Complex numbers from polar to binomic form

How shall we proceed if we want to determine the binomial form of a complex number expressed in the polar form?

Let's see the procedure:

Given now a complex number $z$ in polar form $z = | z |_{α}$ , if we want to find the binomial form we only have to determine $a$ and $b$ , where:

$a$ is the real part and it is: $a = | z | \cos (α)$
$b$ the complex part and it is: $b = | z | \sin (α)$

Example

For example, if we have the complex number in polar form: $6_{225^{\circ}}$ .

We can determine the real part of its binomial form by: $a = 6 \cos (225^{\circ}) = - 3 \sqrt{2}$

And the complex part by: $b = 6 \sin (225^{\circ}) = - 3 \sqrt{2}$

Thus we will write it as $a + i b$ or using the example: $- 3 \sqrt{2} - 3 \sqrt{2}$ (that is a binomial form).

We can say in general terms that in order to translate a complex number in polar form into the binomial form, we only have to use the following formula:

$z_{α} = | z | (\cos α + \sin α \cdot i)$