Hyperbolic Functions

Hyperbolic cosine is $$${\color{red}{{{y}={\cosh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}}}}$$$.hyperbolic sine and hyperbolic cotangent

Hyperbolic sine is $$${\color{blue}{{{y}={\sinh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}}}}$$$.

Hyperbolic tangent is $$${\color{green}{{{y}={\tanh{{\left({x}\right)}}}=\frac{{{\sinh{{\left({x}\right)}}}}}{{{\cosh{{\left({x}\right)}}}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}}}}$$$.

Hyperbolic cotangent is $$$y=\coth\left(x\right)=\frac{\cosh\left(x\right)}{\sinh\left(x\right)}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$$.

Hyperbolic secant is $$${y}=\operatorname{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}=\frac{{2}}{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}$$$ .

Hyperbolic cosecant is $$${y}={\operatorname{csch}{{\left({x}\right)}}}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}}=\frac{{2}}{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}$$$.

There is some similarity between hyperbolic functions and trigonometric.

hyperbolic cosine and hyperbolic tangentDomain of hyperbolic functions is $$${\left(-\infty,\infty\right)}$$$, except for function $$${y}={\coth{{\left({x}\right)}}}$$$ which is undefined when $$${x}={0}$$$.

Formulas that hold for any $$${x}$$$ and $$${y}$$$:

  1. $$${\cosh{{\left({x}\pm{y}\right)}}}={\cosh{{\left({x}\right)}}}{\cosh{{\left({y}\right)}}}\pm{\sinh{{\left({x}\right)}}}{\sinh{{\left({y}\right)}}}$$$.
  2. $$${\sinh{{\left({x}\pm{y}\right)}}}={\sinh{{\left({x}\right)}}}{\cosh{{\left({y}\right)}}}\pm{\cosh{{\left({x}\right)}}}{\sinh{{\left({y}\right)}}}$$$.
  3. $$${{\cosh}}^{{2}}{\left({x}\right)}-{{\sinh}}^{{2}}{\left({x}\right)}={1}$$$.
  4. $$${\cosh{{\left({2}{x}\right)}}}={{\cosh}}^{{2}}{\left({x}\right)}+{{\sinh}}^{{2}}{\left({x}\right)}$$$.
  5. $$${\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$$$.

This formulas can be easily proved using definitions of hyperbolic functions.