Editing Hyperbolic functions (section)

==Comparison with circular functions==

[[File:Circular and hyperbolic angle.svg|right|upright=1.2|thumb|Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of [[sector of a circle|circular sector]] area {{mvar|u}} and hyperbolic functions depending on [[hyperbolic sector]] area {{mvar|u}}.]]
The hyperbolic functions represent an expansion of [[trigonometry]] beyond the [[circular function]]s. Both types depend on an [[argument of a function|argument]], either [[angle|circular angle]] or [[hyperbolic angle]].

Since the [[Circular sector#Area|area of a circular sector]] with radius {{mvar|r}} and angle {{mvar|u}} (in radians) is {{math|1=''r''<sup>2</sup>''u''/2}}, it will be equal to {{mvar|u}} when {{math|1=''r'' = {{sqrt|2}}}}. In the diagram, such a circle is tangent to the hyperbola ''xy'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a [[hyperbolic sector]] with area corresponding to hyperbolic angle magnitude.

The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length {{radic|2}} times the circular and hyperbolic functions.

The hyperbolic angle is an [[invariant measure]] with respect to the [[squeeze mapping]], just as the circular angle is invariant under rotation.<ref>[[Mellen W. Haskell|Haskell, Mellen W.]], "On the introduction of the notion of hyperbolic functions",  [[Bulletin of the American Mathematical Society]] '''1''':6:155–9, [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf full text]</ref>

The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.

The graph of the function {{math|''a'' cosh(''x''/''a'')}} is the [[catenary]], the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.