表の空欄に三角比 ($\sin \theta$, $\cos \theta$, $\tan \theta$) の値を、$\theta = 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ, 180^\circ$ に対してそれぞれ埋める問題です。

1. 問題の内容

表の空欄に三角比 ($\sin \theta$, $\cos \theta$, $\tan \theta$) の値を、$\theta = 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ, 180^\circ$ に対してそれぞれ埋める問題です。

2. 解き方の手順

各角度における $\sin \theta$, $\cos \theta$, $\tan \theta$ 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。

* $\theta = 0^\circ$ のとき:

$\sin 0^\circ = 0$, $\cos 0^\circ = 1$, $\tan 0^\circ = \frac{0}{1} = 0$

* $\theta = 30^\circ$ のとき:

$\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

* $\theta = 45^\circ$ のとき:

$\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\tan 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

* $\theta = 60^\circ$ のとき:

$\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$

* $\theta = 90^\circ$ のとき:

$\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ = \frac{1}{0}$ (定義されない)

* $\theta = 120^\circ$ のとき:

$\sin 120^\circ = \sin (180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}$

$\cos 120^\circ = \cos (180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2}$

$\tan 120^\circ = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}$

* $\theta = 135^\circ$ のとき:

$\sin 135^\circ = \sin (180^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}$

$\cos 135^\circ = \cos (180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{\sqrt{2}}{2}$

$\tan 135^\circ = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1$

* $\theta = 150^\circ$ のとき:

$\sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}$

$\cos 150^\circ = \cos (180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$

$\tan 150^\circ = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

* $\theta = 180^\circ$ のとき:

$\sin 180^\circ = 0$, $\cos 180^\circ = -1$, $\tan 180^\circ = \frac{0}{-1} = 0$

3. 最終的な答え

| $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | $120^\circ$ | $135^\circ$ | $150^\circ$ | $180^\circ$ |

| :---------- | :-------- | :------------------- | :------------------- | :------------------- | :-------- | :------------------- | :------------------- | :------------------- | :-------- |

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

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

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