Question

use a half - angle formula to find the exact value of \\( \sin \frac { 7 \pi } { 8 } \\).
\\( \sin \frac { 7 \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 \(\frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2}\) and \(\frac{7\pi}{8}\) is in the second quadrant (\(\sin x>0\) in the second quadrant), we take the positive sign. Here \(\alpha=\frac{7\pi}{4}\).

Step2: Find the value of \(\cos\frac{7\pi}{4}\)

We know that \(\cos\frac{7\pi}{4}=\cos(2\pi-\frac{\pi}{4})=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).

Step3: Substitute \(\cos\alpha\) into the half - angle formula

Substitute \(\alpha = \frac{7\pi}{4}\) and \(\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\) into \(\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}\).

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Answer:

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