If $A+B+C=0^0\text{ then }\sin A+\sin B+\sin C=$

$$\begin{gathered} \left(\sin A+\sin B\right)+\sin C \\ =2\sin \frac{A+B}{2}\cos \frac{A-B}{2}+2\sin \frac{C}{2}\cos \frac{C}{2} \\ =2\sin \left(\frac{-C}{2}\right)\cos \left(\frac{A-B}{2}\right)+2\sin \frac{C}{2}\cos \frac{C}{2} \\ =-2\sin \frac{C}{2}\cos \left(\frac{A-B}{2}\right)+2\sin \frac{C}{2}\cos \frac{C}{2} \\ =2\sin \frac{C}{2}\left(\cos \frac{C}{2}-\cos \left(\frac{A-B}{2}\right)\right) \\ =2\sin \frac{C}{2}(\cos \left(\frac{A+B}{2}\right)-\cos \left(\frac{A-B}{2}\right)) \\ =-2\sin \frac{C}{2}\left(\cos \left(\frac{A-B}{2}\right)-\cos \left(\frac{A+B}{2}\right)\right) \\ =-2\sin \frac{C}{2}\left(2\sin \frac{A}{2}\sin \frac{B}{2}\right) \\ =-4\sin A/2\sin B/2\sin C/2 \end{gathered}$$