Answer
$$\sin\frac{7\pi}{12}=\frac{\sqrt6+\sqrt2}{4}$$
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 $\frac{\pi}{3}$.
$$\sin\frac{7\pi}{12}$$
Now, we try writing $7\pi$ as the sum of $3\pi$ and $4\pi$.
$$\sin\frac{7\pi}{12}=\sin \Big(\frac{4\pi+3\pi}{12}\Big)=\sin\Big(\frac{4\pi}{12}+\frac{3\pi}{12}\Big)=\sin\Big(\frac{\pi}{3}+\frac{\pi}{4}\Big)$$
We then use the sine sum identity:
$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$
Therefore,
$$\sin\frac{7\pi}{12}=\sin\frac{\pi}{3}\cos\frac{\pi}{4}+\cos\frac{\pi}{3}\sin\frac{\pi}{4}$$
$$\sin\frac{7\pi}{12}=\frac{\sqrt3}{2}\frac{\sqrt2}{2}+\frac{1}{2}\frac{\sqrt2}{2}$$
$$\sin\frac{7\pi}{12}=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}$$
$$\sin\frac{7\pi}{12}=\frac{\sqrt6+\sqrt2}{4}$$