Notes (see the Notes page for further details of the various Notes collections).
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°.
Try this page on Desmos to experiment with plotting points.
Try a join the dots exercise! Polar Curves Join the dots
It is possible to see how polar curves are traced out by using a slider in the domain on the Desmos graphing calculator. Try selecting the image to see r=acoskθ.
Further examples: r=acos2θ, r=a(1-cosθ) r=ae-kθ r2=a2cos2θ
Alternatively, try this Polar Grapher; use the slider to change the angle and you will see how the curve is traced out. Note the value of R is displayed so you can easily see if it is positive or negative.
For a really clear plotter showing the connection between the Cartesian graph of r=f(θ) and the graph in polar coordinates try this Polar Curves and Cartesian Graphs applet. Watch the display carefully as you move the slider; you can easily see when r is negative for example.
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.