Plane trigonometry with practical applications, by Leonard E. Dickson.

Ch. VIII]

FUNCTIONS OF ANY ANGLE

107

83. Trigonometric functions of any angle. The six trigonometric functions of any acute angle were defined in Art. 7. We
shall now define the six trigonometric functions of any angle A,
with the exception of certain of the functions of 0~, 90~, - 180~,
or any multiple of 90~.
Place the angle A in its trigonometric position XOT (so that
its initial side is the horizontal line OX extending to the right from
Y Y
A TV
0iX AxT
FIG. 63 FIG. 64
the origin 0 of a system of rectangular coordinates). Select any
point T, other than 0, on the terminal side of angle A. Let x
be the abscissa of T, y its ordinate, and r its radius vector. Not
only for the cases of the angles A shown in Figs. 63, 64, but for
an angle A of any size and any sign, we make the following
definitions:
sinA=, cosA=-, tan A=- (if x 0),
r r x
csc A=r (if y.0), sec A = (if xo0), cot A= x (if yO).
y x y
Or in words, the sine is the ratio of the ordinate to the radius
vector,..., the cotangent is the ratio of the abscissa to the ordinate
provided the ordinate is not zero.
These ratios are the same, no matter what point T is selected.
The values of the six functions depend only on the angle A.
For the case of an acute angle (Fig. 63), these definitions agree
with those given in Art. 7, where r was called the hypotenuse,
x the adjacent side, and y the opposite side.