Euler’s Formulas

Euler’s formula is our bridge back-and forth between trigonomentric forms (\(\cos\theta\) and \(\sin\theta\)) and complex exponential form (\(e^{j\theta}\)): $$\begin{align} \label{eq:eulers_formula} e^{j\theta} &= \cos\theta + j\sin\theta. \end{align}$$ Here are a few useful identities implied by Euler’s formula. $$\begin{subequations} \label{eq:eulers_formulas} \begin{align} e^{-j\theta} &= \cos\theta - j\sin\theta \label{eq:eulers_formulas_1} \\ \cos\theta &= \Re{(e^{j\theta})} \label{eq:eulers_formulas_2} \\ &= \frac{1}{2}\left(e^{j\theta}+e^{-j\theta}\right) \label{eq:eulers_formulas_3} \\ \sin\theta &= \Im{(e^{j\theta})} \label{eq:eulers_formulas_4} \\ &= \frac{1}{j 2}\left(e^{j\theta}-e^{-j\theta}\right). \label{eq:eulers_formulas_5} \end{align} \end{subequations}$$