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

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 $\frac{\pi}{8}$ is in the first quadrant ($0<\frac{\pi}{8}<\frac{\pi}{2}$), $\sin\frac{\pi}{8}>0$, so we take the positive square root. Let $\alpha=\frac{\pi}{4}$, then $\frac{\alpha}{2}=\frac{\pi}{8}$.

Step2: Substitute $\cos\alpha$ into the formula

We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substitute $\alpha = \frac{\pi}{4}$ into the half - angle formula: [ \begin{align*} \sin\frac{\pi}{8}&=\sqrt{\frac{1-\cos\frac{\pi}{4}}{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:

$\frac{\sqrt{2 - \sqrt{2}}}{2}$