Question

Write the value of $\sin \frac{\mathrm{\pi }}{15}\sin \frac{4\mathrm{\pi }}{15}\sin \frac{3\mathrm{\pi }}{10}$

Answer 1

$$\begin{gathered} \frac{\mathrm{\pi }}{15}=12°,\frac{4\mathrm{\pi }}{15}=48°,\frac{3\mathrm{\pi }}{10}=54° \\ \sin 12°\sin 48°\sin 54° \\ =\frac{1}{2}\left[2\sin 12°\sin 48°\right]\sin 54° \\ =\frac{1}{2}\left[\cos \left(12°-48°\right)-\cos \left(12°+48°\right)\right]\sin 54° \\ =\frac{1}{2}\left[\cos \left(-36°\right)-\cos 60°\right]\sin 54° \\ =\frac{1}{2}\sin 54°\left[\cos 36°-\frac{1}{2}\right] \\ =\frac{1}{2}\left[\sin \left(90°-36°\right)\cos 36°\right]-\frac{1}{4}\sin \left(90°-36°\right) \\ =\frac{1}{2}{\cos }^{2}36°-\frac{1}{4}\cos 36° \\ =\frac{1}{2}{\left(\frac{\sqrt{5}+1}{4}\right)}^2-\left(\frac{\sqrt{5}+1}{16}\right)\left[\cos 36°=\frac{\sqrt{5}+1}{4}\right] \\ =\frac{1}{2}\left(\frac{5+1+2\sqrt{5}}{16}\right)-\left(\frac{\sqrt{5}+1}{16}\right) \\ =\frac{6+2\sqrt{5}}{32}-\frac{\sqrt{5}+1}{16} \\ =\frac{6+2\sqrt{5}-2\sqrt{5}-2}{32} \\ =\frac{4}{32} \\ =\frac{1}{8} \end{gathered}$$