JEE Advance - Mathematics (2011 - Paper 1 Offline - No. 2)
The positive integer value of $$n\, > \,3$$ satisfying the equation $${1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}$$ is
Answer
7
Explanation
We have,
$${1 \over {\sin (\pi /n)}} - {1 \over {\sin (3\pi /n)}} = {1 \over {\sin (2\pi /n)}}$$
$$ \Rightarrow {{\sin (3\pi /n) - \sin (\pi /n)} \over {\sin (\pi /n)\sin (3\pi /n)}} = {1 \over {\sin (2\pi /n)}}{{(2\sin (\pi /n)\cos (2\pi /n))\sin (2\pi /n)} \over {\sin (\pi /n)\sin (3\pi /n)}} = 1$$
$$ \Rightarrow \sin {{4\pi } \over n} = \sin {{3\pi } \over n} \Rightarrow {{4\pi } \over n} + {{3\pi } \over n} = \pi \Rightarrow n = 7$$
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