Polar Coordinates

You CAN Ace Calculus

17trek > calculus > polar coordinates

Topics You Need To Understand For This Page


Page Tools, Related Topics and Links

Search 17Trek Calculus


17calculus is a 17trek subject

ideas to save on books - bags - supplies

effective study techniques - more calculus help

Polar coordinates is one of those topics that can be taught in many different courses. Some students come across the topic in physics for the first time. Sometimes, it's in precalculus or trig. No matter what course you are in right now, you will find everything you need here on polar coordinates. If you've already learned this topic, take some time to review the material on these pages and watch a few videos anyway, so that it is fresh in your mind.

Basic Idea of Polar Coordinates

You should already be familiar with graphing in rectangular coordinates (sometimes called cartesian coordinates). We can use trigonometry to describe the same point(s) or graph another way. Check out the graph on the right.

From basic trig, you know that a point in the plane can be described as \((x,y)\) or as \(( r \cos(\theta), r \sin(\theta) ) \). Comparing these two forms gives you the equations

\( x = r \cos(\theta) \)


\( y = r \sin(\theta) \)

These equations are used to convert between polar coordinates and rectangular coordinates.

Remember from trig that angles can be described in an infinite number of ways, since \( \theta = \theta + 2\pi \) and \(\theta = \theta - 2\pi\). It is always best to use the smallest possible angle in the interval \( (-\pi, \pi]\) or \( [0, 2\pi) \) or whatever is required by the context.


Source: Wikipedia - Polar Coordinate System

Note: These angles are measured in degrees. However, in calculus we almost always specify angles in radians.

One of the biggest differences you will find between trig and polar coordinates is that in trig, r in the above equation is usually \(1\). Trig focuses on the unit circle (when \(r=1\)). However, in polar coordinates we generalize the equations so that r is usually not \(1\).

The positive x-axis is called the polar axis, labeled L in the graph on the right and the point O is called the pole. All angles are measured from the polar axis with positive angles in a counter-clockwise direction.

Polar coordinates are just parametric equations where the parameter is the angle \(\theta\) and r is a function of \(\theta\). It will help you to understand polar coordinates if you have a good understanding of parametrics. Go to the parametrics section for more information.

Okay, it's time to watch some videos.

This first video is really good to give you an overview of polar coordinates.

Here is another good video introduction to polar coordinates. He uses graphs and examples very effectively in this video.

Okay, now you know the basics of polar coordinates and you can work most problems you come across. However, if you really want to understand polar coordinates, then this video clip is good to watch. It gives a more indepth discussion with some very good examples, some unique, which will help you a lot.

next: converting to/from polar coordinates →

Next - - With the basics of polar coordinates under your belt, now it is time to work directly with the points and equations to convert between rectangular and polar coordinates. You will find discussion, videos and practice problems on the next page.

17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades of any individual. We have worked, to the best of our ability, to ensure accurate and correct information and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Real Time Web Analytics

top   -   search   -   page like? 1