How do you find sinhx and coshx when tanhx = 12/13?

[Slimsta]

Homework Statement

find all other hyperbolic function at x for tanhx=12/13

Homework Equations

tanhx = sinhx/coshx
cothx=1/tanhx
etc...

The Attempt at a Solution

the only thing i got is cothx=13/12
all i need to know is how to find sinhx and i will be fine.

 

[Bohrok]

Why not solve for x and then evaluate sinh(x)?

 

[Slimsta]

Why not solve for x and then evaluate sinh(x)?

so x= tanh-1(12/13) ?
then plug it in sinh(x)?

i am not sure if i get it..

 

[Dick]

tanh(x)=sinh(x)/cosh(x)=12/13. cosh(x)^2-sinh(x)^2=1. That's two equations in two unknowns. Can you find the solution?

 

[Bohrok]

$$\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1}} = \frac{12}{13}$$
You can solve for x then plug it into the other hyperbolic trig functions. Or try what Dick suggested, looks like it should be easier.

 

[Slimsta]

yeah i asked a teacher today too and he explained the same thing.
that makes sense.

even though, tanhx = sinhx/coshx = 12/13 then from here i could use sinhx =12 and coshx = 13 right away..
but that's "the cheating way" i guess..

thanks for help!

 

[Dick]

sinh(x)=12 and cosh(x)=13 isn't only the 'cheating way', it's the wrong way. 13^2-12^2 isn't equal to 1.