Đáp án:
\[\left[ \begin{array}{l}
x = \dfrac{\pi }{6} + \arcsin \dfrac{1}{4} + k2\pi \\
x = \dfrac{{7\pi }}{6} - \arcsin \dfrac{1}{4} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\cos 2x = 1 - 2{\sin ^2}x\\
2\cos \left( {2x - \dfrac{\pi }{3}} \right) - 7\sin \left( {x - \dfrac{\pi }{6}} \right) = 0\\
\Leftrightarrow 2\cos \left[ {2.\left( {x - \dfrac{\pi }{6}} \right)} \right] - 7\sin \left( {x - \dfrac{\pi }{6}} \right) = 0\\
\Leftrightarrow 2.\left[ {1 - 2{{\sin }^2}\left( {x - \dfrac{\pi }{6}} \right)} \right] - 7\sin \left( {x - \dfrac{\pi }{6}} \right) = 0\\
\Leftrightarrow 2 - 4{\sin ^2}\left( {x - \dfrac{\pi }{6}} \right) - 7\sin \left( {x - \dfrac{\pi }{6}} \right) = 0\\
\Leftrightarrow 4{\sin ^2}\left( {x - \dfrac{\pi }{6}} \right) + 7\sin \left( {x - \dfrac{\pi }{6}} \right) - 2 = 0\\
\Leftrightarrow \left[ {4{{\sin }^2}\left( {x - \dfrac{\pi }{6}} \right) - \sin \left( {x - \dfrac{\pi }{6}} \right)} \right] + \left[ {8\sin \left( {x - \dfrac{\pi }{6}} \right) - 2} \right] = 0\\
\Leftrightarrow \sin \left( {x - \dfrac{\pi }{6}} \right)\left[ {4\sin \left( {x - \dfrac{\pi }{6}} \right) - 1} \right] + 2.\left[ {4\sin \left( {x - \dfrac{\pi }{6}} \right) - 1} \right] = 0\\
\Leftrightarrow \left[ {4\sin \left( {x - \dfrac{\pi }{6}} \right) - 1} \right].\left[ {\sin \left( {x - \dfrac{\pi }{6}} \right) + 2} \right] = 0\\
\Leftrightarrow \left[ \begin{array}{l}
4\sin \left( {x - \dfrac{\pi }{6}} \right) - 1 = 0\\
\sin \left( {x - \dfrac{\pi }{6}} \right) + 2 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sin \left( {x - \dfrac{\pi }{6}} \right) = \dfrac{1}{4}\\
\sin \left( {x - \dfrac{\pi }{6}} \right) = - 2
\end{array} \right.\\
- 1 \le \sin \left( {x - \dfrac{\pi }{6}} \right) \le 1 \Rightarrow \sin \left( {x - \dfrac{\pi }{6}} \right) = \dfrac{1}{4}\\
\sin \left( {x - \dfrac{\pi }{6}} \right) = \dfrac{1}{4}\\
\Leftrightarrow \left[ \begin{array}{l}
x - \dfrac{\pi }{6} = \arcsin \dfrac{1}{4} + k2\pi \\
x - \dfrac{\pi }{6} = \pi - \arcsin \dfrac{1}{4} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{6} + \arcsin \dfrac{1}{4} + k2\pi \\
x = \dfrac{{7\pi }}{6} - \arcsin \dfrac{1}{4} + k2\pi
\end{array} \right.\,\,\,\,\left( {k \in Z} \right)
\end{array}\)
