Value of Hyperbolic Functions (Calculus)?

Question

tanh(x) = 1/2
How would I go about finding the values of the other hyperbolic functions at x, given this equation? Thanks

Answer 1

sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
= (e^x - e^(-x)) / (e^x + e^(-x))
= 1/2 => 2(e^x - e^(-x)) = (e^x + e^(-x))
=> e^x = 3e^(-x)
=> e^(2x) = 3
=> e^x = √3, e^(-x) = 1/√3.

So sinh(x) = (e^x - e^(-x)) / 2
= (√3 - 1/√3) / 2
= (3 - 1) / 2√3
= 1 / √3

and cosh(x) = (e^x + e^(-x)) / 2
= (√3 + 1/√3) / 2
= (3 + 1) / 2√3
= 2 / √3

which checks out: we needed it to be twice the value for sinh(x).

Answer 2

x = arctanh(.5)

sinhx = sinh(arctanh(.5))
coshx = cosh(arctanh(.5))