What the difference between them

As well as what the difference bettwen coshx and arccosx

The Arcsinh(x) function is also known as the inverse hyperbolic sine function. As its name implies, it is the inverse of the Sinh(x) function, just as the Arcsin(x) function is the inverse of sin(x). Thus if sinh(a) = b, then Arcsinh(b) = 1. The definition of the Sinh(x) function is:


\sinh(x) = \frac 1 2 (e^x - e^{-x)


It can be shown that its inverse is:


Arcsinh(x) = \ln(x + \sqrt{x^2+1})


There's a similar relationship between cosh(x) and Arccosh(x). If cosh(a) = b, then Arccosh(b) = a. The definition of the hyperbolic cosine function is:


\cosh(x) = \frac 1 2 (e^x + e^ {-x})

I asked about arcsin x =?
not arcsinhx

OK - I assumed you were asking about arcsinh, since you mentioned the hyperbolic sine function (sinh). The sine and cosine functions have virtually nothing to do with the sinh and cosh functions. About the only thing similar between them is that sinh and cosh are based on the geometry of a hyperbola (well, catenary actually) in a way that is analogous to how sine and cosine are based on the circle. Hence \cos^2 x + \sin^2x = 1 whereas \cosh^2 x - \sinh^2x =1

Arcsin(x) is the inverse function of sin(x). So if sin(a) = b, then arcsin(b) =a. For example, since sin(pi/2) = 1, you know that arcsin(1) = pi/2. Beyond that, is there anything more specific you would like to know?