Re: Calculating four complex numbers
Hello, Peter!
Perhaps you don't understand DeMoivre's Theorem ... as it applies to roots.
I'll give you a walk-through . . .
Find the FOUR complex numbers such that: \(\displaystyle \,z^4\:=\:-8\,-\,8\sqrt{3}i\)
I have to give my answer in the form \(\displaystyle a\,+\,bi\) with \(\displaystyle a\) and \(\displaystyle b\) fully evaluated.
Do i just apply DeMoivre's Theorem?
I'm not sure how that will give me four complex numbers though
If the polar form is: \(\displaystyle \,z\:=\:r(\cos\theta\,+\,i\cdot\sin\theta)\)
\(\displaystyle \;\;\)then: \(\displaystyle \,z^{\frac{1}{k}} \;=\;r^{\frac{1}{k}}\left[\cos\left(\frac{\theta\,+\,2\pi n}{k}\right)\,+\,i\cdot\sin\left(\frac{\theta\,+\,2\pi n}{k}\right)\right]\;\) . . . for \(\displaystyle n\,=\,0,\,1,\,2,\,\cdots\,k-1\)
We have: \(\displaystyle \,z^4\;=\;16\left(\cos\frac{4\pi}{3}\,+\,i\cdot\sin\frac{4\pi}{3}\right)\) . . . and we want the 4<sup>th</sup> roots.
Then: \(\displaystyle \,z\;=\;16^{\frac{1}{4}}\left[\cos\left(\frac{\frac{4\pi}{3}\,+\,2\pi n}{4}\right)\,+\,i\cdot\sin\left(\frac{\frac{4\pi}{3}\,+\,2\pi n}{4}\right)\right]\)
Simplify: \(\displaystyle \,z\;=\;2\left[\cos\left(\frac{\pi}{3}\,+\,\frac{\pi}{2}n\right)\,+\,i\cdot\sin\left(\frac{\pi}{3}\,+\,\frac{\pi}{2}n\right)\right]\;\) . . . for \(\displaystyle n\,=\,0,\,1,\,2,\,3\)
\(\displaystyle n\,=\,0:\;\;z\:=\:2\left(\cos\frac{\pi}{3} + i\cdot\sin\frac{\pi}{3}\right) \:= \;2\left(\frac{1}{2}\,+\,i\frac{\sqrt{3}}{2}\right) \:=\;1\,+\,i\sqrt{3}\)
\(\displaystyle n\,=\,1:\;\;z\:=\:2\left(\cos\frac{5\pi}{6}\,+\,i\cdot\sin\frac{5\pi}{6}\right)\:=\:2\left(-\frac{\sqrt{3}}{2}\,+\,i\frac{1}{2}\right) \:= \:-\sqrt{3}\,+\,i\)
\(\displaystyle n\,=\,2:\;\;z\:=\:2\left(\cos\frac{4\pi}[3}\,+\,i\cdot\sin\frac{4\pi}{3}\right)\:=\:2\left(-\frac{1}{2}\,-\,i\frac{\sqrt{3}}{2}\right) \:=\: -1\,-\,i\sqrt{3}\)
\(\displaystyle n\,=\,3:\;\;z\:=\:2\left(\cos\frac{11\pi}{6}\,+\,i\cdot\sin\frac{11\pi}{6}\right)\:=\:2\left(\frac{\sqrt{3}}{2}\,-\,i\frac{1}{2}\right) \:=\:\sqrt{3}\,-\,i\)
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