Simple Inverse trigonometry question

[nehcrow]

This is a simple question but I really need to know why:

why is the alternate form of arccosh(x) = log(x + sqrt(x^2 - 1)) and not log(x - sqrt(x^2 - 1))
How do I justify the plus/minus sign?
I need to know this ASAP, thanks.

 

[Mentallic]

This is a simple question but I really need to know why:

why is the alternate form of arccosh(x) = log(x + sqrt(x^2 - 1)) and not log(x - sqrt(x^2 - 1))
How do I justify the plus/minus sign?
I need to know this ASAP, thanks.

If we were to use the minus sign then $$0<x-\sqrt{x^2-1}<1$$ for $$x>1$$ and the log of this is a negative value. To have an inverse function of cosh, we need to choose one of the plus or minus, and we chose to have positive values for the arccosh function so that is why we took the plus.

 

[nehcrow]

Thank you so much!

 

[Mentallic]

You're welcome