11. The sequence $\left\{\alpha_{n}\right\}$ is an arithmetic sequence, $\beta$ is the common difference, the sequence $\left\{\sin \alpha_{n}\right\}$ is a geometric sequence, with common ratio $q$, and $\alpha_{1}, \beta \in \mathbf{R}$, then $q=$ $\qquad$
11. 1 or -1. $\left.\left.-a_{3}\right)\right]$

Since $\alpha_{3}-\alpha_{1}=2 \beta, \alpha_{1}+\alpha_{3}=2 \alpha_{2}$, we have $1-\cos \alpha_{2}=-\cos 2 \alpha_{2}+\cos 2 \beta \Rightarrow \beta=$ $k \pi, k \in \mathbb{Z}$
Therefore, $q=\frac{\sin \alpha_{2}}{\sin \alpha_{1}}=\frac{\sin \left(\alpha_{1}+k \pi\right)}{\sin \alpha_{1}}=(-1)^{k}$, hence $q$ is -1 or 1.