Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : 1-i/1+i

Question

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : \(\frac{1-i}{1+i}\)

Answer 1

Given complex number is z = \(\frac{1-i}{1+i}\) 

We know that the polar form of a complex number Z=x+iy is given by Z=|Z|(cosθ+isinθ) 

Where, 

|Z|=modulus of complex number= \(\sqrt{x^2+y^2}\)

θ =arg(z)=argument of complex number= tan^-1 \(\Big(\frac{|y|}{|x|}\Big)\)

Now for the given problem,

⇒ |z| = \(\sqrt{{0}^2+(1)^2}\) 

⇒ |z| = \(\sqrt{0+1}\) 

⇒ |z| = \(\sqrt{1}\) 

⇒ |z|=1

⇒ θ = tan^-1\(\Big(\frac{1}{0}\Big)\) 

Since x>0,y>0 complex number lies in 1^st quadrant and the value of θ will be as follows 0°≤θ≤90°.

⇒ θ = tan^-1(∞)

⇒ θ = \(\frac{-\pi}{2}\)

⇒  z = \({1}\Big(cos\Big(\frac{-\pi}{2}\Big) + isin\Big(\frac{-\pi}{2}\Big)\Big)\) 

⇒  z = \({1}\Big(cos\Big(\frac{\pi}{2}\Big) - isin\Big(\frac{\pi}{2}\Big)\Big)\)

∴ The Polar form of Z=\(\frac{(1-i)}{(1+i)}\) is z =\({1}\Big(cos\Big(\frac{\pi}{2}\Big) - isin\Big(\frac{\pi}{2}\Big)\Big)\)