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Is there a trigonometric identity for this ?

  1. Nov 24, 2011 #1
    Hi,

    I am trying to figure out what the result is when adding two sinusoids of the same frequency but with different phase and amplitudes. Specifically I want to know if the result is always another sinusoid of the same frequency. For the case of the the same amplitude I have:

    cos(wt) + cos(wt+phi) = 2.cos(wt+phi/2).cos(phi/2) ... this follows from the trig. identity:

    cos(A)+cos(B) = 2.cos[(A+B)/2].cos[(A-B)/2]

    So the addition of two sinusoids of the same frequency and amplitude will always give a sinusoid of the same frequency. What about the more general case:
    cos(wt) + k.cos(wt+phi) where k is a constant. Will the result here also always be a sinusoid of frequency w ? Intuitively I think so, but I havn't been able to derive an expression for the phase and amplitude.
    Does anyone know ?

    Thanks.
     
  2. jcsd
  3. Nov 24, 2011 #2
    cos(wt) + k.cos(wt+phi)=-k sin(phi) sin(wt)+(k cos(phi)+1)cos(wt)
     
  4. Nov 24, 2011 #3

    AlephZero

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    Science Advisor
    Homework Helper

    ... and if you want, you can write any expression of the form
    A sin(wt) + B cos(wt)
    in the form
    sqrt(A^2 + B^2)cos(wt + alpha)
    where
    tan alpha = A/B
     
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