Official solution
Rationalizing the given expression
\[{{\left( {2 + 3i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}\]
For the given expression to be purely imaginary, real part of the above expression should be equal to zero.
\[ \Rightarrow {{2 - 6{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }} = 0\]
\[ \Rightarrow {\sin ^2}\theta = {1 \over 3}\]
\[ \Rightarrow \sin \theta = \pm {1 \over {\sqrt 3 }}\]