# Package or command for inverse hyperbolic trig functions?

What are the commands for all the:

• inverse trig functions (arcsec(x), arcCsc(x), arcCot(x));
• hyperbolic trig functions (sech(x) and csch(x)), and
• inverse hyperbolic trig functions?
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@Ian - thank you for the appropriate tags (this now comes up as the top hit on google when you search the functions + latex)! –  Rax Adaam Feb 8 '13 at 13:07

A bit more about "c or not to c"...

The inverse hyperbolic sine sinh^(-1)z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) and sometimes denoted arcsinhz (Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic sine. The variants Arcsinhz or Arsinhz (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic sine, although this distinction is not always made. Worse yet, the notation arcsinhz is sometimes used for the principal value, with Arcsinhz being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation sinh^(-1)z, sinhz is the hyperbolic sine and the superscript -1 denotes an inverse function, not the multiplicative inverse.

Its principal value of sinh^(-1)z is implemented in Mathematica as ArcSinh[z] and in the GNU C library as asinh(double x).

this from Mathematica online documentation at http://mathworld.wolfram.com/InverseSine.html

So the answer to the "question" is the universal "it depends" with of course the publisher always holding the winning hand!

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After spending some time looking for this, I found this post that suggested defining the new commands for the omitted inverse trig functions.

Here I've augmented that with the full suit of hyperbolic and inverse hyperbolic functions for convenience, as google doesn't turn anything up for this search, nor does the other post come up if one is searching for the inverse hyperbolic functions, specifically.

\documentclass{article}

\usepackage{amsmath}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\arcsec}{arcsec}
\DeclareMathOperator{\arccot}{arcCot}
\DeclareMathOperator{\arccsc}{arcCsc}
\DeclareMathOperator{\arccosh}{arcCosh}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsech}{arcsech}
\DeclareMathOperator{\arccsch}{arcCsch}
\DeclareMathOperator{\arccoth}{arcCoth}

\begin{document}
$\sech x \cschx \arcsec x \arccot x \arccsc x \arccosh x \arcsinh x \arctanh x \arcsech x arccsch x \arccoth x$
\end{document}

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@Christian Thank you for cleaning up the entry, how did you get the proper formatting? I spent an age trying to get it to display properly, but it wouldn't recognize line breaks, so I modified it to make it legible & left it at that. Thank you! –  Rax Adaam Feb 8 '13 at 13:06
You just need to select the code and either click the {} button over the textfield or simply press ctrl+k. This will indent your code by 4 spaces, which then is recognized as code. –  xfoo Feb 8 '13 at 13:15
These definitions are very practical! However, inverse hyperbolic functions don't have 'c' in the name: arsinh, arcosh, artgh... because they give the area of the hyperbolic sector, rather than arc of the triangle. We have trigonometric and hyperbolic functions and their respective inverses, such as "arcus sinus" and "area cosinus hyperbolicus", though pronunciation differs from original Latin. –  Vladimir Jun 14 '13 at 2:58