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Polar Coordinates
Edexcel A Level Further Mathematics – Core Pure Revision Notes
What Are Polar Coordinates?
Instead of describing a point by how far right and how far up it is (Cartesian coordinates x, y), polar coordinates describe it using:
r – the distance from the origin (called the pole). θ – the angle measured anticlockwise from the positive x-axis (the initial line).
So the point (r, θ) in polar coordinates is the same as (r cos θ, r sin θ) in Cartesian.
Converting Between Polar and Cartesian Polar → Cartesian: x = r cos θ y = r sin θ Cartesian → Polar:
r =
p x^2 + y^2 tan θ = y x (check quadrant!)
Also useful: x^2 + y^2 = r^2 and y/x = tan θ.
Note: In polar coordinates, r ≥ 0 unless the question specifies otherwise, and θ is usually taken in the range 0 ≤ θ < 2 π or −π < θ ≤ π.
Sketching Polar Curves
You’ll need to recognise and sketch the standard curves. The approach is always:
- Build a table of (r, θ) values for key angles (0, π/ 6 , π/ 4 , π/ 3 , π/ 2 , π,.. .).
- Note any symmetry (see below).
- Look for when r = 0 (the curve passes through the origin) and when r is maximum.
Standard Curves to Know
Curve Shape r = a Circle of radius a centred at the origin. r = a cos θ Circle of diameter a, centred on the positive x-axis at (a/ 2 , 0). Passes through origin. r = a sin θ Circle of diameter a, centred on the positive y-axis. Passes through origin. r = a(1 + cos θ) Cardioid (heart shape). Max r = 2a at θ = 0; passes through origin at θ = π. r = a + b cos θ Lima¸con. If a > b: dimple-free oval. If a < b: inner loop. If a = b: cardioid. r = a cos(nθ) Rose curve with n petals (if n odd) or 2n petals (if n even). r^2 = a^2 cos(2θ) Lemniscate (figure of eight).
Symmetry Rules
If r(−θ) = r(θ): the curve is symmetric about the initial line (x-axis). If r(π − θ) = r(θ): the curve is symmetric about θ = π/2 (y-axis). If r(θ + π) = r(θ): the curve has rotational symmetry of order 2.
Finding the Area Enclosed by a Polar Curve
This is the key calculation in this topic. The area enclosed between two angles is:
Polar Area Formula
A =
Z (^) β
α
r^2 dθ
This comes from approximating the region with thin sectors of angle dθ and radius r. The area of a thin sector is 12 r^2 dθ.
Worked Example – Area of a Cardioid
Find the total area enclosed by r = a(1 + cos θ).
Due to symmetry about the x-axis, integrate from 0 to π and double:
A = 2 ×
Z (^) π
0
a^2 (1 + cos θ)^2 dθ = a^2
Z (^) π
0
(1 + 2 cos θ + cos^2 θ) dθ
Using cos^2 θ = 12 (1 + cos 2θ):
= a^2
Z (^) π
0
cos 2θ
dθ = a^2
3 θ 2
π
0
3 πa^2 2
