A calculus of the absurd

16.2 Properties

16.2.1 Odd/even nature

Note that as with their analogous trigonometric functions 117117 \(\sin (x)\) is analogous to \(\sinh (x)\) and \(\cos (x)\) is analogous to \(\cosh (x)\) , \(\sinh (x)\) is an odd function, and \(\cosh (x)\) is an even function.

[Proof that sinh(x) is an odd function]Proof that \(\sinh (x)\) is an odd function

\begin{align} \label {proof that sinh is odd} \sinh (-x) & = \frac {e^{(-x)} - e^{-(-x)}}{2} \\ & = \frac {e^{(-x)} - e^{-(-x)}}{2} \\ & = \frac {e^{-x} - e^{x}}{2} \\ & = -\sinh (x) \end{align}

Proof that \(\cosh (x)\) is an odd function

\begin{align} \label {Proof that cosh(x) is an odd function} \cosh (-x) & = \frac {e^{(-x)} + e^{-(-x)}}{2} \\ & = \frac {e^{-x} + e{x}}{2} \\ & = \cosh (x) \end{align}

Proof that \(\tanh (x)\) is an odd function

118118 This can be proved by rewriting \(\tanh (x)\) in terms of \(e^x\), but as we’ve already proved the even/odd nature of \(\cos (x)\) and \(\sin (x)\) it makes sense to use that!

\begin{align} \label {Proof that tanh(x) is an odd function} \tanh (-x) & = \frac {\sinh (-x)}{\cosh (-x)} \\ & = \frac {-\sinh (x)}{\cosh (x)} \\ & = -\frac {\sinh (x)}{\cosh (x)} \\ & = -\tanh (x) \end{align}