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X=r(cos u+i sin u) and y=t(cos v + i sin v)

  1. Jan 25, 2007 #1
    I need help getting this one started... PLEASE...

    Given x=r(cos U + i sin u) and y =t(cos v + i sin v):
    Prove tha tthe modulus of (xy) is the product of their moduli and that the amplitude of (xy) is the sum of their amplitudes.
  2. jcsd
  3. Jan 25, 2007 #2
    I dont really know the answer to this question out-right, though it seems like that they are taking x,y in the complex field to a polar coordinate mapping. When you multiply two elements in this way the length's are a product and the angles add together which can be verified by straight multiplication of the variables xy.

    hince: xy = rt(cos(u+v) + isin(u+v))

    Graphically this looks like x/y lengths multiplied with their angles added together.

    That should give you a good insight into what the modulus/length and the amplitude/direction should be.


    Also: when proving the xy = "" portion I posted, it may be helpful to look up trig identities for cos(x+y) and sin(x+y) as it will allow you to make needed substitutions :P.
  4. Jan 26, 2007 #3
    Maybe use Euler's formlua?
  5. Jan 28, 2007 #4
    The modulus of a complex number is given by multiplying the number by its complex conjugate and taking the square root. So:

    [tex]|x|^2 = x\overline{x}[/tex]
    [tex]|y|^2 = y\overline{y}[/tex]

    [tex]|xy|^2 = xy\overline{xy}=x\overline{x}y\overline{y}=|x|^2|y|^2[/tex]

    This works because complex multiplication is commutative(order doesn't matter). Now just take the square root of both sides and you're done.
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