HMMT 二月 2001 · 团队赛 · 第 8 题

英文解析

8. Express without using trigonometric funccos 5 cos 10 cos 20
   tions.
   a + b a − b
   Solution: We will use the identities cos a + cos b = 2 cos cos and sin a + sin b =
   2 2
   a + b a + b
   2 sin cos . The numerator is (sin 10+sin 80)+(sin 20+sin 70)+(sin 30+sin 60)+(sin 40+
   2 2
   sin 60) = 2 sin 45 (cos 35 + cos 25 + cos 15 + cos 35) = 2 sin 45 ((cos 35 + cos 5) + (cos 25 + cos 15)) =
   √
   4 sin 45 cos 20 (cos 15 + cos 5) = 8 sin 45 cos 20 cos 10 cos 5, so the fraction equals 8 sin 45 = 4 2 .
   ∞
   ∑
   ai