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