What is the Relation between tan and tanh?Date: 08/25/2001 at 03:41:28 From: Barot Ketan Subject: What is the relation between tan and tanh? Hello sir, I want to know the equation type relation between tan and tanh - I mean the relation between tangent and hyperbolic tangent. Date: 08/25/2001 at 05:59:59 From: Doctor Floor Subject: Re: What is the relation between tan and tanh? Hello and thanks for writing, By definition we have: x -x x -x e - e e + e sinh(x) = ---------- and cosh(x) = ---------- 2 2 See, from the Dr. Math archives: Hyperbolic Functions http://mathforum.org/dr.math/problems/weber10.14.98.html Also we have Euler's Equation - see from the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.euler.equation.html ix e = cos x + i*sin x (where i is the imaginary unit "sqrt(-1)"). This gives ix -ix ix -ix e - e e + e sin(x) = ----------- and cos(x) = ----------- 2i 2 Combination gives ix -ix 1 e - e 1 sinh(ix) tanh(ix) tan(x) = - * ----------- = - -------- = -------- i ix -ix i cosh(ix) i e + e From which we conclude the identity i*tan(x) = tanh(ix) If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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