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+yi, 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:
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.