Understanding the Math Behind coth(y) = x

[ognik]

This is one of those 'huh?' moments where I can follow what is said, but don't understand it at all.
From $ coth(y) = x = \frac{{e}^{2y+1}}{{e}^{2y-1}}, $ we extract: $
y=\frac{1}{2} ln \frac{x+1}{x-1} $.
I've looked at graphs and definitions online, I follow what is done (kind of swapping x and y) - but would like to understand the details instead of just parroting it.
So I tried: $\ln\left({x}\right) = \ln\left({{e}^{2y}+1}\right) - \ln\left({{e}^{2y}-1}\right) $ - and am stuck here...

 

[MarkFL]

I would begin with:

$$x=\frac{e^{2y}+1}{e^{2y}-1}$$

and solve for $e^{2y}$ first:

$$x\left(e^{2y}-1\right)=e^{2y}+1$$

$$e^{2y}(x-1)=x+1$$

$$e^{2y}=\frac{x+1}{x-1}$$

Can you proceed?

 

[ognik]

Yes, more than enough, thanks muchly. Just frustrating that I sometimes can't see the bleedin' obvious ...:-(

Do you (or anyone) perhaps have a good way for me to get a lasting 'intuition' about what inverse hyperbolics are? I look at, for example, the well known sin x; it is periodic.

Then, it seems, sinh x is a reflection of sin x about the line y=x. It ends up not very dissimilar from sin x (I looked at 7. The Inverse Trigonometric Functions) , but with a limited range - it is not periodic?

Then arcsin x is again a reflection of sinh x about y=x. It looks closer to what sinx was (Inverse Hyperbolic Functions) , also not periodic?

But what do hyperbolic and inverse hyperbolic functions do - apart from causing me to see double after a while ...Sin is a wave, I can look at ripples in a pond etc. The others?