103 Trigonometry Problems

2. Introductory Problems 65
12. In triangle ABC,
3 sin A + 4 cos B = 6 and 4 sin B + 3 cos A = 1.
Find the measure of angle C.
13. Prove that
tan 3a − tan 2a − tan a = tan 3a tan 2a tan a
for all a ̸= kπ 2
, where k is in Z.
14. Let a, b, c, d be numbers in the interval [0,π] such that
sin a + 7 sin b = 4(sin c + 2 sin d),
cos a + 7 cos b = 4(cos c + 2 cos d).
Prove that 2 cos(a − d) = 7 cos(b − c).
15. Express
sin(x − y) + sin(y − z) + sin(z − x)
as a monomial.
16. Prove that
(4 cos 2 9 ◦ − 3)(4 cos 2 27 ◦ − 3) = tan 9 ◦ .
17. Prove that
(
1 + a ) ( 1 + b ) (
≥ 1 + √ ) 2
2ab
sin x cos x
for all real numbers a,b,x with a,b ≥ 0 and 0