Polar Curve Sketching

Sketching polar curves is an essential skill in AQA Further Mathematics. This lesson covers systematic methods for sketching key families of polar curves: circles, cardioids, limacon curves, rose curves, and spirals.

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1 General Strategy

To sketch a polar curve r = f(theta):

1. Find key values. Evaluate r at theta = 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, pi, and continue if needed.
2. Identify symmetry. If f(theta) = f(-theta), the curve is symmetric about the initial line (x-axis). If f(pi - theta) = f(theta), it is symmetric about theta = pi/2 (y-axis).
3. Find where r = 0. These are points where the curve passes through the pole.
4. Find maximum and minimum r. These give the furthest and nearest points from the pole.
5. Check for negative r. Determine where r < 0 and plot those points in the opposite direction.
6. Plot points and join smoothly.

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2 Circles

r = a (constant)

This is a circle of radius a centred at the origin. All points are at distance a from O regardless of theta.

r = a cos theta

This is a circle of diameter a. Setting r = 0 gives cos theta = 0, so theta = pi/2 and -pi/2. The curve exists for -pi/2 <= theta <= pi/2 (where cos theta >= 0).

The centre is at (a/2, 0) in Cartesian coordinates.

r = a sin theta

This is a circle of diameter a, centred at (0, a/2). Setting r = 0 gives theta = 0 and pi. The curve exists for 0 <= theta <= pi.

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3 Cardioids

r = a(1 + cos theta)

Key values:

theta	0	pi/3	pi/2	2pi/3	pi
r	2a	3a/2	a	a/2	0

The curve is heart-shaped (hence "cardioid"). It:

• Touches the pole at theta = pi
• Has maximum r = 2a at theta = 0
• Is symmetric about the initial line (since cos(-theta) = cos theta)

r = a(1 - cos theta)

This is the reflection of the above through the vertical axis. It touches the pole at theta = 0 and has maximum r = 2a at theta = pi.

r = a(1 + sin theta) and r = a(1 - sin theta)

These are cardioids symmetric about the y-axis (theta = pi/2 line).

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4 Worked Example 1 -- Cardioid

Sketch r = 3(1 + cos theta).

Solution:

theta	0	pi/6	pi/3	pi/2	2pi/3	5pi/6	pi
cos theta	1	sqrt(3)/2	1/2	0	-1/2	-sqrt(3)/2	-1
r	6	5.60	4.5	3	1.5	0.40	0

Properties:

• Symmetric about the x-axis (initial line)
• r = 0 at theta = pi (passes through the pole)
• Maximum r = 6 at theta = 0
• Heart-shaped cardioid with cusp at the pole

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5 Limacon Curves

The general form is r = a + b cos theta (or a + b sin theta).

• If a = b: cardioid (cusp at the pole)
• If a > b > 0: dimpled limacon (no inner loop)
• If 0 < a < b: limacon with inner loop (r goes negative, creating a loop inside)
• If a >= 2b: convex limacon (no dimple)

Worked Example 2 -- Limacon with Inner Loop

Sketch r = 2 + 3 cos theta.

Here a = 2, b = 3. Since a < b, there is an inner loop.