Polar Coordinates

Polar coordinates

Polar coordinates describe a point by its distance from the origin (r) and the angle from the positive x-axis (θ), written (r, θ) — instead of the usual Cartesian (x, y).

Converting between polar and Cartesian

x = r cos θ        y = r sin θ
r = √(x² + y²)     tan θ = y ÷ x

(Take care to place θ in the correct quadrant.)

Polar curves

A polar equation gives r as a function of θ, r = f(θ). Plotting r for different θ produces curves that are often awkward in Cartesian form but elegant in polar. Examples:

• r = a → a circle of radius a centred at the origin.
• r = 2a cos θ → a circle passing through the origin.
• r = a(1 + cos θ) → a cardioid (heart shape).
• r = a θ → a spiral.

Look for symmetry (e.g. about the initial line if replacing θ with −θ leaves the equation unchanged) and where r = 0 (the curve passes through the pole).

Area enclosed by a polar curve

The area swept out between θ = α and θ = β is:

area = ½ ∫[α to β] r² dθ

This is the polar equivalent of finding the area under a curve. Set up the limits from where the curve starts and finishes (often where r = 0).

Tangents

Points where the curve is furthest from or nearest to the pole, or parallel/perpendicular to the initial line, are found using calculus on the parametric forms x = r cos θ, y = r sin θ.

Worked example

Convert the point (r, θ) = (2, π/3) to Cartesian coordinates.

• x = r cos θ = 2 cos(π/3) = 2 × 0.5 = 1.
• y = r sin θ = 2 sin(π/3) = 2 × (√3/2) = √3. So (1, √3). ✓

Common mistakes

• Getting θ in the wrong quadrant when converting from Cartesian.
• Forgetting the ½ and the r² in the polar area formula.
• Choosing the wrong limits (not using where r = 0 for a loop).

Exam tips

• Learn the conversion formulas both ways.
• Use area = ½∫r² dθ with limits from where the curve encloses the region.
• Sketch the curve, checking symmetry and where r = 0.

Key facts to remember

• Polar (r, θ): x = r cos θ, y = r sin θ; r = √(x²+y²), tan θ = y/x.
• Polar curves r = f(θ) (circles, cardioids, spirals); check symmetry and r = 0.
• Area = ½ ∫[α→β] r² dθ.