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Homework Help: Adding Sinusoids of differing Magnitute & Phases

  1. Jul 7, 2009 #1
    1. The problem statement, all variables and given/known data
    Find A and [tex]\theta[/tex] given that:
    [tex]Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)[/tex]
    Could someone elaborate on how to solve this. I mean it looks to me that one simply takes the magnitude of the coefficient and the inverse tangent of the same coefficients. But I feel there needs to be justification as to why we're allowed to do this.

    2. Relevant equations
    not sure.

    3. The attempt at a solution
    But the solution (according to my notes) is: [tex]A = \sqrt{4^2 + 3^2} = 5[/tex] and [tex]\theta^{-1}[/tex] = [tex]\frac{4}{3} = 53.1[/tex]


    Last edited: Jul 7, 2009
  2. jcsd
  3. Jul 7, 2009 #2
    What if instead of

    Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)

    I wanted to write it as,

    Asin(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)

    would the values of A and [tex]\theta[/tex] change at all?
  4. Jul 7, 2009 #3

    Oh, actually I think I know. The x coordinate axis is [tex]Acos(\omega t)[/tex] (negative for the negative x-axis), and the y coordinate axis is [tex]Asin(\omega t)[/tex] (negative for the negative y-axis). So this means the left side value in the equality is the actual vector, and the terms on the right are the x and y components.


  5. Jul 7, 2009 #4


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    Homework Helper

    You need addition and subtraction formulas for sine and cosine (or you can use Euler's identity, but that is more advanced).
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