Question

$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}$ is equal to

• A. $1$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{\sqrt{2}}{7}$

Answer 1

Answer: (C)

$\frac{1}{8}$

$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}$
$=\sin (\frac{\pi }{2}-\frac{3\pi }{7})\sin (\frac{\pi }{2}-\frac{2\pi }{7})\sin (\frac{\pi }{2}-\frac{\pi }{7})\sin \frac{\pi }{2}$
$=\cos \frac{3\pi }{7}\cos \frac{2\pi }{7}\cos \frac{\pi }{7}$
$=-\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}$

$=-\frac{\sin \left(\frac{2^3\pi }{7}\right)}{2^3\sin \left(\frac{\pi }{7}\right)}=-\frac{\sin \left(\frac{8\pi }{7}\right)}{8\sin \left(\frac{\pi }{7}\right)}=\frac{1}{8}$.