Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : -16/1+i√3

Question

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : \(\frac{-16}{1+i\sqrt{3}}\)

Answer 1

Given complex number is z = \(\frac{-16}{1+i\sqrt{3}}\) 

 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| = \(\sqrt{(-4)^2+(4\sqrt{3})^2}\) 

⇒ |z| = \(\sqrt{16+48}\) 

⇒ |z| = \({\sqrt{64}}\) 

⇒ |z|= 8

⇒ θ =tan^-1 \(\Big(\frac{4\sqrt{3}}{4}\Big)\)

Since x>0,y>0 complex number lies in 2^nd quadrant and the value of θ will be as follows 90°≤θ≤180°.

⇒ θ = tan^-1\((\sqrt{3})\)

⇒ θ = \(\frac{2\pi}{3}\)

⇒  z = \(8\Big(cos\Big(\frac{2\pi}{3}\Big) + isin\Big(\frac{2\pi}{3}\Big)\Big)\) 

∴ The Polar form of  z = \(\frac{-16}{1+i\sqrt{3}}\)  is   z = \(8\Big(cos\Big(\frac{2\pi}{3}\Big) + isin\Big(\frac{2\pi}{3}\Big)\Big)\)