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The equation \[z^5 = i\]has $5$ solutions. The unique solution in the third quadrant is $re^{i\theta}$, where $r > 0$ and $0 \leq \theta < 2\pi$. What is the ordered pair $(r, \theta)$?​

 Jan 17, 2019

Best Answer 

 #1
avatar+3976 
+2

\(z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4\)

 

\(\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}\)

.
 Jan 17, 2019
 #1
avatar+3976 
+2
Best Answer

\(z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4\)

 

\(\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}\)

Rom Jan 17, 2019

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