Question

If $\frac{3+2i\sin \theta }{1-2i\sin \theta }$ is a real number and $0<\theta <2\pi$ then $\theta =$

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is A $\pi$

after rationalization , the given number is
$\frac{(3-4{\sin }^2\theta )+8i\sin \theta }{1+4{\sin }^2\theta }$
for real, imaginary part is zero
so $\sin \theta =0$
so $\theta =\pi$