Evaluating The Length of a Geodesic

Question

Using the upper half plane model for Hyperbolic Space, the geodesic between $P=(0,1)$ and $R=(1,1)$ is the segment of the circle $C_1 :(x-1/2)^2+y^2 = 5/4$, thus solving for the length of the curve originating at $R$ and ending at $P$ is done by considering the trasformation $\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 1+ \sqrt{5}/2 cos\theta \\ \sqrt{5}/2 sin\theta \end{pmatrix}$ and then solving the integral $L_{C_1}= \int_R^P \sqrt{\frac{dx^2+dy^2}{y^2}}=\int_{\theta_1}^{\theta_2}\sqrt{csc^2\theta \cdot d\theta^2}=ln\lvert csc \theta - cot \theta \lvert$. Solving for $csc\theta_1$ is done by considering the $(1/2 , 1, \sqrt{5}/2)$ ,triangle generated by the center of $C_1, R,$ and the projection of $R$ onto the $x-$axis. Thus, $csc \theta_1 = \frac{\sqrt{5}}{2}$ and $cot \theta_1 = \frac{1}{2}$ and similarly, $csc \theta_2 = \frac{\sqrt{5}}{2}$ and $cot \theta_2 = \frac{1}{2}$. The issue I am having is that then when plugged back in to solve for the integral, I end up with value $0$, which I know is not correct. What am I doing incorrectly?