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

Ginen,

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

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

solution,

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

We know that$\left| { e }^{ iz } \right| =1\\ \therefore \left| { e }^{ iz } \right| =\left| { e }^{ -r\sin\theta } \right| \longrightarrow \{ always\quad positive\} \\ { e }^{ iz }={ e }^{ -r\sin\theta }$