Ok so in today's lecture on hyperbolic functions, the lecturer drew the well-known graph of the equilateral hyperbola, which shows sinh(a), cosh(a) and the area which is equal to a/2.
However, when I asked him if the hyperbolic tangent appears anywhere in this graph (a la its circular trigonometric cousin, which is the tangent of the unit circle), he didn't have a response and said he'd never thought about that. Can anyone explain if the tangent actually appears somehow in this graph and its geometric meaning related to it (if there is one at all)? I hope I made my question clear enough, but, if this is not the case, I will try to explain it better. Thanks!
The hyperbola $x^2-y^2=1,\,x\ge 1$ can be parameterised as $x=\cosh a,\,y=\pm\sinh a$ so $dy/dx=\pm\coth a$. The normal to the curve then has gradient $\mp\tanh a$.