# Find inverse hyperbolic function

1. Nov 22, 2009

### synergix

1. The problem statement, all variables and given/known data
find f-1(t) and compute f-1(0) where
f(t) = sinh(t)

2. Relevant equations
sinh(t)=( et- e-t)/2

3. The attempt at a solution

t= (ey- e-y)/2

ln2t = y - - y

y = (ln2t)/2

but wikipedia says

what silly mistake have i made this time?

Last edited by a moderator: May 4, 2017
2. Nov 23, 2009

### lanedance

i would check the step where you take the logarithm

ony way that may work, would be to let y = f(t), x=e^t then try and solve for x(y) first

3. Nov 23, 2009

### Bohrok

y = (et- e-t)/2
2y = et- e-t
2yet = e2t - 1 (multiply both sides by et)

Let x = et, then you have a quadratic equation to help you isolate the t.

4. Nov 23, 2009

### synergix

yes! this is how my instructor showed me but we have covered soo much this past couple of weeks I completely forgot about that technique. Thanks!

5. Nov 23, 2009

### synergix

I used z instead of x and I got to
z^2-2tz-1=0
z= t+- sqrt(t^2+1)
z = e^y

so ln both sides

how do i know whether it is positive or negative?

since i can see from the formulas on wikipedia I went ahead with the rest and f-1(0) = ln1 = 0

Last edited: Nov 23, 2009
6. Nov 23, 2009

### lanedance

so is $t-\sqrt{t^2+1}$ greater or less than zero? is this for all t? consider what happens when you take the logarithm, and which is actually a solution to the orginal equation

also, I find the way you swap y & t is quite confusing
$y(t) = sinh(t)$