Find all cube roots of the complex number. Leave answers in trigonometric form.
-$- 27$
A) $3 \operatorname { cis } 30 ^ { \circ } , 3 \operatorname { cis } 150 ^ { \circ } , 3 \operatorname { cis } 270 ^ { \circ }$
B) $3 \operatorname { cis } 60 ^ { \circ } , 3$ cis $180 ^ { \circ } , 3$ cis $300 ^ { \circ }$
C) $3 \operatorname { cis } 180 ^ { \circ } , 3 \operatorname { cis } 0 ^ { \circ }$
D) $3 \operatorname { cis } 120 ^ { \circ } , 3 \operatorname { cis } 180 ^ { \circ } , 3 \operatorname { cis } 300 ^ { \circ }$

Question

Quizplus · Accepted Answer

The complex number $-27$ can be written in trigonometric form as $27(\cos 180^\circ + i\sin 180^\circ)$ because it lies on the negative real axis. To find the cube roots, we divide the angle by 3 and find the cube root of the radius. The radius is $27$, whose cube root is $3$, and dividing $180^\circ$ by $3$ gives $60^\circ$. The cube roots of a complex number are equally spaced around the circle, so we add $120^\circ$ to $60^\circ$ to find the other roots: $60^\circ + 120^\circ = 180^\circ$ and $180^\circ + 120^\circ = 300^\circ$. Thus, the cube roots are $3 \operatorname{cis} 60^\circ$, $3 \operatorname{cis} 180^\circ$, and $3 \operatorname{cis} 300^\circ$.

Find All Cube Roots of the Complex Number −27- 27−27 A) 3cis⁡30∘,3cis⁡150∘,3cis⁡270∘3 \operatorname { cis } 30 ^ { \circ } , 3 \operatorname { cis } 150 ^ { \circ } , 3 \operatorname { cis } 270 ^ { \circ }3cis30∘,3cis150∘,3cis270∘

Find All Cube Roots of the Complex Number $- 27$ A) $3 \operatorname { cis } 30 ^ { \circ } , 3 \operatorname { cis } 150 ^ { \circ } , 3 \operatorname { cis } 270 ^ { \circ }$