# Why are hyperbolic functions defined in terms of exponentials?

 P: 56 Where do the definitions of hyperbolic functions in terms of exponentials come from ?
 P: 56 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 ?
 Emeritus Sci Advisor PF Gold P: 16,091 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.
 P: 56 Why are hyperbolic functions defined in terms of exponentials? 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.
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