与えられた角度 $\theta$ に対して、$\sin \theta$, $\cos \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, $\cot \theta$ の値を求め、表を完成させる問題です。角度 $\theta$ は、$-\frac{\pi}{4}$, $-\frac{\pi}{6}$, $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\frac{3\pi}{4}$, $\pi$ です。ここで、 * $\csc \theta = \frac{1}{\sin \theta}$ * $\sec \theta = \frac{1}{\cos \theta}$ * $\tan \theta = \frac{\sin \theta}{\cos \theta}$ * $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}$

1. 問題の内容

与えられた角度 $\theta$ に対して、$\sin \theta$, $\cos \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, $\cot \theta$ の値を求め、表を完成させる問題です。角度 $\theta$ は、$-\frac{\pi}{4}$, $-\frac{\pi}{6}$, $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\frac{3\pi}{4}$, $\pi$ です。ここで、

* $\csc \theta = \frac{1}{\sin \theta}$

* $\sec \theta = \frac{1}{\cos \theta}$

* $\tan \theta = \frac{\sin \theta}{\cos \theta}$

* $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}$

2. 解き方の手順

まず、基本的な三角関数の値を求めます。

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

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

* $\sin(0) = 0$, $\cos(0) = 1$, $\tan(0) = 0$

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

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

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

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

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

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

* $\sin(\pi) = 0$, $\cos(\pi) = -1$, $\tan(\pi) = 0$

次に、$\csc \theta$, $\sec \theta$, $\cot \theta$ の値を求めます。

* $\csc(-\frac{\pi}{4}) = -\sqrt{2}$, $\sec(-\frac{\pi}{4}) = \sqrt{2}$, $\cot(-\frac{\pi}{4}) = -1$

* $\csc(-\frac{\pi}{6}) = -2$, $\sec(-\frac{\pi}{6}) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$, $\cot(-\frac{\pi}{6}) = -\sqrt{3}$

* $\csc(0)$ は定義されない, $\sec(0) = 1$, $\cot(0)$ は定義されない

* $\csc(\frac{\pi}{6}) = 2$, $\sec(\frac{\pi}{6}) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$, $\cot(\frac{\pi}{6}) = \sqrt{3}$

* $\csc(\frac{\pi}{4}) = \sqrt{2}$, $\sec(\frac{\pi}{4}) = \sqrt{2}$, $\cot(\frac{\pi}{4}) = 1$

* $\csc(\frac{\pi}{3}) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$, $\sec(\frac{\pi}{3}) = 2$, $\cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

* $\csc(\frac{\pi}{2}) = 1$, $\sec(\frac{\pi}{2})$ は定義されない, $\cot(\frac{\pi}{2}) = 0$

* $\csc(\frac{2\pi}{3}) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$, $\sec(\frac{2\pi}{3}) = -2$, $\cot(\frac{2\pi}{3}) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

* $\csc(\frac{3\pi}{4}) = \sqrt{2}$, $\sec(\frac{3\pi}{4}) = -\sqrt{2}$, $\cot(\frac{3\pi}{4}) = -1$

* $\csc(\pi)$ は定義されない, $\sec(\pi) = -1$, $\cot(\pi)$ は定義されない

3. 最終的な答え

表に書き込む値は次の通りです。

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

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

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

| $\csc \theta$ | $-\sqrt{2}$ | $-2$ | undefined | $2$ | $\sqrt{2}$ | $\frac{2\sqrt{3}}{3}$ | $1$ | $\frac{2\sqrt{3}}{3}$ | $\sqrt{2}$ | undefined |

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

| $\sec \theta$ | $\sqrt{2}$ | $\frac{2\sqrt{3}}{3}$ | $1$ | $\frac{2\sqrt{3}}{3}$ | $\sqrt{2}$ | $2$ | undefined | $-2$ | $-\sqrt{2}$ | $-1$ |

| $\tan \theta$ | $-1$ | $-\frac{\sqrt{3}}{3}$ | $0$ | $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | undefined | $-\sqrt{3}$ | $-1$ | $0$ |

| $\cot \theta$ | $-1$ | $-\sqrt{3}$ | undefined | $\sqrt{3}$ | $1$ | $\frac{\sqrt{3}}{3}$ | $0$ | $-\frac{\sqrt{3}}{3}$ | $-1$ | undefined |