The Fourier Transform of the Sine and Cosine Functions

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On this page, the Fourier Transforms for the sinusois sine and cosine function are determined. The result is easily obtained using the Fourier Transform of the complex exponential.

We'll look at the cosine with frequency f=A cycles/second. This cosine function can be rewritten, thanks to Euler, using the identity:

cosine function as sum of complex exponentials
[Equation 1]

Along with the linearity property of the Fourier transform, the Fourier transform can be easily found:

calculation of cosine fourier transform
[Equation 2]

The integrals from the last lines in equation [2] are easily evaluated using the results of the previous page. Equation [2] states that the fourier transform of the cosine function of frequency A is an impulse at f=A and f=-A. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A.

The Fourier Transform for the sine function can be determined just as quickly using Euler's identity for the sine function:

cosine function as sum of complex exponentials
[Equation 3]

The result is:

calculation of sinusoidal sine fourier transform
[Equation 4]

Note that the Fourier Transform of the real function, sin(t) has an imaginary Fourier Transform (no real part). This is characteristic of odd functions.


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