hyperbolic tangent vs tangent

Question

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent?

The reason I ask this question is to get a geometrical intuition of the notion of rapidity in relativity: $$\phi = \mbox{tanh}^{-1}(v/c)$$

Answer 1

(just comments transfered to answer) s

$ \phi$ cannot be represented by an angle.

if only because it has a range $ − \infty < \phi < \infty $ , while angles have a range $ − \pi < x < \pi $ )

$ \phi$ can be represented by an area
see en.wikipedia.org/wiki/Hyperbolic_function and math.stackexchange.com/a/451372/88985 .

$ \phi$ is twice the area between:

• the x-axis,
• the line conecting the origin and the point (1,tanh) and
• the hyperbola,

I agree it is all not very intuitive, but then you could also see the normal trigonomic functions as functions of the area of the enclosed circle sector instead as function of the arc length.