The value of sin (π/(14)) sin ((3π)/(14)) sin ((5π)/(14)) is


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Question Number 90628 by Josephbaraka@gmail.com last updated on 25/Apr/20

$The value of \sin \frac{π}{14} \sin \frac{3 π}{14} \sin \frac{5 π}{14} is$
Commented by jagoll last updated on 25/Apr/20

$x = \frac{π}{14} \Rightarrow 7 x = \frac{π}{2}$ $w = \sin x \sin 3 x \sin 5 x$ $2 w \cos x = \sin 2 x \sin 5 x \sin 3 x$ $4 w \cos x = (\cos 3 x - \cos 7 x) \sin 3 x$ $4 w \cos x = \cos 3 x \sin 3 x$ $8 w \cos x = \sin 6 x$ $w = \frac{\sin 6 x}{8 \cos x} = \frac{\cos (\frac{π}{14})}{\cos (\frac{π}{14})} = \frac{1}{8}$ $\sin 6 x = \sin (\frac{3 π}{7}) = \cos (\frac{π}{14})$
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