Relationship between e, sin and cos

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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 }