Question

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

A
ercosθ
B
ercosθ
C
ersinθ
D
ersinθ

Solution

## The correct option is D $$e^{-rsin\theta}$$Ginen,$$z=r{ e }^{ i\theta }$$To find$$\left| { e }^{ iz } \right|$$ solution, for given,$$z=r{ e }^{ i\theta }\\ z=r(\cos\theta +i\sin\theta )\left\{ { e }^{ i\theta }=(\cos\theta +i\sin\theta ) \right\} \\ iz=ir(\cos\theta +i\sin\theta )\\ iz=r(i\cos\theta +{ i }^{ 2 }\sin\theta )\\ iz=r(-\sin\theta +i\cos\theta )\\ iz=-r\sin\theta +ir\cos\theta \\ { e }^{ iz }={ e }^{ -r\sin\theta +ir\cos\theta }\\ \left| { e }^{ iz } \right| =\left| { e }^{ -r\sin\theta } \right| \left| { e }^{ ir\cos\theta } \right|$$ We know that$$\left| { e }^{ iz } \right| =1\\ \therefore \left| { e }^{ iz } \right| =\left| { e }^{ -r\sin\theta } \right| \longrightarrow \{ always\quad positive\} \\ { e }^{ iz }={ e }^{ -r\sin\theta }$$Mathematics

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