Question

The numerical value of $\sin \frac{\pi }{18}.\sin \frac{5\pi }{18}.\sin \frac{7\pi }{18}$ is equal to

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

Answer 1

Answer: (B)

$\frac{1}{8}$

Given expression is

$\sin \frac{\pi }{18}.\sin \frac{5\pi }{18}.\sin \frac{7\pi }{18}$

$=\cos \frac{8\pi }{18}\cos \frac{4\pi }{18}\cos \frac{2\pi }{18}$

$($multiply and divide by $2\sin \frac{2\pi }{18})$

$=\frac{\cos \frac{8\pi }{18}\cos \frac{4\pi }{18}2\sin \frac{4\pi }{18}}{2^2\sin \frac{2\pi }{18}}=\frac{2\cos \frac{8\pi }{18}\sin \frac{8\pi }{18}}{2^3\sin \frac{2\pi }{18}}$

$=\frac{\sin \frac{16\pi }{18}}{2^3\sin \frac{2\pi }{18}}=\frac{1}{8}$