$\begin{aligned} & \sin x-\sin 2 x+\sin 3 x=0 \\ & \sin x-2 \sin x \cdot \cos x+3 \sin x-4 \sin ^3 x=0 \\ & 4 \sin x-4 \sin ^3 x-2 \sin x \cdot \cos x=0 \\ & 2 \sin x\left(1-\sin ^2 x\right)-\sin x \cdot \cos x=0 \\ & 2 \sin x \cdot \cos ^2 x-\sin x \cdot \cos x=0 \\ & \sin x \cdot \cos x(2 \cos x-1)=0 \\ & \sin x=0, \cos x=0, \cos x=\frac{1}{2} \\ & x=0, \frac{\pi}{3} \quad x \in\left[0, \frac{\pi}{2}\right)\end{aligned}$