8. Let the equation $a \cos x + b \sin x + c = 0 \left(a^{2} + b^{2} \neq 0\right)$ have two distinct real roots $\alpha$ and $\beta$ in $[0, \pi]$, find the value of $\sin (\alpha+\beta)$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
8. Let $\cos x=u, \sin x=v$, then the original equation can be transformed into $\left\{\begin{array}{l}a u+b v+c=0 \\ u^{2}+v^{2}=1(v \geqslant 0)\end{array}\right.$ In the right figure, the line (1) and the semicircle (2) must intersect at two different points $A$ and $B$, and the inclinations of $O A$ and $O B$ are $\alpha$ and $\beta$ (or $\beta, \alpha$), respectively. Draw $O C \perp A B$ intersecting the semicircle at $C$, then the inclination angle of $O C$ is $\frac{\alpha+\beta}{2}$.
Also, $k_{\alpha C} \cdot k_{A B}=-1$,
which means $\tan \frac{\alpha+\beta}{2} \cdot\left(-\frac{a}{b}\right)=-1$,
thus $\tan \frac{\alpha+\beta}{2}=\frac{a}{b}$,
$$
\sin (\alpha+\beta)=\frac{2 \tan \frac{\alpha+\beta}{2}}{1+\tan ^{2} \frac{\alpha+\beta}{2}}=\frac{2 \cdot \frac{b}{a}}{1+\left(\frac{b}{a}\right)^{2}}=\frac{2 a b}{a^{2}+b^{2}} .
$$