# arcosh function

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.

Why is it called the *arcosh* rather than the *arccosh*? See here.

## Equation and graph

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

$$ x = \cosh y $$

then:

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

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 $y=x$.

## Logarithm formula for arcosh

arcosh can be calculated directly, using a logarithm function, like this:

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

Here is a proof of the logarithm formula for arcosh. It is very similar to the proof for arsinh.

We will use:

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

The cosh of *u* will be *x*, because cosh is the inverse of arcosh:

$$ x = \cosh {u} $$

One form of the formula for cosh is:

$$ \cosh{u} = \frac{e^{2u}+1}{2e^{u}} $$

This gives us:

$$ x = \cosh {u} = \frac{e^{2u}+1}{2e^{u}} $$

Multiplying both sides by $2e^{u}$ gives:

$$ 2 x e^{u} = e^{2u} + 1 $$

This is a quadratic in $e^{u}$ (using the fact that $e^{2u}=(e^{u})^2$):

$$ 0 = (e^{u})^2 -2 x (e^{u})+1 $$

We use the quadratic formula with $a=1$, $b=-2x$, $c=1$:

$$ e^{u} = \frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{2x\pm\sqrt{4{x^2}-4}}{2} $$

Simplifying and taking the positive solution (since $e^{u}$ must be positive) gives:

$$ e^{u} = x+\sqrt{x^2-1} $$

Taking the logarithm of both sides gives:

$$ ln(e^{u}) = u = ln(x+\sqrt{x^2-1}) $$

And since *u* is $\operatorname{arcosh}{x}$ this gives:

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

## See also

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