Phương trình ${\sin ^2}x + \sqrt 3 \sin x\cos x = 1$có bao nhiêu nghiệm thuộc $\left[ {0;2\pi } \right]?$

Ta có : ${\sin ^2}x + \sqrt 3 \sin x\cos x = 1$$\Leftrightarrow \dfrac{{1 - \cos 2x}}{2} + \dfrac{{\sqrt 3 }}{2}\sin 2x = 1$

$\begin{array}{l} \Leftrightarrow \dfrac{{\sqrt 3 }}{2}\sin 2x - \dfrac{1}{2}\cos 2x = \dfrac{1}{2} \\\Leftrightarrow \dfrac{1}{2}\cos 2x - \dfrac{{\sqrt 3 }}{2}\sin 2x = \dfrac{1}{2}\\ \Leftrightarrow \cos \dfrac{\pi }{3}.\cos 2x - \sin \dfrac{\pi }{3}\sin 2x = \dfrac{1}{2}\\ \Leftrightarrow \cos \left( {2x + \dfrac{\pi }{3}} \right) = \cos \dfrac{\pi }{3} \\\Leftrightarrow \left[ \begin{array}{l}2x + \dfrac{\pi }{3} = \dfrac{\pi }{3} + k2\pi \\2x + \dfrac{\pi }{3} =  - \dfrac{\pi }{3} + m2\pi \end{array} \right. \\\Leftrightarrow \left[ \begin{array}{l}x = k\pi \\x =  - \dfrac{\pi }{3} + m\pi \end{array} \right.\left( {k,\,m \in \mathbb{Z}} \right)\end{array}$

Vì $x \in \left[ {0;2\pi } \right]$ nên ta có

+ $0 \le k\pi  \le 2\pi  \Leftrightarrow 0 \le k \le 2$$\Leftrightarrow \left[ \begin{array}{l}k = 0 \Rightarrow x = 0\\k = 1 \Rightarrow x = \pi \\k = 2 \Rightarrow x = 2\pi \end{array} \right.$ 

+ $0 \le  - \dfrac{\pi }{3} + m2\pi  \le 2\pi$$\Leftrightarrow \dfrac{1}{6} \le m \le \dfrac{7}{6} \Leftrightarrow m = 1 \Rightarrow x = \dfrac{{2\pi }}{3}.$

Vậy có bốn nghiệm thuộc $\left[ {0;2\pi } \right]$