Question

Solve the equation
$\left(vi\right)3{\cos }^{2}x-2\sqrt{3}\sin x\cos x-3{\sin }^{2}x=0$

Answer 1

$3{\cos }^{2}x-2\sqrt{3}\sin x\cos x-3{\sin }^{2}x=0$

$\Rightarrow 3({\cos }^{2}x-{\sin }^{2}x)-2\sqrt{3}\sin x\cos x=0$

$\Rightarrow 3\cos 2x-\sqrt{3}\sin 2x=0$

$\Rightarrow \sqrt{3}\sin 2x=3\cos 2x$

$\Rightarrow \tan 2x=\sqrt{3}$

$\Rightarrow \tan 2x=\tan \frac{\pi }{3}$

We know the general solution of $\tan x=\tan \alpha$ is $x=n\pi +\alpha ,n\in Z$

Thus, the general solution is
$2x=n\pi +\frac{\pi }{3}\Rightarrow x=\frac{n\pi }{2}+\frac{\pi }{6}$

Hence, the general solution is $x=\frac{n\pi }{2}+\frac{\pi }{6},n\in Z$