与えられた三角関数の値を求める問題です。もし値が存在しない場合は×と答えます。

1. 問題の内容

与えられた三角関数の値を求める問題です。もし値が存在しない場合は×と答えます。

2. 解き方の手順

三角関数の定義、単位円、三角関数の周期性、および以下の公式を利用します。

* $\sin(\theta + 2\pi) = \sin(\theta)$

* $\cos(\theta + 2\pi) = \cos(\theta)$

* $\tan(\theta + \pi) = \tan(\theta)$

* $\sin(-\theta) = -\sin(\theta)$

* $\cos(-\theta) = \cos(\theta)$

* $\tan(-\theta) = -\tan(\theta)$

(1) $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$

(2) $\sin\frac{7\pi}{6} = \sin(\pi + \frac{\pi}{6}) = -\sin\frac{\pi}{6} = -\frac{1}{2}$

(3) $\sin\pi = 0$

(4) $\cos\frac{5\pi}{4} = \cos(\pi + \frac{\pi}{4}) = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$

(5) $\cos\frac{5\pi}{3} = \cos(2\pi - \frac{\pi}{3}) = \cos(-\frac{\pi}{3}) = \cos\frac{\pi}{3} = \frac{1}{2}$

(6) $\cos\frac{7\pi}{6} = \cos(\pi + \frac{\pi}{6}) = -\cos\frac{\pi}{6} = -\frac{\sqrt{3}}{2}$

(7) $\tan\frac{2\pi}{3} = \tan(\pi - \frac{\pi}{3}) = -\tan\frac{\pi}{3} = -\sqrt{3}$

(8) $\tan\frac{5\pi}{4} = \tan(\pi + \frac{\pi}{4}) = \tan\frac{\pi}{4} = 1$

(9) $\tan\pi = 0$

(10) $\sin(-\frac{\pi}{6}) = -\sin\frac{\pi}{6} = -\frac{1}{2}$

(11) $\sin(-\frac{5\pi}{4}) = -\sin\frac{5\pi}{4} = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2}$

(12) $\sin(-\frac{5\pi}{3}) = -\sin\frac{5\pi}{3} = -(-\frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{2}$

(13) $\cos(-\frac{\pi}{4}) = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$

(14) $\cos(-\frac{5\pi}{6}) = \cos\frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$

(15) $\cos(-\frac{4\pi}{3}) = \cos\frac{4\pi}{3} = \cos(\pi + \frac{\pi}{3}) = -\cos\frac{\pi}{3} = -\frac{1}{2}$

(16) $\tan(-\frac{4\pi}{3}) = -\tan\frac{4\pi}{3} = -\tan(\pi + \frac{\pi}{3}) = -\tan\frac{\pi}{3} = -\sqrt{3}$

(17) $\tan(-\frac{11\pi}{6}) = -\tan\frac{11\pi}{6} = -\tan(2\pi - \frac{\pi}{6}) = -(-\tan\frac{\pi}{6}) = \tan\frac{\pi}{6} = \frac{\sqrt{3}}{3}$

(18) $\tan(-\frac{3\pi}{2}) = -\tan\frac{3\pi}{2} = -\tan(\pi + \frac{\pi}{2})$。$\tan\frac{\pi}{2}$は定義されないので、$\tan(-\frac{3\pi}{2})$も定義されません。

3. 最終的な答え

(1) $\frac{\sqrt{3}}{2}$

(2) $-\frac{1}{2}$

(3) $0$

(4) $-\frac{\sqrt{2}}{2}$

(5) $\frac{1}{2}$

(6) $-\frac{\sqrt{3}}{2}$

(7) $-\sqrt{3}$

(8) $1$

(9) $0$

(10) $-\frac{1}{2}$

(11) $\frac{\sqrt{2}}{2}$

(12) $\frac{\sqrt{3}}{2}$

(13) $\frac{\sqrt{2}}{2}$

(14) $-\frac{\sqrt{3}}{2}$

(15) $-\frac{1}{2}$

(16) $-\sqrt{3}$

(17) $\frac{\sqrt{3}}{3}$

(18) $\times$