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Thread: [SOLVED] Complex Numbers

  1. #1
    Member Awsom Guy's Avatar
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    [SOLVED] Complex Numbers

    Hi everybody,

    Question,

    Find the argument of z for each of the following in the interval [0,2pie]. (Give exact answers where possible)

    z=3+2i

    the answer is 0.588

    please show the working out aswell. Thanks everyone.
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  2. #2
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    $\displaystyle \arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2}
    {3}} \right)$
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  3. #3
    Member Awsom Guy's Avatar
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    umm..

    Quote Originally Posted by Plato View Post
    $\displaystyle \arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2}
    {3}} \right)$
    could you explian it further
    thanks
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  4. #4
    Flow Master
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    Quote Originally Posted by Awsom Guy View Post
    could you explian it further
    thanks
    Plot z = 3 + 2i on an Argand diagram. Simple trigonometry gives the result previously stated.
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  5. #5
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Awsom Guy View Post
    Hi everybody,

    Question,

    Find the argument of z for each of the following in the interval [0,2pie]. (Give exact answers where possible)

    z=3+2i

    the answer is 0.588

    please show the working out aswell. Thanks everyone.
    see this Complex Argument -- from Wolfram MathWorld
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  6. #6
    MHF Contributor

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    The real question appears to be: Do you know the definition of "argument" of a complex number?
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  7. #7
    Member Awsom Guy's Avatar
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    Confused

    Quote Originally Posted by Amer View Post
    thats good but could you please show me how to work this one out because I am sort of confused on this bit
    Thanks
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  8. #8
    Flow Master
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    Quote Originally Posted by Awsom Guy View Post
    thats good but could you please show me how to work this one out because I am sort of confused on this bit
    Thanks
    You've been given all the tools necessary (and I assume you have class notes and a textbook as well). It's time you started using those tools - please show your work and clearly state where you're stuck.
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  9. #9
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Awsom Guy View Post
    thats good but could you please show me how to work this one out because I am sort of confused on this bit
    Thanks
    complex numbers written in general as

    $\displaystyle z=x+iy$ right


    then the argument of z given by

    $\displaystyle \tan \theta = \frac{y}{x}$ since the argument is $\displaystyle \theta $ values so to find theta values take arctan ( $\displaystyle tan^{-1} $ ) for both sides you will have

    $\displaystyle \theta = \tan^{-1} \frac{y}{x} $ the argument of z denoted by arg(z)

    note:- argument of z have infinitely many values

    the principle argument of z denoted by Arg(z) is the unique value of $\displaystyle \theta $ such that $\displaystyle -\pi < \theta <= \pi $ as you can see the principle argument take one value

    you can say that

    $\displaystyle arg(z)=Arg(z)+2n\pi$ n integar number


    Examples:-

    Find the argument and the principle argument to the following:

    $\displaystyle 1)z=\sqrt{3}+i $ y=1 and x=sqrt{3} so

    $\displaystyle Arg(z)=\tan^{-1} \frac{1}{\sqrt{3}}$

    $\displaystyle Arg(z)=\frac{\pi}{6}$

    $\displaystyle arg(z)=\frac{\pi}{6}+2n\pi$


    $\displaystyle 2)z=1+\sqrt{3}i $

    $\displaystyle Arg(z)=\tan^{-1}\frac{\sqrt{3}}{1}$

    $\displaystyle Arg(z)=\frac{\pi}{3}$

    $\displaystyle arg(z)=\frac{\pi}{3}+2n\pi$


    $\displaystyle 3)z=-2+2i$

    $\displaystyle Arg(z)=\tan^{-1}\frac{2}{-2}$

    $\displaystyle Arg(z)=\tan^{-1}(-1)$
    the tan is negative in the second quarter and in the third quarter but the principle argument take the values in the interval $\displaystyle \left(-\pi,\pi\right]$

    or to make it easier to determine the angle write it

    $\displaystyle Arg(z)=-\tan^{-1}(1)$
    so

    $\displaystyle Arg(z)=\frac{-\pi}{4}$

    $\displaystyle arg(z)=\frac{-\pi}{4}+2n\pi$



    $\displaystyle 4)z=-5+7i $

    $\displaystyle Arg(z)=\tan^{-1} \frac{7}{-5} $ to make it easier to determine the angle

    $\displaystyle Arg(z)=-\tan^{-1} \frac{7}{5} $

    $\displaystyle Arg(z)=-54.462 $

    $\displaystyle arg(z)=-54.462+2n\pi$

    I think it is clear now
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