Introduction Hyperbolic Inverses—Logarithms
Each of the six hyperbolic functions has an inverse relation. These are known as the hyperbolic arcsine, hyperbolic arccosine, hyperbolic arctangent, hyperbolic arccosecant, hyperbolic arcsecant, and hyperbolic arccotangent. In formulas and equations, they are abbreviated arcsinh or sinh ^{–1} , arccosh or cosh ^{–1} , arctanh or tanh ^{–1} , arccsch or csch ^{–1} , arcsech or sech ^{–1} , and arccoth or coth ^{–1} respectively. These relations become functions when their domains are restricted as shown in the graphs of Figs. 47 through 412.
The Natural Logarithm
Now it is time to learn a little about logarithms. It is common to write “the natural logarithm of x ” as “ln x .” This function is the inverse of the base e exponential function. The natural logarithm function and the base e exponential function “undo” each other. Suppose x and v are real numbers, and y and u are positive real numbers. If e ^{x} = y , then x = ln y , and if ln u = v , then u = e ^{v} .
The natural logarithm function is useful in expressing the inverse hyperbolic functions, just as the exponential function can be used to express the hyperbolic functions.
You can find the natural logarithm of a specific number using a calculator. Enter the number for which you want to find the natural logarithm, and then hit the “ln” key. Beware: the logarithm of 0 or any negative real number is not defined in the set of real numbers.
Hyperbolic Inverses As Logarithms
You can find hyperbolic inverses of specific quantities using a calculator that has the “ln” function. Here are the expressions for the hyperbolic inverses, in terms of natural logarithms. (The ½ power represents the square root.)
arcsinh x = ln [ x + ( x ^{2} + 1) ^{1/2} ]
arccosh x = ln [x + ( x ^{2} – 1) ^{1/2} ]
arctanh x = 0.5 ln [(1 + x )/(1 – x )]
arccsch x = ln [ x ^{–1} + ( x ^{–2} + 1) ^{1/2} ]
arcsech x = ln [ x ^{–1} + ( x ^{–2} – 1) ^{1/2} ]
arctanh x = 0.5 ln [( x + 1)/( x – 1 )]
In these expressions, the values 0.5 represent exactly ½. The formulas are a little bit messy, but if you plug in the numbers and take your time doing the calculations, you shouldn’t have trouble. Be careful about the order in which you perform the operations. Perform the operations in the innermost sets of parentheses or brackets first, and then work outward.
Let’s see what the graphs of the inverse hyperbolic functions look like.
Arcsine, Arccosine, and Arctangent
Hyperbolic Arcsine
Figure 47 is a graph of the function y = arcsinh x (or y = sinh ^{–1} x ). Its domain and range both encompass the entire set of real numbers.
Hyperbolic Arccosine
Figure 48 is a graph of the function y = arccosh x (or y = cosh ^{–1} x ). The domain includes real numbers x such that x ≥ 1. The range of the hyperbolic arccosine function is limited to the nonnegative reals, that is, to real numbers y such that y ≥ 0.

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