First slide

Multiple and sub- multiple Angles

Question

The value of $\sin (\frac{π}{18}) \sin (\frac{5 π}{18}) \sin (\frac{7 π}{18})$ , is

Moderate

A

$1 / 2$
B

$1 / 4$
C

$1 / 8$
D

$1 / 16$

Solution

We have
$\sin \frac{π}{18} \sin \frac{5 π}{18} \sin \frac{7 π}{18}$
$= \cos (\frac{π}{2} - \frac{π}{18}) \cos (\frac{π}{2} - \frac{5 π}{18}) \cos (\frac{π}{2} - \frac{7 π}{18})$
$\begin{array}{l} = \cos \frac{8 π}{18} \cos \frac{4 π}{18} \frac{2 π}{8} \cos \frac{2 π}{8} \\ = \cos \frac{π}{9} \cos \frac{2 π}{9} \cos \frac{4 π}{9} = \frac{\sin (2^{3} \cdot π / 9)}{2^{3} \frac{1}{\sin π / 9}} = \frac{1}{2^{3}} = \frac{1}{8} \end{array}$

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