Polar coordinates are simply a way of defining the position of a point in 2 dimensions. The distance from the origin (r) and the angle made with the x axis (measure in an anti-clockwise direction) define the position of the point. The point in the above diagram is a distance 3 from the origin and the angle made with the x axis is 30°.
This tutorial from analyzemath.com will help you understand how to plot polar coordinates; it will also help you understand why there is more than one way to specify a particular point.
For example, try plotting (3, 60), (3,-300) Note the other suggestions in the tutorial.
For some excellent resources on polar curves see these from the mathcentre.
Explore polar curves using David Little’s applet.
You can very easily experiment with families of polar curves using the excellent Desmos graphing calculator. Click on the image below and experiment with the sliders.
Desmos polar curve, click on the image to experiment
It is possible to see how polar curves are traced out by using a slider in the domain on the Desmos graphing calculator. Try this example showing r=acoskθ.
Further examples: r=acos2θ, r=a(1-cosθ) r=ae-kθ
See also The Polar Gallery from mathsdemos.org. If you scroll down the page you will see that you can download a set of 14 Excel files which will allow you to experiment with several families of polar curves.
