The inverse hyperbolic cosine function, in modern notation written as arcosh(x) or arccosh(x) or cosh^{-1}x, gives the value t (hyperbolic angle), so that:

The inverse hyperbolic cosine function accepts values not smaller than 1, because . Since the hyperbolic cosine is defined through the natural exponential function , its inverse can be defined through the natural logarithm function, using the following formula, for real x, with x≥1:

Properties

The derivative of the inverse hyperbolic cosine function is:

The integral of the inverse hyperbolic cosine function is given by: