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If $K=\sin\large\frac{5\pi}{8}$$\sin \large\frac{5\pi}{18}$$\sin \large\frac{7\pi}{18}$,then the numerical value of $K$ is _____

$\begin{array}{1 1}(A)\;\large\frac{1}{2}&(B)\;1\\(C)\;\large\frac{1}{8}&(D)\;\sqrt 2\end{array} $

1 Answer

Toolbox:
  • $\sin a.\sin b=\large\frac{1}{2}$$(\cos (a-b)-\cos(a+b))$
Given :
$K=\sin\large\frac{5\pi}{8}$$\sin \large\frac{5\pi}{18}$$\sin \large\frac{7\pi}{18}$
$\Rightarrow\sin 10^{\large\circ}\sin 50^{\large\circ}\sin 70^{\large\circ}$
$\Rightarrow \large\frac{1}{2}$$(\cos 40^{\large\circ}-\cos 60^{\large\circ}).\sin 70^{\large\circ}$
$\Rightarrow \large\frac{1}{2}$$(\cos 40^{\large\circ}-\large\frac{1}{2})$$.\sin 70^{\large\circ}$
$\Rightarrow \large\frac{-1}{2}(\large\frac{1}{2}-$$\cos 40^{\large\circ}).\sin 70^{\large\circ}$
$\Rightarrow \large\frac{-1}{4}$$(1-2\cos 40^{\large\circ}).\sin 70^{\large\circ}$
$\Rightarrow \large\frac{-1}{4}$$(1-2\sin 50^{\large\circ}).\sin 70^{\large\circ}$
$\cos (\large\frac{\pi}{2}$$-\theta)=\sin \theta$
$\Rightarrow \large\frac{-1}{4}$$[\sin 70^{\large\circ}+(\cos 120^{\large\circ}-\cos 20^{\large\circ})]$
$\Rightarrow \large\frac{-1}{4}$$[\sin 70^{\large\circ}+(\large\frac{-1}{2}$$-\cos 20^{\large\circ})]$
$\Rightarrow \large\frac{-1}{4}$$[\sin 70^{\large\circ}-\cos 20^{\large\circ}-\large\frac{1}{2}]$
$\Rightarrow \large\frac{-1}{4}$$[\sin 70^{\large\circ}-\sin 70^{\large\circ}-\large\frac{1}{2}]$
$\Rightarrow -\large\frac{1}{4}\times (-\large\frac{1}{2})=\frac{1}{8}$
Hence (C) is the correct answer.
answered Apr 22, 2014 by sreemathi.v
 

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