Question

If $k=sin\frac{\pi }{18}.sin\frac{5\pi }{18}.sin\frac{7\pi }{18}$ then the numerical value of k is

• A. $\frac{1}{4}$
• B. $\frac{1}{8}$
• C. $\frac{1}{16}$
• D. $Noneofthese$

Answer 1

The correct option is B

$\frac{1}{8}$

$Wehavek=sin\frac{\pi }{18}sin\frac{5\pi }{18}sin\frac{7\pi }{18}=cos(\frac{\pi }{2}-\frac{\pi }{18})cos(\frac{\pi }{2}-\frac{5\pi }{18})cos(\frac{\pi }{2}-\frac{7\pi }{18})=cos\frac{\pi }{9}cos\frac{2\pi }{9}cos\frac{4\pi }{9}=\frac{sin{2}^3\frac{\pi }{9}}{2^{3}sin\frac{\pi }{9}}=\frac{sin\frac{8\pi }{9}}{8sin\frac{\pi }{9}}\frac{sin(\pi -\frac{\pi }{9})}{8sin\frac{\pi }{9}}=\frac{1}{8}$