I'm learning about the Fourier Series and have a few question. First, we have 2 methods to find the Fourier coefficient of the Fourier Series. 1. Use the formula Cn below. note that: x=2∏ft 2. Expand the function. For Example I have y(t)=3cos(2∏9t)+4cos(2∏8t), let use the Expansion method y(t)=3cos(2∏9t)+4cos(2∏8t) <---- Already expand we have y(j2∏9t)=3 at f=9 Hz Fourier coefficient=3/2 y(j2∏8t)=4 at f=8 Hz Fourier coefficient=4/2 Note:I devided by 2 for two side analysis. This method makes sense to me. but when it comes to use of formula Cn, it becomes nonsense. note that: x=2∏ft My question is how come multiply the whole function with sine & cosine at some particular frequency and integrate it. it turns into the Fourier coefficient. I mean why is that actually work? It's an area, how can that related to the coefficient. I want some intuition. Thank you!