Question

Find sum of all possible values of $\theta$ in the interval $\left(-\frac{\pi }{2},\pi \right)$ for which $\frac{3+2i\sin \theta }{1-2i\sin \theta }$ is purely imaginary;

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\mathrm{\pi }$
• C. $\frac{2\mathrm{\pi }}{3}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is C

$\frac{2\mathrm{\pi }}{3}$

Find the sum of all possible values of $\theta$:

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

Rationalize it:

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

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

Now, for $z$ to be purely imaginary, $Re\left(z\right)=0$

$\Rightarrow$$\frac{3-4{\sin }^2\theta }{1+4{\sin }^2\theta }=0$

$\Rightarrow$ ${\sin }^2\theta =\frac{3}{4}$

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

As $\theta \in \left(-\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)=\pm \frac{\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3}$

Sum of all values of $\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}$

Hence, Option ‘C’ is Correct.