Toán Lớp 11: 2cos(2x-$\pi$ /3)-7sin(x-$\pi$ /6)=0
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Toán Lớp 11: 2cos(2x-$\pi$ /3)-7sin(x-$\pi$ /6)=0
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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)\]
\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}\)