Answer
$$\sin\frac{5\pi}{12}=\frac{\sqrt2+\sqrt6}{4}$$
Work Step by Step
*STRATEGY: In this type of exercise, we would try to rewrite the angle in terms of already familiar angles, including $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$
$$\sin\frac{5\pi}{12}$$
We try separating $5\pi$ into $2\pi$ and $3\pi$. In detail,
$$\sin\frac{5\pi}{12}=\sin\Big(\frac{2\pi+3\pi}{12}\Big)=\sin\Big(\frac{2\pi}{12}+\frac{3\pi}{12}\Big)=\sin\Big(\frac{\pi}{6}+\frac{\pi}{4}\Big)$$
Now we use the sine sum identity:
$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$
Therefore,
$$\sin\frac{5\pi}{12}=\sin\frac{\pi}{6}\cos\frac{\pi}{4}+\cos\frac{\pi}{6}\sin\frac{\pi}{4}$$
$$\sin\frac{5\pi}{12}=\frac{1}{2}\frac{\sqrt2}{2}+\frac{\sqrt3}{2}\frac{\sqrt2}{2}$$
$$\sin\frac{5\pi}{12}=\frac{\sqrt2}{4}+\frac{\sqrt6}{4}$$
$$\sin\frac{5\pi}{12}=\frac{\sqrt2+\sqrt6}{4}$$