Question

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

• A. $6$
• B. $4$
• C. $7$
• D. $8$

Answer 1

The correct option is B $4$

Given equation is
${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0\Rightarrow {\sin }^{2}x-\left(\frac{\sqrt{3}+1}{2}\right)\sin x+\frac{\sqrt{3}}{4}=0\Rightarrow 4{\sin }^{2}x-2\sqrt{3}\sin x-2\sin x+\sqrt{3}=0\Rightarrow 2\sin x(2\sin x-\sqrt{3})-1(2\sin x-\sqrt{3})=0\Rightarrow (2\sin x-\sqrt{3})(2\sin x-1)=0\Rightarrow \sin x=\frac{1}{2},\frac{\sqrt{3}}{2}\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{3},\frac{2\pi }{3}(\because x\in [-\pi ,\pi \left]\right)$

Hence, the number of solutions is $4.$