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Using Euler’s Formula to Derive Sine and Cosine

November 24, 2021

Introduction

Cosine and sine can be written as the sum of two complex exponentials. Euler’s formula states

(1) $\begin{equation*}e^{j\omega} = \text{cos}(\omega) + j \text{sin}(\omega).\end{equation*}$

More posts on complex numbers and complex sinusoids:

Cosine

To relate cosine add $e^{-j\omega}$ to (1),

(2) $\begin{equation*}e^{j\omega} + e^{-j\omega} = \text{cos}(\omega) + j \text{sin}(\omega) + \text{cos}(\omega) - j \text{sin}(\omega).\end{equation*}$

Simplifying the right side of (2),

(3) $\begin{equation*}e^{j\omega} + e^{-j\omega} = 2\text{cos}(\omega).\end{equation*}$

Dividing (3) by 2,

(4) $\begin{equation*}\text{cos}(\omega) = \frac{1}{2}\left(e^{j\omega} + e^{-j\omega}\right).\end{equation*}$

Sine

To relate sine subtract $e^{-j\omega}$ from (1),

(5) $\begin{equation*}e^{j\omega} - e^{-j\omega} = \text{cos}(\omega) + j \text{sin}(\omega) - \text{cos}(\omega) + j \text{sin}(\omega).\end{equation*}$