Bounds on Theta Examples

What bounds on θ produce the single petal of the graph

r = cos(2θ)

shown below?

We'll start out with something that seems reasonable, like 0 ≤ θ ≤ π, and see what we find.

That's definitely not right. We didn't find all of the petal we wanted, and we got a bunch of other junk. See if we can decrease the upper bound on θ until we find the top half of the petal we want. 

How about ?

Still not right. We have the upper half of the petal, but we also have half a petal that we don't want:

This seems like we have twice as much graph as we want, we'll try cutting the upper bound of θ in half again. 

If   ,

we find

That's the upper half of the petal we want. To find the lower half of the petal, let the lower bound for θ reflect its upper bound. 

When we find the piece of the graph we want: