Phương trình \(3{\sin ^2}x...

C

- Hướng dẫn giải

Ta có: \(3{\sin ^2}x - 7\sin x\cos x - 10{\cos ^2}x = 0\)

\(\begin{array}{l}\Leftrightarrow 3{\sin ^2}x - 10\sin x\cos x + 3\sin x\cos x - 10{\cos ^2}x = 0\\\Leftrightarrow \sin x\left( {3\sin x - 10\cos x} \right) + \cos x\left( {3\sin x - 10\cos x} \right) = 0\end{array}\)

\(\Leftrightarrow \left( {3\sin x - 10\cos x} \right)\left( {\sin x + \cos x} \right) = 0\)

\(\Leftrightarrow \left[ \begin{array}{l}3\sin x = 10\cos x\\\sin x = - \cos x\end{array} \right. \) \(\Leftrightarrow \left[ \begin{array}{l}\tan x = \dfrac{{10}}{3}\\\tan x = - 1\end{array} \right. \) \(\Leftrightarrow \left[ \begin{array}{l}x = - \dfrac{\pi }{4} + k\pi \\x = \arctan \left( {\dfrac{{10}}{3}} \right) + k\pi \end{array} \right.\;\left( {k \in \mathbb{Z}} \right)\)