返回题库

HMMT 二月 2001 · 团队赛 · 第 8 题

HMMT February 2001 — Team Round — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Express without using trigonometric func- cos 5 cos 10 cos 20 tions. ∞ ∑ ai
解析
  1. Express without using trigonometric func- cos 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