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Thread: Polar Representation

  1. #1
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    Polar Representation

    Find the polar represenatation z= -1 + i.

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

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


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

    A walkthrough of another problem could help me:

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

    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 :
    $\displaystyle z=|z| e^{i \theta}=|z| (\cos \theta+i \sin \theta) \quad \quad (1)$

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

    So factor by $\displaystyle \sqrt{2}$ :

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

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

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

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

    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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