Definitions of Hyperbolic Functions
Hyperbolic functions are a family of functions that are analogous to the ordinary trigonometric (or circular) functions, but they are defined using the hyperbola rather than the circle.
The following table gives the Hyperbolic Functions: sinh, csch, cosh, sech, tanh, coth. Scroll down the page for more examples and solutions.
Definitions
The basic hyperbolic functions are defined in terms of the exponential function \(e^x\):
1. Hyperbolic Sine (sinh):
\(\text{sinh }x=\frac{e^x-e^{-x}}{2}\)
2. Hyperbolic Cosine (cosh):
\(\text{cosh }x=\frac{e^x+e^{-x}}{2}\)
3. Hyperbolic Tangent (tanh):
\(\text{tanh }x=\frac{\text{sinh }x}{\text{cosh }x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\)
4. Hyperbolic Cotangent (coth):
\(\text{coth }x=\frac{\text{cosh }x}{\text{sinh }x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}\)
5. Hyperbolic Secant (sech):
\(\text{sech }x=\frac{1}{\text{cosh }x}=\frac{2}{e^x+e^{-x}}\)
6. Hyperbolic Cosecant (csch):
\(\text{csch }x=\frac{1}{\text{sinh }x}=\frac{2}{e^x-e^{-x}}\)
Key Identities
1. Pythagorean Identities:
\(\text{cosh}^2x-\text{sinh}^2x=1\)
\(1-\text{tanh}^2x=\text{sech}^2x\)
\(\text{coth}^2x-1=\text{csch}^2x\)
2. Sum and Difference Formulas:
\(\text{sinh}\left( x\pm y \right)=\text{sinh }x\text{ cosh }y\pm \text{cosh }x\text{ sinh }y\)
\(\text{cosh}\left( x\pm y \right)=\text{cosh }x\text{ cosh }y\pm \text{sinh }x\text{ sinh }y\)
\(\text{tanh}\left( x\pm y \right)=\frac{\text{tanh }x\pm \text{tanh }y}{1\pm \text{tanh }x \text{ tanh }y}\)
3. Double Angle Formulas:
\(\text{sinh}\left( 2x \right) = 2\text{ sinh }x\text{ cosh }x\)
\(\text{cosh}\left( 2x \right) = \text{cosh}^2x+\text{sinh}^2x=2\text{ cosh}^2x-1\)
Videos
Defining the Hyperbolic Functions
Overview of hyperbolic function.
The lesson defines the hyperbolic functions, shows the graphs of the hyperbolic functions, and gives the properties of hyperbolic functions.
Hyperbolic Identities
How to use hyperbolic identities to simplify a hyperbolic function?
Derivatives of Hyperbolic Functions
Formulas for the derivatives of the hyperbolic functions and how to use the formulas to find a few derivatives.
Try out our new and fun Fraction Concoction Game.
Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.