Inverse Hyperbolic Functions

Definition

Inverse hyperbolic functions are the inverse operations of the hyperbolic functions, which include the hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), hyperbolic secant (sech), hyperbolic cosecant (csch), and hyperbolic cotangent (coth). These inverse functions allow us to undo the transformations performed by the original hyperbolic functions, providing a way to solve problems involving hyperbolic functions.

5 Must Know Facts For Your Next Test

1. Inverse hyperbolic functions are denoted with the prefix 'arc' (e.g., arcsinh, arccosh, arctanh).
2. The inverse hyperbolic functions are used to solve problems involving exponential growth and decay, as well as to find the angle or argument of a hyperbolic function.
3. The inverse hyperbolic functions can be used to find the original value of a hyperbolic function, just as the inverse trigonometric functions can be used to find the original angle of a trigonometric function.
4. Inverse hyperbolic functions are often used in physics, engineering, and other scientific fields to model and analyze phenomena involving exponential growth or decay.
5. The graphs of the inverse hyperbolic functions are the mirror images of the graphs of the corresponding hyperbolic functions, reflecting the undoing of the original transformation.

Review Questions

• Explain the relationship between hyperbolic functions and inverse hyperbolic functions.
  • The inverse hyperbolic functions are the inverse operations of the hyperbolic functions. Just as the inverse trigonometric functions undo the transformations performed by the trigonometric functions, the inverse hyperbolic functions undo the transformations performed by the hyperbolic functions. This allows us to solve problems involving hyperbolic functions by finding the original value or argument that was transformed by the hyperbolic function.
• Describe how the inverse hyperbolic functions are used to model and analyze phenomena involving exponential growth or decay.
  • The inverse hyperbolic functions are particularly useful in modeling and analyzing phenomena that exhibit exponential growth or decay, such as radioactive decay, population growth, and the behavior of certain physical systems. By using the inverse hyperbolic functions, we can determine the original value or argument that was transformed by the exponential function, allowing us to better understand and predict the behavior of these systems.
• Analyze the graphical properties of the inverse hyperbolic functions and explain how they relate to the graphs of the corresponding hyperbolic functions.
  • The graphs of the inverse hyperbolic functions are the mirror images of the graphs of the corresponding hyperbolic functions. This reflects the undoing of the original transformation performed by the hyperbolic function. For example, the graph of '$\arcsinh(x)$' is the mirror image of the graph of '$\sinh(x)$', with the '$x$-axis' and '$y$-axis' interchanged. This property of the inverse hyperbolic functions allows us to easily visualize the relationship between the original function and its inverse, and to use the graphs to solve problems involving hyperbolic functions.

Related terms

Hyperbolic Functions: The hyperbolic functions are a set of mathematical functions that are analogous to the trigonometric functions, but are defined in terms of the exponential function rather than the circular functions.

Inverse Functions: An inverse function is a function that undoes the operation of another function, so that if you apply the original function and then the inverse function, you end up back where you started.

Exponential Functions:

Exponential functions are functions in which the independent variable appears as the exponent, and the dependent variable is the result of raising a constant to that power.