Find the values of : 8 - Vidyadip

Question

Find the values of : $\sin \frac{\pi}{8}$

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Answer

We know that $\sin ^2 \theta=\frac{1-\cos 2 \theta}{2}$
Substituting $\theta=\frac{\pi}{8}$, we get
$ \sin ^2 \frac{\pi}{8}=\frac{1-\cos \frac{\pi}{4}}{2}$
$=\frac{1-\frac{1}{\sqrt{2}}}{2}$
$=\frac{\sqrt{2}-1}{2 \sqrt{2}}$
$\therefore \sin \frac{\pi}{8}=\sqrt{\frac{\sqrt{2}-1}{2 \sqrt{2}}} \ldots\left[\because \sin \frac{\pi}{8} \text { is positive }\right]$
$\therefore \sin \frac{\pi}{8}=\sqrt{\frac{\sqrt{2}-1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}}$
$=\sqrt{\frac{2-\sqrt{2}}{4}}$
$\therefore \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$
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