My question is very simple. I want to show the curve
$$\varphi(t)=((2\cos t+1)\cos t,(2\cos t+1)\sin t)$$
Where $0\le t\le 2\pi$, has multiple points.
The book I'm reading says $P=(0,0)$ is a multiple point such that for every value of $t$ for which $\cos t=\frac{1}{2}$ is taken to $P$ by $\varphi$.
If I didn't know this information beforehand, how can I find the multiple points of this curve?
Note that in polar coordinates, the curve has equation $$ r= 2\cos \theta+1 $$ So $r=0$ (or equivalently, $P=(0,0)$ is reached) when $$ 2\cos \theta+1=0 $$ i.e., when $$ \cos\theta=-\frac{1}{2} $$