Answer
$-\displaystyle \frac{\sqrt{6}}{6}$
Work Step by Step
Letting $A=2x, $
we express $x=\displaystyle \frac{A}{2},\quad \displaystyle \sin x=\sin\frac{A}{2}$
$( \sin x$ is negative, as $x$ terminates in Q.III.)
Half-Angle identity:
$\displaystyle \sin\frac{A}{2}=-\sqrt{\frac{1-\cos A}{2}}=-\sqrt{\frac{1-(\frac{2}{3})}{2}}$
$=-\displaystyle \sqrt{\frac{3-2}{6}}=-\frac{1}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}}$
$=-\displaystyle \frac{\sqrt{6}}{6}$
