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

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

(B) $\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$

JEE Main · 2016 · 03 Apr 2016 · Offline · Complex Numbers

A value of

θ

for which

\frac{2 + 3 i s i n θ}{1 - 2 i s i n θ}

is purely imaginary, is:

Answer: (C)

Multiply top and bottom by the conjugate of the denominator:

\frac{( 2 + 3 i s i n θ ) ( 1 + 2 i s i n θ )}{1 + 4 s i n ^{2} θ} = \frac{( 2 - 6 s i n ^{2} θ ) + 7 i s i n θ}{1 + 4 s i n ^{2} θ}

. The number is purely imaginary when its real part vanishes,

2 - 6 sin^{2} θ = 0

, so

sin^{2} θ = \frac{1}{3}

and

sin θ = \frac{1}{3}

. A value of

θ

is therefore

sin^{- 1} (\frac{1}{3})

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option