\[
32\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)
\]
Next, using DeMoivre's find the 5 th root of
\[
\begin{array}{c}
16+16 \sqrt{3} i= \\
32\left(\cos 60^{\circ}+i \sin 6.0^{\circ}\right)
\end{array}
\]
Find \( r \) and \( \theta \) where \( \theta \) is in the third quadrant.
\[
\begin{array}{c}
r^{5}=32 \\
5 \theta=60^{\circ}, 420^{\circ}, 780^{\circ}, 1140^{\circ}, 1500^{\circ} \\
r=2 \quad \theta=[?] \circ
\end{array}
\]
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Answer

\( \theta = 240^{\circ} \)

Steps

Step 1 ：\( r = 2 \)

Step 2 ：\( 5\theta = 60^{\circ}, 420^{\circ}, 780^{\circ}, 1140^{\circ}, 1500^{\circ} \)

Step 3 ：\( \theta = 240^{\circ} \)