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Euler's Theorem Converting from Trignometric to Exponential Form

  1. Aug 27, 2008 #1
    r(cos u + i sin u)


    t(cos v + i sin v)

    How do I convert these into exponential form using Euler's Theorem?
     
  2. jcsd
  3. Aug 27, 2008 #2

    tiny-tim

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    Welcome to PF!

    Hi jaylwood! Welcome to PF! :smile:

    cos u + i sin u = eiu :smile:

    (I don't understand why you're not recognising that? :confused:)
     
  4. Aug 27, 2008 #3
    okay here is the problem i have. Given x = r(cos u + i sin u) and y = t(cos v + i sin v)
    Prove that the amplitude of (xy) is the sum of their amplitudes. I don't understand where to go with it.
     
  5. Aug 27, 2008 #4

    tiny-tim

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    ah … so that's the problem!

    ok … x = r eiu, y = t eiv

    so multiply them, and you get xy = … ? :smile:
     
  6. Aug 27, 2008 #5
    rt eiu eiv What do i do to simplify that? Or reconvert it back to trignometric form?
     
  7. Aug 27, 2008 #6

    tiny-tim

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    oh come on …

    eiu eiv = … ? :smile:
     
  8. Aug 27, 2008 #7
    ei(u+v)
     
  9. Aug 28, 2008 #8

    tiny-tim

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    amplitude …

    (just got up … :zzz:)

    That's right! :smile:

    So the amplitude of xy is … ?
     
  10. Aug 28, 2008 #9
    u+v ? but what happens to the rt?
     
    Last edited: Aug 28, 2008
  11. Aug 28, 2008 #10

    tiny-tim

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    Yes! :smile:

    (it's that easy :biggrin:)

    Any other problems?
     
  12. Aug 28, 2008 #11
    what happens to the rt?
     
  13. Aug 28, 2008 #12

    tiny-tim

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    They're just ordinary numbers.

    Treat them as usual …

    xy = rt ei(u+v) :smile:
     
  14. Aug 28, 2008 #13
    So what would be my final answer?
     
  15. Aug 28, 2008 #14

    tiny-tim

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    Well, the question was …
    … so the answer is that the amplitude of their sum is u + v, which is the sum of their amplitudes! :smile:

    (which is why you didn't need to bother with x and t at the end :wink:)
     
  16. Aug 28, 2008 #15
    Thank you so much.
     
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