The hyperbolic sine function is a one-to-one function and thus has an inverse. As usual, the graph of the inverse hyperbolic sine function

also denoted by by reflecting the graph of about the line

For all inverse hyperbolic functions but the inverse hyperbolic cotangent and the inverse hyperbolic cosecant, the domain of the real function is connected.

Inverse hyperbolic sine (if the domain is the whole real line)

$$arcsinhx=ln(x+\sqrt{x^2+1}$$

Inverse hyperbolic cosine (if the domain is the closed interval

.

$$arccoshx=ln(x+\sqrt{x^2-1})$$

Inverse hyperbolic tangent [if the domain is the open interval (−1, 1)]

$$arctanhx=\frac{1}{2}ln\left(\frac{1+x}{1-x}\right)$$

Inverse hyperbolic cotangent [if the domain is the union of the open intervals (−∞, −1) and (1, +∞)]

$$arccothx=\frac{1}{2}ln\left(\frac{x+1}{x-1}\right)$$

Inverse hyperbolic cosecant (if the domain is the real line with 0 removed)