Answer
$-\frac{1}{2}\sqrt{2-\sqrt{2}}$
Work Step by Step
Use the half-angle formula, $\sin\frac{u}{2}=\pm\sqrt{\frac{1-\cos u}{2}}$. Note that $\frac{9\pi}{8}$ is in Quadrant III, where sine is negative, so we take the negative square root.
$\sin \frac{9\pi}{8}$
$=\sin \frac{\frac{9\pi}{4}}{2}$
$=-\sqrt{\frac{1-\cos \frac{9\pi}{4}}{2}}$
$=-\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}$
$=-\sqrt{\frac{(1-\frac{\sqrt{2}}{2})*2}{2*2}}$
$=-\sqrt{\frac{2-\sqrt{2}}{4}}$
$=-\frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}$
$=-\frac{1}{2}\sqrt{2-\sqrt{2}}$
