Sketching polar curves

Objective. Use symmetry, zeros and maxima to sketch curves.

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Learn: the key idea

Sketching a polar curve r = f(θ) uses features such as symmetry about the initial line (when f(−θ)=f(θ)), the values of θ giving r=0 (where the curve passes through the pole), and the maximum value of r (showing how far the curve extends). Tabulating r at key angles such as 0, π/6, π/4, π/3, π/2 and so on builds up the shape systematically before joining points smoothly. For example, r = 2(1+cos θ) is a cardioid, symmetric about the initial line, with r=4 at θ=0 and r=0 at θ=π.

Worked example

For the polar curve r = 3 cos(2θ), find the values of θ in [0, π] for which r = 0, which are useful in sketching the curve.

1 $Set r=0: 3 cos(2θ) = 0, so cos(2θ) = 0$
2 $Solve: 2θ = π/2, 3π/2, 5π/2, ... within the relevant range$
3 $For θ in [0,π], 2θ is in [0,2π], giving 2θ = π/2 or 3π/2$
4 $So θ = π/4 or θ = 3π/4$

Answer: θ = π/4 and θ = 3π/4

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Common mistake. Students plot negative r values by reflecting through the origin incorrectly, or ignore that r cannot be negative in many syllabus conventions. Always check your convention and plot r=0 points carefully, as they indicate the curve touches the pole.

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