MHB Understanding the Math Behind coth(y) = x

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The discussion revolves around understanding the mathematical relationship expressed by coth(y) = x and its transformation into y = (1/2) ln((x+1)/(x-1)). Participants express frustration in grasping the underlying concepts, particularly how to manipulate the equation to isolate e^(2y). There is a comparison made between hyperbolic functions and trigonometric functions, noting that while sine is periodic, hyperbolic sine is not, leading to confusion about their properties and applications. The need for a deeper intuition about inverse hyperbolic functions is highlighted, emphasizing the challenge in visualizing their behavior compared to familiar trigonometric functions. Overall, the thread reflects a quest for clarity in understanding hyperbolic functions and their inverses.
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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...
 
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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?
 
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?
 
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