use a half - angle formula to find the exact value of \\( \\sin \\frac { 5 \\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{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}}). [ \begin{align*} \sin\frac{5\pi}{8}&=\sqrt{\frac{1-(-\frac{\sqrt{2}}{2})}{2}}\ &=\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{\frac{2+\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})