I hope everyone in this community is staying safe, well and isolated. In this unprecedented situation I am starting to learn about some non-Euclidean geometry and explore down a fractal.
In the comments and answers to my question, I hope to find a visual way and get hints toward understanding the derivation of the following hyperbolic trigonometric formulas: $$ \cos \alpha = \frac{\tanh \frac12 s}{\tanh s} \qquad\qquad \cosh\frac12 s = \frac{\cos \frac12 \alpha}{\sin\alpha} $$ where $s$ is the side, and $\alpha$ the angle, of an equilateral hyperbolic triangle.
Is there a visual way to construct an understanding for the latter formula? I included a visual from the Wikipedia on the Poincaré disk model (or conformal disk model) which shows the hyperbolic model in green over the unit disk and curves being projected onto the disk. Can this visual (or any other visual) be used to come up with the trigonometry of the hyperbolic plane?
Sorry if this question is not specific but my end goal for this exploration is to construct a tiling of the hyperbolic plane.