Question

If $\sin \theta =\frac{1}{2},\tan \theta =\frac{1}{\sqrt{3}},\forall n\in I$, then most general values of $\theta$ are:

• A. $2n\pi +\frac{\pi }{6},\forall n\in I$
• B. $2n\pi +\frac{\pi }{4},\forall n\in I$
• C. $2n\pi +\frac{\pi }{3},\forall n\in I$
• D. $2n\pi +\frac{\pi }{2},\forall n\in I$

Answer 1

The correct option is A $2n\pi +\frac{\pi }{6},\forall n\in I$

$\because \sin \theta =\frac{1}{2}=\sin \frac{\pi }{6}$
$\Rightarrow \theta =\frac{\pi }{6},\pi -\frac{\pi }{6}$
and $\tan \theta =\frac{1}{\sqrt{3}}=\tan \left(\frac{\pi }{6}\right)$
$\Rightarrow \theta =\frac{\pi }{6},\pi +\frac{\pi }{6}$
$\therefore$ Common value of $\theta$ is $\frac{\pi }{6}$
$\therefore$ General value of $\theta$ is $2n\pi +\frac{\pi }{6},\forall n\in I$