Question

4 (c) \\( \cos \left( \frac { 7 \pi } { 8 } \
ight) \\)

Explanation:

Step1: Use the cosine half - angle formula

The cosine half - angle formula is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). We can write \(\frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2}\), so \(\alpha=\frac{7\pi}{4}\).

Step2: Determine the sign

Since \(\frac{7\pi}{8}\) is in the second quadrant (\(\frac{\pi}{2}<\frac{7\pi}{8}<\pi\)), and \(\cos x<0\) in the second quadrant.

Step3: Calculate \(\cos\frac{7\pi}{4}\)

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

Step4: Apply the half - angle formula

\(\cos\frac{7\pi}{8}=-\sqrt{\frac{1+\cos\frac{7\pi}{4}}{2}}=-\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}=-\sqrt{\frac{2+\sqrt{2}}{4}}=-\frac{\sqrt{2+\sqrt{2}}}{2}\)

Answer:

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