Smallest value of 'a' for which the graph of the curve $r = 4\sin2\theta$ is complete

Question

Recently, I asked for the smallest value of 'a' for which the graph of the curve $r = 5\sin\theta$ is complete. This turned out to be $\pi$ and not $2\pi$, because $\sin(\theta+\pi) = -\sin(\theta)$ (Give the smallest value for 'a' to complete the graph). Now it turns out that the graph of the curve $r = 4\sin2\theta$ is complete for $a = 2\pi$. Now my question is why? I know that $\sin2\theta = 2\sin\theta \cos\theta$, but I still can't relate this property to that. Thanks in advance!

Answer 1

In this case, you have $\sin 2(\theta+\pi)=\sin(2\theta+2\pi)=\sin2\theta$, so geometrically, increasing the polar angle by $\pi$ induces a symmetry w.r.t. the origin.

Answer 2

The answer is similar. There is a chance that $\theta+\pi$ and $\theta$ plot the same point. Let's see if that's the case. Note that $$r(\theta)=4\sin 2\theta=4\sin(2\theta+2\pi)=4\sin(2(\theta+\pi))=r(\theta+\pi)$$ Which means that the $r$ is the same, but the angles differ by $\pi$. Hence the points plotted are not the same, but opposite.