Polar Coordinates

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Edexcel IAL Further Pure Mathematics 2 (WFM02)

Polar Coordinates

The big idea: A polar curve tells you how far you are from the pole in each direction θ. Area comes from adding thin sectors, ½r²δθ, and tangents come from converting back to x and y.

Summary — What This Chapter Covers

x = r cos θ, y = r sin θ, r² = x² + y²; in this course r ≥ 0.
Know the standard shapes: r = a, θ = α, r = 2a cos θ, r = kθ, r = a(1 ± cos θ), r = a(p + q cos θ), r = a cos 2θ, r² = a² cos 2θ.
Area = ½ ∫ r² dθ between the limits that sweep out the region once.
Tangent parallel to the initial line: d(r sin θ)/dθ = 0. Perpendicular: d(r cos θ)/dθ = 0.

1. Polar Coordinates and Equations

Conversions

Moving between polar and Cartesian

A point (r, θ) is a distance r from the pole O in the direction θ measured anticlockwise from the initial line. Convert with x = r cos θ, y = r sin θ. For example, r = 2a cos θ gives r² = 2ar cos θ, i.e. x² + y² = 2ax: a circle with centre (a, 0) and radius a.

2. Sketching Polar Curves

Sketching

Shapes worth knowing

Tabulate r for key values of θ (0, π/2, π, …), and use symmetry: a curve in cos θ only is symmetrical about the initial line. Only plot where r ≥ 0.

Equation	Shape
r = a(1 + cos θ)	cardioid, cusp at the pole, r = 2a at θ = 0
r = a(p + q cos θ), p ≥ 2q	egg shape, no dimple
r = a(p + q cos θ), q < p < 2q	dimpled at θ = π
r = a cos 2θ (r ≥ 0)	two loops, along θ = 0 and θ = π
r = kθ	spiral

3. Area Enclosed by a Polar Curve

Area

½ ∫ r² dθ

Area A = ½ ∫ from α to β of r² dθ

Worked Example

Find the area enclosed by r = 2(1 + cos θ).

Practice Question 1

Find the area of one loop of r = a cos 2θ.

cos² and sin²

Use cos²θ = ½(1 + cos 2θ) and sin²θ = ½(1 − cos 2θ) before integrating r².

4. Tangents Parallel or Perpendicular to the Initial Line

Tangents

Converting to y and x

Conditions Parallel to the initial line: dy/dθ = 0, where y = r sin θ Perpendicular: dx/dθ = 0, where x = r cos θ

Worked Example

Practice Question 2

Memorise

Conversions

x = r cos θ, y = r sin θ

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Polar Coordinates