Topic 5 of 8 · Form 6
Polar coordinates
A point is given by (r, θ): x = r cos θ, y = r sin θ, r² = x² + y².
Notes for this topic 1 file, in order
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Polar form
A point is given by (r, θ): x = r cos θ, y = r sin θ, r² = x² + y².
Sketching curves
Common curves: circles (r = a, r = 2a cos θ), cardioids (r = a(1 + cos θ)), spirals (r = aθ), roses (r = a sin nθ). Find r at key angles, symmetry and where r = 0 (tangents at the pole).
Area
Area bounded by a polar curve between θ = α and θ = β: A = ½ ∫ r² dθ.
Worked example: Find the area enclosed by the cardioid r = a(1 + cos θ).
A = ½ ∫₀^(2π) a²(1 + 2cos θ + cos²θ) dθ = ½a²[2π + 0 + π] = 3πa²/2
Converting between forms
Worked example: Convert r = 4 cos θ to Cartesian form.
r² = 4r cos θ → x² + y² = 4x → (x − 2)² + y² = 4, a circle centre (2, 0), radius 2
Tangents
Tangents parallel to the initial line where dy/dθ = 0 (with y = r sin θ); perpendicular where dx/dθ = 0.
Check yourself
- Convert (x, y) = (1, √3) to polar coordinates.
- Find the area of the region enclosed by r = 2 for 0 ≤ θ ≤ π/2.
- Convert r = 2 sin θ to Cartesian form.
Show answers
- (2, π/3)
- ½ × 4 × π/2 = π
- x² + (y − 1)² = 1
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