Question

The value of 108 $\sin \frac{\mathrm{\pi }}{9}-144{\sin }^3\frac{\mathrm{\pi }}{9}$ is ____________.

Answer 1

$$\begin{gathered} 108\sin \frac{\mathrm{\pi }}{9}-144{\sin }^3\frac{\mathrm{\pi }}{9} \\ =36\left(3\sin \frac{\mathrm{\pi }}{9}-4{\sin }^3\frac{\mathrm{\pi }}{9}\right) \\ =36\left(\sin 3\left(\frac{\mathrm{\pi }}{9}\right)\right)\left[\mathrm{using}\mathrm{identity}\because 3\sin x-4{\sin }^{3}x=\sin 3x\right] \\ =36\left(\sin \frac{\mathrm{\pi }}{3}\right) \\ =36\left(\frac{\sqrt{3}}{2}\right) \\ =18\sqrt{3} \\ \therefore 108\sin \frac{\mathrm{\pi }}{9}-144{\sin }^3\frac{\mathrm{\pi }}{9}\mathrm{is}18\sqrt{3} \end{gathered}$$