Question

The solution set of the equation $4sin \theta cos ⁡ \theta -2cos ⁡ \theta -2\sqrt{3}sin ⁡ \theta +\sqrt{3}=0$ in the interval $\left(0 , 2 \pi \right)$ is

• A. $\left\{\frac{3 \pi }{4} , \frac{7 \pi }{4}\right\}$
• B. $\left\{\frac{\pi }{3} , \frac{5 \pi }{3}\right\}$
• C. $\left\{\frac{3 \pi }{4} , \pi , \frac{\pi }{3} , \frac{5 \pi }{3}\right\}$
• D. $\left\{\frac{\pi }{6} , \frac{5 \pi }{6} , \frac{11 \pi }{6}\right\}$

Answer 1

Answer: (D)

$4sin \theta cos ⁡ \theta -2cos ⁡ \theta -2\sqrt{3}sin ⁡ \theta +\sqrt{3}=0$
$2cos \theta \left(2 sin ⁡ \theta - 1\right)-\sqrt{3}\left(2 sin ⁡ \theta - 1\right)=0$
$\left(2 cos \theta - \sqrt{3}\right)\left(2 sin ⁡ \theta - 1\right)=0$
$\Rightarrow sin \theta =\frac{1}{2},cos ⁡ \theta =\frac{\sqrt{3}}{2}$
$\theta =\frac{\pi }{6},\frac{5 \pi }{6},\frac{11 \pi }{6}$