Hyperbolic trigonometric functions

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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}}
Sinh-cosh.png Tanh-coth.png
sinh and cosh tanh and coth
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