Hyperbolic angle
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A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1.
The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is log x.
Note that unlike circular angle, hyperbolic angle is unbounded, as is the function log x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle is considered to be negative when 0 < x < 1.
The hyperbolic functions sinh, cosh, and tanh use the hyperbolic angle as their independent variable since their values may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle. Thus this parameter becomes one of the most useful in the calculus of a real variable.
[edit] Reference
John Stillwell (1998) Numbers and Geometry exercise 9.5.3, p. 298, Springer-Verlag ISBN 0-387-98289-2.
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