Question

If z=r$e^{i\theta }$,then |$e^{iz}$|=

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

Answer 1

The correct option is B

$e^{-rsin\theta }$

if z=r$e^{i\theta }$=r(cos$\theta$ + isin$\theta$)

$\Rightarrow$ iz=ir(cos$\theta$ + isin$\theta$)=-rcos$\theta$ + irsin$\theta$

or $e^{iz}$ = $e^{-rcos\theta +irsin\theta }$=$e^{-sin\theta }{e}^{ricos\theta }$

or |$e^{iz}$|=$|e^{-rsin\theta }||e^{ircos\theta }|$

=$e^{-rsin\theta }$$[co{s}^2\left(rcos\theta \right)+si{n}^2\left(rcos\theta \right){]}^2$=$e^{-rsin\theta }$