If k = sin frac{pi }{{18}},.,sin frac{{5pi }} | vedclass

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Question

(b) We have $k = \sin \frac{\pi }{{18}}\sin \frac{{5\pi }}{{18}}\sin \frac{{7\pi }}{{18}}$
$ = \cos \left( {\frac{\pi }{2} - \frac{\pi }{{18}}} \righ

Vedclass · Accepted Answer

$\frac{1}{8}$

Vedclass · Answer

(b) We have $k = \sin \frac{\pi }{{18}}\sin \frac{{5\pi }}{{18}}\sin \frac{{7\pi }}{{18}}$
$ = \cos \left( {\frac{\pi }{2} - \frac{\pi }{{18}}} \right)\cos \left( {\frac{\pi }{2} - \frac{{5\pi }}{{18}}} \right)\cos \left( {\frac{\pi }{2} - \frac{{7\pi }}{{18}}} \right)$
$ = \cos \frac{\pi }{9}\cos \frac{{2\pi }}{9}\cos \frac{{4\pi }}{9} = \frac{{\sin {2^3}\frac{\pi }{9}}}{{{2^3}\sin \frac{\pi }{9}}} = \frac{{\sin \frac{{8\pi }}{9}}}{{8\sin \frac{\pi }{9}}}$
$ = \frac{{\sin \left( {\pi - \frac{\pi }{9}} \right)}}{{8\sin \frac{\pi }{9}}} = \frac{1}{8}$.

If $k = \sin \frac{\pi }{{18}}\,.\,\sin \frac{{5\pi }}{{18}}\,.\,\sin \frac{{7\pi }}{{18}},$ then the numerical value of $k$ is

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