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

108

TRIGONOMETRY

[Art. 83

When A is equal to 0~, 90~, 180~, or any multiple of 90~, one of
the numbers x and y is zero and cannot be used as a divisor. It
was therefore necessary in our definition of tan A as y/x to exclude the cases in which x = 0. No definition is given of the
tangent of 90~, 270~, or of any odd multiple of 90~; the tangent
is said to be undefined for these angles. Accordingly we avoid the
symbols tan 90~, tan 270~, and similarly csc 0~, cot 0~, sec 90~,
csc 180~, cot 180~, sec 270~, together with the symbols obtained
by increasing or decreasing these angles by multiples of 360~. The
fact that there exists no entirely satisfactory definition of these
excluded symbols will become evident when we have studied the
graphs of the functions (Arts. 107-8).
Angles - 340~, 20~, 380~ and 740~ have the same terminal side
when placed in their trigonometric positions, and hence by definition have the same sines, the same cosines, etc. So always, the
trigonometric functions of angles which differ by any multiple
of 360~ have the same values.
EXAMPLE 1. Find the trigonometric functions of angle 120~.
Solution. Place angle 120~ in its trigonometric position XOT (Fig. 65). For convenT ience take OT = r = 2. Since ZBOT = 60~,
i' 1\ ~ we complete the equilateral triangle OTB, and
see that OC and TC are of lengths 1 and
/ i.F\ i /3 respectively. Hence x = - 1, y = + 3,
-/__t"1^^.-l so that
C O
FoC l f i>sin 120 ~=- cos 120 ~=
FIG. 65 2 2
tan 120~= -=-3.
EXAMPIE 2. Given tan A = 4/3, find sin A and cos A.
Solution. The ratio of the ordinate y to the abscissa x is 4/3. Since it is a
question of a ratio and not of actual lengths, we may take x = = 3 to avoid
fractions. First, let x = + 3. Then
y = + 4, r2 = 32 + 42, r = 5,

and angle A is in the first quadrant (as in Fig. 63). Thus sin A = 4/5,