Find the modulus and argument of the following complex numbers and hen

1.	Find the modulus and argument of the following complex numbers and hence express each of them in the polar form : 1 + i
Answer» Given complex number is Z=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\| = \(\sqrt{(1)^2+(1)^2}\) ⇒ \|z\| = \(\sqrt{1+1}\) ⇒ \|z\| = \(\sqrt{2}\) ⇒ θ = tan^-1\(\Big(\frac{1}{1}\Big)\) Since x>0,y>0 complex number lies in 1^stquadrant and the value of θ will be as follows 0°≤θ≤90°. ⇒ θ = tan^-1(1) ⇒ θ = \(\frac{\pi}{4}\) ⇒ z = \(\sqrt{2}\Big(cos\Big(\frac{\pi}{4}\Big) + isin\Big(\frac{\pi}{4}\Big)\Big)\) ∴ The Polar form of Z=1+i is z =\(\sqrt{2}\Big(cos\Big(\frac{\pi}{4}\Big) + isin\Big(\frac{\pi}{4}\Big)\Big)\)

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

Answer»

Given complex number is Z=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| = \(\sqrt{(1)^2+(1)^2}\)

⇒ |z| = \(\sqrt{1+1}\)

⇒ |z| = \(\sqrt{2}\)

⇒ θ = tan^-1\(\Big(\frac{1}{1}\Big)\)

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

⇒ θ = tan^-1(1)

⇒ θ = \(\frac{\pi}{4}\)

⇒ z = \(\sqrt{2}\Big(cos\Big(\frac{\pi}{4}\Big) + isin\Big(\frac{\pi}{4}\Big)\Big)\)

∴ The Polar form of Z=1+i is z =\(\sqrt{2}\Big(cos\Big(\frac{\pi}{4}\Big) + isin\Big(\frac{\pi}{4}\Big)\Big)\)