For each formula I will explicitly state if some change of sign occurs or not (the different sign is marked in green). Most of the formulas that follow correspond precisely to a trig formula or they differ by at most a change of sign. For the moment we have to postpone this discussion to the end of Calc3 or Calc4, but still we should be aware of the fact that the impressive similarity between trig formulas and hyperbolic formulas is not a pure coincidence. Roughly speaking ordinary trigonometric functions are trigonometric functions of purely real numbers, and hyperbolic functions are trigonometric functions of purely imaginary numbers. Unfortunately this can be completely understood only if you have some knowledge of the complex numbers. For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same dignity as trigonometric functions.
