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

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.