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 $(r, \theta)$?
+6250 \(z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4\)
\(\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}\)
.+6250 \(z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4\)
\(\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}\)
