use a half - angle formula to find the exact value of \\( \\sin \\frac { 7 \\pi } { 8 } \\).\n\\( \\sin…

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}}). [ \begin{align*} \sin\frac{7\pi}{8}&=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{2-\sqrt{2}}{4}}\ &=\frac{\sqrt{2 - \sqrt{2}}}{2} \end{align*} ]

Answer:

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