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Orthogonality of Sin&Cos

  1. Jun 28, 2011 #1
    Could someone kindly explain whether the 90 degree phase difference between sine & cosine functions contribute to the fact that they are orthogonal? I just studied fourier series and treating sines and cosines as vectors is fine for my brain to handle, but I can't tell whether the phase difference of 90 degrees is a coincidence relative to their orthogonality in an infinite dimension space. Just trying to understand what I learn.

  2. jcsd
  3. Jun 28, 2011 #2


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    I would answer that with a no, but I'm open to a contrary view. Phase shifting doesn't explain the orthogonality of {sin(nx)} on (0,pi), for example. To really get a grip on it I think you need to look at Sturm-Liouville theory about eigenvalue problems and orthogonality of eigenfunctions. A good place to look is


    The sequence {sin(nx)} are eigenfunctions of the S-L problem

    y'' + λy = 0
    y(0)=0, y(pi)=0
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