Question

Represent the given complex numbers in polar form $\frac{1}{1+i}$.

Answer 1

Given,

$\frac{1}{1+i}$

$=\frac{1}{1+i}\times \frac{1-i}{1-i}$

$z=\frac{1}{2}-i\frac{1}{2}$

$r=\sqrt{{\left(\frac{1}{2}\right)}^2+{(-\frac{1}{2})}^2}$

$\therefore r=\frac{1}{\sqrt{2}}$

$\theta ={\tan }^{-1}\frac{b}{a}={\tan }^{-1}\left(\frac{-\frac{1}{2}}{\frac{1}{2}}\right)={\tan }^{-1}(-1)=-\frac{\pi }{4}$

$\therefore \theta =180-45={135}^{\circ }=\frac{3\pi }{4}$

$z=\frac{1}{\sqrt{2}}(\cos \frac{3\pi }{4}-i\sin \frac{3\pi }{4})$