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DSP for Beginners: Simple Explanations for Complex Numbers! The second edition includes a new chapter on complex sinusoids.

Using Euler’s Formula to Derive Sine and Cosine
November 24, 2021

Table of Contents


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:


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*}


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*}

Simplifying the right side of (5),

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

Dividing (6) by 2j,

(7)   \begin{equation*}\text{sin}(\omega) = \frac{1}{2j}\left(e^{j\omega} - e^{-j\omega}\right).\end{equation*}


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