Difference between revisions of "Hyperbolic trigonometric functions"

Revision as of 13:30, December 9, 2008

The hyperbolic trigonometric functions are analogs of the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle. They are defined as:

Hyperbolic sine: $sinh(x)={\frac {e^{x}-e^{-x}}{2}}$
Hyperbolic cosine: $cosh(x)={\frac {e^{x}+e^{-x}}{2}}$
Hyperbolic tangent: $tanh(x)={\frac {sinh(x)}{cosh(x)}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$
Hyperbolic cosecant: $csch(x)={\frac {1}{sinh(x)}}={\frac {2}{e^{x}-e^{-x}}}$
Hyperbolic secant: $sech(x)={\frac {1}{cosh(x)}}={\frac {2}{e^{x}+e^{-x}}}$
Hyperbolic cotangent: $coth(x)={\frac {cosh(x)}{sinh(x)}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}$

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 *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 secant: <math>sech(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 cotangent: <math>coth(x) = \frac{cosh(x)}{sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}</math>
 [[Category:Mathematics]]