Wave Walker DSP

DSP Algorithms for RF Systems

Most Popular

Buy the Book!

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

bookCover2Ed
examples1
examples2
eulersFormula
radians
negativeFrequency
adding
posNegAngles
complexSinusoidPhase
Using Euler’s Formula to Derive Sine and Cosine
November 24, 2021

Table of Contents

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*}

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*}

Conclusion

More posts on complex numbers and complex sinusoids:

Leave a Reply

God, the Lord, is my strength; he makes my feet like the deer’s; he makes me tread on my high places. Habakkuk 3:19

This website participates in the Amazon Associates program. As an Amazon Associate I earn from qualifying purchases.

© 2021-2023 Wave Walker DSP