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R method

  1. Dec 6, 2006 #1
    Can someone tell me why Asin(ax)+Bcos(ax) always gives another sinusoidal wave?
     
  2. jcsd
  3. Dec 6, 2006 #2

    arildno

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    Because you can rewrite that expression into a sinusoidal form.
     
  4. Dec 7, 2006 #3
    and can that answer be translated into a more mathematical reason?
     
  5. Dec 7, 2006 #4
    Try it out...
    put A = C*cos(d) B = C*sin(d) where C and d are arbitrary constants. Of course you can put sin in the place of cos and vice-versa, and still get a sinusoidal wave.
     
  6. Dec 7, 2006 #5
    [tex]A\sin ax + B \cos ax = R \sin (ax+b)[/tex]

    Expanding the right hand side gives you [tex]R\sin ax \cos b + R \cos ax \sin b[/tex]

    This gives [tex]R \cos b = A[/tex] and [tex]R \sin b = B[/tex], therefore [tex]b = \tan^{-1}\frac{B}{A}[/tex].

    A similar method gives you R in terms of A and B, thus turning your sum of trig functions into another, single, trig function. :)
     
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