Đề bài

Phương trình \(\sin 3x = \cos x\) có các nghiệm là: A. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{8} + k\frac{\pi }{2}}\\{x = \frac{\pi }{4} + k\pi }\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)                        B. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{8} + k\frac{\pi }{2}}\\{x =  - \frac{\pi }{4} + k\pi }\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\) C. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{8} + k\pi }\\{x = \frac{\pi }{4} + k\frac{\pi }{2}}\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)                  D. \(\left[ {\begin{array}{*{20}{c}}{x =  - \frac{\pi }{8} + k\frac{\pi }{2}}\\{x =  - \frac{\pi }{4} + k\pi }\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)

Lời giải chi tiết

Ta có:\(\sin 3x = \cos x \Leftrightarrow \sin 3x = \sin \left( {\frac{\pi }{2} - x} \right) \Leftrightarrow \left[ \begin{array}{l}3x = \frac{\pi }{2} - x + k2\pi \\3x = \pi  - \left( {\frac{\pi }{2} - x} \right) + k2\pi \end{array} \right.\)\( \Leftrightarrow \left[ \begin{array}{l}4x = \frac{\pi }{2} + k2\pi \\2x = \frac{\pi }{2} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{8} + k\frac{\pi }{2}\\x = \frac{\pi }{4} + k\pi \end{array} \right.\)  \(\left( {k \in \mathbb{Z}} \right)\)Đáp án đúng là A.