arcosh function

Martin McBride, 2020-02-03
Tags arcosh
Categories hyperbolic functions

The arcosh function is a hyperbolic function. It is the inverse of the cosh, and is also known as the inverse hyperbolic cosine function.

Equation and graph

The arcosh function is defined as the inverse of cosh, ie if:

$$ x = \cosh y $$


$$ y = \operatorname{arcosh} x $$

There is a also a formula for finding arcosh directly:

$$ \operatorname{arcosh} x= \ln{x+{\sqrt {x^{2}-1}}} $$

Here is a graph of the function:

The function is valid for x >= 1.

arcosh as inverse of cosh

This animation illustrates the relationship between the cosh function and the arcosh function:

The first, blue, curve is the cosh function.

The grey dashed line is the line $y=x$.

The second, red, curve is the arcosh function. As with any inverse function, it is identical to the original function reflected in the line $y=x$.

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