Hyperbolic Functions

Hyperbolic cosine is `color(red)(y=cosh(x)=(e^x+e^(-x))/2)`.hyperbolic sine and hyperbolic cotangent

Hyperbolic sine is `color(blue)(y=sinh(x)=(e^x-e^(-x))/2)`.

Hyperbolic tangent is `color(green)(y=tanh(x)=(sinh(x))/(cosh(x))=(e^x-e^(-x))/(e^x+e^(-x)))`.

Hyperbolic cotangent is `y=coth(x)=(cosh(x))/(sinh(x)=(e^x+e^(-x))/(e^x-e^(-x)))`.

Hyperbolic secant is `y=text(sech)(x)=1/(cosh(x))=2/(e^x+e^(-x))` .

Hyperbolic cosecant is `y=csch(x)=1/(sinh(x))=2/(e^x-e^(-x))`.

There is some similarity between hyperbolic functions and trigonometric.

hyperbolic cosine and hyperbolic tangentDomain of hyperbolic functions is `(-oo,oo)`, except for function `y=coth(x)` which is undefined when `x=0`.

Formulas that hold for any `x` and `y`:

  1. `cosh(x+-y)=cosh(x)cosh(y)+-sinh(x)sinh(y)`.
  2. `sinh(x+-y)=sinh(x)cosh(y)+-cosh(x)sinh(y)`.
  3. `cosh^2(x)-sinh^2(x)=1`.
  4. `cosh(2x)=cosh^2(x)+sinh^2(x)`.
  5. `sinh(2x)=2sinh(x)cosh(x)`.

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