Answer
$\frac{1}{2}\sqrt{2+\sqrt{3}}$
Work Step by Step
Use the half-angle formula, $\sin\frac{u}{2}=\pm\sqrt{\frac{1-\cos u}{2}}$. Note that $75^\circ$ is in Quadrant I, where sine is positive, so we take the positive square root.
$\sin 75^\circ$
$=\sin \frac{150^\circ}{2}$
$=\sqrt{\frac{1-\cos 150^\circ}{2}}$
$=\sqrt{\frac{1-(-\frac{\sqrt{3}}{2})}{2}}$
$=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$
$=\sqrt{\frac{(1+\frac{\sqrt{3}}{2})*2}{2*2}}$
$=\sqrt{\frac{2+\sqrt{3}}{4}}$
$=\frac{\sqrt{2+\sqrt{3}}}{\sqrt{4}}$
$=\frac{1}{2}\sqrt{2+\sqrt{3}}$
