Relationship between e, sin and cos

Converting from e to sin/cos

It is often useful when doing signal processing to understand the relationship between e, sin and cos. Sometimes difficult calculations involving even or odd functions of $e$ can be greatly simplified by using the relationship to simplify things. The relationship is as follows:

$e^{j\theta }=cos(\theta )+j*sin(\theta ).$

Converting from sin/cos to e

The reverse conversion is also often helpful:

$cos(\theta )={\frac {e^{j\theta }+e^{-j\theta }}{2}}$

$sin(\theta )={\frac {e^{j\theta }-e^{-j\theta }}{2j}}$

We can test to see that this works as follows:

${e^{j\theta }}$	$=cos(\theta )+j*sin(\theta )$
	$={\frac {e^{j\theta }+e^{-j\theta }}{2}}+j*{\frac {e^{j\theta }-e^{-j\theta }}{2j}}$
	$={\frac {e^{j\theta }+e^{-j\theta }}{2}}+{\frac {e^{j\theta }-e^{-j\theta }}{2}}$
	$={\frac {(e^{j\theta }+e^{-j\theta })+(e^{j\theta }-e^{-j\theta })}{2}}$
	$={\frac {2*e^{j\theta }}{2}}$
$e^{j\theta }$	$=e^{j\theta }$

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