A calculus of the absurd

14.7.2 Writing complex numbers in terms of the exponential function

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. ¹⁰⁸¹⁰⁸ 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}