Not an answer - just want to note that the curve has more hidden branches. They can be seen looking at the parametric equation
$$
x=\frac{\sin(\varphi)}\varphi,\quad y=\frac{1-\cos(\varphi)}\varphi
$$
where $\varphi=2\theta$

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/oSy4a.png