Answer
$3(1-(1/2)^{1/3}).$
Work Step by Step
Let $u=\cos \theta$, then $du=-\sin\theta d\theta$ and $u:1\to 1/2$
\begin{align*}
\int_{0}^{\pi/3} \sin \theta \cos^{-2/3} \theta d \theta &=-\int_{1}^{1/2} u^{-2/3} du\\
&=-\frac{1}{1/3 }u^{1/3} |_1^{1/2}\\
&=3(1-(1/2)^{1/3}).
\end{align*}
