The real value of $\mathrm{\theta }$ for which the expression$\frac{3+2\mathrm{isin}\mathrm{\theta }}{1-2\mathrm{isin}\mathrm{\theta }}$is purely imaginary is

$\frac{(3+2\mathrm{isin}\mathrm{\theta })(1+2i\sin \mathrm{\theta })}{1+4{\sin }^2\mathrm{\theta }}$is purely imaginery then

 Real part of  $\frac{(3+2\mathrm{isin}\mathrm{\theta })(1+2i\sin \mathrm{\theta })}{1+4{\sin }^2\mathrm{\theta }}$ =0
$\Rightarrow 3-4{\sin }^2\mathrm{\theta }=0\Rightarrow \mathrm{\theta }=\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{3}$