Find the indicated roots. Write the answer in polar form.
-Cube roots of $27 \left( \cos 321 ^ { \circ } + i \sin 321 ^ { \circ } \right)$
A) $3 \left( \cos 107 ^ { \circ } + i \sin 107 ^ { \circ } \right) , 3 \left( \cos 227 ^ { \circ } + i \sin 227 ^ { \circ } \right) , 3 \left( \cos 347 ^ { \circ } + i \sin 347 ^ { \circ } \right)$
B) $3 \left( \cos 197 ^ { \circ } + i \sin 197 ^ { \circ } \right) , 3 \left( \cos 257 ^ { \circ } + i \sin 257 ^ { \circ } \right) , 3 \left( \cos 137 ^ { \circ } + i \sin 137 ^ { \circ } \right)$
C) $3 \left( \cos 107 ^ { \circ } + i \sin 107 ^ { \circ } \right) , 3 \left( \cos 167 ^ { \circ } + i \sin 167 ^ { \circ } \right) , 3 \left( \cos 47 ^ { \circ } + i \sin 47 ^ { \circ } \right)$
D) $3 \left( \cos 107 ^ { \circ } + i \sin 107 ^ { \circ } \right) , 3 \left( \cos 227 ^ { \circ } + i \sin 227 ^ { \circ } \right) , 3 \left( \cos 227 ^ { \circ } + i \sin 227 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

To find the cube roots of $27(\cos 321^\circ + i\sin 321^\circ)$, we use De Moivre's theorem. The magnitude of the root is the cube root of 27, which is 3. The angles are obtained by adding $360^\circ$ to the original angle and dividing by 3. The original angle is $321^\circ$, so the angles for the roots are $107^\circ$, $227^\circ$, and $347^\circ$, corresponding to the three cube roots.

Find the Indicated Roots 27(cos⁡321∘+isin⁡321∘)27 \left( \cos 321 ^ { \circ } + i \sin 321 ^ { \circ } \right)27(cos321∘+isin321∘)

Find the Indicated Roots $27 \left( \cos 321 ^ { \circ } + i \sin 321 ^ { \circ } \right)$