Question

Let $z=\frac{\cos \theta +i\sin \theta }{cos\theta -i\sin \theta },\frac{\pi }{4}<\theta <\frac{\pi }{2}$ Then arg $z$ is

• A. $2\theta$
• B. $2\theta -\pi$
• C. $\pi +2\theta$
• D. None of these

Answer 1

The correct option is A $2\theta$

$\cos \theta +i\sin \theta =e^{i\theta },\cos \theta -i\sin \theta =e^{-i\theta }$

$z=\frac{e^{i\theta }}{e^{-i\theta }}=e^{2i\theta }$

As $\frac{\pi }{4}<\theta <\frac{\pi }{2}$

$\frac{\pi }{2}<2\theta <\pi$

Therefore, $arg\left(z\right)=2\theta$