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

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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*}


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

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