Question

how do i test for symmetry? r=16cos3(theta)

Answer 1

symmetry around what?

Answer 2

http://www.most.gov.mm/techuni/media/EM_02011_chap5b.pdf

Answer 3

symmetry with respect to theta=pie/2, the polar axis, and the pole

Answer 4

thanks but how do i actually work it to find the answer?

Answer 5

This question will, unfortunately, take more time to answer than I have available. I can give you the following transformations and an example of what to do for one of them, and hopefully you can see what you have to do for the others.

A polar function is symmetric about the polar axis if, when you replace (r, theta) with EITHER (r,-theta) or (-r, pi-theta), you get the original function back.
Here, if (r, theta) is on your graph, then\[(r, \theta) \rightarrow r=16 \cos 3 \theta \]\[\rightarrow r=16 \cos 3 (- \theta))\]\[\rightarrow (r, -\theta)\]is on your graph (because cosine is even (i.e. cos(theta) = cos(-theta)).

Answer 6

The function's symmetric about the pole if when you replace\[(r, \theta)\] with (-r, theta) OR (r, pi+ theta), you get the original function back. So here,\[(r, \theta) \rightarrow r=16\cos 3 \theta\]\[\rightarrow r=16 \cos 3 (\pi+\theta)\]\[\rightarrow r=-16\cos 3 \theta\ne 16\cos3\theta\]So neither (-r theta) or (r, pi + theta) lie on the graph (note, when I'm giving you two points to choose from in your test, these points on the graph are identical (you just take a different route getting there (draw it if you can't see it))).

Answer 7

If the function's symmetric about pi/2, then replacing (r, theta) with either (r,pi-theta) or (-r, -theta) should get you back to (r, theta). Here,\[(r, \pi-\theta) \rightarrow 16 \cos 3(\pi-\theta))=-16\cos 3 \theta=-r \ne r\]i.e. if you replace with pi-theta, you don't get back to (r, theta), you get to (-r, theta). So this isn't symmetric about pi/2.

There is a mistake in the reasoning above. I should have said,\[(r,\pi+\theta) \rightarrow 16\cos 3 (\pi + \theta)= -16\cos 3 \theta = -r \ne r\]

Answer 8

So it's only symmetric about the polar axis.