Hyperbolic functions are different versions of trigonometric functions . They are defined using a hyperbola instead of a circle . Also, as the derivatives of sin(t ) and cos(t ) are cos(t ) and –sin(t ) , the derivatives of sinh(t ) and cosh(t ) are cosh(t ) and sinh(t ) .
The basic hyperbolic functions are:
hyperbolic sine "sinh " (),hyperbolic cosine "cosh " (),Using these, you can get:
hyperbolic tangent "tanh " (),hyperbolic cotangent "coth " (),hyperbolic secant "sech " (),hyperbolic cosecant "csch " or "cosech " ()and these have similar logic to the normal trigonometric functions.
The inverse hyperbolic functions are:
inverse hyperbolic sine "arsinh " (also "sinh−1 ", "asinh " or sometimes "arcsinh ")inverse hyperbolic cosine "arcosh " (also "cosh−1 ", "acosh " or sometimes "arccosh ")inverse hyperbolic tangent "artanh " (also "tanh−1 ", "atanh " or sometimes "arctanh ")inverse hyperbolic cotangent "arcoth " (also "coth−1 ", "acoth " or sometimes "arccoth ")inverse hyperbolic secant "arsech " (also "sech−1 ", "asech " or sometimes "arcsech ")inverse hyperbolic cosecant "arcsch " (also "arcosech ", "csch−1 ", "cosech−1 ","acsch ", "acosech ", or sometimes "arccsch " or "arccosech ")Definitions
With hyperbolic angle u , the hyperbolic functions sinh and cosh can be defined using the exponential function eu . In the figure
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{\displaystyle A=(e^{-u},e^{u}),\ B=(e^{u},\ e^{-u}),\ OA+OB=OC}
Exponential definitions
sinh x is half the difference of ex e −x cosh x is the average of ex e −x Hyperbolic sine: the odd part of the exponential function, that is,
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{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}
Hyperbolic cosine: the even part of the exponential function, that is,
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{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}
sinh , cosh and tanh csch , sech and coth Hyperbolic tangent:
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{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}.}
Hyperbolic cotangent: for x ≠ 0
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{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}.}
Hyperbolic secant:
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{\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}.}
Hyperbolic cosecant: for x ≠ 0
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{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.}
