Hyperbolic Functions

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.

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Hyperbolic Identities
How to use hyperbolic identities to simplify a hyperbolic function?

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Derivatives of Hyperbolic Functions
Formulas for the derivatives of the hyperbolic functions and how to use the formulas to find a few derivatives.

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