hyperbole trig functions

**taurus** · October 10th 2007, 04:40 AM

i know sinh(x) = (e^x - e^-x)/2

but what is sinhInverse(x) equal to?

**topsquark** · October 10th 2007, 05:21 AM

Originally Posted by taurus

i know sinh(x) = (e^x - e^-x)/2

but what is sinhInverse(x) equal to?

First of all, the terminology is "hyperbolic" not "hyperbole."

We have a function
$y = \frac{e^x - e^{-x}}{2}$

To find the inverse we need to switch the roles of x and y:
$x = \frac{e^y - e^{-y}}{2}$

and solve for y:
$2x = e^y - e^{-y}$ <-- Multiply through by $e^y$

$2xe^y = e^{2y} - 1$

$e^{2y} - 2xe^y - 1 = 0$

This is a quadratic in $e^y$ , so
$e^y = \frac{2x \pm \sqrt{4x^2 + 4}}{2}$

$e^y = x \pm \sqrt{x^2 + 1}$

$y = ln \left ( x \pm \sqrt{x^2 + 1} \right )$

We discard the "-" solution since the arguement of ln cannot be negative. ( $x - \sqrt{x^2 + 1}$ is negative everywhere.)

So finally
$y = ln \left ( x + \sqrt{x^2 + 1} \right )$

This is the inverse function to sinh(x). (The dotted line in the graph below is the line y = x.)

-Dan

**taurus** · October 10th 2007, 05:51 AM

just a question
i got a TI 84 plus silver edition graphics calculator, does anyone know where the sinh button is if at all?

**topsquark** · October 10th 2007, 06:30 AM

Originally Posted by taurus

just a question
i got a TI 84 plus silver edition graphics calculator, does anyone know where the sinh button is if at all?

Try this download.

-Dan

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