Question

use a half - angle formula to find the exact value of \\( \sin \frac { \pi } { 8 } \\).
\\( \sin \frac { \pi } { 8 } = \\)

Explanation:

Step1: Recall the half - angle formula for sine

The half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$.
Since $\frac{\pi}{8}$ is in the first quadrant ($0<\frac{\pi}{8}<\frac{\pi}{2}$), $\sin\frac{\pi}{8}>0$, so we take the positive square root. Let $\alpha=\frac{\pi}{4}$, then $\frac{\alpha}{2}=\frac{\pi}{8}$.

Step2: Substitute $\cos\alpha$ into the formula

We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substitute $\alpha = \frac{\pi}{4}$ into the half - angle formula:

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Answer:

$\frac{\sqrt{2 - \sqrt{2}}}{2}$