\(\begin{array}{l}\cos 2x + \sin x = 0 \Leftrightarrow \sin x =  - \cos 2x\\ \Leftrightarrow  - \sin x = \cos 2x \Leftrightarrow \sin \left( { - x} \right) = \cos 2x \Leftrightarrow \cos \left( {\frac{\pi}{2} + x} \right) = \cos 2x\\ \Leftrightarrow \left[ \begin{array}{l}2x = \frac{\pi }{2} + x + k2\pi \\2x =  - \frac{\pi }{2} - x + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\3x =  - \frac{\pi }{2} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x =  - \frac{\pi }{6} + \frac{{k2\pi }}{3}\end{array} \right.\,\,\left( {k \in Z} \right)\end{array}\)

Xét nghiệm \(x=\frac{\pi }{2}+k2\pi \in \left[ 0;2\pi  \right)\overset{k\in Z}{\mathop{\Leftrightarrow }}\,0\le \frac{\pi }{2}+k2\pi <2\pi \overset{k\in Z}{\mathop{\Leftrightarrow }}\,0\le \frac{1}{2}+2k<2\Leftrightarrow -\frac{1}{4}\le k<\frac{3}{4}\overset{k\in Z}{\mathop{\Leftrightarrow }}\,k=0.\)

Khi k = 0 ta có nghiệm \(x=\frac{\pi }{2}.\)

Xét nghiệm \(x =  - \frac{\pi }{6} + \frac{{k2\pi }}{3} \in \left[ {0;2\pi } \right)\mathop  \Leftrightarrow \limits^{k \in Z} 0 \le  - \frac{\pi }{6} + \frac{{k2\pi }}{3} < 2\pi  \Leftrightarrow 0 \le  - \frac{1}{6} + \frac{{2k}}{3} < 2\mathop  \Leftrightarrow \limits^{k \in Z} \frac{1}{4} \le k < \frac{{13}}{4}\mathop  \Leftrightarrow \limits^{k \in Z} \left\{ \begin{array}{l}k = 1\\k = 2\\k = 3\end{array} \right.\)

Khi k = 1 ta có nghiệm \(x=-\frac{\pi }{6}+\frac{2\pi }{3}=\frac{\pi }{2}\)

Khi k = 2 ta có nghiệm \(x=-\frac{\pi }{6}+\frac{4\pi }{3}=\frac{7\pi }{6}\)

Khi k = 3 ta có nghiệm \(x=-\frac{\pi }{6}+\frac{6\pi }{3}=\frac{11\pi }{6}\)

Vậy nghiệm của phương trình thuộc \(\left[ 0;2\pi  \right)\) là: \(\left\{ \frac{\pi }{2};\frac{7\pi }{6};\frac{11\pi }{6} \right\}\)

 Chọn D