Representations of Trigonometric and Hyperbolic Functions in Terms of Sector Areas
A sector of angle of a unit circle has an area equal to radians. So half the area can serve as the argument for the trigonometric functions via parametric equations for and . The two constructions shown are consistent with the trigonometric identities and . (As a consequence, circular functions are alternatively called trigonometric functions.)[more]
An analogous set of relations exists for the hyperbolic functions, based on the unit hyperbola . The asymptote is shown as a dashed line. The corresponding area is the sector swept out by a path from the origin following the hyperbola beginning on the axis at . Half this area, designated by , can then serve as the argument in parametric representations of the hyperbolic functions. The integral over the area can be evaluated to give , consistent with and . The two hyperbolic constructions are consistent with the identities and . The construction for is not as neat as its analog for .
The ranges of and are constrained to fit within the scale of the graphics, but the behavior at extrapolated values should be evident.[less]
The thumbnail and the third snapshot show the comparative behavior of corresponding trigonometric and hyperbolic functions.
 S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier Academic Press, 2008, pp. 71–72, 188–189.