Question

Find the general solution of ${(1-2\cos \theta )}^2+{(\tan \theta +\sqrt{3})}^2=0$.

• A. $2n\pi -\frac{\pi }{3}$
• B. $n\pi -\frac{\pi }{3}$
• C. $2n\pi +\frac{\pi }{3}$
• D. none of these

Answer 1

The correct option is A $2n\pi -\frac{\pi }{3}$

${(1-2\cos \theta )}^2+{(\tan \theta +\sqrt{3})}^2=0$

$\Rightarrow {(1-2\cos \theta )}^2=0$ and ${(\tan \theta +\sqrt{3})}^2=0$

$\Rightarrow \cos \theta =\frac{1}{2}$ and $\tan \theta =-\sqrt{3}$

$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$ and $\theta =n\pi +(-\frac{\pi }{3})$

Hence general solution is $\theta =2n\pi -\frac{\pi }{3}$