Plane trigonometry, by S.L. Loney.

[Exs. XX.] TRIGONOMETRICAL EQUATIONS.

139

27. sin 2a + sin 2t + sin 2y
=2 (sin a + sin 3 + sin y) (1 + cos a + cos 3 + cos y)
if a+p3+y=0.
28. Verify that
sin3 a sin (b - c) + sin3 b sin (c - a) + sin3 c sin (a - b)
+ sin (a + b + c) sin (b - c) sin (c - a) sin (a - b) =0.
If A, B, C, and D be any angles prove that
29. si sinsin sin in (A - B) + sin B sin C sin (B - C)
+ sin C sin A sin (C - A) + sin (A - B) sin (B - C) sin (C - A)= 0.
30. sin (A - B) cos(A + B)+ sin (B - C) cos (B + C)
+ sin (C- D) cos (C + D) + sin (D - A) cos (D +A) =0.
31. sin (A +B - 2C) cos B- sin (A + C - 2B) cos C
=sin (B — C){cos(B+C-A) +cos(C+A -B)+cos (A+B- C)}.
32. sin(A+B 6'+D)+sin(A+B- C-D)+sin(A+B-C'+D)
+sin (A + B +C —D)=4 sin (A +B)cos CcosD.
33. If any theorem be true for values of A, B, and C such that
A+B+C=180~,
prove that the theorem is still true if we substitute for A, B, and C
respectively the quantities
A B C
(1) 90 -, 90~- - and 90~ -,
or (2) 180~- 2A, 180- 2B, and 180~- 2C.
If x+y+z=xyz prove that
3x-x3 3y-y3 3z -z3 3x-x 3 3y-y3 3z-z3
34: i +
34 1-32 1- 3y2 -_ 3z2 1-3x 'l-3y -3z2
and 35. x(l-y2)(1-z2) +y(l-z2) ( -x) + (1 - x2) (l-y2)=4xyz.
128. The Addition and Subtraction Theorems may be
used to solve some kinds of trigonometrical equations.
Ex. Solve the equation
sin x + sin 5x = sin 3x.