(1) $\cos\frac{2}{9}\pi + \cos\frac{4}{9}\pi + \cos\frac{5}{9}\pi + \cos\frac{7}{9}\pi$ の値を求める。 (2) $\sin\frac{13}{14}\pi + \cos\frac{11}{14}\pi + \sin\frac{5}{7}\pi - \sin\frac{\pi}{14}$ の値を求める。

1. 問題の内容

(1) $\cos\frac{2}{9}\pi + \cos\frac{4}{9}\pi + \cos\frac{5}{9}\pi + \cos\frac{7}{9}\pi$ の値を求める。

(2) $\sin\frac{13}{14}\pi + \cos\frac{11}{14}\pi + \sin\frac{5}{7}\pi - \sin\frac{\pi}{14}$ の値を求める。

2. 解き方の手順

(1)

$\cos\frac{5}{9}\pi = \cos(\pi - \frac{4}{9}\pi) = -\cos\frac{4}{9}\pi$

$\cos\frac{7}{9}\pi = \cos(\pi - \frac{2}{9}\pi) = -\cos\frac{2}{9}\pi$

したがって、

$\cos\frac{2}{9}\pi + \cos\frac{4}{9}\pi + \cos\frac{5}{9}\pi + \cos\frac{7}{9}\pi = \cos\frac{2}{9}\pi + \cos\frac{4}{9}\pi - \cos\frac{4}{9}\pi - \cos\frac{2}{9}\pi = 0$

(2)

$\sin\frac{13}{14}\pi = \sin(\pi - \frac{\pi}{14}) = \sin\frac{\pi}{14}$

$\cos\frac{11}{14}\pi = \cos(\frac{\pi}{2} + \frac{2}{14}\pi) = \cos(\frac{\pi}{2} + \frac{\pi}{7}) = -\sin\frac{\pi}{7}$

$\sin\frac{5}{7}\pi = \sin(\pi - \frac{2}{7}\pi) = \sin\frac{2}{7}\pi$

したがって、

$\sin\frac{13}{14}\pi + \cos\frac{11}{14}\pi + \sin\frac{5}{7}\pi - \sin\frac{\pi}{14} = \sin\frac{\pi}{14} - \sin\frac{\pi}{7} + \sin\frac{2}{7}\pi - \sin\frac{\pi}{14} = - \sin\frac{\pi}{7} + \sin\frac{2}{7}\pi$

$\sin\frac{2\pi}{7} - \sin\frac{\pi}{7} = 2\cos(\frac{3\pi}{14})\sin(\frac{\pi}{14})$

$\cos\frac{11}{14}\pi = \cos(\pi - \frac{3}{14}\pi) = - \cos\frac{3}{14}\pi = - \sin(\frac{\pi}{2} - \frac{3}{14}\pi) = - \sin(\frac{7\pi - 3\pi}{14}) = -\sin(\frac{4\pi}{14}) = -\sin(\frac{2\pi}{7})$

よって、$\cos\frac{11}{14}\pi = - \sin\frac{2\pi}{7}$

$\sin\frac{13}{14}\pi + \cos\frac{11}{14}\pi + \sin\frac{5}{7}\pi - \sin\frac{\pi}{14} = \sin\frac{\pi}{14} - \sin\frac{2\pi}{7} + \sin\frac{5\pi}{7} - \sin\frac{\pi}{14}$

$= -\sin\frac{2\pi}{7} + \sin\frac{5\pi}{7} = -\sin\frac{2\pi}{7} + \sin(\pi - \frac{2\pi}{7}) = -\sin\frac{2\pi}{7} + \sin\frac{2\pi}{7} = 0$

3. 最終的な答え

(1) 0

(2) 0