Answer
$$\sin\frac{5\pi}{12}=\frac{\sqrt6+\sqrt2}{4}$$
Work Step by Step
$$\sin\frac{5\pi}{12}$$
We see that: $$\frac{5\pi}{12}=\frac{3\pi}{12}+\frac{2\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6}$$
Therefore, $$\sin\frac{5\pi}{12}=\sin\Big(\frac{\pi}{4}+\frac{\pi}{6}\Big)$$
Apply the addition formula for sine here, we have:
$$\sin\frac{5\pi}{12}=\sin\frac{\pi}{4}\cos\frac{\pi}{6}+\cos\frac{\pi}{4}\sin\frac{\pi}{6}$$
$$\sin\frac{5\pi}{12}=\frac{\sqrt2}{2}\times\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\times\frac{1}{2}$$
$$\sin\frac{5\pi}{12}=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}$$
$$\sin\frac{5\pi}{12}=\frac{\sqrt6+\sqrt2}{4}$$
