Giải phương trình: sin (3x + 2pi/3) + sin (x - 7pi/5) = 0

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