7.2b Euler's Theorem

Euler's Formula

Euler’s formula states that for a phase angle $\theta$,
$e^{j\theta }=\cos \theta +j\sin \theta .$
This theorem is used to convert complex exponentials to rectangular form.

Properties

(Sine Representation) $\sin \theta =\frac{e^{j\theta }-e^{-j\theta }}{2j}$.

We start by expressing $e^{j\theta }=\cos \theta +j\sin \theta$. Then,

$$\begin{array}{rl}{e}^{j\theta }-e^{-j\theta }& =\cos (\theta )+j\sin (\theta )-\cos (-\theta )-j\sin (-\theta )\\ & =\cos (\theta )+j\sin (\theta )-\cos (\theta )+j\sin (\theta )\\ & =2j\sin \theta .\end{array}$$

Therefore, $\sin \theta =\frac{e^{j\theta }+e^{-j\theta }}{2i}$.

(Cosine Representation) $\cos \theta =\frac{e^{j\theta }+e^{-j\theta }}{2}$.

We start by expressing $e^{j\theta }=\cos \theta +j\sin \theta$. Then,

$$\begin{array}{rl}{e}^{j\theta }+e^{-j\theta }& =\cos (\theta )+j\sin (\theta )+\cos (-\theta )+j\sin (-\theta )\\ & =\cos (\theta )+j\sin (\theta )+\cos (\theta )-j\sin (\theta )\\ & =2\cos \theta .\end{array}$$

Therefore, $\cos \theta =\frac{e^{j\theta }+e^{-j\theta }}{2}$.