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

Ch. X]

FUNCTIONS OF SEVERAL ANGLES

147

the third equation following by division. Applying this principle of
"composition and division" to the law of sines
a sin A
b sin B'
we get
a+b sinA+sinB
a-b sin A-sin B
Replacing the second member by its value in Ex. 9, we have the law of
tangents (Art. 88).
If A, B, C are angles of a triangle, prove that
16.* sinA + sinB + sin C = 4cos I A cos I Bcos C.
17.* sinA + sinB - sin C = 4sin A sin Bcos C.
18.* cos A + cosB + cosC = + 4sin A sin Bsin C.
19.* cos A - cos B + cosC = 4cos i Asin B cos C.
20.* sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
21.* sin 4A + sin 4B + sin 4C = - 4 sin 2A sin 2B sin 2C.
105. Trigonometric equations. Unlike an identity, an equation
is true only for special values. The exercises below include only
equations which can be solved by methods wholly similar to those
employed in the following four illustrative examples. More difficult equations are solved graphically in Arts. 109, 112.
EXAMPLE 1. Find the positive angles <360~ for which
2 sin2x + 3 cos x = 2.
Solution. Replacing sin2 x by 1 - cos2 x, we get 3 cos x - 2 cos2 x = 0.
Hence cos x = 0 or 3/2. The second value must be discarded since cos x
cannot exceed 1 numerically. Thus cos x = 0, x = 90~ or 270~.
EXAMPLE 2. Find the positive angles <360~ for which
cos x + cos 2x + cos 3x = 0.
Solution. By formula (24),
cos 3x + cos x = 2 cos 2x cos x.
Hence the proposed equation becomes
cos 2x (2 cos x + 1) = 0.
Hence either cos 2x = 0 or 2 cos x = -1. The positive solutions <360~ of the
latter are x = 120~, 240~. Next, cos 2x = 0 only when 2x is 90~ or 270~ or