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I know the plot of $\tanh$ activation function looks like. I also know that its output has a range of $[-1, 1]$. Furthermore, I also know the it is defined as follows

$$ \tanh(x) = \frac{\sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$

  1. What exactly is the meaning of term "hyperbolic"?
  2. What is the interpretation of the word hyperbolic in this context? How is the equation above derived?
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  • $\begingroup$ The answer by Tophat is a good one, but I'll add a fun fact. Rotations trade a bit of the component in one dimension to another, turning "x"ness into "y"ness. In the theory of special relativity, you can also trade a bit of the component of a spacial dimension with the time dimension, turning length into duration. This is why "clocks slow down" and "objects contract" when moving relative to an observer. The difference, though, is that the latter uses hyperbolic trig functions, while the former uses standard (circular) trig functions. $\endgroup$
    – Tac-Tics
    Commented Mar 6, 2018 at 17:25

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The hyperbolic trig functions follow the equation for a Rectangular hyperbola, which is something you should be familiar from analytical geometry. The recentgular hyperbola is defined by $x^2 - y^2 = 1$ and if you let $x = cosh(t)$ and $y = sinh(t)$ and plug it into the rectangular hyperbola equation you can verify this fact quite easily.

From this idea you can derive the rest of the hyperbolic trig functions. (Also they reason they are given trig names is because they resemble the properties trig functions have).

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