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{{Redirect|Hyperbolic curve|the geometric curve|Hyperbola}}
{{Anchor|Sinh|Cosh|Tanh|Sech|Csch|Coth}}
[[File:sinh cosh tanh.svg|
In [[mathematics]], '''hyperbolic functions''' are analogues of the ordinary [[trigonometric function]]s, but defined using the [[hyperbola]] rather than the [[circle]]. Just as the points {{math|(cos ''t'', sin ''t'')}} form a [[unit circle|circle with a unit radius]], the points {{math|(cosh ''t'', sinh ''t'')}} form the right half of the [[unit hyperbola]]. Also, similarly to how the derivatives of {{math|sin(''t'')}} and {{math|cos(''t'')}} are {{math|cos(''t'')}} and {{math|–sin(''t'')}} respectively, the derivatives of {{math|sinh(''t'')}} and {{math|cosh(''t'')}} are {{math|cosh(''t'')}} and {{math|+sinh(''t'')}} respectively.
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The following inequality is useful in statistics:
<math>\operatorname{cosh}(t) \leq e^{t^2 /2}</math> <ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation
It can be proved by comparing term by term the Taylor series of the two functions.
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