AP Calculus BC glossary

Polar symmetry

Also called: Symmetry of a polar curve

Polar symmetry means a curve r equals f of theta repeats across a line or about the pole. Replacing theta by negative theta without changing the equation shows symmetry about the polar axis, replacing theta by pi minus theta tests the vertical line, and replacing r by negative r tests the pole.

The polar axis test asks whether f(−θ)=f(θ)f(-\theta) = f(\theta), which is why r=2cos⁡θr = 2\cos\theta and the cardioid r=1+cos⁡θr = 1 + \cos\theta are symmetric about the horizontal axis. The vertical line θ=π2\theta = \frac{\pi}{2} test asks whether f(π−θ)=f(θ)f(\pi - \theta) = f(\theta), satisfied by r=2sin⁡θr = 2\sin\theta. The pole test passes if the equation survives replacing rr by −r-r or θ\theta by θ+π\theta + \pi.

Symmetry does real work on limits of integration. The cardioid r=1+cos⁡θr = 1 + \cos\theta is symmetric about the polar axis, so sweeping θ\theta from 00 to π\pi covers half the region and the enclosed area is twice that sweep. The same reasoning isolates a single petal of r=sin⁡(2θ)r = \sin(2\theta), traced once as θ\theta runs from 00 to π2\frac{\pi}{2}.

A=2⋅12∫0π(1+cos⁡θ)2 dθ=3π2A = 2 \cdot \frac{1}{2}\int_{0}^{\pi} \left(1 + \cos\theta\right)^{2}\,d\theta = \frac{3\pi}{2}

The mistake

Deciding a curve has no symmetry because a test fails. The tests are sufficient, not necessary, since one curve has many polar equations. The rose r=sin⁡(2θ)r = \sin(2\theta) fails the −r-r test, because −r=sin⁡(2θ)-r = \sin(2\theta) is a different equation, yet replacing θ\theta by θ+π\theta + \pi leaves it unchanged, so the curve is symmetric about the pole after all.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)