Question

If $k=\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right),$ then the numerical value of $k$ is

Answer 1

Given: $k=\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)$

$\Rightarrow k=\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$

$\Rightarrow k=\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$

$\Rightarrow k=\sin {10}^{\circ }\sin ({60}^{\circ }-{10}^{\circ })\sin ({60}^{\circ }+{10}^{\circ })$

$\Rightarrow k=\frac{1}{4}\sin {30}^{\circ }$ $\left(\begin{array}{c}\because \sin \theta \sin ({60}^{\circ }-\theta )\\ \sin ({60}^{\circ }+\theta )=\frac{1}{4}\sin 3\theta \end{array}\right)$

$\Rightarrow k=\frac{1}{4}\times \frac{1}{2}$

$\therefore k=\frac{1}{8}$