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Why do we use sine and cosine in fourier transform together? [closed]

I think it is possible to use only cosine function, but why the formula is used with sine? I am trying to understand but don't know what to do after Fourier transform with imaginary part and real part.

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You use a pair of sines and cosines to encode the phase of the signal. Alternatively, you can also write them as a complex exponential, and then the phase of the wave gets encoded into the arg of the coefficient. – Mikola Jul 6 2010 at 5:08
"We" don't. I always use a Fourier transform of the form $$\int_{-\infty}^\infty f(x)e^{-2\pi itx}\,dx.$$ – Robin Chapman Jul 6 2010 at 5:46
For instance, if you use only cosines, then what is your Fourier series for $x\mapsto \sin(2\pi x)$ going to be? – Yemon Choi Jul 6 2010 at 5:55
@Robin Chapman - I mean the same thing – maximus Jul 6 2010 at 6:07
My current feeling is that this question would be better suited to one of the forums/websites mentioned in the FAQ mathoverflow.net/faq – Yemon Choi Jul 6 2010 at 6:07
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closed as not a real question by Steve Huntsman, Robin Chapman, Yemon Choi, Harald Hanche-Olsen, Andrey RekaloJul 6 2010 at 16:13

I'm not sure which Fourier transform you mean, but I have only seen Fourier series written where the phase, e.g. $e^{2\pi i n x}$ for some integer n, is decomposed into its real and imaginary parts $cos(2\pi n x) +i sin(2\pi n x)$. Since sine and cosine are related to each other by translation, $sin (\pi/2 - x) = cos (x)$ and similarly $cos (\pi/2 - x) = sin (x)$ (this is a classic trig identity and is an instance of the sum angle formula where one of the angles is $\pi/2$ and one is -x). You can rewrite the above expression in terms of only sines or only cosines. The choice of sine or cosine is up to the author and purely a matter of preference. There is no "best way" to write the Fourier series.

Writing Fourier series in terms of sines and cosines seems to be antiquated within mathematics; I don't know if it is still popular in physics or engineering. Most modern books on Fourier series (e.g. any of Stein's books on Fourier analysis) write formulas in terms of the complex exponentials above. This is in part because the Fourier transform on the real line is written similarly, all the necessary information is encoded in the exponential like orthogonality relations and most likely because it's easier to write.

As far as what to do "after Fourier transform", I don't know what you mean. Could you please explain?

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Thank you for the answer. By "after Fourier transform" I mean actually After getting fourier coefficients, so it means after calculating the formula I will get some coeffitients of cosine and sine separately. – maximus Jul 6 2010 at 6:06
Using translation to write everything in terms of cos or sin brings no advantage and, as you say, makes working with orthogonality relations much harder than it needs to be. At the big picture level, one really needs to be using complex exponentials in order to see various duality phenomena, and using sin plus certain translates of it just obscures things. – Yemon Choi Jul 6 2010 at 6:06

Fourier Transforms require imaginary and real parts for the reasons discussed in comments. However, there are transforms that only have real parts such as the Discrete Cosine Transform.

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