# Why are hyperbolic functions defined in terms of exponentials?

## Main Question or Discussion Point

Where do the definitions of hyperbolic functions in terms of exponentials come from ?

Maybe somebody tried e^i*x=cos x+i*sin x and e^-i*x=cos x-i*sin x with i*x instead of x and got functions that acted a lot like regular trig functions so they named them a lot like trig functions, like cos(i*x)=cosh x.
Could that be where those definitions come from ?
If so,do they come from somewhere else also ?

Hurkyl
Staff Emeritus
Gold Member
AFAIK, the hyperbolic trigonometric functions were originally defined geometrically, analogously to the ordinary (circular) trigonometric functions, just based on the rectangular hyperbola rather than the circle.

Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.

AlephZero