A calculus of the absurd

16.3 Relationship to trig functions

Using Euler’s formula, it is possible to write both \(\cos (\theta )\) and \(\sin (\theta )\) in terms of \(e^{x}\). As \(e^{i\theta } = \cos (\theta ) + i\sin (\theta )\), and \(e^{-ix} = \cos (-\theta ) + i\sin (-\theta ) = \cos (\theta ) - i\sin (\theta )\) we can either add or subtract these two quantities in order to write both trigonometric functions in terms of \(e\).

For \(\cos (x)\), we can add \(e^{ix}\) and \(e^{-ix}\).

\begin{align} e^{ix} + e^{-ix} & = \cos (\theta ) - i\sin (\theta ) + \cos (\theta ) + i\sin (\theta ) \\ & = 2\cos (\theta ) \end{align}

Thus we can say that

\begin{equation} \cos (x) = \frac {e^{ix} + e^{-ix}}{2} \end{equation}

for all values of x. 122122 Which looks remarkably like a hyperbolic function!.

We can do a similar thing for \(\sin (x)\).

\begin{align} e^{ix} - e^{-ix} & = \cos (\theta ) - i\sin (\theta ) - (\cos (\theta ) + i\sin (\theta )) \\ & = -2i\sin (\theta ) \end{align}

Which means that

\begin{equation} \sin (x) = \frac {e^{ix} - e^{-ix}}{-2i} \end{equation}

If, in either of these equations we set \(x=i\theta \) we can write \(\sin (i\theta )\) and \(\cos (i\theta )\) in terms of \(\sinh (x)\) and \(\cosh (x)\), respectively.

\begin{align} \sin (i\theta ) & = \frac {e^{i(i\theta )} - e^{-i(i\theta )}}{-2i} \\ & = \frac {e^{-\theta } - e^{\theta }}{-2i} \\ & = i\frac {e^\theta - e^{-\theta }}{2} \text { multiplying by $\frac {i}{i}=1$} \\ & = i\sinh (\theta ) \end{align}

\begin{align} \cos (i\theta ) & = \frac {e^{i(i\theta )} + e^{-i(i\theta )}}{2} \\ & = \frac {e^{-\theta } + e^{\theta }}{2} \\ & = \cosh (\theta ) \end{align}