Complex Numbers And Quadratic Equations question 836
Question: The amplitude of $ {e^{{e^{-i\theta }}}} $ is equal to [RPET 1997]
Options:
A) $ \sin \theta $
B) $ -\sin \theta $
C) $ {e^{\cos \theta }} $
D) $ {e^{\sin \theta }} $
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ z={e^{{e^{-i\theta }}}}={e^{\cos \theta -i\sin \theta }} $ $ ={e^{\cos \theta }}{e^{-i\sin \theta }} $ $ z={e^{\cos \theta }}[\cos (\sin \theta )-i\sin (\sin \theta )] $ $ z={e^{\cos \theta }}\cos (\sin \theta )-i{e^{\cos \theta }}\sin (\sin \theta ) $ $ amp(z)={{\tan }^{-1}}[ -\frac{{e^{\cos \theta }}\sin (\sin \theta )}{{e^{\cos \theta }}\cos (\sin \theta )} ] $ $ ={{\tan }^{-1}}[\tan (-\sin \theta )]=-\sin \theta $ .
BETA
