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Math Help - DeMoivre's theorem

  1. #1
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    DeMoivre's theorem

    I have this problem and I want to make sure I understand it.

    Use DeMoivre's theorem to write [3cis(pi/2)][squared] in the form a + bi without trigonometric functions.
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  2. #2
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    Hello, Mike!

    Use DeMoivre's theorem to write \left[3\,\text{cis}\left(\frac{\pi}{2}\right)\right]^2 in the form a + bi

    We don't really need DeMoivre's Theorem for this one, but here goes . . .

    We have: . \bigg[3\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)\bigg]^2\;=\;3^2\bigg[\cos\left(2\!\cdot\!\frac{\pi}{2}\right) + i\sin\left(2\!\cdot\!\frac{\pi}{2}\right)\bigg]

    . . . . . . = \;9\left[\cos\pi + i\sin\pi\right] \;=\;9[-1 + i(0)] \;=\;\boxed{-9}

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    We could have squared it without DeMoivre's Theorem.

    3\,\text{cis}\frac{\pi}{2}\:=\:3\left(\cos\frac{\p  i}{2} + i\sin\frac{\pi}{2}\right) \:=\:3[0 + i(1)] \:=\:3i

    . . Then: . (3i)^2\:=\:-9

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