平方根と分数で表すことができる有名角の三角比の値を計算して、整理しました。
目次
有名角の三角比
鋭角の有名角（$30^{\circ}$, $45^{\circ}$, $60^{\circ}$）
| $\theta$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ |
| $\sin \theta$ | $\displaystyle \frac{1}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{\sqrt{3}}{2}$ |
| $\cos \theta$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{1}{2}$ |
| $\tan \theta$ | $\displaystyle \frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ |
$0^{\circ}$ 〜 $180^{\circ}$ までの有名角
| $\theta$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ | $120^{\circ}$ | $135^{\circ}$ | $150^{\circ}$ | $180^{\circ}$ |
| $\sin \theta$ | $0$ | $\displaystyle \frac{1}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $1$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{1}{2}$ | $0$ |
| $\cos \theta$ | $1$ | $\displaystyle \frac{\sqrt{3}}{2}$ | $\displaystyle \frac{\sqrt{2}}{2}$ | $\displaystyle \frac{1}{2}$ | $0$ | $\displaystyle -\frac{1}{2}$ | $\displaystyle -\frac{\sqrt{2}}{2}$ | $\displaystyle -\frac{\sqrt{3}}{2}$ | $-1$ |
| $\tan \theta$ | $0$ | $\displaystyle \frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | × | $-\sqrt{3}$ | $-1$ | $\displaystyle \frac{1}{\sqrt{3}}$ | $0$ |
$15^{\circ}$ と $18^{\circ}$ の倍数の三角比
$15^{\circ}$, $18^{\circ}$, $36^{\circ}$
$\theta = 15^{\circ}$
$\displaystyle \sin 15^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$
$\displaystyle \cos 15^{\circ} = \frac{\sqrt{6}+\sqrt{2}}{4}$
$\displaystyle \tan 15^{\circ} = 2-\sqrt{3}$
$\theta = 18^{\circ}$
$\displaystyle \sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$
$\displaystyle \cos 18^{\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$
$\displaystyle \tan 18^{\circ} = \frac{1}{\sqrt{5+2\sqrt{5}}}$
$\theta = 36^{\circ}$
$\displaystyle \sin 36^{\circ} = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$
$\displaystyle \cos 36^{\circ} = \frac{\sqrt{5}+1}{4}$
$\displaystyle \tan 36^{\circ} = \sqrt{5-2\sqrt{5}}$
$54^{\circ}$, $72^{\circ}$, $75^{\circ}$
$\theta = 54^{\circ}$
$\displaystyle \sin 54^{\circ} = \frac{\sqrt{5}+1}{4}$
$\displaystyle \cos 54^{\circ} = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$
$\displaystyle \tan 54^{\circ} = \frac{\sqrt{25+10\sqrt{5}}}{5}$
$\theta = 72^{\circ}$
$\displaystyle \sin 72^{\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$
$\displaystyle \cos 72^{\circ} = \frac{\sqrt{5}-1}{4}$
$\displaystyle \tan 72^{\circ} = \sqrt{5+2\sqrt{5}}$
$\theta = 75^{\circ}$
$\displaystyle \sin 75^{\circ} = \frac{\sqrt{6}+\sqrt{2}}{4}$
$\displaystyle \cos 75^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$
$\displaystyle \tan 75^{\circ} = 2+\sqrt{3}$
