Is There a Faster Method for Solving Hyperbolic Function Problems?

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

The attempt is in the picture. Is this the right method? Is there any faster method without cumbersome calculations?

 

[Mark44]

Homework Statement

View attachment 214910

Homework Equations

The Attempt at a Solution

The attempt is in the picture. Is this the right method? Is there any faster method without cumbersome calculations?

From this page, https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions, I see that $\sinh(\cosh^{-1}(x) = \sqrt{x^2 - 1}$, for |x| > 1.

 

[Clara Chung]

From this page, https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions, I see that $\sinh(\cosh^{-1}(x) = \sqrt{x^2 - 1}$, for |x| > 1.

Thank you. I get the answer.
x=sinh(-arccosh(x+2))
=-sinh(arccosh(x+2))
=-root(x^2+4x+3)
And the website is very helpful

 

[LCKurtz]

@Clara Chung: The problem is, if you look at the graph it looks like there is a solution $x=-\frac 3 4$. And you can check that works exactly in your last equation of your original post but not your root solution.

 

[Clara Chung]

@Clara Chung: The problem is, if you look at the graph it looks like there is a solution $x=-\frac 3 4$. And you can check that works exactly in your last equation of your original post but not your root solution.

So it is x^2=x^2+4x+3
X=-3/4