sin22.5°、cos22.5°、tan22.5°はどんな数？ ~ 数学について考えてみる

2022年2月3日

sin22.5°、cos22.5°、tan22.5°はどんな数？

PLUMBAGO

　 $22.5°\ (=\dfrac{\pi}{8})$ のとき三角関数がどんな式で表されるのかを調べてみました。

$\sin22.5°$

　半角の公式

\sin^{2} \frac{θ}{2} = \frac{1 - \cos θ}{2}

$\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2}$

より

\begin{aligned} \sin^{2} 22.5 ° & = \frac{1 - \cos 45 °}{2} \\ = \frac{1 - \frac{\sqrt{2}}{2}}{2} \\ = \frac{2 - \sqrt{2}}{4} \\ \sin 22.5 ° & = \underline{\frac{\sqrt{2 - \sqrt{2}}}{2}} & (∵ \sin 22.5 ° > 0) \end{aligned}

$\begin{align*}\sin^2 22.5°&=\frac{1-\cos45°}{2}\\[0.5em]&=\frac{1-\cfrac{\sqrt{2}}{2}}{2}\\[0.5em]&=\frac{2-\sqrt{2}}{4}\\[0.5em]\sin22.5°&=\underline{\frac{\sqrt{2-\sqrt{2}}}{2}}&(\because\sin22.5°>0)\end{align*}$

$\cos22.5°$

　半角の公式

\cos^{2} \frac{θ}{2} = \frac{1 + \cos θ}{2}

$\cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}$

より

\begin{aligned} \cos^{2} 22.5 ° & = \frac{1 + \cos 45 °}{2} \\ = \frac{1 + \frac{\sqrt{2}}{2}}{2} \\ = \frac{2 + \sqrt{2}}{4} \\ \cos 22.5 ° & = \underline{\frac{\sqrt{2 + \sqrt{2}}}{2}} & (∵ \cos 22.5 ° > 0) \end{aligned}

$\begin{align*}\cos^2 22.5°&=\frac{1+\cos45°}{2}\\[0.5em]&=\frac{1+\cfrac{\sqrt{2}}{2}}{2}\\[0.5em]&=\frac{2+\sqrt{2}}{4}\\[0.5em]\cos22.5°&=\underline{\frac{\sqrt{2+\sqrt{2}}}{2}}&(\because\cos22.5°>0)\end{align*}$

$\tan22.5°$

\tan θ = \frac{\sin θ}{\cos θ}

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

より

\begin{aligned} \tan 22.5 ° & = \frac{\sin 22.5 °}{\cos 22.5 °} \\ = \frac{\frac{\sqrt{2 - \sqrt{2}}}{2}}{\frac{\sqrt{2 + \sqrt{2}}}{2}} \\ = \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2 + \sqrt{2}}} \times \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2 + \sqrt{2}}} \\ = \frac{\sqrt{2}}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}} \\ = \underline{\sqrt{2} - 1} \end{aligned}

$\begin{align*}\tan22.5°&=\frac{\sin22.5°}{\cos22.5°}\\[0.5em]&=\frac{\cfrac{\sqrt{2-\sqrt{2}}}{2}}{\cfrac{\sqrt{2+\sqrt{2}}}{2}}\\[0.5em]&=\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}×\frac{\sqrt{2+\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\\[0.5em]&=\frac{\sqrt{2}}{2+\sqrt{2}}×\frac{2-\sqrt{2}}{2-\sqrt{2}}\\[0.5em]&=\underline{\sqrt{2}-1}\end{align*}$

　それぞれの近似値は以下のようになります。