If z-rei - then find the value of -eiz-

Z=re^i θ = r(cos θ + i sin θ) ⇒ iz = ir(cos θ + i sin θ) = r sin θ + i r cos θ ⇒ e^iz=e^( r sin θ + ir cos θ) =e^ r sin θ e^ri cos θ ⇒ |e^iz| =e^ r sin θ|e^iα| ...

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Byju's Answer

Standard XIII

Mathematics

Euler's Representation

If z=rei θ, t...

Question

If $z = r e^{i θ}$ , then find the value of $| e^{i z} |$

e^{i θ}

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e^{- i sin θ}

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e^{- r sin θ}

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e^{- r cos θ}

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Solution

The correct option is C $e^{- r sin θ}$
$z = r e^{i θ} = r (cos θ + i sin θ)$
$\Rightarrow i z = i r (cos θ + i sin θ)$
$= - r sin θ + i r cos θ$
$\Rightarrow e^{i z} = e^{(- r sin θ + i r cos θ)}$
$= e^{- r sin θ} e^{r i cos θ}$
$\Rightarrow | e^{i z} | = e^{- r sin θ} | e^{i α} |$ , where $α = r cos θ$
$= e^{- r sin θ} [{cos}^{2} α + {sin}^{2} α]^{\frac{1}{2}}$
$= e^{- r sin θ}$

Suggest Corrections

If z=reiθ, then find the value of |eiz|

The correct option is C e−rsin θz=reiθ=r(cos θ+isin θ) ⇒iz=ir(cos θ+isin θ) =−r sin θ+ir cos θ ⇒eiz=e(−r sin θ+ir cos θ) =e−rsin θeri cos θ ⇒|eiz|=e−rsin θ|eiα|, where α=r cos θ =e−rsin θ[cos2α+sin2α]12 =e−rsin θ

If $z = r e^{i θ}$ , then find the value of $| e^{i z} |$