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• A-Level Further Maths

Polar Coordinates

The Cartesian system in two-dimensions models points in terms of x and y. The polar system, however, models a point as a distance form the pole, r (generally the origin) at a certain angle from the initial line, θ (typically the positive horizontal axis). Yes, this is like the modulus-argument form of complex numbers and Argand diagrams.

From the diagram, we can derive equations to convert between polar and Cartesian systems:

r cos(θ) = x
r sin(θ) = y

Where θ is given by:

θ = arctan(y/x)

And r is defined using Pythagoras' theorem:

r² = x² + y²

Sketching Polar Curves

To sketch a polar curve, use a graphical calculator or draw u a table of values for regular intervals of θ. This can be done quickly using the table function on the CASIO ClassWiz fx-991EX, and we recommend using π/6 as an interval.

The curve in this example is known as a cardioid, due to its dimple. This is common for equations in the form r = a(p+qcos(θ)), but only if qp < 2q. When p ≥ 2q, there is no dimple, making it more egg-shaped: