Giải phương trình \(\left( {\sin 2x + \cos {\rm{2}...

C

- Hướng dẫn giải

Lời giải chi tiết:

\(\begin{array}{l}\,\,\,\,\,\,\,\left( {\sin 2x + \cos {\rm{2}}x} \right)\cos x + 2\cos 2x - \sin x = 0\\ \Leftrightarrow \sin 2x.\cos x + \cos 2x.\cos x + 2.\cos 2x - \sin x = 0\\ \Leftrightarrow 2\sin x.{\cos ^2}x + \left( {\cos x + 2} \right).\cos 2x - \sin x = 0\\ \Leftrightarrow \sin x\left( {2{{\cos }^2}x - 1} \right) + \left( {\cos x + 2} \right).\cos 2x = 0\\ \Leftrightarrow \sin x.\cos 2x + \left( {\cos x + 2} \right).\cos 2x = 0\\ \Leftrightarrow \cos 2x.\left( {\sin x + \cos x + 2} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}\cos 2x = 0\\\sin x + \cos x + 2 = 0\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}2x = \frac{\pi }{2} + k\pi \\\sqrt 2 .\sin \left( {x + \frac{\pi }{4}} \right) =  - 2\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{4} + \frac{{k\pi }}{2}\,\,\,\left( {k \in \mathbb{Z}} \right)\\\sin \left( {x + \frac{\pi }{4}} \right) =  - \sqrt 2 \,\,\left( {Loai} \right)\end{array} \right.\end{array}\)

Vậy \(S = \left\{ {\frac{\pi }{4} + \frac{{k\pi }}{2},\,\,k \in \mathbb{Z}} \right\}\).

Chọn C.