• # question_answer $\frac{3+2i\sin \theta }{1-2i\sin \theta }$ will be purely imaginary, if $\theta =$ [IIT 1976; Pb. CET 2003] A) $2n\pi \pm \frac{\pi }{3}$ B)   $n\pi +\frac{\pi }{3}$ C) $n\pi \pm \frac{\pi }{3}$ D) None of these [Where $n$ is an integer]

$\frac{3+2i\sin \theta }{1-2i\sin \theta }$ will be purely imaginary, if the real part vanishes, i.e., $\frac{3-4{{\sin }^{2}}\theta }{1+4{{\sin }^{2}}\theta }=0$ Þ  $3-4{{\sin }^{2}}\theta =0$  (only if $\theta$ be real) Þ  $\sin \theta =\pm \frac{\sqrt{3}}{2}=\sin \left( \pm \frac{\pi }{3} \right)$ Þ  $\theta =n\pi +{{(-1)}^{n}}\left( \pm \frac{\pi }{3} \right)=n\pi \pm \frac{\pi }{3}$