Giải phương trình:

\(\sin \left( {\frac{x}{3} + \frac{\pi }{2}} \right) = \frac{{\sqrt 3 }}{2}\);

Do \(\sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2}\) nên \(\sin \left( {\frac{x}{3} + \frac{\pi }{2}} \right) = \frac{{\sqrt 3 }}{2}\)\( \Leftrightarrow \sin \left( {\frac{x}{3} + \frac{\pi }{2}} \right) = \sin \frac{\pi }{3}\)

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

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

\( \Leftrightarrow \left[ \begin{array}{l}x = - \frac{\pi }{2} + k6\pi \\x = \frac{\pi }{2} + k6\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\).