Complex Numbers And Quadratic Equations question 116
Question: If $ z=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6} $ then [AMU 2002]
Options:
A) $ |z|=1,argz=\frac{\pi }{4} $
B) $ |z|=1,argz=\frac{\pi }{6} $
C) $ |z|=\frac{\sqrt{3}}{2},argz=\frac{5\pi }{24} $
D) $ |z|=\frac{\sqrt{3}}{2},argz={{\tan }^{-1}}\frac{1}{\sqrt{2}} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ z=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}=\frac{\sqrt{3}}{2}+\frac{i}{2} $
$ \therefore |z|=\sqrt{\frac{3}{4}+\frac{1}{4}}=1 $ and $ arg(z)={{\tan }^{-1}}( \frac{y}{x} )={{\tan }^{-1}}( \frac{1/2}{\sqrt{3}/2} )={{\tan }^{-1}}( \frac{1}{\sqrt{3}} ) $
$ \Rightarrow arg(z)={{\tan }^{-1}}( \tan \frac{\pi }{6} )=\frac{\pi }{6} $ .
BETA
