Complex Numbers And Quadratic Equations question 90
Question: The amplitude of the complex number $ z=\sin \alpha +i(1-\cos \alpha ) $ is
Options:
A) $ 2\sin \frac{\alpha }{2} $
B) $ \frac{\alpha }{2} $
C) $ \alpha $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ z=\sin \alpha +i(1-\cos \alpha ) $
Þ $ amp(z)={{\tan }^{-1}}( \frac{1-\cos \alpha }{\sin \alpha } )={{\tan }^{-1}}( \frac{2{{\sin }^{2}}\frac{\alpha }{2}}{2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}} ) $ $ ={{\tan }^{-1}}\tan ( \frac{\alpha }{2} )=\frac{\alpha }{2} $ .
BETA
