The value of 108 sin

Question:

The value of $108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is

 

Solution:

$108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$

$=36\left(3 \sin \frac{\pi}{9}-4 \sin ^{3} \frac{\pi}{9}\right)$

$=36\left(\sin 3\left(\frac{\pi}{9}\right)\right)$    $\left[\right.$ using identity $\left.\because 3 \sin x-4 \sin ^{3} x=\sin 3 x\right]$

$=36\left(\sin \frac{\pi}{3}\right)$

$=36\left(\frac{\sqrt{3}}{2}\right)$

$=18 \sqrt{3}$

$\therefore 108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is $18 \sqrt{3}$

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