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Symmetry of Polar Graphs - Problem 2

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.

I want to talk about symmetry about the y axis for polar graphs. Now we’ve already discussed one test for symmetry about the y axis, but it turns out there is another one.

The point that’s the reflection of point p around the y axis can be represented in a number of ways. One of the ways is with the coordinates r, pi minus theta. If this angle is theta, and this point’s the reflection of point p, then it makes sense that this angle here is theta. And so the angle of this point in polar coordinates, would be pi minus theta. This angle would be pi minus theta. So the point r, pi minus theta is the reflection of r theta across theta equals pi over 2, which is the y axis.

Now let’s see if we can use this to develop a test. I’ve got a graph of a curve that looks symmetric about the y axis. And I have r theta, a point on that curve, across along with r theta minus, pi minus theta. I want to see if this curve is symmetric about the y axis.

So my test is going to be, you want to make sure that the equation is true for the point r pi minus theta. And the way you use that test is you plug pi minus theta in for theta. I get 2 sine of pi minus theta. I want to see if I can get the coordinate r out of this.

Now remember that sine, there’s an identity called the sine supplementary angle identity, it says that sine of pi minus theta equals sine of theta. In other words, if two angles are supplements, they have the same sine value. Well that means that this equals 2 sine theta. And that’s exactly what r is. And so that proves that this equation does work for pi minus theta r. And that means that this curve is symmetric about the y axis.

Now why would we need two tests for symmetry about the y axis? It turns out that, one of the tests may not work and so you may have to use the other one just to be sure. And sometimes the tests for symmetry don’t work. The equations don’t test as symmetrical but then when you graph them, they are symmetrical. This is one of the weird things that can happen with polar equations.

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