A calculus of the absurd

16.2 Properties

16.2.1 Odd/even nature

Note that as with their analogous trigonometric functions 100100 \(\sin (x)\) 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

101101 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}