Phương trình sin 3x = cos x có các nghiệm là:

A. \(\left[ \begin{array}{l}x = \frac{\pi }{8} + k\frac{\pi }{2}\\x = \frac{\pi }{4} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).

B. \(\left[ \begin{array}{l}x = \frac{\pi }{8} + k\frac{\pi }{2}\\x = - \frac{\pi }{4} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).

C. \(\left[ \begin{array}{l}x = \frac{\pi }{8} + k\pi \\x = \frac{\pi }{4} + k\frac{\pi }{2}\end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).

D. \(\left[ \begin{array}{l}x = - \frac{\pi }{8} + k\frac{\pi }{2}\\x = - \frac{\pi }{4} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).

Đáp án đúng là: A

Ta có sin 3x = cos x

\( \Leftrightarrow \sin 3x = \sin \left( {\frac{\pi }{2} - x} \right)\)

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

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

\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{8} + k\frac{\pi }{2}\\x = \frac{\pi }{4} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).