Polar coordinates and polar curves

Polar coordinates

A point is (r, θ), where r is the distance from the pole O and θ is measured anticlockwise from the initial line (the positive x-axis). In F2, r ≥ 0: only sketch the parts of a curve where r is not negative.

Converting: x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x (check the quadrant).

Converting equations

Multiply through by r if it helps: r = 4cos θ gives r² = 4r cos θ, so x² + y² = 4x, a circle with centre (2, 0) and radius 2.

r = 2sec θ gives r cos θ = 2, the line x = 2. More generally r = p sec(α - θ) gives r cos(θ - α) = p: a straight line whose nearest point to O is at distance p in the direction θ = α. θ = α is a half-line from O.

Standard curves

Sketch by finding r at θ = 0, π/2, π, 3π/2, noting where r = 0, and using symmetry: if r(-θ) = r(θ) the curve is symmetric about the initial line.

Standard polar curves

Sketching tips

The scheme of work calls a sketch an essential part of any solution strategy, to be used whenever appropriate. Mark where the curve meets the initial line and the line θ = π/2, and label the pole. For roses, the loops are where r > 0; tangents at the pole are the values of θ where r = 0.

Worked example

Find the Cartesian equation of r = 4cos θ and describe the curve.

1. Multiply by r: r² = 4r cos θ.
2. Replace r² by x² + y² and r cos θ by x: x² + y² = 4x.
3. Complete the square: (x - 2)² + y² = 4.

Answer: A circle with centre (2, 0) and radius 2, passing through the pole

Check yourself

1. Find the Cartesian equation of r = 2sec θ.
   • y = 2
   • x² + y² = 4
   • x = 2
   • y = 2x
   Show answer

   x = 2. r = 2/cos θ, so r cos θ = 2, that is x = 2.

2. Find the x-coordinate of the point with polar coordinates (4, 2π/3).
   Show answer

   -2. x = r cos θ = 4cos(2π/3) = 4 × (-1/2) = -2.

3. What is the curve r = 3(1 + cos θ)?
   • A circle of radius 3
   • A convex limaçon
   • A cardioid with a cusp at the pole
   • A rose with 3 loops
   Show answer

   A cardioid with a cusp at the pole. r = a(1 + cos θ) is a cardioid; r = 0 when θ = π, giving the cusp at O.

4. Which describes the curve r = 4 + 2cos θ?
   • A limaçon with a dimple
   • A convex limaçon (egg shape), since 4 ≥ 2 × 2
   • A cardioid
   • A circle of radius 4
   Show answer

   A convex limaçon (egg shape), since 4 ≥ 2 × 2. For r = a + b cos θ: a ≥ 2b gives a convex curve. Here a = 4 and b = 2.

5. How many loops does the curve r = 5sin 3θ have, for 0 ≤ θ < 2π with r ≥ 0?
   Show answer

   3. sin 3θ ≥ 0 for 0 ≤ θ ≤ π/3, 2π/3 ≤ θ ≤ π and 4π/3 ≤ θ ≤ 5π/3: three loops.

6. What is the name of the curve r² = a² cos 2θ?
   • Cardioid
   • Lemniscate
   • Spiral
   • Limaçon
   Show answer

   Lemniscate. r² = a² cos 2θ is a figure-of-eight curve called a lemniscate; it exists where cos 2θ ≥ 0.

7. Find the polar equation of x² + y² = 6y.
   • r = 6cos θ
   • r = 6
   • r² = 6sin θ
   • r = 6sin θ
   Show answer

   r = 6sin θ. r² = 6r sin θ, so r = 6sin θ (dividing by r ≠ 0; the pole is still on the curve at θ = 0).

8. The line r = 2sec(π/4 - θ) has Cartesian equation x + y = k. Find k, to 3 decimal places.
   Show answer

   2.828. r cos(θ - π/4) = 2. Expanding: r cos θ cos(π/4) + r sin θ sin(π/4) = 2, so (x + y)/√2 = 2 and x + y = 2√2 = 2.828.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.