\(\begin{array}{l}\cos (\pi-\alpha)=-\cos \alpha=\frac{2}{3} \\ \pi\lt \alpha\lt \frac{3 \pi}{2} \Rightarrow \sin \alpha\lt 0 \\ \sin ^{2} \alpha+\cos ^{2} \alpha=1 \Rightarrow \sin ^{2} \alpha=1-\cos ^{2} \alpha=1-\frac{4}{9}=\frac{5}{9} \\ \Rightarrow \sin \alpha=-\frac{\sqrt{5}}{3} \\ \sin 2 \alpha=2 \sin \alpha \cos \alpha=2\left(-\frac{\sqrt{5}}{3}\right)\left(-\frac{2}{3}\right)=\frac{4 \sqrt{5}}{9}\end{array}\)