## Polar form

Polar form of complex numbers is another way of representing complex number.

z= a+ib is the rectangular form of a complex number.

z = r(cos θ + i sin θ) is the polar form of the complex number.

Here 'r' represents the absolute value and 'θ' represents argument of the complex number.

Representation:

In the above diagram x- axis is the real axis and y-axis is the imaginary axis.  We will express real and imaginary part of the complex number in the form of r and θ where r is the modulus and θ is the argument of the complex number. θ is the angle made by the complex number with the real ( x) axis.

In the above diagram, we know that

r² = a² + b²

By the trigonometric ratios we know that

cos θ = a/r and sin θ = b/r.

we can rewrite it as

r cosθ = a  and  r sinθ = b.

Substituting the value of a and b in the rectangular coordinate form we get

z = r cosθ + i r sinθ

z  = r(cosθ + i sinθ)

where r = ∣z∣ = √(a² + b²) and

θ= tan ⁻¹ (b/a)

Note:  If a > 0 then θ= tan ⁻¹ (b/a).

If  a < 0 then θ= tan ⁻¹ (b/a) + π

or

θ= tan ⁻¹ (b/a) + 180°

Example:

Express z= 1+i in the form of polar.

Solution:

Let us find the value of r.

In the given complex number a = 1 and b = 1

So the value of r is  √(a² + b²)

r = √(1² + 1²)

= √2

and cos θ = a/r and sin θ = b/r.

cos θ = 1/√2   and sin θ = 1/√2

which implies θ = tan ⁻¹ (b/a)

= tan ⁻¹ (1/1).

=  tan ⁻¹ (1).

=  π/4.

So the polar form is z = 1/√2(cos π/4 + i sin π/4).

Example:

Convert in to rectangular form.

z = 2(cos 30° + i sin 30° )

Solution:

z  = 2 cos30°  + i 2sin 30°

Here

a  = r cosθ and b = r sin θ.

So a  = 2 cos 30° and b = 2 sin 30°

a  =  2(0.87)    and b = 2(0.5)

a  =  1.74        and b = 1

The rectangular form is z = 1.74 + i

Students can through the examples and explanation on their own. If you are having any doubt you can contact us through mail, we will help you to clear your doubts.

We welcome your valuable suggestions for the improvement of our site. Please use the box given below to express your comments.