Question

The value of $\sin \frac{\pi }{7}\sin \frac{2\pi }{7}\sin \frac{3\pi }{7},$ is

• A. $1/8$
• B. $\sqrt{7}/8$
• C. $\sqrt{7}/2$
• D. $\sqrt{7}/16$

Answer 1

The correct option is D $\sqrt{7}/16$

$sin\frac{\pi }{7}sin\frac{2z}{7}sin\frac{3z}{7}$

$=\frac{1}{2}(cos\frac{\pi }{7}-cos\frac{3\pi }{7})sin\frac{3z}{7}$

$=\frac{1}{2}[sin\frac{3z}{7}cos\frac{\pi }{7}-sin\frac{3z}{7}cos\frac{3z}{7}]$

$=\frac{1}{2}\left[\right[\frac{1}{2}sin\frac{2z}{7}+sin\frac{4z}{7}\left]\right]-\frac{1}{2}[sin0+sin\frac{6z}{7}]$

$=\frac{1}{2}\times \frac{1}{2}[sin\frac{2z}{7}+sin\frac{4z}{7}-sin\frac{6z}{7}]$

$=\frac{1}{8}\left(1.32287565553\right)$

$=\mathrm{0.16.5}\approx \frac{\sqrt{7}}{16}$