Question

use a half - angle formula to find the exact value of \\( \cos \frac { 7 \pi } { 8 } \\).
\\( \cos \frac { 7 \pi } { 8 } = \square \\)

Explanation:

Step1: Recall the half - angle formula

The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). For \(\cos\frac{7\pi}{8}\), we have \(\alpha=\frac{7\pi}{4}\), and \(\frac{\alpha}{2}=\frac{7\pi}{8}\). Since \(\frac{7\pi}{8}\) is in the second quadrant (\(\frac{\pi}{2}<\frac{7\pi}{8}<\pi\)), \(\cos\frac{7\pi}{8}<0\).

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

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

Step3: Substitute into the half - angle formula

Substitute \(\alpha = \frac{7\pi}{4}\) into \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}\) (negative because of the quadrant).

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Answer:

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