Is the Polar Curve r=cos(a/2) Symmetric About the Y-Axis?

[hyper]

Hello, this question is about symmetry of polar coordinates.

For a polar-curve to be symmetric around the x-axis we require that if (r,a) lies on the graph then (r,-a) or (-r,Pi-a) lies on the graph.

To be symmetric about the y-axis we require that (-r,-a) or (r,Pi-a) lies on the graph.

Now let's look at the graph r=cos(a/2)

Since cos(-a/2)=cos(a/2) then the curve is symmetric about the x-axis. Of the requirements of symmetri I wrote earlier it doesn't seem to be symmetric about the y-axis. But when I draw it, symmetry about the y-axis occours. How can this be shown mathematically?


hyper

 

[D H]

The function in question obviously repeats over a range of 4*pi. What does the graph of the function over the range theta=0 to 2*pi versus theta=2*pi to 4*pi tell you about symmetry with respect to the y axis?