Giải phương trình: \(\sin \left( {3x + \frac{{2\pi }}{3}} \right) + \sin \left( {x - \frac{{7\pi }}{5}} \right) = 0\).

\(\sin \left( {3x + \frac{{2\pi }}{3}} \right) + \sin \left( {x - \frac{{7\pi }}{5}} \right) = 0\)

⇔ \(\sin \left( {3x + \frac{{2\pi }}{3}} \right) - \sin \left( {x - \frac{{7\pi }}{5}} \right) = 0\)

⇔ \(\sin \left( {3x + \frac{{2\pi }}{3}} \right) = \sin \left( {x - \frac{{7\pi }}{5}} \right)\)

⇔ \(\left[ \begin{array}{l}3x + \frac{{2\pi }}{3} = x - \frac{{7\pi }}{5} + k2\pi \\3x + \frac{{2\pi }}{3} = x - \left( {x - \frac{{7\pi }}{5}} \right) + k2\pi \end{array} \right.\)

⇔ \(\left[ \begin{array}{l}x = - \frac{{8\pi }}{{15}} + k\pi \\x = \frac{{11\pi }}{{60}} + \frac{{k\pi }}{2}\end{array} \right.\left( {k \in \mathbb{Z}} \right)\)

Vậy phương trình đã cho có nghiệm là \(\left[ \begin{array}{l}x = - \frac{{8\pi }}{{15}} + k\pi \\x = \frac{{11\pi }}{{60}} + \frac{{k\pi }}{2}\end{array} \right.\left( {k \in \mathbb{Z}} \right)\).