Hyperbolic function (1 Viewer)

[addikaye03]

Express the following as algebriac functions of x:

a) sinh^2(arctanh(x))

b) tanh(arccosh(x))

They might be easy, i might be missing something

 

[shaon0]

Express the following as algebriac functions of x:

a) sinh^2(arctanh(x))

b) tanh(arccosh(x))

They might be easy, i might be missing something

Let A=atanh(x) and u=sinhA:
sinhA=u
sqrt(coshA^2-1)=u
coshA=sqrt(u^2+1)
u/sqrt(u^2+1)=tanh(A)=x
u^2=x^2u^2+x^2
u^2(1-x^2)=x^2
u=x/sqrt(1-x^2)

(sinh(atanh(x)))^2=x^2/(1-x^2)

Same as above:
tanh(A)=u
coshA=x
sqrt(1+sinh(A)^2)=x
sinhA=sqrt(x^2-1)
u=tanh(A)=sqrt(x^2-1)/x

tanh(acosh(A))=sqrt(x^2-1)/x

 

[addikaye03]

Let A=atanh(x) and u=sinhA:
sinhA=u
sqrt(coshA^2-1)=u
coshA=sqrt(u^2+1)
u/sqrt(u^2+1)=tanh(A)=x
u^2=x^2u^2+x^2
u^2(1-x^2)=x^2
u=x/sqrt(1-x^2)

(sinh(atanh(x)))^2=x^2/(1-x^2)

Same as above:
tanh(A)=u
coshA=x
sqrt(1+sinh(A)^2)=x
sinhA=sqrt(x^2-1)
u=tanh(A)=sqrt(x^2-1)/x

tanh(acosh(A))=sqrt(x^2-1)/x

Thanks mate, makes clear sense now.

 

[Drongoski]

shaonO has done a very good job.

2nd one can be done like this as well (essentially similar to shaonO's):

 

[gurmies]

There's actually a "triangle" in hyperbolic space which can be used quite easily with these questions. I believe it has applications in relativity, but it also works here. c^2 = a^2 - b^2, where a > b.

 

[shaon0]

There's actually a "triangle" in hyperbolic space which can be used quite easily with these questions. I believe it has applications in relativity, but it also works here. c^2 = a^2 - b^2, where a > b.

Kinda looks like the ellipse equation.

 

[gurmies]

An alternative would be to find sech using the relationship between tanh and sech and then to proceed with sinh and cosh

 

[addikaye03]

An alternative would be to find sech using the relationship between tanh and sech and then to proceed with sinh and cosh

That's how i originally went about solving them. Drongoski, your solution for Q2 is epic! That's very elegant

 

[shaon0]

That's how i originally went about solving them. Drongoski, your solution for Q2 is epic! That's very elegant

That's how my uni tutor does it. I was going to do that method but it would've been hard to understand without LaTeX. It's analoguous to what Gurmies said.