Editing Hyperbolic angle (section)

{{short description|Argument of the hyperbolic functions}}
[[Image:Hyperbolic sector.svg|thumb|200px|right|The curve represents ''xy'' = 1. A hyperbolic angle has magnitude equal to the area  of the corresponding [[hyperbolic sector]], which is in ''standard position'' if {{math|''a'' {{=}} 1}}]]

In [[geometry]], '''hyperbolic angle''' is a [[real number]] determined by the [[area]] of the corresponding [[hyperbolic sector]] of ''xy'' = 1 in Quadrant I of the [[Cartesian plane]]. The hyperbolic angle parametrizes the [[unit hyperbola]], which has [[hyperbolic functions]] as coordinates. In mathematics, hyperbolic angle is an [[invariant measure]] as it is preserved under [[hyperbolic rotation]].

The hyperbola ''xy'' = 1 is [[rectangular hyperbola|rectangular]] with semi-major axis <math>\sqrt 2</math>, analogous to the circular [[angle]] equaling the area of a [[circular sector]] in a circle with radius <math>\sqrt 2</math>.

Hyperbolic angle is used as the [[dependent and independent variables|independent variable]] for the [[hyperbolic functions]] sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a [[hyperbolic sector#Hyperbolic triangle|hyperbolic triangle]].
The parameter thus becomes one of the most useful in the [[calculus]] of [[real number|real]] variables.