Question

If $k=\sin \pi /18\sin 5\pi /18\sin 7\pi /18$, then the numerical value of $k$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{18}$

Answer 1

The correct option is C $\frac{1}{8}$

$k=\sin \pi /8\sin 5\pi /18\sin 5\pi /18\sin 7\pi /18$

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

$=\frac{1}{2}[\cos {40}^{\circ }-\cos {60}^{\circ }]\sin {70}^{\circ }$

$=\frac{1}{2}\cos {40}^{\circ }\sin {70}^{\circ }-\frac{1}{4}\sin {70}^{\circ }$

$=\frac{1}{4}[\sin {110}^{\circ }+\sin {30}^{\circ }]-\frac{1}{4}\sin {70}^{\circ }$

$=\frac{1}{4}\sin ({180}^{\circ }-{70}^{\circ })+\frac{1}{4}\times \frac{1}{2}-\frac{1}{4}\sin {70}^{\circ }=\frac{1}{8}$