Phương trình sau \(\sin \left( {3x + \frac{\pi }{3}} \right) = - \frac{{\sqrt 3 }}{2}\) có bao nhiêu nghiệm thuộc khoảng \(\left( {0;\frac{\pi }{2}} \right)\)?

* Hướng dẫn giải

\(\sin \left( {3x + \frac{\pi }{3}} \right) =  - \frac{{\sqrt 3 }}{2} \Leftrightarrow \left[ \begin{array}{l}3x + \frac{\pi }{3} =  - \frac{\pi }{3} + k2\pi \\3x + \frac{\pi }{3} = \frac{{4\pi }}{3} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}3x =  - \frac{{2\pi }}{3} + k2\pi \\3x = \pi  + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{{2\pi }}{9} + k\frac{{2\pi }}{3}\\x = \frac{\pi }{3} + k\frac{{2\pi }}{3}\end{array} \right.\)

Vì \(x \in \left( {0;\frac{\pi }{2}} \right)\)  nên \(x = \frac{\pi }{3};\,\,\,x = \frac{{4\pi }}{9}\)