Representation of a Complex Number in Polar Form

Representation of a Complex Number in Polar Form:

Let P (x, y) be any point in the argand plane corresponding to a complex number z = x + i y such that it makes an angle θ with positive direction of X-axis then the polar form of z = x + i y is-

z = r (cos θ + i sin θ) where r = | z | = √(x² + y²) and tan θ = | y/x |

The argument of z depends upon the Quadrant in which the complex number lies.

Case I- If z lies in the first quadrant then the argument of (z) = θ.

Case II- If z lies in the second quadrant then the argument of (z) = 180° – θ.

Case III- If z lies in the third quadrant then the argument of (z) = 180° + θ or θ – 180°.

Case IV- If z lies in the fourth quadrant then the argument of (z) = 360° – θ or -θ.

Example- Express the following complex number in Polar Form √3 – i

Solution- Let z = √3 – i
Now, r = | z | = √[(√3)² + (-1)²] = √(3 + 1) = 2

and tan θ = | y/x | = | -1/√3 | = 1/√3
∴ θ = 30°

Since z lies in the fourth quadrant,
∴ arg (z) = α = -θ = -30°

∴ The required polar form of z = √3 – i is,
z = r [cos α + i sin α]
⇒ z = 2 [cos (-30°) + i sin (-30°)]
⇒ z = 2 [cos 30° – i sin 30°]

Representation of Complex Numbers	Newton’s Law of Universal Gravitation
Demoivre’s Theorem	Zeroth Law of Thermodynamics
Cube Roots of Unity	Seebeck Effect
Square Roots of Complex Numbers	Units, Dimensions and Vectors– NIOS