First slide

Multiple and sub- multiple Angles

Question

Let $f (θ) = s i n θ s i n 3 θ s i n 5 θ$ , then $f (π / 14)$ is equal to

Moderate

A

1/8
B

1/4
C

1/7
D

1/14

Solution

As $\frac{π}{14} - \frac{π}{2} - \frac{3 π}{7}, \frac{3 π}{14} - \frac{π}{2} - \frac{2 π}{7}$
and $\frac{5 π}{14} = \frac{π}{2} - \frac{π}{7}$ , we can write
$\begin{array}{l} f (\frac{π}{14}) = \cos \frac{3 π}{7} \cos \frac{2 π}{7} \cos \frac{π}{7} \\ \Rightarrow 2 \sin \frac{π}{7} f (\frac{π}{14}) = \sin \frac{2 π}{7} \cos \frac{2 π}{7} \cos \frac{3 π}{7} \\ \Rightarrow 4 \sin \frac{π}{7} f (\frac{π}{14}) = \sin \frac{4 π}{7} \cos \frac{3 π}{7} \\ \Rightarrow 8 \sin (\frac{π}{7}) f (\frac{π}{14}) = \sin (\frac{7 π}{7}) + \sin (\frac{π}{7}) = \sin \frac{π}{7} \\ \Rightarrow f (\frac{π}{14}) = \frac{1}{8} \end{array}$

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