103 Trigonometry Problems

5
Solutions to Advanced Problems
1. Two exercises on sin k ◦ sin(k + 1) ◦ :
(a) [AIME2 2000] Find the smallest positive integer n such that
(b) Prove that
Solution: Note that
Thus
1
sin 45 ◦ sin 46 ◦ + 1
sin 47 ◦ sin 48 ◦ +···+ 1
sin 133 ◦ sin 134 ◦
= 1
sin n ◦ .
1
sin 1 ◦ sin 2 ◦ + 1
sin 2 ◦ sin 3 ◦ +···+ 1
sin 89 ◦ sin 90 ◦
cos 1◦
=
sin 2 1 ◦ .
sin 1 ◦ = sin[(x + 1) ◦ − x ◦ ]
= sin(x + 1) ◦ cos x ◦ − cos(x + 1) ◦ sin x ◦ .
sin 1 ◦
sin x ◦ sin(x + 1) ◦ = cos x◦ sin(x + 1) ◦ − sin x ◦ cos(x + 1) ◦
sin x ◦ sin(x + 1) ◦
= cot x ◦ − cot(x + 1) ◦ .