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Reduce a trig.function

  1. Jan 28, 2010 #1
    "Reduce" a trig.function

    1. The problem statement, all variables and given/known data

    Z(t) = the real part of :
    (-5.8 + 2.2i)exp(iwt)

    1. Reform it into: Acos(wt + ø)
    2. Then reform it into Bsin(wt)+Dcos(wt)




    2. Relevant equations



    3. The attempt at a solution

    I found the 2nd step to be much easier as I just have to use eulers formula to remove the exponential then just multiply it with the complex number that is before it and take the real part of it, leaving me with Z(t) = -5.8cos(wt) -2.2sin(wt)

    The first step should be equally easy but for some odd reason (my lack of knowledge in trig) I get an angle that is wrong.

    Acos(wt +ø ) = Asin(wt)cos(ø) + Acos(wt)sin(ø) =>

    -5.8cos(wt) - 2.2sin(wt) = Asin(wt)cos(ø) + Acos(wt)sin(ø) =>
    (cause cos and sin are lin.independant?)

    -5.8cos(wt) = Acos(wt)sin(ø) => A = -5.8/sin(ø)
    -Same logic- =>A = -2.2/cos(ø) =>

    -2.2/cos(ø) = -5.8/sin(ø) or: -2.2tan(ø) = -5.8
    => ø = 1.21
    Which means A = -5.8/sin(1.21) = -6.2
    A = -2.2/cos(1.21) = +.35

    ??

    That is the inconsistency and I do not know why, I am quite sure everything in the math is correct :S
     
  2. jcsd
  3. Jan 28, 2010 #2
    Re: "Reduce" a trig.function

    just split it up into -5.8exp(iwt) +2.2exp(iwt).
    exp(iwt) = cos(wt) + isin(wt).
    the real part of this formula is the cos(wt)
    so you will have -5.8cos(wt) + 2.2cos(wt)
    that gives -3.6cos(wt), and your answer I believe
     
  4. Jan 28, 2010 #3
    Re: "Reduce" a trig.function

    But

    (-5.8 + 2.2i)exp(iwt) = (-5.8 + 2.2i)(cos(wt) + isin(wt)
    = -5.8cos(wt) -5.8isin(wt) +2.2icos(wt) + 2.2(i*i)sin(wt)
    = (-5.8cos(wt) -2.2sin(wt)) + i(2.2cos(wt) -5.8sin(wt))

    Where the first part is the real part, leaving me with both a sine and cosine and unfortunantly not a sum of cosine ;(
     
  5. Jan 28, 2010 #4
    Re: "Reduce" a trig.function

    gah i did make a mistake. the two imaginary terms multilpy and become a real term.
     
  6. Jan 28, 2010 #5
    Re: "Reduce" a trig.function

    so what we have here now is -5.8cos(wt) -2.2sin(wt) and your question is how can you put that in the form of Acos(wt + phi)??
     
  7. Jan 28, 2010 #6
    Re: "Reduce" a trig.function

    sine and cosine are out of phase by pi/2, so you could technically just add pi/2 or subtract p/2 and you would have either both sines or both cosines. But then you would have a phase on one and no phase on the other, making them unable to be summed. so i dont know, check out some trig identities
     
  8. Jan 28, 2010 #7
    Re: "Reduce" a trig.function

    Actually plotting the function lets me "see" by how much it is out of phase so numerically I can approximate the Cos(wt + ø) but analytically? (There is an answer, so says the book)

    How do you find the amplitude of 2 signals being added? -5.8cos(wt) -2.2sin(wt)

    Still not able to find the phase.
     
  9. Jan 28, 2010 #8
    Re: "Reduce" a trig.function

    Consider the following:
    [tex]
    \cos(x - \theta) = \cos x \cos\theta + \sin x \sin\theta
    [/tex]

    Therefore, let's multiple by a constant R.
    [tex]\begin{array}{1}
    R \cos(x - \theta) = R\cos(x)\cos(\theta) + R\sin(x)sin(\theta) \\
    R\cos(x - \theta) = (R\cos(\theta))\cos x + (R\sin(\theta))\sin x \\
    R\cos(x - \theta) = a\cos x + b\sin x
    [/tex]
    Where
    [tex]
    a = R\cos\theta \ , \ b = R\sin\theta
    [/tex]
    Obviously, [tex]R = \sqrt{a^2 + b^2}[/tex]

    So, you have a and b. Solve for theta.
     
  10. Jan 28, 2010 #9

    Dick

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    Homework Helper

    Re: "Reduce" a trig.function

    You started out with the right general idea. But cos(wt+phi)=cos(wt)*cos(phi)-sin(wt)*sin(phi). It looks like you used the sin addition formula instead of cos. And your inconsistency at the end is just a mistake. -5.8/sin(1.21)=(-6.20) and -2.2/cos(1.21)=(-6.23). NOT +.35. The reason the two values of 'A' are slightly different is that you rounded off the angle to 1.21.
     
    Last edited: Jan 28, 2010
  11. Jan 28, 2010 #10
    Re: "Reduce" a trig.function

    Ah yes thank you it all makes sense, I used that formula for the sum of an angle because the book insists that the following are equivalent, which does not agree with the formula for the cosine of sum of angles.

    (1) Z(t) = Acos(wt +ø)
    (2) Z(t) = Asin(wt)cos(ø) + Acos(wt)sin(ø)
    (3) Z(t) = Bsin(wt) + Dcos(wt)
     
  12. Jan 28, 2010 #11

    Dick

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    Homework Helper

    Re: "Reduce" a trig.function

    (1) is only the same as (2) if you pick different phi's for each one.
     
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