If z = r e^ i ,then | e^ iz |= - Vidyadip

MCQ

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

A
${e^{r\sin \theta }}$
✓
${e^{ - r\sin \theta }}$
C
${e^{ - r\cos \theta }}$
D
${e^{r\cos \theta }}$

✓

Answer

Correct option: B.

${e^{ - r\sin \theta }}$

b
(b)If $z = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )$
==> $iz = ir(\cos \theta + i\sin \theta ) = - r\sin \theta + ir\cos \theta $
or ${e^{iz}} = {e^{( - r\sin \theta + ir\cos \theta )}} = {e^{ - \sin \theta }}{e^{ri\cos \theta }}$
or $|{e^{iz}}| = |{e^{ - r\sin \theta }}||{e^{ri\cos \theta }}|$$ = {e^{ - r\sin \theta }}|{e^{ir\,\cos \theta }}|$
$ = {e^{ - r\sin \theta }}{[\{ {\cos ^2}(r\cos \theta ) + {\sin ^2}(r\cos \theta )\} ]^{1/2}} = {e^{ - r\sin \theta }}$

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