The-value-of-sin-pi-14-sin-3pi-14-sin-5pi-14-is-

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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