For $x\in \left(0,\pi \right)$, the equation $\sin x+2\sin 2x-\sin 3x=3$ has

$\sin x+2\sin 2x-\sin 3x=3$     $\Rightarrow \sin x-4\sin x\cos x-3\sin x+4{\sin }^{3}x=3$

$\Rightarrow -2-4\cos x+4{\sin }^{2}x=3\cos ecx \left(dividing throughout by \sin x\right)$

$\Rightarrow 2-4{\cos }^{2}x-4\cos x=3\cos ecx$

$\Rightarrow 3-4{\left(\cos x-\frac{1}{2}\right)}^2=3\cos ecx$

L.H.S $<3$ ; R.H.S $\ge 3$, No solution.