Question

The most general values of $\theta$ satisfying the equation $(1+2\sin \theta {)}^2+(\sqrt{3}\tan \theta -1{)}^2=0$ are given by

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

here, $\sin \theta =–\frac{1}{2}$ and $\tan \theta =\frac{1}{\sqrt{3}}$
As, $a^2+b^2=0\Rightarrow a=b=0$
$\sin \theta =-\frac{1}{2}\Rightarrow \theta =m\pi +(-1{)}^n[-\frac{\pi }{6}]$
And $\tan \theta =\frac{1}{\sqrt{3}}\Rightarrow \theta =m\pi +\frac{\pi }{6}$
For common values, m must be odd
Ie, m = 2n + 1
$\Rightarrow \theta =2n\pi +\frac{7\pi }{6}$