103 Trigonometry Problems

3
Advanced Problems
1. Two exercises on sin k ◦ sin(k + 1) ◦ :
(a) Find the smallest positive integer n such that
1
sin 45 ◦ sin 46 ◦ + 1
sin 47 ◦ sin 48 ◦ +···+ 1
sin 133 ◦ sin 134 ◦ = 1
sin n ◦ .
(b) Prove that
1
sin 1 ◦ sin 2 ◦ + 1
sin 2 ◦ sin 3 ◦ +···+ 1 cos 1◦
sin 89 ◦ =
sin 90◦ sin 2 1 ◦ .
2. Let ABC be a triangle, and let x be a nonnegative real number. Prove that
a x cos A + b x cos B + c x cos C ≤ 1 2 (ax + b x + c x ).
3. Let x,y,z be positive real numbers.