103 Trigonometry Problems

68 103 Trigonometry Problems
(b) cos A + cos B + cos C ≤ 3 2 .
28. Let ABC be a triangle. Prove that
(a) cos A cos B cos C ≤ 1 8 ;
(b) sin A sin B sin C ≤ 3√ 3
8 ;
(c) sin A + sin B + sin C ≤ 3√ 3
2 .
(d) cos 2 A + cos 2 B + cos 2 C ≥ 3 4 ;
(e) sin 2 A + sin 2 B + sin 2 C ≤ 9 4 ;
(f) cos 2A + cos 2B + cos 2C ≥− 3 2 ;
(g) sin 2A + sin 2B + sin 2C ≤ 3√ 3
2 .
29. Prove that
for all x ̸= kπ 6
, where k is in Z.
tan 3x
( π
) ( π
)
tan x = tan 3 − x tan
3 + x
30. Given that
find n.
(1 + tan 1 ◦ )(1 + tan 2 ◦ ) ···(1 + tan 45 ◦ ) = 2 n ,
31. Let A = (0, 0) and B = (b, 2) be points in the coordinate plane. Let ABCDEF
be a convex equilateral hexagon such that ̸ FAB = 120 ◦ , AB ‖ DE, BC ‖
EF, and CD ‖ FA, and the y coordinates of its vertices are distinct elements
of the set {0, 2, 4, 6, 8, 10}. The area of the hexagon can be written in the form
m √ n, where m and n are positive integers and n is not divisible by the square
of any prime. Find m + n.