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. \(\left[ {\begin{array}{*{20}{c}}{x = - \frac{\pi }{8} + k\frac{\pi }{2}}\\{x = - \frac{\pi }{4} + k\pi }\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)
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Sử dụng công thức \(\cos x = \sin \left( {\frac{\pi }{2} - x} \right)\)
Sử dụng kết quả \(\sin x = \sin \alpha \Leftrightarrow \left[ \begin{array}{l}x = \alpha + k2\pi \\x = \pi - \alpha + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
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.