Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
The amplitude of $e ^{e-1 \theta}$ is equal to
A
$\sin \theta$
✓
$-\sin \theta$
C
$e^{\cos \theta}$
D
$e^{\sin \theta}$
✓
Answer
Correct option: B.
$-\sin \theta$
(B) Let $z = e ^{ e ^{ e \theta}}= e ^{\cos \theta- i \sin \theta}= e ^{\cos \theta} e ^{- i \sin \theta}$ $= e ^{\cos \theta}[\cos (\sin \theta)- i \sin (\sin \theta)]$ $= e ^{\cos \theta} \cos (\sin \theta)- ie ^{\cos \theta} \sin (\sin \theta)$ $\therefore \operatorname{amp}(z)=\tan ^{-1}\left[-\frac{ e ^{\cos \theta} \sin (\sin \theta)}{ e ^{\cos \theta} \cos (\sin \theta)}\right]$ $=\tan ^{-1}[\tan (-\sin \theta)]$ $=-\sin \theta$
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