Question

If $z=r{e}^{i\theta }$, then the value of $|e^{iz}|$ is equal to

• A. $e^{rcos\theta }$
• B. $e^{-rcos\theta }$
• C. $e^{rsin\theta }$
• D. $e^{-rsin\theta }$

Answer 1

The correct option is D $e^{-rsin\theta }$

Ginen,

$z=r{e}^{i\theta }$

To find$\left|e^{iz}\right|$

solution,

for given,$z=r{e}^{i\theta }z=r(\cos \theta +i\sin \theta )\{e^{i\theta }=(\cos \theta +i\sin \theta \left)\right\}iz=ir(\cos \theta +i\sin \theta )iz=r(i\cos \theta +i^2\sin \theta )iz=r(-\sin \theta +i\cos \theta )iz=-r\sin \theta +ir\cos \theta e^{iz}=e^{-r\sin \theta +ir\cos \theta }\left|e^{iz}\right|=\left|e^{-r\sin \theta }\right|\left|e^{ir\cos \theta }\right|$

We know that$\left|e^{iz}\right|=1\therefore \left|e^{iz}\right|=\left|e^{-r\sin \theta }\right|⟶\left\{alwayspositive\right\}{e}^{iz}=e^{-r\sin \theta }$