Fourier Series coefficient question

1. Sep 15, 2012

okami11408

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

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!