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Question

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


A
ercosθ
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B
ercosθ
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C
ersinθ
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D
ersinθ
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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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