Complex Numbers And Quadratic Equations question 660
Question: The principle value of the $ \arg (z) $ and $ |z| $ of the complex number $ z=1+\cos ( \frac{11\pi }{9} )+i\sin ( \frac{11\pi }{9} ) $ are respectively.
Options:
A) $ \frac{11\pi }{8},2\cos ( \frac{\pi }{18} ) $
B) $ -\frac{7\pi }{18},-2\cos ( \frac{11\pi }{18} ) $
C) $ \frac{2\pi }{9},2\cos ( \frac{7\pi }{18} ) $
D) $ -\frac{\pi }{9},-2\cos ( \frac{\pi }{18} ) $
Show Answer
Answer:
Correct Answer: B
Solution:
$ z=1+cos\frac{11\pi }{9}+i\sin \frac{11\pi }{9} $ $ Re(z) > 0 and Im( z ) < 0 $ , so the number lies in the fourth quadrant. Also $ z=2\cos \frac{11\pi }{18}{ \cos \frac{11\pi }{18}+i\sin \frac{11\pi }{18} } $ $ =2\cos \frac{11\pi }{18}{ \cos ( -\frac{7\pi }{18} )+i\sin ( -\frac{7\pi }{18} ) } $
$ \therefore \arg (z)=-\frac{7\pi }{18} $ $ | z |=| 2\cos \frac{11\pi }{18} |=-2\cos \frac{11\pi }{18} $
BETA
