Polar Coordinates

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°.

Desmos - polar coordinates
Try this page on Desmos to experiment with plotting points. 

For some excellent resources on polar curves see these from the mathcentre. (Notes & examples)

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 domain and 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 

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.
Polar Grapher

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.

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