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Inverse hyperbolic cosine

The inverse hyperbolic cosine can be expresses with logarithms as

 

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Explanation

We start with the hyperbolic cosine

Substitution of e^θ = k gives

We continue only with the positive solution and substitute k = e^θ so that

Now we take the logarithm on both sides

After substitution of θ = arccosh x we get

 

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Example 1

You can see that cosh⁻¹(1) = 0, because

cosh⁻¹(1) = ln (1 + √0)  = ln (1) = 0

 

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