use the half - angle formula to find the exact value.\n sin\frac{pi}{8}=square

Explanation:

Step1: Recall half - angle formula

The half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$. We want to find $\sin\frac{\pi}{8}$, so $\alpha=\frac{\pi}{4}$.

Step2: Determine the sign

Since $\frac{\pi}{8}$ is in the first - quadrant ($0<\frac{\pi}{8}<\frac{\pi}{2}$), $\sin\frac{\pi}{8}>0$.

Step3: Substitute $\cos\alpha$ value

We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substituting $\alpha = \frac{\pi}{4}$ into the half - angle formula $\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}$, we get $\sin\frac{\pi}{8}=\sqrt{\frac{1-\cos\frac{\pi}{4}}{2}}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}$.

Step4: Simplify the expression

First, simplify the fraction inside the square - root: $\frac{1-\frac{\sqrt{2}}{2}}{2}=\frac{2 - \sqrt{2}}{4}$. Then $\sin\frac{\pi}{8}=\sqrt{\frac{2-\sqrt{2}}{4}}=\frac{\sqrt{2 - \sqrt{2}}}{2}$.

Answer:

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