Question
A value of $$\theta $$ for which $$\frac{{2 + 3i\sin \theta }}{{1 - 2i\sin \theta }}$$ is purely imaginary, is:
A.
$${\sin ^{ - 1}}\left( {\frac{{\sqrt 3 }}{4}} \right)$$
B.
$${\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)$$
C.
$$\frac{\pi }{3}$$
D.
$$\frac{\pi }{6}$$
Answer :
$${\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)$$
Solution :
Rationalizing the given expression
$$\frac{{\left( {2 + 3i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)}}{{1 + 4{{\sin }^2}\theta }}$$
For the given expression to be purely imaginary, real part of the above expression should be equal to zero.
$$\eqalign{
& \Rightarrow \,\,\frac{{2 - 6{{\sin }^2}\theta }}{{1 + 4{{\sin }^2}\theta }} = 0 \cr
& \Rightarrow \,\,{\sin ^2}\theta = \frac{1}{3} \cr
& \Rightarrow \,\,\sin \theta = \pm \frac{1}{{\sqrt 3 }} \cr} $$