Remember the Argand diagram in which the point (a, b) corresponds to the complex number z = a + ib

When the complex number is written as a + ib where a and b are real numbers, this is known as the Cartesian form.

This point (a,b) can also be specified by giving the distance, r, of the point from the origin and the angle, , between the line joining the point to the origin and the positive x-axis.

By some simple trigonometry it follows that a = r cos and b = r sin

Thus the complex number z can be written as r cos + i r sin

This is known as the polar form of a complex number.
r is called the modulus of z and is the argument of z.

The argument of a complex number is the angle between the positive x-axis and the line representing the complex number on an Argand diagram. It is denoted arg (z).

Polar form of a complex number z = a + ib is z = r(cosΘ + isinΘ)

where angle Θ is known as argument(or amplitude) of z, written as arg(z) or amp(z) and is determined by

          tanΘ = (b/a) or Θ = tan^-1(b/a).

The unique value of Θ such that -π < Θ ≤ π for which a = rcosΘ and b = rsinΘ, is known as principle value of argument.

The general value of the amplitude is (2nπ + Θ), where n is an integer and Θ is the principal value of arg(z).

NOTE: Θ = Theta    and    π = pie