# Finding relation between angle and length in a kinematics problem

The diagram of my kinematics problem is as of above. I am having trouble finding the proper relation between the length $\overline{CD}$ and the angle $\varrho$. Note that the radius need not be 2 a shown in the image, but can be arbitrary. Using some trigonometry I did however come to a solution, which seems too complicated since this is a kinematics problem, and the focus is not too much on mathematics: $$\overline{CD}=\frac{R}{2}\left | \cos\varrho \right |+R\sqrt{1-\frac{1}{4}sin^{2}\varrho }$$ Anyone else have a simpler solution?

• Essentially, you’re trying to express the equation of the circle in polar coordinates. See en.wikipedia.org/wiki/Polar_coordinate_system#Circle. – amd Oct 26 '15 at 18:56
• Your formula is nearly fine: just drop the absolute value around $\cos\rho$ and it's OK. It can be derived from the cosine rule and I don't think there is a simpler one. – Aretino Oct 26 '15 at 20:35
• @aretino I cannot just leave it out, since the range for $\varphi$ is from zero to Pi – Emir Šemšić Oct 26 '15 at 22:14
• For $\rho=\pi$ you have $\cos\rho=-1$ and $\sin\rho=0$ so that from your formula WITHOUT absolute value one gets $CD=-R/2+R=R/2$, which is the correct value. With absolute value you would obtain the wrong result $CD=3R/2$. – Aretino Oct 27 '15 at 14:35