What is an Inverse Hyperbolic Function?

Inverse hyperbolic functions are mathematical functions that operate inversely to hyperbolic functions. Just like inverse trigonometric functions relate to trigonometric functions, inverse hyperbolic functions relate to hyperbolic functions such as hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh).

For example, the inverse hyperbolic sine function, denoted as sinh⁻¹(x) or arcsinh(x), is the function that undoes the operation of the hyperbolic sine function. In other words,

If y = sinh⁻¹(x), then x = sinh(y).

Similarly, the inverse hyperbolic cosine function, denoted as cosh⁻¹(x) or arccosh(x), undoes the operation of the hyperbolic cosine function.

If y = cosh⁻¹(x), then x = cosh(y).

And the inverse hyperbolic tangent function, denoted as tanh⁻¹(x) or arctanh(x), undoes the operation of the hyperbolic tangent function.

If y = tanh⁻¹(x), then x = tanh(y).

These functions are useful in various branches of mathematics, physics, engineering, and other fields where hyperbolic functions appear. They have applications in areas such as signal processing, control theory, and differential equations.

Inverse hyperbolic functions, such as arcsinh and arcosh, are practical tools in real-life scenarios despite their mathematical appearance. They assist in solving complex problems in physics, engineering, and economics by handling exponential and hyperbolic relationships efficiently. These functions act as “undo” buttons for hyperbolic functions, making them invaluable in various practical applications.