A plane curve is given by $\gamma (\theta)=(r \cos \theta , r \sin \theta )$, where r is a smooth function of $\theta$ (so that $(r, \theta )$ are the polar coordinates of $\gamma (\theta )$). Under what conditions is $\gamma$ regular? Find all functions $r(\theta )$ for which $\gamma$ is unit-speed. Show that, if $\gamma $ is unit-speed, the image of $\gamma $ is a circle; what is its radius?
I have done the following:
$\gamma$ is regular if the following condition doesn't hold $$r' (\theta )=r(\theta )=0$$
$\gamma$ is unit-speed when $r(\theta )=\pm \sin (\theta+c)$, where $c$ a constant.
If $\gamma$ is unit-speed, then $r(\theta)=\pm \sin (\theta +c)$, so $$\gamma (\theta)=(\pm \sin (\theta+c) \cos \theta, \pm \sin (\theta+c)\sin \theta)$$
Does this mean that we have two $\gamma$ ? So do we have to show for both of them, one for $+$ and one for $-$, that its image is a circle? One for only one of them?
You can choose $C=c+\pi$, and then $\sin{(\theta+C)}=-\sin{(\theta+c)}$, so having the $\pm$ there doesn't create any more solutions if you allow any $c \in [0,2\pi)$.