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Math Help - Polar Representation

  1. #1
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    Polar Representation

    Find the polar represenatation z= -1 + i.

    Solution: |z| = \sqrt{2} and \theta = \frac{3\pi}{4}. Thus,

    -1 + i = \sqrt{2}[ cos\frac{3\pi}{4} + i sin\frac{3\pi}{4}]


    It's been awhile since I have taken calculus, let alone any geometry. Could someone please explain to me how to find the \theta in this answer?

    A walkthrough of another problem could help me:

    Find the polar representation for: (2-i)^2

    Thanks for any help!
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  2. #2
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    Hello !
    Quote Originally Posted by shadow_2145 View Post
    Find the polar represenatation z= -1 + i.
    A complex number is :
    z=|z| e^{i \theta}=|z| (\cos \theta+i \sin \theta) \quad \quad (1)

    If z=x+iy, the modulus of z is |z|=\sqrt{x^2+y^2}
    Here, z=-1+i \implies x=-1 \text{ and } y=1 \implies |z|=\sqrt{(-1)^2+1}=\sqrt{2}

    So factor by \sqrt{2} :

    z=-1+i=\underbrace{\sqrt{2}}_{|z|} \left(\frac{-1}{\sqrt{2}}+i \frac{1}{\sqrt{2}} \right)

    According to relation (1), \cos \theta+i \sin \theta=\frac{-1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}

    When 2 complex numbers are equal, their real parts are equal and their imaginary parts are equal.

    Therefore \cos \theta=\frac{-1}{\sqrt{2}}=- ~\frac{\sqrt{2}}{2} and \sin \theta=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}

    And here, simple trigonometry can help you.

    When you get used to it, you can skip most of the steps
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  3. #3
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    I see! Easy to forget the simple things sometimes. Thanks, I appreciate it.
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