\(\sin \left( {2x + \frac{\pi }{3}} \right) = \sin \left( {3x - \frac{\pi }{6}} \right)\)

 \( \Leftrightarrow \left[ \begin{array}{l}2x + \frac{\pi }{3} = 3x - \frac{\pi }{6} + k2\pi \\2x + \frac{\pi }{3} = \pi - \left( {3x - \frac{\pi }{6}} \right) + k2\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}2x - 3x = - \frac{\pi }{6} - \frac{\pi }{3} + k2\pi \\2x + 3x = \pi + \frac{\pi }{6} - \frac{\pi }{3} + k2\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\5x = \frac{{5\pi }}{6} + k2\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = \frac{\pi }{6} + k\frac{{2\pi }}{5}\end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\).