5. Given the equation $a \cos x + b \sin x + c = 0$ has two distinct roots $\alpha, \beta$ in the interval $0 < x < \pi$, then $\sin (\alpha + \beta)=$
5. $\frac{2 a b}{a^{2}+b^{2}} \quad a \cos \alpha+b \sin \alpha+c=0$ (1) $\quad a \cos \beta+b \sin \beta+c=0$ (2), (1) - (2) gives $-2 a \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}+$ $2 b \sin \frac{\alpha-\beta}{2} \cos \frac{\alpha+\beta}{2}=0$ so $\tan \frac{\alpha+\beta}{2}=\frac{b}{a}$ so $\sin (\alpha+\beta)=\frac{2 \tan \frac{\alpha+\beta}{2}}{1+\tan ^{2} \frac{\alpha+\beta}{2}}=\frac{2 a b}{a^{2}+b^{2}}$