Polar Form of a Complex Number

Cartesian coordinates are simple enough, but what about polar coordinates? We have covered this topic in the past, but we can take things a step further by finding polar forms of complex numbers. But how exactly do we do this, and why might this be an important step? Let's find out:

Reviewing our terms

Before we get started, let's take a second to review a few important terms:

First of all, how are polar coordinates different compared to Cartesian coordinates? Essentially, a set of Cartesian coordinates is the result of two measures: Up and down. But what might happen if we drew a diagonal line straight from the origin to the coordinates? The endpoint might be the same, but the line now measures a single distance from the origin rather than two distances (up and down). The single, diagonal line represents our polar coordinates.

Recall that we express our polar coordinates with an angle and a distance. This is notably different compared to Cartesian coordinates, which we represent with x and y, in polar we instead use r (radius) and theta (angle).

While the x and y coordinates represent the distances from the axes, the distance value in a set of polar coordinates represents the distance to the origin (0,0). The angle is measured from the x-axis.

We also need to quickly review the concept of a complex number.

Recall that a complex number takes the following form: $a+bi$

In this expression, a and b are real numbers and i represents the "imaginary unit." This imaginary unit is the square root of -1. We can plot an imaginary unit on the complex plane. Instead of having x and y axes, our new plane has an imaginary axis instead of a y-axis and a real axis instead of an x-axis.

Going back to our complex number $a+bi$, we can plot this as the ordered pair (a,b) on our complex plane.

When we do this, the distance from the origin to our complex number from the origin is called the "modulus." We also call this the absolute value.

We can express this absolute value as:

$\left|z\right|=|a+bi|$

This also equals:

$\sqrt{(a-0)2^{}+(b-0)2^{}}or\sqrt{a^2+b^2}$

Finding polar coordinates for our complex numbers

The rectangular form of our complex number is represented in this format:$z=a+bi$

But we can also represent this complex number in a different way called the "polar form."

Take a look at this diagram:

The y-axis is our imaginary axis (i), while the x-axis is our real axis (a).

Polar coordinates have two values: The angle θ and the distance from the origin (r).

All we need to do is find these two values for our complex number, and we will have our polar coordinates.

But how exactly do we accomplish this? In order to get our values, we need to refer back to the handy Pythagorean Theorem.

Take a look at the above diagram. What kind of shape do we notice? That's right -- a triangle. And whenever we see triangles, we know that we can apply the Pythagorean Theorem to help us solve all kinds of problems. Finding the polar coordinates of a complex number is certainly no exception.

Based on the above diagram, we can plug our values into the Pythagorean Theorem: $r^2=a^2+b^2$

Let's also bring our trigonometric ratios into the mix:

We know that the cosine of our angle θ is adjacent/hypotenuse or $a/r$. In other words:

$\cos \theta =a/r$

We also know that the sine of our angle θ is opposite/hypotenuse or $b/r$. In other words:

$sinTopics related to the Polar Form of a Complex Number Transformation of Graphs using Matrices - Dilation Complex Numbers Simplifying Absolute Value Expressions Flashcards covering the Polar Form of a Complex Number Precalculus Flashcards CLEP Precalculus Flashcards Practice tests covering the Polar Form of a Complex Number Precalculus Diagnostic Tests Pair your student with a suitable tutor who understands how to find the polar form of a complex number Finding the polar form of a complex number relies on skills your student already knows. But what if they need a refresher on complex numbers and polar coordinates? Without a solid understanding of these foundational concepts, it's easy to feel lost during class time. Tutoring gives them the opportunity to review past concepts and learn at a manageable yet productive pace. Students are also free to ask as many questions as they like during these 1-on-1 sessions. Reach out to our Educational Directors today and let Varsity Tutors pair your student with a suitable tutor. Subjects Near MeSeries 28 Test Prep1st Grade French Tutors4th Grade Science TutorsPMP TutorsOrganismal Biology TutorsActuarial Exam P Test Prep3rd Grade French TutorsMathematical Optimization TutorsPS Exam - Professional Licensed Surveyor Principles of Surveying Exam Test PrepComputer Science TutorsSalesforce Admin Test PrepSeries 3 Test PrepECAA Courses & ClassesThermochemistry TutorsUK A Level Sports Science TutorsIB Film TutorsMandarin Chinese 3 TutorsCIA - Certified Internal Auditor Test PrepSHRM-SCP - Society for Human Resource Management- Senior Certified Professional TutorsSAT Subject Test in Spanish Courses & ClassesPopular CitiesNew York City TutoringDayton TutoringMadison TutoringSyracuse TutoringSan Antonio TutoringColumbus TutoringAnn Arbor TutoringDetroit TutoringLas Vegas TutoringCincinnati TutoringPopular SubjectsSpanish Tutors in SeattleMCAT Tutors in HoustonSSAT Tutors in New York CityBiology Tutors in PhoenixSAT Tutors in BostonSpanish Tutors in San Francisco-Bay AreaEnglish Tutors in PhoenixMCAT Tutors in BostonFrench Tutors in Los AngelesReading Tutors in Boston;Download our free learning tools apps and test prep books$