Inverse Hyperbolic Functions: Properties & Applications

Instructor: Russell Frith
Inverse hyperbolic functions are named the same as inverse trigonometric functions with the letter 'h' added to each name. In this lesson, properties and applications of inverse hyperbolic functions are introduced.

What Are Inverse Hyperbolic Functions?

Suppose you are tasked with taking your dog for a walk and your dog stubbornly refuses. As your dog resists, you pull harder and harder on his leash. Finally, you're in a pitched tug-of-war with the struggling pooch and you proceed to walk down the sidewalk with the leash pulled taut and your dog standing fast in the middle of your driveway, which is perpendicular to the sidewalk. Eventually you overpower your dog and he starts to drag in an unusual trajectory as shown in the figure:


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Such a path is called a tractrix . If the dog's initial position is at (0,a) then in terms of Cartesian coordinates, the position of the dog walker with respect to the position of the dog is:


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One term used in this equation of motion is the inverse hyperbolic secant function. This function is a member of a class of functions known as inverse hyperbolic functions. This lesson presents essential details on some of these functions.

Examples of inverse hyperbolic functions

The inverse hyperbolic sine function (arcsinh(x)) is written as


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The graph of this function is:


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Both the domain and range of this function are the set of real numbers. This function may be reformulated in terms of natural log. Start with


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To find the inverse solve for x and then interchange x and y.


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The derivative of arcsinh(x) may be found by differentiating the natural log representation.


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The inverse hyperbolic cosine function (arccosh(x)) is written as


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This function is not one-to-one. The inverse of cosh(x) is obtained if and only if the restricted version of this function is used:


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The graph of this function is:


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This function may be reformulated in terms of natural log. Start with


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The derivative of arccosh(x) may be found by differentiating the natural log representation.


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The inverse hyperbolic tangent function (arctanh(x)) is defined as


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The graph of this function is:

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This function may be reformulated in terms of the natural log function.


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Using the natural log formulation, one can obtain the derivative as:


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