12. Given an acute angle $\alpha$ satisfies $\cos \alpha=\frac{3}{5}, \cos (\alpha+\beta)=-\frac{5}{13},$ the quadrant in which $\beta$ lies is
$\qquad$ .
12. First or third.

From $\cos \alpha=\frac{3}{5}$ we get $\sin \alpha=\frac{4}{5}$. When $\cos (\alpha+\beta)=-\frac{5}{13}, \sin (\alpha+\beta)=\frac{12}{13}$, then $\cos \beta=\frac{33}{65}$, $\sin \beta=\frac{56}{65}$, so $\beta$ is in the first quadrant. When $\cos (\alpha+\beta)=-\frac{5}{13}, \sin (\alpha+\beta)=\frac{12}{13}$, then $\cos \beta=-\frac{63}{65}, \sin \beta=$ $-\frac{16}{65}$.