Question

use a half - angle formula to find the exact value of \\( \sin \frac { 5 \pi } { 8 } \\).
\\( \sin \frac { 5 \pi } { 8 } = \square \\)

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 \(\frac{5\pi}{8}\) is in the second quadrant (\(\frac{\pi}{2}<\frac{5\pi}{8}<\pi\)), \(\sin\frac{5\pi}{8}>0\). Let \(\alpha=\frac{5\pi}{4}\), then \(\frac{\alpha}{2}=\frac{5\pi}{8}\).

Step2: Find the value of \(\cos\alpha\)

We know that \(\cos\frac{5\pi}{4}=\cos(\pi+\frac{\pi}{4})=-\cos\frac{\pi}{4}=-\frac{\sqrt{2}}{2}\).

Step3: Substitute into the half - angle formula

Substitute \(\cos\alpha =-\frac{\sqrt{2}}{2}\) into \(\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}\).

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Answer:

\(\frac{\sqrt{2+\sqrt{2}}}{2}\)