Question with hyperbolic functions

label Calculus
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There's a question asking that if tanhx = 12/13, then find the other hyperbolic functions at x.  I've gotten to sinhx/coshx = 12/13, but I'm stuck on where to go next.  Any pointers on how to proceed?

Jan 24th, 2015

tanh(x) =
sinh(x) / cosh(x) =
((1/2) * (e^(x) - e^(-x))) / ((1/2) * (e^(x) + e^(-x))) =>
(e^(x) - e^(-x)) / (e^(x) + e^(-x))

12 / 13 = (e^(x) - e^(-x)) / (e^(x) + e^(-x))
12 * (e^(x) + e^(-x)) = 13 * (e^(x) - e^(-x))
12 * e^(x) + 12 * e^(-x) = 13 * e^(x) - 13 * e^(-x))
25 * e^(-x) = e^(x)
25 = e^(2x)
ln(25) = 2x
2 * ln(5) = 2x
x = ln(5)

sinh(x) =
(1/2) * (e^(x) - e^(-x)) =>
(1/2) * (5 - (1/5)) =>
(1/2) * (24/5) =>
12/5

cosh(x) =
(1/2) * (e^(x) + e^(-x)) =
(1/2) * (26/5) =>
13/5

coth(x) =
1/tanh(x) =
13/12

sech(x) =
1/cosh(x) =
5/13

csch(x) =
1/sinh(x) =
5/12



Always Remember this formulas  sinh(x) = (1/2) * (e^(x) - e^(-x)) and cosh(x) =  (1/2) * (e^(x) + e^(-x))

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Jan 24th, 2015

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