PUMaC 2013 · 团队赛 · 第 16 题
PUMaC 2013 — Team Round — Problem 16
题目详情
- Is cos 1 rational? Prove. 2
解析
- Is cos 1 rational? Prove. SOLUTION: ◦ 2 Assume cos 1 ∈ Q . From cos 2 x = 2 cos x − 1, cos x ∈ Q implies cos 2 x ∈ Q . In particular, ◦ ◦ ◦ cos 8192 ∈ Q , where cos 8192 = sin 2 . Next, from compound angle formula, sin x, sin y, cos x, cos y ∈ Q implies sin ( x ± y ) , cos ( x ± y ) ∈ ◦ ◦ Q . Then, all the following are rationals from sin 2 , cos 2 ∈ Q : ◦ ◦ ◦ ◦ sin 4 , cos 4 , sin 64 , cos 64 , √ 3 ◦ and so is sin 60 = , which is a contradiction. 2 6