The number of solutions of the equation $(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^{2} x=-\frac{4}{1+\sqr

Question

The number of solutions of the equation $(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^{2} x=-\frac{4}{1+\sqrt{3}}, x \in\left[-2 \pi, \frac{5 \pi}{2}ight]$ is\

Professor PC Thomas Classes · Answer

(a): Given,\ $(4-\sqrt{3}) \sin x-2 \sqrt{3} \cos ^{2} x=-\frac{4}{1+\sqrt{3}}$,\ $x \in\left[-2 \pi, \frac{5 \pi}{2}ight]$\ $\Rightarrow \quad(4-\sqrt{3}) \sin x-2 \sqrt{3}\left(1-\sin ^{2} xight)=2(1-\sqrt{3})$\ $\Rightarrow \quad 2 \sqrt{3} \sin ^{2} x+4 \sin x-\sqrt{3} \sin x-2=0$\ $\Rightarrow \quad(2 \sin x-1)(\sqrt{3} \sin x+2)=0$\ $\Rightarrow \quad \sin x=\frac{1}{2}$\ $\left[\because \sin x 
eq \frac{-2}{\sqrt{3}}ight]$\ $\Rightarrow \quad x=\frac{-11 \pi}{6}, \frac{-7 \pi}{6}, \frac{\pi}{6}, \frac{5 \pi}{6}, \frac{13 \pi}{6}$\ ∴ Number of solutions $=5$\