Hyperbolic function

Hyperbolic functions are functions similar to ordinary trigonometric functions, but based on the unit hyperbola ($x^{2}-y^{2}=1$) rather than the unit circle ($x^{2}+y^{2}=1$). The most common are hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), their reciprocals hyperbolic cosecant (csch), hyperbolic secant (sech), and hyperbolic cotangent (coth), and their inverses.

Identities

Basic identities

• $\sinh(x)={\frac {e^{x}-e^{-x}}{2}}$

• $\cosh(x)={\frac {e^{x}+e^{-x}}{2}}$

• $\tanh(x)={\frac {\sinh(x)}{\cosh(x)}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$

• ${\text{csch}}(x)={\frac {2}{e^{x}-e^{-x}}}$

• ${\text{sech}}(x)={\frac {2}{e^{x}+e^{-x}}}$

• $\coth(x)={\frac {\cosh(x)}{\sinh(x)}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}$

Relations to trig functions

• $\sinh(x)=-i\sin(xi)$

• $\cosh(x)=\cos(xi)$

• $\tanh(x)=-i\tan(xi)$

• ${\text{csch}}(x)=i\csc(xi)$

• ${\text{sech}}(x)=\sec(xi)$

• $\coth(x)=i\cot(xi)$

Pythagorean identities

• $\cosh ^{2}(x)-\sinh ^{2}(x)=1$
• $\tanh ^{2}(x)+{\text{sech}}^{2}(x)=1$
• $\coth ^{2}(x)-{\text{csch}}^{2}(x)=1$

Other

• $\sinh {\big (}\ln(\phi ){\big )}={\frac {1}{2}}$ , where ϕ represents the Golden ratio

See also

• Trigonometric functions
• Hyperbola