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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, as shown in plot 1.
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.
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 plot 2 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.
Krista King Math - Overview of Polar Coordinates
Here is another good video introduction to polar coordinates. He uses graphs and examples very effectively in this video.
PatrickJMT - Introduction to Polar Coordinates
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.
MIT OCW - Understand Polar Coordinates
Graphing In Polar Coordinates
Krista King Math - How do you sketch a polar curve (cardioid)? [6min-32secs]
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.
next: converting to/from polar coordinates →