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Math Help - Number systems

  1. #1
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    Number systems

    Not sure if this is the right place to post this but could someone help get me started and point me in the right direction.

    Given:
    x = r(cos u + i sin u) and y = t(cos v + i sin v)


    Prove that the modulus of (xy) is the product of their moduli. Show each step of your proof.

    Prove that the amplitude of (xy) is the sum of their amplitudes. Show each step of your proof.


    If you know of any place I could visit to understand this better I would be grateful.


    Thanks
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  2. #2
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    Hi

    x = r(cos u + i sin u) and y = t(cos v + i sin v)
    xy = rt(cos u cos v - sin u sin v + i(sin u cos v + sin v cos u)
    xy = rt(cos(u+v) + i sin(u+v))
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    x = r(cos u + i sin u) and y = t(cos v + i sin v)
    For the next line I got
    xy = rt * (cos u cos v + i cos u sin v + i sin u cos v + i^2 sin u sin v) I understand that i * i = -1 but why is - sin u sin v now after cos u cos v? in your second line? I know I must be missing a rule thats causes it, so could you please mind me.
    xy = rt(cos u cos v - sin u sin v + i(sin u cos v + sin v cos u)
    xy = rt(cos(u+v) + i sin(u+v))
    Thanks for the help just got back a break for holidays so thats why I haven't responded before.
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  4. #4
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    Anyone?
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  5. #5
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    Trigonometry

    Hello Tally
    Originally Posted by running-gag
    Hi

    x = r(cos u + i sin u) and y = t(cos v + i sin v)
    For the next line I got
    xy = rt * (cos u cos v + i cos u sin v + i sin u cos v + i^2 sin u sin v) I understand that i * i = -1 but why is - sin u sin v now after cos u cos v? in your second line? I know I must be missing a rule thats causes it, so could you please mind me.
    xy = rt(cos u cos v - sin u sin v + i(sin u cos v + sin v cos u)
    xy = rt(cos(u+v) + i sin(u+v))
    I'm not quite sure what your problem is here, because you have all the steps. As you said, i^2 = -1, so i^2\sin u \sin v = - \sin u \sin v, which is real. So, simply collect together the reals and the imaginaries:

    rt(\cos u \cos v + i\cos u \sin v + i\sin u \cos v - \sin u \sin v)

     = rt(\cos u \cos v - \sin u \sin v + i(\cos u \sin v + \sin u \cos v))

    just as you would collect together 'like terms' in any other algebraic expression.

    Then you can use the addition formulae to get the result you want.

    Grandad
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  6. #6
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    Thank you Grandad; it was the grouping of the real and imaginary numbers that completely slipped my mind.

    Thank you for jogging my memory.
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