Polar Coordinates — Question 5

Question 5

Problem

A polar curve is unchanged when $\theta$ is replaced by $-\theta$. What symmetry does this reveal, and why?

Question 5 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. The points at angles $\theta$ and $-\theta$ reflect across the polar axis, so the curve is symmetric about the $x$-axis.