If (( )/(18)) ((5 )/(18)) ((7 )/(18))=K, then the value of ((10K )/(3)) is:

Question

If $\sin (\frac{\pi }{18})\sin (\frac{5\pi }{18})\sin (\frac{7\pi }{18})=K$, then the value of $\sin (\frac{10K\pi }{3})$ is:

Accepted Answer

Let

$K=\sin (\frac{\pi }{18})\sin (\frac{5\pi }{18})\sin (\frac{7\pi }{18})$

We need to find

$\sin (\frac{10K\pi }{3})$

First, write the angles in degrees:

$\frac{\pi }{18}={10}^{\circ },\, \frac{5\pi }{18}={50}^{\circ },\, \frac{7\pi }{18}={70}^{\circ }$

So,

$K=\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$

Now use the standard identity:

$\sin 3x=4\sin x\sin ({60}^{\circ }+x)\sin ({60}^{\circ }-x)$

Taking $x={10}^{\circ }$,

$\sin {30}^{\circ }=4\sin {10}^{\circ }\sin {70}^{\circ }\sin {50}^{\circ }$

Since

$\sin {30}^{\circ }=\frac{1}{2}$

we get

$\frac{1}{2}=4\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }$

Hence,

$\sin {10}^{\circ }\sin {50}^{\circ }\sin {70}^{\circ }=\frac{1}{8}$

Therefore,

$K=\frac{1}{8}$

Now substitute in the required expression:

$\sin (\frac{10K\pi }{3})=\sin (\frac{10}{3}\cdot \frac{1}{8}\pi )=\sin (\frac{10\pi }{24})=\sin (\frac{5\pi }{12})$

Now,

$\frac{5\pi }{12}={75}^{\circ }$

So,

$\sin {75}^{\circ }=\sin ({45}^{\circ }+{30}^{\circ })$

Using the formula $\sin (A+B)$:

$\sin {75}^{\circ }=\sin {45}^{\circ }\cos {30}^{\circ }+\cos {45}^{\circ }\sin {30}^{\circ }$

$=\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\cdot \frac{1}{2}$

$=\frac{\sqrt{3}+1}{2\sqrt{2}}$

So the value is

$\frac{\sqrt{3}+1}{2\sqrt{2}}$

Hence, the correct option is A.

Solutions · 1