R method

  • Thread starter mtanti
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  • #1
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Can someone tell me why Asin(ax)+Bcos(ax) always gives another sinusoidal wave?
 

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  • #2
arildno
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Because you can rewrite that expression into a sinusoidal form.
 
  • #3
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and can that answer be translated into a more mathematical reason?
 
  • #4
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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.
 
  • #5
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[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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