The hyperbolic sine function is a one-to-one function, and thus has an inverse. As usual, we obtain the graph of the inverse hyperbolic sine function (also denoted by ) by reflecting the graph of about the line *y*=*x*:

Since 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 , and try to solve for *x*:

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

So using the quadratic formula, we obtain

Since for all *x*, and since for all *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

Here it is: Express the inverse hyperbolic cosine functions in terms of the logarithmic function!

Click here to see the answer, and to continue.

Fri Jul 19 11:01:21 MDT 1996

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