$\theta$ が与えられたときに、$\sin{\theta}$、$\cos{\theta}$、$\tan{\theta}$ の値を求める問題です。 (1) $\theta = \frac{4}{3}\pi$ (2) $\theta = \frac{7}{4}\pi$ (3) $\theta = 4\pi$ (4) $\theta = -\frac{7}{6}\pi$

1. 問題の内容

$\theta$ が与えられたときに、$\sin{\theta}$、$\cos{\theta}$、$\tan{\theta}$ の値を求める問題です。

(1) $\theta = \frac{4}{3}\pi$

(2) $\theta = \frac{7}{4}\pi$

(3) $\theta = 4\pi$

(4) $\theta = -\frac{7}{6}\pi$

2. 解き方の手順

(1) $\theta = \frac{4}{3}\pi$ の場合:

$\frac{4}{3}\pi = \pi + \frac{1}{3}\pi$ なので、第3象限の角です。

$\sin{\frac{4}{3}\pi} = -\sin{\frac{\pi}{3}} = -\frac{\sqrt{3}}{2}$

$\cos{\frac{4}{3}\pi} = -\cos{\frac{\pi}{3}} = -\frac{1}{2}$

$\tan{\frac{4}{3}\pi} = \frac{\sin{\frac{4}{3}\pi}}{\cos{\frac{4}{3}\pi}} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}$

(2) $\theta = \frac{7}{4}\pi$ の場合:

$\frac{7}{4}\pi = 2\pi - \frac{1}{4}\pi$ なので、第4象限の角です。

$\sin{\frac{7}{4}\pi} = -\sin{\frac{\pi}{4}} = -\frac{\sqrt{2}}{2}$

$\cos{\frac{7}{4}\pi} = \cos{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}$

$\tan{\frac{7}{4}\pi} = \frac{\sin{\frac{7}{4}\pi}}{\cos{\frac{7}{4}\pi}} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1$

(3) $\theta = 4\pi$ の場合:

$4\pi = 2 \times 2\pi$なので、$\theta = 0$と同じになります。

$\sin{4\pi} = \sin{0} = 0$

$\cos{4\pi} = \cos{0} = 1$

$\tan{4\pi} = \frac{\sin{4\pi}}{\cos{4\pi}} = \frac{0}{1} = 0$

(4) $\theta = -\frac{7}{6}\pi$ の場合:

$-\frac{7}{6}\pi = -\pi - \frac{\pi}{6}$ なので、第2象限の角です。

$-\frac{7}{6}\pi = -210^\circ$

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

$\sin{(-\frac{7}{6}\pi)} = \sin{(-\frac{7}{6}\pi + 2\pi)} = \sin{(\frac{5}{6}\pi)} = \sin{(\pi - \frac{\pi}{6})} = \sin{(\frac{\pi}{6})} = \frac{1}{2}$

$\cos{(-\frac{7}{6}\pi)} = \cos{(\frac{5}{6}\pi)} = \cos{(\pi - \frac{\pi}{6})} = -\cos{(\frac{\pi}{6})} = -\frac{\sqrt{3}}{2}$

$\tan{(-\frac{7}{6}\pi)} = \frac{\sin{(-\frac{7}{6}\pi)}}{\cos{(-\frac{7}{6}\pi)}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

3. 最終的な答え

(1) $\sin{\frac{4}{3}\pi} = -\frac{\sqrt{3}}{2}$, $\cos{\frac{4}{3}\pi} = -\frac{1}{2}$, $\tan{\frac{4}{3}\pi} = \sqrt{3}$

(2) $\sin{\frac{7}{4}\pi} = -\frac{\sqrt{2}}{2}$, $\cos{\frac{7}{4}\pi} = \frac{\sqrt{2}}{2}$, $\tan{\frac{7}{4}\pi} = -1$

(3) $\sin{4\pi} = 0$, $\cos{4\pi} = 1$, $\tan{4\pi} = 0$

(4) $\sin{(-\frac{7}{6}\pi)} = \frac{1}{2}$, $\cos{(-\frac{7}{6}\pi)} = -\frac{\sqrt{3}}{2}$, $\tan{(-\frac{7}{6}\pi)} = -\frac{\sqrt{3}}{3}$