Question

Find real $\theta$ such that $\left(\frac{3+2i\sin \theta }{1-2i\sin \theta }\right)$ is purely imaginary

• A. $n\pi \pm \pi /2$ (n integer)
• B. $n\pi \pm \pi /6$ (n integer)
• C. $n\pi \pm \pi /3$ (n integer)
• D. $n\pi \pm \pi /4$ (n integer)

Answer 1

The correct option is C $n\pi \pm \pi /3$ (n integer)

$z=\frac{3+2i\sin \theta }{1-2i\sin \theta }$

$=\frac{(3+2i\sin \theta )(1+2i\sin \theta )}{1+4{\sin }^2\theta }$..............$\left(rationalisingthedenominator\right)$

$=\frac{3+6i\sin \theta +2i\sin \theta -4{\sin }^2\theta }{1+4{\sin }^2\theta }$

$=\frac{3-4{\sin }^2\theta +8i\sin \theta }{1+4{\sin }^2\theta }$

$z$ is purely imaginary when $(3-4{\sin }^2\theta )=0$

$\Rightarrow \sin \theta =\pm \frac{\sqrt{3}}{2}$

Therefore, $\Rightarrow \theta =n\pi \pm \frac{\pi }{3}$
Ans: C