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Math Help - Complex number question, involving argument and modulus

  1. #1
    LHS
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    Complex number question, involving argument and modulus



    If anyone could help me out here, in particular if you could explain what AB^2 is referring too. Thanks!
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  2. #2
    Super Member malaygoel's Avatar
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    Let z=2e^{i\theta_1}

    w=5e^{i\theta_2}

    we have now
    \frac{w}{z}=2.5e^{i(\theta_2-\theta_1)}

    hence angle between OB and OC is \theta_1

    which gives \theta_1=\frac{\pi}{4}

    Now, we have,
    AB^2=39

    I think AB here represents the distance between A and B.
    5^2+2^2-2.5.2.cos(\theta_2-\theta_1)=39

    29-20cos(\theta_2-\frac{\pi}{4})=39

    there is only one unknown, hence it can be solved.
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  3. #3
    LHS
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    Thank you for your help!
    But unfortunately your solution is outside the level of my current course (I started complex numbers and my further course last week), I do not recognise what you are doing with e. Do you know any solutions with more basic math?
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  4. #4
    MHF Contributor
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    Hi

    arg\left(\frac{w}{z}\right) = arg(w) - arg(z)

    But
    arg\left(\frac{w}{z}\right) = (e_x,OC) [2\pi]

    And
    arg(w) - arg(z) = (e_x,OB) - (e_x,OA) = (OA,OB) [2\pi]

    Therefore
    (e_x,OC) = (OA,OB) [2\pi]

    To find (OA,OB) you can use the formula ABČ = OAČ + OBČ - 2.OA.OB.cos(AOB)
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  5. #5
    LHS
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    Ah I understand a little better now, so the angle AOB is 2pi/3?
    How do you suggest finding the lines OA or OB angle with the x axis?
    Unfortunately as I said earlier, I don't really understand your lines 2-4.
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  6. #6
    MHF Contributor
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    Quote Originally Posted by LHS View Post
    Ah I understand a little better now, so the angle AOB is 2pi/3?
    Yes
    Quote Originally Posted by LHS View Post
    How do you suggest finding the lines OA or OB angle with the x axis?
    The argument of a complex number z is one measurement of the angle between the x axis (positive values) and the line connecting O to the point represented by the complex number z

    This is why arg(w) = (ex,OB)
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  7. #7
    LHS
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    Right, ok, I see, thank for you your help!
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