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

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!

• Apologies, I wrote down the wrong equation. It should be $r = 4\sin2\theta$ – Stallmp Jul 12 at 10:11

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.
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.