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Complex Numbers

If α =π/15 ...

Question

If $α = \frac{π}{15}$ ,then $cos 2 α cos 4 α cos 8 α cos 14 α = - \frac{1}{2 k}$ . Find the value of $k$ .

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Solution

$cos 2 α cos 4 α cos 8 α cos 14 α = cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15} cos \frac{14 π}{15}$
$= cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15} cos (π - \frac{π}{15}) = - cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15}$
$= \frac{- 2 cos \frac{π}{15} cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15}}{2 sin \frac{π}{15}}$
$= \frac{- 2 sin \frac{2 π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15}}{2 sin \frac{π}{15}} = \frac{- sin \frac{4 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15}}{4 sin \frac{π}{15}}$
$= \frac{- sin \frac{8 π}{15} cos \frac{8 π}{15}}{8 sin \frac{π}{15}} = \frac{- sin \frac{14 π}{15}}{16 sin \frac{π}{15}} = \frac{- sin (π - \frac{π}{15})}{16 sin \frac{π}{15}} = \frac{- sin \frac{π}{15}}{16 sin \frac{π}{15}} = \frac{- 1}{8} ∴ k = 4$

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