If I have the sinh of x, what can I do to sinh(x) to get its arcsinh?

Question

I am trying to learn to do math functions using a basic calculator. I know to find sinh, I use the equation sinh(x)=(e^x - e^[-x]) / 2. Now, what can I do to this result to find the arcsinh of x?

And is the procedure the same with the other geometric functions (sosinh and tangent)?

So how would I find the natural log (ln) of a number using a basic calculator?

Answer 1

sinh x=(e^x-e^-x)/2
If you call sin h x = y and e^x=z(always positive)
2y = z-1/z so z^2-2yz -1=0
so z= (2y +sqrt(4y^2+4)/2 = y +sqrt(1+y^2)
and x= ln[y+sqrt(1+y^2) ] inverse function
Interchange x and y and you get
arcsinh x= ln [x+sqrt(1+x^2)]