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Thread: Complex numbers

  1. #1
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    Complex numbers

    Any help on these two would be appreciated:

    1) Express the complex number -2[sqrt)2+2[sqrt]2i in the trigonometric form r(cos[theta]+isin[theta]) or rcis(theta).

    2) Express the complex number 12|cos(pi/3)+isin(pi/3)| in the form a+bi without trigonometric functions.
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  2. #2
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    Hello, Mike!

    I connect the two forms of the numbers with this diagram.
    Code:
            |                 *
            |              *  |
            |       r   *     |
            |        *        |y
            |     *           |
            |  * θ            |
          - * - - - - - - - - * -
                     x
    1) Express the complex number $\displaystyle \text{-}2\sqrt{2} +2\sqrt{2}i$ in the form $\displaystyle r(\cos\theta + i\sin\theta)$

    Going from $\displaystyle x + iy$ to $\displaystyle r(\cos\theta + i\sin\theta)$, use: .$\displaystyle \begin{Bmatrix}r^2 & = & x^2+y^2\\ \tan\theta & = & \frac{y}{x}\end{Bmatrix}$


    We are given: .$\displaystyle x \,=\,\text{-}2\sqrt{2},\;y \,=\,2\sqrt{2}$

    We have: .$\displaystyle r^2 \:=\:(\text{-}2\sqrt{2})^2 + (2\sqrt{2})^2\:=\:8 + 8 \:=\:16\quad\Rightarrow\quad r \,=\,\pm4$

    . . And: .$\displaystyle \tan\theta \:=\:\frac{2\sqrt{2}}{\text{-}2\sqrt{2}}\:=\:-1\quad\Rightarrow\quad\theta \:=\:\frac{3\pi}{4},\:\text{-}\frac{\pi}{4}$

    We know the point in in Quadrant 2, so we'll use $\displaystyle \theta = \frac{3\pi}{4},\;r = 4$

    . . Answer: .$\displaystyle 4\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)$



    2) Express the complex number $\displaystyle 12\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right)$ in the form $\displaystyle a+bi$

    Going from $\displaystyle r(\cos\theta + i\sin\theta)$ to $\displaystyle x + iy$, use: .$\displaystyle \begin{Bmatrix}x \:=\:r\cos\theta \\ y \:=\:r\sin\theta\end{Bmatrix}$


    We are given: .$\displaystyle r \,=\,12,\;\theta\,=\,\frac{\pi}{3}$

    We have: .$\displaystyle \begin{array}{cc}x \:=\:12\cos\frac{\pi}{3}\:=\:12\left(\frac{1}{2}\r ight) \:=\:6 \\
    y \:=\:12\sin\frac{\pi}{3} \:=\:12\left(\frac{\sqrt{3}}{2}\right)\:=\:6\sqrt{ 3} \end{array}$

    . . Answer: .$\displaystyle 6 + 6\sqrt{3}i$

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  3. #3
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    Thanks, that should help me with the other ones I'm working on. Thanks a lot.
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