Giải phương trình:

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

Do $\sin \left( { - \frac{\pi }{4}} \right) = - \frac{{\sqrt 2 }}{2}$ nên $\sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right) = - \frac{{\sqrt 2 }}{2}$ $\Leftrightarrow \sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right) = \sin \left( { - \frac{\pi }{4}} \right)$

$\Leftrightarrow \left[ \begin{array}{l}\frac{x}{2} + \frac{\pi }{4} = - \frac{\pi }{4} + k2\pi \\\frac{x}{2} + \frac{\pi }{4} = \pi - \left( { - \frac{\pi }{4}} \right) + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)$

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

$\Leftrightarrow \left[ \begin{array}{l}x = - \pi + k4\pi \\x = 2\pi + k4\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)$.