17. 已知 $\cos \alpha=-\frac{3}{4}, \sin \beta=\frac{2}{3}, \alpha$ 是第三象限角, $\beta \in\left(\frac{\pi}{2}, \pi\right)$.

(1)求 $\sin 2 \alpha$ 的值;

(2)求 $\cos (2 \alpha+\beta)$ 的值.
(1) $\because \alpha$ 是第三象限角， $\cos \alpha=-\frac{3}{4}$ ，

$\therefore \sin \alpha=-\sqrt{1-\cos ^{2} \alpha}=-\frac{\sqrt{7}}{4}$,

$\therefore \sin 2 \alpha=2 \sin \alpha \cos \alpha=2 \times\left(-\frac{\sqrt{7}}{4}\right) \times\left(-\frac{3}{4}\right)=\frac{3 \sqrt{7}}{8}$.

(2) $\because \beta \in\left(\frac{\pi}{2}, \pi\right), \sin \beta=\frac{2}{3}$,
$\therefore \cos \beta=-\sqrt{1-\sin ^{2} \beta}=-\frac{\sqrt{5}}{3}$.

又 $\because \cos 2 \alpha=2 \cos ^{2} \alpha-1=2 \times \frac{9}{16}-1=\frac{1}{8}$ ，

$\therefore \cos (2 \alpha+\beta)=\cos 2 \alpha \cos \beta-\sin 2 \alpha \sin \beta$

$=\frac{1}{8} \times\left(-\frac{\sqrt{5}}{3}\right)-\frac{3 \sqrt{7}}{8} \times \frac{2}{3}=-\frac{\sqrt{5}+6 \sqrt{7}}{24}$.