Hyperbolic FunctionsDate: 10/14/98 at 00:04:55 From: Anne Weber Subject: Trig Functions using exponentials A. What do cosh(x) and sinh(x) stand for? Why are they called this? These functions occur naturally in the physical sciences. Give one example of where they might occur. B. What is the value of sinh(2)? C. What's the definition of cosh(x) and sinh(x) in terms of exponential functions? D. Use you answer in part C to prove that tanh^2(x) + sech^2(x) = 1 if: sinh(x) 1 tanh(x) = ------- and sech(x) = ------- cosh(x) cosh(x) Thanks in advance, Anne Weber Date: 10/14/98 at 04:37:02 From: Doctor Pat Subject: Re: Trig Functions using exponentials Anne, Cosh(x) (pronounced so it rhymes with "gosh") and sinh(x) (sounds like cinch, or some say sink) are called hyperbolic functions because they can be defined on a unit hyperbola, just as the others are called circular functions, because they are defined on a unit circle. An example is a hanging cable, the catenary arch. Sinh(2) is easily found from the definition below, so I will use it there. The definitions are: x -x x -x e - e e + e sinh(x) = ------------ and cosh(x) = ----------- 2 2 Then when x = 2, we get sinh(2) = (e^2 - e^(-2))/2 which is approximately 3.62. I'm going to leave the last part for you to do. You should be able to use the definitions to get this last piece. Just use the definitions and substitute in the exponentials and then simplify. Good luck, - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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