三角関数の表を完成させる問題です。具体的には、$\theta$ が $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, 2\pi$ であるときの、$\sin \theta$, $\cos \theta$, $\tan \theta$ の値を求めます。

1. 問題の内容

三角関数の表を完成させる問題です。具体的には、$\theta$ が $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, 2\pi$ であるときの、$\sin \theta$, $\cos \theta$, $\tan \theta$ の値を求めます。

2. 解き方の手順

各$\theta$の値に対して、$\sin \theta$, $\cos \theta$, $\tan \theta$ の値を計算します。

$\tan \theta = \frac{\sin \theta}{\cos \theta}$ であることを利用します。

$\theta = 0$:

$\sin 0 = 0$

$\cos 0 = 1$

$\tan 0 = \frac{\sin 0}{\cos 0} = \frac{0}{1} = 0$

$\theta = \frac{\pi}{6}$:

$\sin \frac{\pi}{6} = \frac{1}{2}$

$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$

$\tan \frac{\pi}{6} = \frac{\sin \frac{\pi}{6}}{\cos \frac{\pi}{6}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

$\theta = \frac{\pi}{4}$:

$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

$\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

$\theta = \frac{\pi}{3}$:

$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$

$\cos \frac{\pi}{3} = \frac{1}{2}$

$\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$

$\theta = \frac{\pi}{2}$:

$\sin \frac{\pi}{2} = 1$

$\cos \frac{\pi}{2} = 0$

$\tan \frac{\pi}{2} = \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} = \frac{1}{0}$ (定義されない)

$\theta = \pi$:

$\sin \pi = 0$

$\cos \pi = -1$

$\tan \pi = \frac{\sin \pi}{\cos \pi} = \frac{0}{-1} = 0$

$\theta = 2\pi$:

$\sin 2\pi = 0$

$\cos 2\pi = 1$

$\tan 2\pi = \frac{\sin 2\pi}{\cos 2\pi} = \frac{0}{1} = 0$

3. 最終的な答え

| $\theta$ | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\pi$ | $2\pi$ |

|---|---|---|---|---|---|---|---|

| $\sin \theta$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$ | $0$ | $0$ |

| $\cos \theta$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$ | $-1$ | $1$ |

| $\tan \theta$ | $0$ | $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | 定義されない | $0$ | $0$ |