Difference between revisions of "Hyperbolic trigonometric functions"

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The '''hyperbolic trigonometric functions''' are analogs of the standard [[trigonometric function]]s using a [[hyperbola]] as the defining [[conic section]] rather than a [[circle]]. They are defined as:
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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:
  
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*Hyperbolic sine: <math>sinh(x) = \frac{e^x - e^{-x}}{2}</math>
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*Hyperbolic sine: <math>\sinh(x) = \frac{e^x - e^{-x}}{2}</math>
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*Hyperbolic cosine: <math>cosh(x) = \frac{e^x + e^{-x}}{2}</math>
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*Hyperbolic cosine: <math>\cosh(x) = \frac{e^x + e^{-x}}{2}</math>
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*Hyperbolic tangent: <math>tanh(x) = \frac{sinh(x)}{cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}</math>
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*Hyperbolic tangent: <math>\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}</math>
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*Hyperbolic cosecant: <math>csch(x) = \frac{1}{sinh(x)} = \frac{2}{e^x - e^{-x}}</math>
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*Hyperbolic cosecant: <math>\text{csch}(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}}</math>
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*Hyperbolic secant: <math>sech(x) = \frac{1}{sinh(x)} = \frac{2}{e^x + e^{-x}}</math>
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*Hyperbolic secant: <math>\text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}</math>
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*Hyperbolic cotangent: <math>coth(x) = \frac{cosh(x)}{sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math>
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*Hyperbolic cotangent: <math>\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math>
  
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[[Category:Mathematics]]
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==Graphs==
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{| class="wikitable"
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|-
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|<!--column1-->[[Image:Sinh-cosh.png]]
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|<!--column2-->[[Image:Tanh-coth.png]]
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|-
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|<!--column1-->sinh and cosh
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|<!--column2-->tanh and coth
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|}<!--end wikitable-->
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==Identities==
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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,
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:<math>
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\cos^2{x}+\sin^2{x} = 1
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</math>
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has the corresponding hyperbolic identity,
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:<math>
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\cosh^2{x}-\sinh^2{x} = 1
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</math>
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Other identities include:
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:<math>
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\sinh{2x} =2\sinh{x}\cosh{x}
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</math>
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:<math>
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\cosh{2x} = \cosh^2{x}+\sinh^2{x}
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</math>
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Also, note that:
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:<math>
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\sinh{iz} = i\sin{z}
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</math>
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:<math>
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\cosh{iz}=\cos{z}
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</math>
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==References==
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{{reflist}}
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==See also==
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*[[Trigonometric function]]
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[[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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh(x) = \frac{e^x - e^{-x}}{2}}
  • Hyperbolic cosine: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh(x) = \frac{e^x + e^{-x}}{2}}
  • Hyperbolic tangent: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}}
  • Hyperbolic cosecant: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{csch}(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}}}
  • Hyperbolic secant: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}}
  • Hyperbolic cotangent: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}}

Graphs

Sinh-cosh.png Tanh-coth.png
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2{x}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\sinh^2{x}} . For example, the trigonometric identity,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2{x}+\sin^2{x} = 1 }

has the corresponding hyperbolic identity,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh^2{x}-\sinh^2{x} = 1 }

Other identities include:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh{2x} =2\sinh{x}\cosh{x} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh{2x} = \cosh^2{x}+\sinh^2{x} }

Also, note that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sinh{iz} = i\sin{z} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh{iz}=\cos{z} }

References

  1. ↑ Hyperbolic functions from mathworld.wolfram.com

See also

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