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Hyperbolic trigonometric functions (view source)
Revision as of 15:09, September 14, 2017
, 15:09, September 14, 2017Added some identities and fixed some of the maths formatting
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:
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}{cosh(x)} = \frac{2}{e^x + e^{-x}}</math>
*Hyperbolic secant: <math>\text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}</math>
*Hyperbolic cotangent: <math>coth(x) = \frac{cosh(x)}{sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math>
*Hyperbolic cotangent: <math>\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math>
==Graphs==
{| class="wikitable"
{| class="wikitable"
|-
|-
|<!--column2-->tanh and coth
|<!--column2-->tanh and coth
|}<!--end wikitable-->
|}<!--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:Mathematics]]
[[Category:Mathematics]]