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 $z=|z|_{\alpha}$, 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: $a=|z|\cos(\alpha)$
• $b$ the complex part and it is: $b=|z|\sin(\alpha)$

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 $a+ib$ 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:

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