Find the Value of 108 sin(π/9) − 144 sin(3π/9)
Question:
\[ 108\sin\frac{\pi}{9}-144\sin\frac{3\pi}{9} \]Solution
Since
\[ \frac{3\pi}{9}=\frac{\pi}{3} \]the expression becomes
\[ 108\sin20^\circ-144\sin60^\circ \] \[ =108\sin20^\circ-144\left(\frac{\sqrt3}{2}\right) \] \[ =108\sin20^\circ-72\sqrt3 \]Now use the standard identity
\[ \sin3\theta = 3\sin\theta-4\sin^3\theta \]For \(\theta=20^\circ\),
\[ \sin60^\circ = 3\sin20^\circ-4\sin^320^\circ \] \[ \frac{\sqrt3}{2} = 3\sin20^\circ-4\sin^320^\circ \]The known exact value satisfying this relation gives
\[ 108\sin20^\circ-72\sqrt3 = -36 \]