Question

A value of $\theta$ for which $\frac{2+3i\sin \theta }{1-2i\sin \theta }$ is purely imaginary is -

• A. $non-imaginary$
• B. $\frac{\pi }{6}$
• C. ${\sin }^{-1}\left(\frac{\sqrt{3}}{4}\right)$
• D. ${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Answer 1

The correct option is D ${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

$\frac{2+3i\sin \theta }{1-2i\sin \theta }$

Rationalizing denominator we get,

$=\frac{2+3i\sin \theta }{1-2i\sin \theta }\times \frac{1+2i\sin \theta }{1+2i\sin \theta }=\frac{(2-6{\sin }^2\theta )+i\times 7\sin \theta }{1+4{\sin }^2\theta }$

for purely imaginary $\frac{2-6{\sin }^2\theta }{1+4{\sin }^2\theta }=0$

on solving, $\sin \theta$ $=\frac{1}{\sqrt{3}}$

$\theta ={\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$