\(5+2\sqrt{3}sinx.cosx-3\sqrt{3}sinx=2cos^2x+3cosx\)
\(\Leftrightarrow5+2\sqrt{3}sinx.cosx-3\sqrt{3}sinx=2-2sin^2x+3cosx\)
\(\Leftrightarrow2sin^2x-3\sqrt{3}sinx+3+2\sqrt{3}sinx.cosx-3cosx=0\)
\(\Leftrightarrow\left(2sinx-\sqrt{3}\right)\left(sinx-\sqrt{3}\right)+\sqrt{3}cosx\left(2sinx-\sqrt{3}\right)=0\)
\(\Leftrightarrow\left(2sinx-\sqrt{3}\right)\left(sinx+\sqrt{3}cosx-\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\frac{\sqrt{3}}{2}\\\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx=\frac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\frac{\sqrt{3}}{2}\\sin\left(x+\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\end{matrix}\right.\)
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