Difference between revisions of "Hyperbolic trigonometric functions"
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(New page: The '''hyperbolic trigonometric functions''' are analogs of the standard trigonometric functions using a hyperbola rather than a circle. They are defined as: *Hyperbolic sine: <math>sinh...) |
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| − | The '''hyperbolic trigonometric functions''' are | + | The '''hyperbolic trigonometric functions''', also referred to as simply "hyperbolic functions," are analogous to the standard [[trigonometric function]]s using a [[hyperbola]] as the defining [[conic section]] rather than a [[circle]].<ref>[http://mathworld.wolfram.com/HyperbolicFunctions.html Hyperbolic functions] from mathworld.wolfram.com</ref> This has the effect of removing any [[imaginary number|i's]] that appear in the [[complex number|complex]] definition of the standard trigonometric functions. As such, they tend to [[differentiation|differentiate]] in an analogous way to standard trigonometric functions, up to perhaps a negative sign. They are defined as: |
| − | *Hyperbolic sine: <math>sinh(x) = \frac{e^x - e^{-x}}{2}</math> | + | *Hyperbolic sine: <math>\sinh(x) = \frac{e^x - e^{-x}}{2}</math> |
| − | *Hyperbolic cosine: <math>cosh(x) = \frac{e^x + e^{-x}}{2}</math> | + | *Hyperbolic cosine: <math>\cosh(x) = \frac{e^x + e^{-x}}{2}</math> |
| − | *Hyperbolic tangent: <math>tanh(x) = \frac{sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}</math> | + | *Hyperbolic tangent: <math>\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}</math> |
| − | *Hyperbolic cosecant: <math>csch(x) = \frac{1}{sinh(x)} = \frac{2}{e^x - e^{-x}}</math> | + | *Hyperbolic cosecant: <math>\text{csch}(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}}</math> |
| − | *Hyperbolic secant: <math>sech(x) = \frac{1}{ | + | *Hyperbolic secant: <math>\text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}</math> |
| − | *Hyperbolic | + | *Hyperbolic cotangent: <math>\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math> |
| − | [[Category: | + | ==Graphs== |
| + | {| class="wikitable" | ||
| + | |- | ||
| + | |<!--column1-->[[Image:Sinh-cosh.png]] | ||
| + | |<!--column2-->[[Image:Tanh-coth.png]] | ||
| + | |- | ||
| + | |<!--column1-->sinh and cosh | ||
| + | |<!--column2-->tanh and coth | ||
| + | |}<!--end wikitable--> | ||
| + | |||
| + | ==Identities== | ||
| + | The hyperbolic trigonometric functions have many identities that are similar to those of [[trigonometric functions]]. These can be remembered by replacing instances of <math>\sin^2{x}</math> with <math>-\sinh^2{x}</math>. For example, the trigonometric identity, | ||
| + | |||
| + | :<math> | ||
| + | \cos^2{x}+\sin^2{x} = 1 | ||
| + | </math> | ||
| + | |||
| + | has the corresponding hyperbolic identity, | ||
| + | |||
| + | :<math> | ||
| + | \cosh^2{x}-\sinh^2{x} = 1 | ||
| + | </math> | ||
| + | |||
| + | Other identities include: | ||
| + | |||
| + | :<math> | ||
| + | \sinh{2x} =2\sinh{x}\cosh{x} | ||
| + | </math> | ||
| + | |||
| + | :<math> | ||
| + | \cosh{2x} = \cosh^2{x}+\sinh^2{x} | ||
| + | </math> | ||
| + | |||
| + | Also, note that: | ||
| + | |||
| + | :<math> | ||
| + | \sinh{iz} = i\sin{z} | ||
| + | </math> | ||
| + | |||
| + | :<math> | ||
| + | \cosh{iz}=\cos{z} | ||
| + | </math> | ||
| + | |||
| + | ==References== | ||
| + | {{reflist}} | ||
| + | |||
| + | ==See also== | ||
| + | *[[Trigonometric function]] | ||
| + | |||
| + | [[Category:Trigonometry]] | ||
Latest revision as of 22:59, March 1, 2021
The hyperbolic trigonometric functions, also referred to as simply "hyperbolic functions," are analogous to the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle.[1] This has the effect of removing any i's that appear in the complex definition of the standard trigonometric functions. As such, they tend to differentiate in an analogous way to standard trigonometric functions, up to perhaps a negative sign. They are defined as:
- Hyperbolic sine:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cosecant:
- Hyperbolic secant:
- Hyperbolic cotangent:
Contents
Graphs
|
|
| sinh and cosh | tanh and coth |
Identities
The hyperbolic trigonometric functions have many identities that are similar to those of trigonometric functions. These can be remembered by replacing instances of 
with 
. For example, the trigonometric identity,
has the corresponding hyperbolic identity,
Other identities include:
Also, note that:
References
- ↑ Hyperbolic functions from mathworld.wolfram.com
