Question

The value of $\sin \frac{\pi }{5}\sin \frac{2\pi }{5}\sin \frac{3\pi }{5}\sin \frac{4\pi }{5}$ is

• A. $\frac{5}{16}$
• B. $\frac{11}{16}$
• C. $\frac{13}{16}$
• D. $\frac{3}{16}$

Answer 1

The correct option is A $\frac{5}{16}$

$\frac{\pi }{5},\frac{4\pi }{5}$ and $\frac{2\pi }{5},\frac{3\pi }{5}$ are
Supplementary angles, so
$\sin \frac{4\pi }{5}=\sin (\pi -\frac{\pi }{5})=\sin \frac{\pi }{5}$
Similarly,
$\sin \frac{3\pi }{5}=\sin \frac{2\pi }{5}$

So,
$\sin \frac{\pi }{5}\sin \frac{2\pi }{5}\sin \frac{3\pi }{5}\sin \frac{4\pi }{5}={\sin }^2\frac{\pi }{5}{\sin }^2\frac{2\pi }{5}.....\left(i\right)$
We know that,
$\sin \frac{\pi }{5}=\frac{\sqrt{10-2\sqrt{5}}}{4}$
$\sin \frac{2\pi }{5}=\sin (\frac{\pi }{2}-\frac{\pi }{10})=\cos \frac{\pi }{10}=\frac{\sqrt{10+2\sqrt{5}}}{4}$

From$\left(i\right)$${\sin }^2\frac{\pi }{5}{\sin }^2\frac{2\pi }{5}={\left(\frac{\sqrt{10-2\sqrt{5}}}{4}\right)}^2{\left(\frac{\sqrt{10+2\sqrt{5}}}{4}\right)}^2={\left(\frac{\sqrt{100-20}}{16}\right)}^2={\left(\frac{4\sqrt{5}}{16}\right)}^2=\frac{5}{16}$