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Its easy to check that every matrix in $SL_2(\mathbb{R})$ can be wrtten uniquely as $$\begin{pmatrix} \lambda & \alpha \\\ 0 & \lambda^{-1}\end{pmatrix}\begin{pmatrix} \cos{\theta} & -\sin{\theta} \\\ \sin{\theta} & \cos{\theta} \end{pmatrix}$$ where $\lambda>0$. This is exactly written then as a coset representative of the quotient you wrote down above. You can arrive at the hyperbolic metric by following the definition of the pushforward metric from a left invariant metric on $SL_2(\mathbb{R})$. Notice $(\alpha,\lambda)$ is a point in the upperhalf plane.