Question

Total number of solution of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ in $x\in [-\pi ,\pi ]$ is :

• A. $6$
• B. $4$
• C. $7$
• D. $8$

Answer 1

The correct option is B $4$

${\cos }^{2}x+\left(\frac{\sqrt{3}+1}{2}\right)\sin x-\frac{\sqrt{3}}{4}-1=0\Rightarrow {\sin }^{2}x-\left(\frac{\sqrt{3}+1}{2}\right)\sin x+\frac{\sqrt{3}}{4}=0\Rightarrow 4{\sin }^{2}x-2\sqrt{3}\sin x-2\sin x+\sqrt{3}=0\Rightarrow 2\sin x(2\sin x-\sqrt{3})-1(2\sin x-\sqrt{3})=0\Rightarrow (2\sin x-\sqrt{3})(2\sin x-1)=0$
$\Rightarrow \sin x=\frac{1}{2},\frac{\sqrt{3}}{2}\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{3},\frac{2\pi }{3}$
$\therefore$ There are four solutions in $[-\pi ,\pi ]$.