Question

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

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

Answer 1

The correct option is B $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Rationalizing the given expression
$\frac{(2+3isin\theta )(1+2isin\theta )}{1+4si{n}^2\theta }$
For the given expression to be purely imaginary, real part of the above expression should be equal to zero.
$\Rightarrow \frac{2-6si{n}^2\theta }{1+4si{n}^2\theta }=0\Rightarrow si{n}^2\theta =\frac{1}{3}\Rightarrow sin\theta =\pm \frac{1}{\sqrt{3}}$