Question

Write the value of sin $\frac{\mathrm{\pi }}{12}$ sin $\frac{5\mathrm{\pi }}{12}$.

Answer 1

sin $\frac{\mathrm{\pi }}{12}$ sin $\frac{5\mathrm{\pi }}{12}$

$$\begin{gathered} =\frac{1}{2}\times 2\left(\sin \frac{\mathrm{\pi }}{12}\right)\left(\sin \frac{5\mathrm{\pi }}{12}\right) \\ =\frac{1}{2}\left[\cos \left(\frac{\mathrm{\pi }}{12}-\frac{5\mathrm{\pi }}{12}\right)-\cos \left(\frac{\mathrm{\pi }}{12}+\frac{5\mathrm{\pi }}{12}\right)\right]\left[\because 2\sin A\sin B=\cos (A-B)-\cos (A+B)\right] \\ =\frac{1}{2}\left[\cos \left(-\frac{\mathrm{\pi }}{3}\right)-\cos \frac{\mathrm{\pi }}{2}\right] \\ =\frac{1}{2}\left(\frac{1}{2}-0\right) \\ =\frac{1}{4} \end{gathered}$$