Hyperbolic functions

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:^[1]

• hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),^[2]
• hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),

Using these, you can get:^[3]

• hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
• hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),
• hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),
• hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)

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")^[4][5]
• 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 e^u.^[1][3] In the figure ${\displaystyle A=(e^{-u},e^{u}),\ B=(e^{u},\ e^{-u}),\ OA+OB=OC}$ .

Exponential definitions

sinh x is half the difference of e^x and e^−x

cosh x is the average of e^x and e^−x

• Hyperbolic sine: the odd part of the exponential function, that is, $${\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}$$
• Hyperbolic cosine: the even part of the exponential function, that is, $${\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}$$

sinh, cosh and tanh

csch, sech and coth

• Hyperbolic tangent: $${\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}.}$$
• Hyperbolic cotangent: for x ≠ 0, $${\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}.}$$
• Hyperbolic secant: $${\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}.}$$
• Hyperbolic cosecant: for x ≠ 0, $${\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.}$$

Related pages

• e (mathematical constant)
• Trigonometric functions