Hyperbolic capabilities come about within the alternatives of numerous linear differential equations (by way of example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates.
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coth x = cosh x sinh x = e x + e − x e x − e − x = displaystyle coth x= frac cosh x sinh x = frac e^ x +e^ -x e^ x -e^ -x =