$\begin{aligned} & \sin \frac{5 \pi}{12}=\sin 75^{\circ} \\ &=\sin \left(45^{\circ}+30^{\circ}\right) \\ &=\sin 45^{\circ} \cos 30^{\circ}+\cos 45^{\circ} \sin 30^{\circ} \\ &=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \cdot \frac{1}{2}=\frac{1}{\sqrt{2}}\left(\frac{\sqrt{3}+1}{2}\right) \\ &=\frac{\sqrt{3}+1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{4} \end{aligned}$