find the exact value of sin(13π/8).\na. -√((2 + √2)/4)\nb. √((2 + √2)/4)\nc. -√((2 - √2)/4)\nd. √((2 - √2)/4)

Answer:

C. $-\sqrt{\frac{2 - \sqrt{2}}{4}}$

Explanation:

Step1: Rewrite the angle

$\sin(\frac{13\pi}{8})=\sin(\frac{16\pi - 3\pi}{8})=\sin(2\pi-\frac{3\pi}{8})$

Step2: Use the sine - angle formula

Since $\sin(2\pi - \alpha)=-\sin\alpha$, then $\sin(2\pi-\frac{3\pi}{8})=-\sin(\frac{3\pi}{8})$

Step3: Express $\frac{3\pi}{8}$ in half - angle form

$\frac{3\pi}{8}=\frac{\frac{3\pi}{4}}{2}$, and the half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$. Here $\alpha=\frac{3\pi}{4}$, and $\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}$

Step4: Calculate $\sin\frac{3\pi}{8}$

$\sin\frac{3\pi}{8}=\sqrt{\frac{1-\cos\frac{3\pi}{4}}{2}}=\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{2 + \sqrt{2}}{4}}$

Step5: Find $\sin\frac{13\pi}{8}$

Since $\sin\frac{13\pi}{8}=-\sin\frac{3\pi}{8}$, then $\sin\frac{13\pi}{8}=-\sqrt{\frac{2 - \sqrt{2}}{4}}$ (after rationalizing the denominator and considering the sign).