How did this page come up with the Fourier coefficients? It basically jump after coming up with this $$A_n=\sqrt{a^2_n+b^2_n} , \quad\phi_n=\arctan\left(\frac{a_n}{b_n}\right).$$
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Are you asking how they derived it in terms of the integral? – Piotr Benedysiuk Jul 03 '16 at 19:53
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Wikipedia is not really a textbook, it doesn't have to prove everything, just provide the proper sources for any claim. I suggest you look at a textbook (calculus textbook or a more advanced textbook about Fourier series) – Yuriy S Jul 03 '16 at 22:01
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As it stands, the question is not good. If you just want a proof, then get rid of the condition that it should be the one supposedly used on the Wikipedia page. – Yuriy S Jul 03 '16 at 22:03