Giải phương trình:

\(\sin \left( {3x + \frac{\pi }{4}} \right) = - \frac{1}{2}\);

\(\sin \left( {3x + \frac{\pi }{4}} \right) = - \frac{1}{2}\)

\( \Leftrightarrow \sin \left( {3x + \frac{\pi }{4}} \right) = \sin \left( { - \frac{\pi }{6}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}3x + \frac{\pi }{4} = - \frac{\pi }{6} + k2\pi \\3x + \frac{\pi }{4} = \pi - \left( { - \frac{\pi }{6}} \right) + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}3x = - \frac{\pi }{6} - \frac{\pi }{4} + k2\pi \\3x = \pi + \frac{\pi }{6} - \frac{\pi }{4} + k2\pi \end{array} \right.\)

\[ \Leftrightarrow \left[ \begin{array}{l}3x = - \frac{{5\pi }}{{12}} + k2\pi \\3x = \frac{{11\pi }}{{12}} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = - \frac{{5\pi }}{{36}} + k\frac{{2\pi }}{3}\\x = \frac{{11\pi }}{{36}} + k\frac{{2\pi }}{3}\end{array} \right.\left( {k \in \mathbb{Z}} \right)\].

Vậy phương trình đã cho có các nghiệm là \[x = - \frac{{5\pi }}{{36}} + k\frac{{2\pi }}{3}\] và \[x = \frac{{11\pi }}{{36}} + k\frac{{2\pi }}{3}\] với k ∈ ℤ.