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Fourier analysis, or one of the main applications for Fourier analysis, is to analyze something that's complicated by way of sines and cosines. So say that you were analyzing a sound coming out of a French horn and the horn's sound waveform is a spiky-looking thing, you can use Fourier analysis to approximate it using sines and cosines and it will allow you to analyze the different components, such as the frequencies, in the particular sound that's being reproduced.
So without a concrete example of this, the concept is hard to understand. If you're learning this in math class and they just show you the formula, which is really disgusting-- usually the formula, you look at it and say, I have no idea what this applies to. I don't know why I'm learning it, and I really don't care. And you have to do proofs based on this formula. You have to analyze and pick apart this formula. You have no idea what it does. So all these people in my math class were sitting here looking at this strange weird formula and having no idea what it did. And I had just taken a physics class ... a class on music in computers that had shown me exactly how to use this formula. So I went in my math class, you know, I aced the entire section on the test because I had a concrete example in my head. I knew what I was looking at, you know, because I compared the formula to something that was concrete. It was beautiful. I was like, oh, yeah, I completely understand this. And people would look at me like I was crazy.
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