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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.

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

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

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

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.

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

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:

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.



Areas Enclosed by Polar Curves

The area of a sector of a polar curve can be calculated using integration. However, simply integrating r will not work. Instead:

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

Of course, you an also calculate areas between polar curves. To do this, you need to find the angle at which they intersect.




Tangents to Polar Curves

To find tangents to a polar curve, you need to convert it into Cartesian form (one equation for x and one for y, both in terms of θ), using the formula at the top of this notes sheet. Then, you can differentiate parametrically.


Standard results are:

Polar coordinates. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.