foe x belongs to (0,pie) the eqn sinx + 2sin2x - sin3x =3

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{sinx}+2\sin \left(2\mathrm{x}\right)-\sin \left(3\mathrm{x}\right)=3 \\ \mathrm{Maximum} \mathrm{value} \mathrm{of} \mathrm{sinx} \mathrm{is} \mathrm{when} \mathrm{x}=\frac{\mathrm{\pi }}{2} \\ \Rightarrow 2\mathrm{x}=\mathrm{\pi } \mathrm{and} \sin \left(\mathrm{\pi }\right)=0 \\ 3\mathrm{x}=\frac{3\mathrm{\pi }}{2} \\ \sin \left(3\mathrm{x}\right)=-1 \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=1+0-1=2\ne 3 \mathrm{i}.\mathrm{e}. \mathrm{R}.\mathrm{H}.\mathrm{S}. \\ \mathrm{So} \mathrm{now} \mathrm{we} \mathrm{maximize} \sin \left(2\mathrm{x}\right) \\ \Rightarrow 2\mathrm{x}=\frac{\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{x}=\frac{\mathrm{\pi }}{4} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\sin \left(\frac{\mathrm{\pi }}{4}\right)+2\sin \left(\frac{\mathrm{\pi }}{2}\right)-\sin \left(\frac{3\mathrm{\pi }}{4}\right) \\ =\frac{1}{\sqrt{2}}+2-\frac{1}{\sqrt{2}} \\ =2\ne 3 \mathrm{i}.\mathrm{e}. \mathrm{R}.\mathrm{H}.\mathrm{S}. \\ \mathrm{Hence} \mathrm{given} \mathrm{equation} \mathrm{has} \mathrm{no} \mathrm{solution}. \end{gathered}$$

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