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Symmetry of Polar Graphs - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

There are many polar curves that are symmetric. The symmetry of polar graphs about the x-axis can be determined using certain methods. Knowing the polar graph symmetry can help us calculate the area inside a polar curve. Some curves that can have symmetry of polar graphs are circles, cardioids and limacon, and roses and conic sections.

How do you recognize the symmetry of a polar graph just by looking at its equation? I want to first look at symmetry across the x axis.
So I've plotted 2 points and I want to suggest that this point point q is the reflection of point p across the x axis. And notice how we get this reflection. We take the coordinates of p our theta and all we do is switch the sine of theta, negative theta. And this point is a reflection of point p.
So let's use that idea to come up with a test for symmetry. Here's a curve a polar curve that's that is symmetric about the x axis. I know it's symmetric but I'm going to I'm going to come up with a test for symmetry and see if it works on this guy.
Notice I've got the graph drawn here and I have a point r theta along with it's reflection r negative theta drawn on the graph. The test that I'm going to use is the equation for the graph has to be true for r negative theta. so what I'm going to do is I'm going to plug negative theta into the equation and see if it's true.
So let me start with the right hand side. 2 over 1 minus cosine of negative theta. Now because cosine's an even function, the cosine of negative theta equals the cosine of theta. So I get 1 minus cosine theta in the denominator. Now this is exactly r. So what this shows me is that negative theta r does actually work so this curve passes the test. And any time you want to test for symmetry across the x axis, see if the equation is true for r negative theta. Plug in negative theta and see if you can get r.

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