Inverse Hyperbolic Functions

Since $\cosh x$ is defined in terms of the exponential function, you should not be surprised that its inverse function can be expressed in terms of the logarithmic function:

Let's set $\displaystyle y=\cosh(x) $, keep in mind that we restrict to $x\geq 0$, and try to solve for x:

&\Longleftrightarrow&(e^x)^2-2y (e^x)+1=0

This is a quadratic equation with ex instead of x as the variable. y will be considered a constant.

So using the quadratic formula, we obtain


Since $x\geq 0$ we have that $e^x\geq 1$ for all x, and since $y-\sqrt{y^2-1}$ fails to exceed 1 for some y, we have to discard the solution with the minus sign, so


and consequently


Read that last sentence again slowly!

We have found out that

Helmut Knaust

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