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# Inverse hyperbolic trigonometric functions

The inverses of the hyperbolic trigonometric functions (hyperbolic functions) are the **area hyperbolic functions**. The names hint at the fact that they give the area of a sector of the unit hyperbola *x* 2 − *y* 2 = 1 in the same way that the inverse circular trigonometric functions (inverse trigonometric functions) give the length of an arc of the unit circle *x* 2 + *y* 2 = 1.

*The abbreviations *arcsinh*, *arccosh*, etc., are commonly used, even though they are misnomers, since the prefix *arc* is the abbreviation for *arcus*, while the prefix *ar* stands for *area*.*^{[1]}
^{[2]}
^{[3]}

In computer science this is often shortened to `asinh`, `acosh`, etc. The notation sinh −1(*x*), cosh −1(*x*), etc., are also used, despite the fact that care must be taken to avoid misinterpretations of the superscript `−1` as a power as opposed to a shorthand for inverse (e.g., cosh −1(*x*) versus cosh(*x*) −1).

The values of area hyperbolic functions (inverse hyperbolic functions) are hyperbolic areas (area of a sector of the unit hyperbola).

## Contents

## Area hyperbolic sine

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## Area hyperbolic cosine

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## Area hyperbolic tangent

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## Area hyperbolic cosecant

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## Area hyperbolic secant

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## Area hyperbolic cotangent

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## See also

## Notes

- ↑ As stated by Jan Gullberg,
*Mathematics: From the Birth of Numbers*(New York: W. W. Norton & Company, 1997), ISBN 039304002X, p. 539:Another form of notation, arcsinh

*x*, arccosh*x*,*etc.*, is a practice to be condemned as these functions have nothing whatever to do with__arc__, but with__ar__ea, as is demonstrated by their full Latin names,

arsinh*area sinus hyperbolicus*

arcosh*area cosinus hyperbolicus, etc.* - ↑ As stated by Eberhard Zeidler, Wolfgang Hackbusch and Hans Rudolf Schwarz, translated by Bruce Hunt,
*Oxford Users' Guide to Mathematics*(Oxford: Oxford University Press, 2004), ISBN 0198507631, Section 0.2.13: "The inverse hyperbolic functions", p. 68: "The Latin names for the inverse hyperbolic functions are area sinus hyperbolicus, area cosinus hyperbolicus, area tangens hyperbolicus and area cotangens hyperbolicus (of*x*). ..." This aforesaid reference uses the notations arsinh, arcosh, artanh, and arcoth for the respective inverse hyperbolic functions. - ↑ As stated by Ilja N. Bronshtein, Konstantin A. Semendyayev, Gerhard Musiol and Heiner Muehlig,
*Handbook of Mathematics*(Berlin: Springer-Verlag, 5th ed., 2007), ISBN 3540721215, doi:10.1007/978-3-540-72122-2, Section 2.10: "Area Functions", p. 91:The

*area functions*are the inverse functions of the hyperbolic functions, i.e., the*inverse hyperbolic functions*. The functions sinh*x*, tanh*x*, and coth*x*are strictly monotone, so they have unique inverses without any restriction; the function cosh*x*has two monotonic intervals so we can consider two inverse functions. The name*area*refers to the fact that the geometric definition of the functions is the area of certain hyperbolic sectors (...)