9. Given $\sin \alpha+\sin (\alpha+\beta)+\cos (\alpha+\beta)=\sqrt{3}, \beta \in\left[\frac{\pi}{4}, \pi\right]$, find the value of $\beta$.


9. From the given, $(1+\cos \beta-\sin \beta) \cdot \sin \alpha+(\sin \beta+\cos \beta) \cdot \cos \alpha=\sqrt{3}$. Let $u=\sin \alpha, v=\cos \alpha$, then $l:(1+\cos \beta-\sin \beta) \cdot u+(\sin \beta+\cos \beta) \cdot \cos \alpha-\sqrt{3}=0 . \odot O: u^{2}+v^{2}=1$. Since $l$ and $\odot O$ have a common point, we know $\frac{|-\sqrt{3}|}{\sqrt{(1+\cos \beta-\sin \beta)^{2}+(\sin \beta+\cos \beta)^{2}}} \leq 1$, which means $\cos \beta \geqslant \sin \beta$, and $\beta \in\left[\frac{\pi}{4}, \pi\right]$, thus $\cos \beta=\sin \beta$, hence $\beta=\frac{\pi}{4}$.