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Math Help - Equation and parameter

  1. #1
    Super Member dhiab's Avatar
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    Equation and parameter

    Solve in C :
    t is real parameter
    Find the modulus and argument of this solutions .
    Last edited by dhiab; July 31st 2009 at 11:42 PM.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Hint: let z=r ( cos \theta+i sin \theta)
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  3. #3
    MHF Contributor red_dog's Avatar
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    \Delta=b^2-4ac=4-\frac{4}{\cos^2t}=\frac{4(\cos^2t)}{\cos^2t}=-4\tan^2t

    z_{1,2}=\frac{2\pm 2i\tan t}{2}=1\pm i\tan t

    z_1=\frac{1}{\cos t}\left(\cos t+i\sin t\right)

    z_2=\frac{1}{\cos t}\left(\cos t-i\sin t\right)=\frac{1}{\cos t}\left(\cos(2\pi-t)+i\sin(2\pi-t)\right)

    If t\in\left[\left.0,\frac{\pi}{2}\right)\right.\Rightarrow|z_1  |=|z_2|=\frac{1}{\cos t}

    If t\in\left(\left.\frac{\pi}{2},\pi\right]\right.\Rightarrow|z_1|=|z_2|=-\frac{1}{\cos t}

    \arg z_1=t, \ \arg z_2=2\pi-t
    Last edited by red_dog; August 1st 2009 at 12:04 AM.
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  4. #4
    Super Member dhiab's Avatar
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    Quote Originally Posted by red_dog View Post
    \Delta=b^2-4ac=4-\frac{4}{\cos^2t}=\frac{4(\cos^2t)}{\cos^2t}=-4\tan^2t

    z_{1,2}=\frac{2\pm 2i\tan t}{2}=1\pm i\tan t

    z_1=\frac{1}{\cos t}\left(\cos t+i\sin t\right)

    z_2=\frac{1}{\cos t}\left(\cos t-i\sin t\right)=\frac{1}{\cos t}\left(\cos(2\pi-t)+i\sin(2\pi-t)\right)

    If t\in\left[\left.0,\frac{\pi}{2}\right)\right.\Rightarrow|z_1  |=|z_2|=\frac{1}{\cos t}

    If t\in\left(\left.0,\frac{\pi}{2}\right]\right.\Rightarrow|z_1|=|z_2|=-\frac{1}{\cos t}

    \arg z_1=t, \ \arg z_2=2\pi-t
    Hello THANK YOU but I'have this remak :

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  5. #5
    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by dhiab View Post
    Hello THANK YOU but I'have this remak :

    In my country \arg z is called the reduced argument and \arg z\in[0,2\pi]

    Arg (z) means all the arguments and Arg (z)=\arg z+2k\pi

    In this case 0\leq t\leq \pi, \ t\neq\frac{\pi}{2}

    -\pi\leq-t\leq 0 and -t is not the reduced argument. Then we have to reduce it by adding some periods of sine and cosine. It is sufficiently to add one period. So \arg z=2\pi-t
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