:PROPERTIES: :ID: 007fa3a4-0821-46f9-8e68-ea079320b036 :mtime: 20231026011312 :ctime: 20231026011307 :END: #+title: hyperbolic trigonometric functions #+filetags: :public:project: * Definitions Hyperbolic Functions ** Sinh \[sinh(x) = \frac{e^{x}-e^{-x}}{2}\] ** Cosh \[cosh(x) = \frac{e^{x}+e^{-x}}{2}\] ** Tanh \[tanh(x) = \frac{sinh(x)}{cosh(x)}\] ** Sech \[sech(x) = \frac{1}{cosh(x)}\] ** Cosech \[Cosec(x) = \frac{1}{sinh(x)}\] ** Coth \[coth(x) = \frac{1}{tanh(x)}\] * Relations between hyperbolic functions and trigonometric functions \[cos(ix) = cosh(x)\] \[sin(ix) = i sinh(x)\] \[tan(ix) = i tanh(x)\] Hyperbolic functions are related to [[id:ab249b72-fa90-4826-b9e4-c1e7d967ac82][complex trig functions]] * Properties of Hyperbolic Functions - $cosh$ is an even function - $sinh$ is an odd function - $tanh$ is an odd function * Algebraic Identities of hyperbolic functions \[(cosh(x))^{2} - (sinh(x))^{2} = 1\] \[cosh(A + B) = cosh(A) cosh(B) + sinh(A) sinh(B)\] In general, to convert a circular identity to a trigonometric identity, replace $cos$ with $cosh$, replace $sin$ with $sinh$, and if there is a product of 2 $sinh$, flip the sign. \[cosh(0) = 1\] \[sinh(0) = 0\] * Plots of hyperbolic functions #+BEGIN_SRC sage plot(cosh(x), (x , -1, 1)) #+END_SRC * Restrictions on hyperbolic functions \[1 \le cosh(x)\] \[-1 < tanh(x) < 1\] * Inverse Hyperbolic Functions \[arcsinh(x) = \log(x + \sqrt{x^{2} + 1})\] \[arccosh(x) = \pm \log(x + \sqrt{x^{2} - 1})\] Note that $arccosh$ has two answers because $cosh$ is even, i.e. $cosh(-x) = cosh(x)$.