x" etc. The student should distinguish between sin a a;, and
sin a; 2 .)
DEM. In Fig. 4, let x represent any arc as AP, less than 90. Then PD = sin *,
OD or Pd = cos#, AT = tana?, OT = seca, at = cotz, OZ = cosecz, AD = versin *,
and ad = co versin x.
= ; but op
(1). In the right-angled triangle POD
PD 3 + OD 2 OP 2 > 01> sin 2 a; + cos 2 a; = 1,
since OP = radius = 1.
(2). From the similar triangles POD and
AT _ PD f> _ sin a;
OA OD' ~cosa;.
(8). From the similar triangles POd and
at Pd cos x
= 7r-3, Or COtiC -: .
Oa Od sin x
(4). Multiplying (2) and (3) together,
sin a; cos a; 1
tan#cot2:= : = 1, 01 tan;? =
cos x sin x cot*
(5). From the similar triangles OTA and
' * ' ~~ COS X
DEFINITIONS AND FUNDAMENTAL RELATIONS.
(6). From the similar triangles Ota and OPd,
Of OP 1
TT- = X-TJ or cosec# = - .
Oa Od' sin a-
(7). From the right-angled triangle OAT,
OT a = OA a -f AT 2 , or sec'z = 1 + tan 1 *.
(8). From the right-angled triangle
0^ = 2 + otf 2 , or cosec' 2 ^ = 1 4- cot a z.
(9). AD = AO - OD, or vers x = 1 - cos x.
(10). ad = aO Od, or covers x 1 sin x.
Thus the fundamental relations of the functions are established for an are
less than 90. But it will readily appear that the relations are the same for any
other arc. For example, let x = AP' be any arc between 90 and 180. Then
the triangle P'D'O gives sin a # + cos 2 z 1, since P'D' = sin #, and OD' = cos x.
The similar triangles P'D'O and OAT' give ^ = ^~ t or tan x = B 2^.. anc | t i ic .
similar triangles P'd'O and t'aO give cot x = - - . In like manner let the
student observe the relations when x = APP'P", or an arc between 180 and
270. So also when x = APP'P"P'", or an arc between 270 and 360.
22. COR. 1. The tangent and cotangent of the same angle are
reciprocals of each other ; so also are the secant and cosine, and the
cosecant and the sine. Thus, if tana; 3, cotx ; since cot a; =
- . If sectf = 2, cosa; = 4-: since sec a; = - , or cosz = - .
tana: cos a; sec a;
23. COR. 2. Sines and cosines cannot exceed' I. Tangents and
cotangents can have any values from fo^oo. Secants and cosecants
can have any values between*! and*-ao . Versed-sines and cover sed-
gines can have any values betiveen and 2. These conclusions' will
readily appear from the definitions, and an inspection of Fig. 4.
SIGNS OF THE TRIGONOMETRICAL FUNCTIONS.
24. Prop. Angles (or arcs) considered as generated from right
to left being called positive * and marked +, those considered as gen-
erated from left to right are to be called negative and marked .
* This ie purely an arbitrary convention. We might equally well reverse It
DEM. This is a direct application of the significance of the 4- and signs.
See Complete School Algebra, pp. 20-23.) Thus, in Pig. 5, if the angle AOP,
considered as generated by the revolution
of a line from the position OA in the direc-
tion of the arrow-head (from right to left),
is called positive and marked + , an angle
generated by the motion of a line from the
position OA in the opposite direction (from
left to right), as the angle AOP" thus gen-
erated, is to be considered negative and
marked . Let it be carefully observed
that it is the assumed direction of the
motion of the generatrix that determines
the sign of the angle (or arc). Two lines
meeting at a common point may be con-
sidered as designating either a -f or a
angle, according to the direction of motion
assumed. Thus the lines OA and OP', Fig.
5, may form the positive angle measured
Fia. 5. by the arc APP', or the negative, salient
angle measured by the negative arc AP'"P"P'. Q. E. D.
25. Prop. Radius being considered as always extending in the
same direction, viz., from the centre toivard the circumference, is
26. Prop. The sign of the sine of an angle between and 180
being +, that of an angle between 180 and 360 is .
DEM. In Fig. 5. we observe that the sines of all angles terminating in the 1st
and 2d quadrants, . ., between and 180, may be considered as measured
from the primary diameter AB, upward, while those of angles terminating in the
3d and 4th quadrants, i. e., between 180 and 360, are reckoned downward
(rom the same line; hence, the former being called + , the latter should be ,
is tlie two species are estimated in opposite directions. Q. E. D.
A more elegant conception is to consider the sine as projected upon the diam-
eter vertical to that passing through the origin, as aC ; whence Qd is the sine
of AOP (or arc AP). Now this line evidently is when the angle is ; and as the
angle increases, the sine increases, being generated from upward, and hence
is called + . This is the same conception as we use in the case of the cosine.
Adopting it, we see that sines reckoned from O upward are +, and downward
. Cosines reckoned from to the right, are + , and to the left, .
27. COR. The cosecant of an arc has the same sign as its sine^
eince coseca; = ; and as 1, being the radius, is +, the sign of
is the same as the sign of sin x.
DEFINITIONS AND FUNDAMENTAL RELATIONS.
28. Prop. The sign of the cosine of an angle between and
90, and between 270 and 360, is +, while that of an angle between
90 and 270 is -.
DEM. In Fig. 5, we observe that the cosines of all angles terminating in the
1st and 4th quadrants, may be considered as estimated from the centre towarc.
the right, as OD, OD'"; while correspondingly, the cosines of angles terminating
in the 3d and 3d quadrants will be estimated from the centre toward the left, aa
OD', OD". Hence, by reason of this opposition of direction, the former are
called +, and the latter . Q. E. D.
29. COR. The secant of an angle has the same sign as its cosine,
since these functions are reciprocals of each other. (See #7.)
30. Prop. The sign of the tangent of an angle between and
90, and also between 180 and 270, is + ; while that of an angle
between 90 and 180, and between 270 and 300, is -.
DEM. Since tan x = ?!B_ w hen sin x and cos x have like signs, tan a; is + , by
the rules of division; and when sin * and cos a- have different signs,* tan a is .
Now, in the 1st and 3d quadrants* the signs of sin a: and cos a are alike, hence
in these quadrants tana; is plus; but in 2d and 4th quadrants sin a: and cos a
have unlike signs, and consequently in these tan x is . Q. E. D.
31. COR. The sign of the cotangent is the same as the sign of the
tangent of the same angle, since cot x = .
32. Prop. Versed-sine and coversed-sine are always +.
DEM. Vers x = 1 - cos x and as cos a; cannot exceed 1, 1 cos a? is al
ways +. In like manner, covers a; = 1 sin x\ and as sin a; cannot exceed 1,
1 sin x is always 4- . Q. E. D.
SCH. 1. It is essential that the law of the signs, as explained above, be well
understood, and the facts fixed in memory. Fig. 6 will aid the student in fixing
the law in the memory. Having this constantly be-
fore the mind, and remembering that tan and cot
are + when sin and cos have like signs, and when
tbey have unlike, and that cos and sec have like
eigns, as also sin and cosec, or, more simply, that
sin 1 1 1
tan = , cot = - , sec = , and cosec = ,
cos tan cos sm
the student cannot fail to know the sign of a func-
tion at a glance.
It will be of service to remember that versed-sine
and coversed-sine, and all the functions of angles
of the 1st quadrant, are -t- ; but that of the other
functions than the versed-sine and coversed-sine, of
angles terminating in the other quadrants, but two are + in each quadrant