Hw1 Problem 9

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Problem 9

(+) Find all solution in $\mathbb{C}$ of the given equations.

Solution
(a) $z^4 = -1$

(1)

\begin{align} |z|^4(cos4\theta +isin4\theta)=1(-1+0i)\\ |z|=1 \quad cos4\theta=-1 \quad sin4\theta=0\\ 4\theta=\pi+n2\pi\\ \theta=\frac{\pi}{4}+n\frac{\pi}{2}\\ \theta=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\\ z_1=cos\frac{\pi}{4} + isin\frac{\pi}{4}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\\ z_2=cos\frac{3\pi}{4} + isin\frac{3\pi}{4}=-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\\ z_3=cos\frac{5\pi}{4} + isin\frac{5\pi}{4}=-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\\ z_4=cos\frac{7\pi}{4} + isin\frac{7\pi}{4}=\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\\ \end{align}

(b) $z^6 = 1$

(2)

\begin{align} |z|^6(cos6\theta +isin6\theta)=1(1+0i)\\ |z|=1 \quad cos6\theta=1 \quad sin6\theta=0\\ 6\theta=0+n2\pi\\ \theta=0+n\frac{\pi}{3}\\ \theta=0, \frac{\pi}{3}, \frac{2\pi}{3},\pi, \frac{4\pi}{3}, \frac{5\pi}{3}\\ z_1=cos0 + isin0=1+0i=1\\ z_2=cos\frac{\pi}{3} + isin\frac{\pi}{3}=\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ z_3=cos\frac{2\pi}{3} + isin\frac{2\pi}{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ z_4=cos\pi + isin\pi=-1+0i=-1\\ z_5=cos\frac{4\pi}{3} + isin\frac{4\pi}{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i\\ z_6=cos\frac{5\pi}{3} + isin\frac{5\pi}{3}=\frac{1}{2}-\frac{\sqrt{3}}{2}i\\ \end{align}