I have this curve in polar coordinates:
$$\left\{\begin{array}{c} x=r_1\cos\theta+r_2\cos\tfrac\theta 2 \\ y=r_1\sin\theta+r_2\sin\tfrac\theta 2\end{array} \right.$$ since this curve is some sort of a circle I know that simple derivation of the polar equations is incorrect. Can you please tell how to do this?
Note that $\theta$ is not the polar angle but parameter instead.
\begin{align*} x' &= -r_{1} \sin \theta-\frac{r_{2}}{2} \sin \frac{\theta}{2} \\ y' &= r_{1} \cos \theta+\frac{r_{2}}{2} \cos \frac{\theta}{2} \\ \end{align*}
Equation of tangent:
$$\frac{y-r_{1} \sin \theta-r_{2} \sin \frac{\theta}{2}} {x-r_{1} \cos \theta-r_{2} \cos \frac{\theta}{2}} =\frac{ r_{1} \cos \theta+\frac{r_{2}}{2} \cos \frac{\theta}{2}} {-r_{1} \sin \theta-\frac{r_{2}}{2} \sin \frac{\theta}{2}}$$