Hyperbolic functions

Definition

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They include $\sinh(x)$, $\cosh(x)$, $\tanh(x)$, and their reciprocals.

5 Must Know Facts For Your Next Test

1. $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$ are the definitions of the hyperbolic sine and cosine functions.
2. The identities $\cosh^2(x) - \sinh^2(x) = 1$ and $1 - \tanh^2(x) = \text{sech}^2(x)$ mirror trigonometric identities.
3. Hyperbolic functions have derivatives: $(\sinh(x))' = \cosh(x)$ and $(\cosh(x))' = \sinh(x)$.
4. $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$ is the definition of the hyperbolic tangent function.
5. Hyperbolic functions can be used to solve certain types of differential equations.

Review Questions

• What are the definitions of $\sinh(x)$ and $\cosh(x)$ in terms of exponential functions?
• State the identity that relates $\cosh^2(x)$ and $\sinh^2(x)$.
• What are the derivatives of $\sinh(x)$ and $\cosh(x)?

Related terms

Exponential Function: A function where an independent variable is in the exponent, typically written as $e^x$.

Trigonometric Functions: Functions related to angles, most commonly sine ($\sin$), cosine ($\cos$), and tangent ($\tan$).

$\text{Inverse Hyperbolic Functions}$: $\text{Functions that reverse hyperbolic functions; examples include }\text{$\text{arsinh}(x), \text{arcosh}(x), \text{artanh}(x)}.$$