Phương trình lượng giác sin 3x - 4sin x.cos 2x = 0 có các nghiệm là:
A. $\left[ \begin{array}{l}x = k2\pi \\x = \pm \frac{\pi }{3} + n\pi \end{array} \right.$.
B. $\left[ \begin{array}{l}x = k\pi \\x = \pm \frac{\pi }{6} + n\pi \end{array} \right.$.
C. $\left[ \begin{array}{l}x = k\frac{\pi }{2}\\x = \pm \frac{\pi }{4} + n\pi \end{array} \right.$.
D. $\left[ \begin{array}{l}x = k\frac{{2\pi }}{3}\\x = \pm \frac{{2\pi }}{3} + n\pi \end{array} \right.$.
A. $\left[ \begin{array}{l}x = k2\pi \\x = \pm \frac{\pi }{3} + n\pi \end{array} \right.$.
B. $\left[ \begin{array}{l}x = k\pi \\x = \pm \frac{\pi }{6} + n\pi \end{array} \right.$.
C. $\left[ \begin{array}{l}x = k\frac{\pi }{2}\\x = \pm \frac{\pi }{4} + n\pi \end{array} \right.$.
D. $\left[ \begin{array}{l}x = k\frac{{2\pi }}{3}\\x = \pm \frac{{2\pi }}{3} + n\pi \end{array} \right.$.
Chọn B
$\sin 3x - 4\sin x.\cos 2x = 0 \Leftrightarrow 3\sin x - 4{\sin ^3}x - 4\sin x\left( {1 - 2{{\sin }^2}x} \right) = 0$
$4{\sin ^3}x - \sin x = 0 \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\2{\sin ^2}x = \frac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\\cos 2x = \frac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = k\pi \\x = \pm \frac{\pi }{6} + n\pi \end{array} \right.,\,\left( {k,n \in \mathbb{Z}\,} \right)$
$\sin 3x - 4\sin x.\cos 2x = 0 \Leftrightarrow 3\sin x - 4{\sin ^3}x - 4\sin x\left( {1 - 2{{\sin }^2}x} \right) = 0$
$4{\sin ^3}x - \sin x = 0 \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\2{\sin ^2}x = \frac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\sin x = 0\\\cos 2x = \frac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = k\pi \\x = \pm \frac{\pi }{6} + n\pi \end{array} \right.,\,\left( {k,n \in \mathbb{Z}\,} \right)$