All A level specifications in further mathematics include Polar Coordinates. There are many excellent resources to help students become familiar with curves where r is given as a function of θ.
Try this page on Desmos to experiment with plotting points.
For a really clear plotter on GeoGebra from J Mulholland, showing the connection between the Cartesian graph of r=f(θ) and the graph in polar coordinates, try this Cartesian and Polar Grapher. Watch the display carefully as you move the slider; for example, you can easily see when r is negative.
We can also use WolframAlpha to demonstrate the values of theta generating which parts of the curve.
We can also use WolframAlpha to plot a polar curve specifying a range for θ.
It is possible to see how polar curves are traced out by using a slider in the domain on Desmos. Experiment with the sliders for this polar curve, acoskθ.
Further examples: r=acos2θ, r=a(1-cosθ) r=ae-kθ r2=a2cos2θ
Cardioids a+bsinθ and a + bcosθ
When do you get a dimple?
When do you get an inner loop?
Try this Polar Grapher on Desmos; 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.
Further Resources
Learn Desmos – Polar Graphing
- GeoGebra – polar areas: Polar Area, MEI Area under polar curves.
- Desmos – Polar Area Shader
- From the mathcentre, see the Quick Reference leaflet and Teach Yourself document.
- Polar graph paper, scroll down the page on Mathsbits.
- Just the Maths – Polar Curves notes.
- TES – Polar Curves, SRWhitehouse, introduce students to sketching and identifying polar curves.
- TES – Areas bounded by polar curves, SRWhitehouse, examples and exercise with brief solutions.
Lumen Learning
Graphs and Symmetry of Polar Curves
7.3 Polar Coordinates. Authored by: Ryan Melton. License: CC BY: Attribution