Editing Hyperbolic coordinates (section)

==Trigonometry==
[[File:Cartesian_hyperbolic_triangle.svg|thumb|right|250px|Right triangles with legs proportional to sinh and cosh]]
The [[hyperbolic function]]s sinh, cosh, and tanh can be illustrated with hyperbolic coordinates. Let
:<math>A = (e^t, e^{-t}), \ B=(e^{-t}, e^t), \ C = (e^t + e^{-t}) .</math>
Then BCAO  forms a [[rhombus]] with diagonals intersecting at <math>M = (\frac{e^t + e^{-t} }{2},\ \frac{e^t + e^{-t} }{2} ) </math>. The hyperbolic cosine is defined as <math>\cosh t = \frac{e^t + e^{-t} }{2},</math> so ''M'' = ( cosh ''t'', cosh ''t'').

The semi-diagonal MA is [[equipollence (geometry)|equipollent]] to <math>(\frac{-e^{-t} + e^t }{2}, \ \frac{e^t - e^{-t} }{2}) = (- \sinh t,\ \sinh t) </math>. Evidently the diagonals divide the rhombus into four congruent right triangles. 

The angle MOA is the [[hyperbolic angle]] parameter ''t'' of cosh and sinh, and <math>\tanh t = \frac{\sinh t}{\cosh t}</math> and has a value in (–1, 1).