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.)

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.