11)
$2\sin^2x+3\sin x\cos x-3\cos^2x=1$
- Với $\cos x=0$: $2\sin^2x=1$ (loại)
- Với $\cos x\ne 0\to x\ne \dfrac{\pi}{2}+k\pi$
Chia hai vế cho $\cos^2x$:
$2\tan^2x+3\tan x-3=1+\tan^2x$
$\to \tan^2x+3\tan x-4=0$
$\to \left[\begin{matrix} \tan x=1\\ \tan x=-4\end{matrix}\right.$
$\to \left[\begin{matrix} x=\dfrac{\pi}{4}+k\pi \\ x=\arctan(-4)+k\pi\end{matrix}\right.$ (TM)
12)
$3\sin^2x+8\sin x\cos x+(8\sqrt3-9)\cos^2x=0$
- Với $\cos x=0$: $3\sin^2x=0$ (loại)
- Với $\cos x\ne 0\to x\ne \dfrac{\pi}{2}+k\pi$
Chia hai vế cho $\cos^2x$:
$3\tan^2x+8\tan x+8\sqrt3-9=0$
$\Delta'=4^2-3(8\sqrt3-9)=43-24\sqrt3=(3\sqrt3-4)^2>0$
$ \sqrt{\Delta'}=3\sqrt3-4$
$\to \left[\begin{matrix} \tan x=\dfrac{-4+3\sqrt3-4}{3}=\dfrac{3\sqrt3-8}{3}\\ \tan x=\dfrac{ -4-3\sqrt3+4}{3}=-\sqrt3\end{matrix}\right.$
$\to \left[\begin{matrix} x=\arctan\dfrac{3\sqrt3-8}{3}+k\pi\\ x=-\dfrac{\pi}{3}+k\pi\end{matrix}\right.$ (TM)
