Question

The value of $\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}$ is

• A. $\frac{1}{16}$
• B. $\frac{1}{8}$
• C. $\frac{1}{2}$
• D. none of these

Answer 1

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

$\sin \left(\frac{\pi }{14}\right)\sin \left(\frac{3\pi }{14}\right)\sin \left(\frac{5\pi }{14}\right)$

$\sin \left(\frac{3\pi }{14}\right)=\cos (\frac{\pi }{2}-\frac{3\pi }{14})$

$=\cos \left(\frac{4\pi }{14}\right)$

$\sin \left(\frac{5\pi }{14}\right)\cos (\frac{\pi }{2}-\frac{5\pi }{14})$

$=\cos \left(\frac{2\pi }{14}\right)$

$\therefore \sin \left(\frac{\pi }{14}\right)\cos \left(\frac{4\pi }{14}\right)\cos \left(\frac{2\pi }{14}\right)$

Multiply and divide by $\cos \left(\frac{\pi }{14}\right)$

$2\cos \left(\frac{\pi }{14}\right)\sin \left(\frac{\pi }{14}\right)\cos \left(\frac{4\pi }{14}\right)\cos \left(\frac{2\pi }{14}\right)\frac{1}{\cos \left(\frac{\pi }{14}\right)}$

$\frac{1}{2\cos \left(\frac{\pi }{14}\right)}\left[\frac{2}{2}\sin \left(\frac{2\pi }{14}\right)\cos \left(\frac{2\pi }{14}\right)\cos \left(\frac{4\pi }{14}\right)\right]$

$\frac{1}{\cos \left(\frac{\pi }{14}\right)}[\frac{2}{2}\sin \left(\frac{4\pi }{14}\right).\cos \left(\frac{4\pi }{14}\right)]$

$\frac{1}{8\cos \left(\frac{\pi }{14}\right)}\sin \left(\frac{8\pi }{14}\right)$

$\frac{1}{8\cos \left(\frac{\pi }{14}\right)}.\sin (\frac{\pi }{2}+\frac{\pi }{14})$

$\frac{1}{\cos \left(\frac{\pi }{14}\right)}.\cos \left(\frac{\pi }{14}\right)$

$=\frac{1}{8}$