Polar form is a way of representing complex numbers by graphing them. But before determining how to find the polar form, we must first establish what a complex number is. A complex number is a trigonometry concept that means a “combination of real numbers and imaginary numbers.” Real numbers can be positive or negative. They are essentially any quantifiable number that, when squared, results in a positive value.

Imaginary numbers are numbers that don't compute on a calculator. When squared, they result in a negative value. Because these numbers are imaginary, they cannot be expressed with numerical values. Instead, the unit used for imaginary numbers is *i*, the square root of -1. Here are some examples of imaginary numbers: 5i, 3.6i, −14.3i, (√7)i, 2,891i. All of these numbers have been multiplied by the value of *i* or √−1.

Complex numbers have a real part and an imaginary part. Complex number *z* is used to represent the combined result of the real and imaginary parts. Here are some examples of complex numbers:

## Graphing a Complex Number

Graphical representation of a complex number can be expressed in two forms: rectangular form and polar form. In rectangular form, complex numbers are graphed on a complex plane (otherwise known as a quadrant of a graph) and plotted as an ordered pair (x and y values). Since real numbers are expressed as x-values, the horizontal axis of the graph is called the real axis. Imaginary numbers are expressed as y-values, making the vertical axis the imaginary axis. The rectangular form of a complex number is expressed with the equation .

The graph shows a complex number graphed in rectangular coordinates. Because the real number is a positive x-value, it is graphed in the positive real axis.

## Finding the Polar Form

Finding the polar form of a complex number is determining the distance (expressed by *r*) and the angle (expressed by *θ*) that is formed along the real axis. Together, these numbers are called the polar coordinates.

So rather than forming a rectangular representation of the complex number, the polar form represents complex numbers by forming a vector drawn opposite to the x-value of the real number. See an example of a complex number graphed in polar form below. The x-values represent real numbers, the y-values represent imaginary numbers, and the angled vector, x+y*i*, represents the complex number:

Using Pythagoras’ theorem, we know that the following trigonometric function is true:

We can also use the following trigonometric ratios to find the polar form of the complex number:

**sine=**

**cosine=**

**tangent=**

Using the equation for the rectangular form of a complex number, substitute your *a* and *b* with your x and y-values:

Combine these two equations to get the polar form of a complex number.

Finding the polar form of complex numbers is a hard concept to master. However, once you master it you’ll see how useful it is in helping you solve and graph other trigonometry concepts and theorems.