More than just a few problems that happen to pop up in the textbook, I mean.
You mean cosh and sinh?
I'm not really sure, I don't remember doing it very often. I remember them more from diff eq more than anything. I have friends who claim they did though, when I brought up the fact that I had no idea how to work with them. I wouldn't forget about them, but I also wouldn't spend a lot of time working with them. Learn the e^ definitions and call it a day unless you feel inclined to go further. My 2 cents.
I am not sure what you mean by "used" in Calculus III. The point of any Calculus course is to teach concepts and methods that can be used in other courses! and the hyperbolic functions are definitely used in other courses. They act a great deal like the ordinary sine and cosine functions: The general solution to the very fundamental differential equation [itex]d^2 y/dx^2+ y= 0[/itex] is [itex]y(x)= A cos(x)+ B sin(x)[/itex] and, similarly, the general solution to the equally fundamental differential equation [itex]d^2y/dx^2- y= 0[/itex] is [itex]y(x)= A cosh(x)+ B sinh(x)[/itex]. Just as for trig functions we have [itex]sin^2(x)+ cos^2(x)= 1[/itex] so or hyperbolic functions we have [itex]cosh^2(x)- sinh^2(x)= 1[/itex]. And when you study "functions of a complex variable" you learn the cos(x) and cosh(x), as well as sin(x) and sinh(x) are really just the "real" and "imaginary" parts of the same function.
The treatment of hyperbolic sine and cosine in Calculus III is (or "was", as I remember) very brief. Any use of cosh or sinh in the course might account for about 2 or 3 days in the semester sized course.
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