Inverse hyperbolic functions

Definition

Inverse hyperbolic functions are the inverses of the hyperbolic functions, such as sinh, cosh, and tanh. They are used to solve equations involving hyperbolic functions.

5 Must Know Facts For Your Next Test

1. The inverse hyperbolic sine function is denoted as $\sinh^{-1}(x)$ or $\text{arsinh}(x)$ and is defined by $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$.
2. The inverse hyperbolic cosine function is denoted as $\cosh^{-1}(x)$ or $\text{arcosh}(x)$ and is defined only for $x \geq 1$ by $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$.
3. The inverse hyperbolic tangent function is denoted as $\tanh^{-1}(x)$ or $\text{artanh}(x)$ and is defined for $-1 < x < 1$ by $\tanh^{-1}(x) = \frac{1}{2}\ln(\frac{1+x}{1-x})$.
4. Inverse hyperbolic functions can be derived from their corresponding exponential definitions and logarithmic identities.
5. They have applications in various fields including calculus, complex analysis, and physics.

Review Questions

• What is the definition of the inverse hyperbolic sine function?
• For which values of x is the inverse hyperbolic cosine function defined?
• Express the inverse hyperbolic tangent function in terms of natural logarithms.

Related terms

Hyperbolic Functions: Functions similar to trigonometric functions but based on hyperbolas instead of circles. Includes sinh, cosh, tanh.

$e^x$ (Exponential Function): A fundamental mathematical function where the base e (approximately equal to 2.71828) is raised to the power of x.

$\ln(x)$ (Natural Logarithm): The logarithm to the base e; it represents the time needed to reach a certain level of continuous growth.