## 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 | = √(x^{2} + y^{2}) 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 – iSolution- Let z = √3 – iNow, r = | z | = √[(√3) ^{2} + (-1)^{2}] = √(3 + 1) = 2and 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°] |