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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