What are polar coordinates, and how can we use them to solve STEP and other advanced mathematics examinations problems?

Polar Bearings

Age 16 to 18 Challenge Level:
What on earth are polar coordinates, and why would you want to use them?

Maltese Cross

Age 16 to 18 Challenge Level:
Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Polar Flower

Age 16 to 18 Challenge Level:
This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.