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Question:

If $\operatorname{Sin} X=\frac{1}{6}$, find the value of $\sin 3 x$

 

Solution:

$\operatorname{Sin} X=\frac{1}{6}$

Given: $\operatorname{Sin} X=\frac{1}{6}$

To find: $\sin 3 x$

We know that,

$\sin 3 x=3 \sin x-\sin ^{3} x$

Putting the values, we get

$\sin 3 x=3 \times\left(\frac{1}{6}\right)-\left(\frac{1}{6}\right)^{3}$

$\sin 3 x=\frac{1}{6}\left[3-\left(\frac{1}{6}\right)^{2}\right]$

$\sin 3 x=\frac{1}{6}\left[3-\frac{1}{36}\right]$

$\sin 3 x=\frac{1}{6}\left[\frac{108-1}{36}\right]$

$\sin 3 x=\frac{107}{216}$

 

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