Question

If $z=\frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta },\frac{\pi }{4}<\theta <\frac{\pi }{2}$ ,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 C $\pi -2\theta$

$z=\frac{cos\theta +isin\theta }{cos\theta -isin\theta }$
$=\frac{e^{i\theta }}{e^{-i\theta }}$ ...(using Euler's form).
$=e^{2i\theta }$
$=cos\left(2\theta \right)+isin\left(2\theta \right)$
Since $\frac{\pi }{4}<\theta <\frac{\pi }{2}$
$\frac{\pi }{2}<2\theta <\pi$
$arg\left(z\right)=ta{n}^{-1}\left(tan\right(2\theta \left)\right)$ and $2\theta$ is in $2^{nd}$ quadrant.
Hence $arg\left(z\right)=\pi -2\theta$