Find all cube roots of the complex number. Leave answers in trigonometric form.
-125
A) $5 \operatorname { cis } 45 ^ { \circ } , 5 \operatorname { cis } 165 ^ { \circ } , 5 \operatorname { cis } 325 ^ { \circ }$
B) $5 \operatorname { cis } 90 ^ { \circ } , 5 \operatorname { cis } 210 ^ { \circ } , 5 \operatorname { cis } 330 ^ { \circ }$
C) $5 \operatorname { cis } 0 ^ { \circ } , 5 \operatorname { cis } 120 ^ { \circ } , 5 \operatorname { cis } 240 ^ { \circ }$
D) $5 \operatorname { cis } 0 ^ { \circ } , 5 \operatorname { cis } 60 ^ { \circ } , 5 \operatorname { cis } 120 ^ { \circ }$

Question

Quizplus · Accepted Answer

The complex number 125 can be written as $125 \operatorname{cis} 0^\circ$ because it is a real number with magnitude 125 and angle 0 degrees. To find its cube roots, we take the cube root of the magnitude and divide the angle by 3. The cube root of 125 is 5, and dividing 0 degrees by 3 gives 0 degrees. The other cube roots are found by adding $360^\circ/3 = 120^\circ$ successively. Thus, the cube roots are $5 \operatorname{cis} 0^\circ$, $5 \operatorname{cis} 120^\circ$, and $5 \operatorname{cis} 240^\circ$.

Find All Cube Roots of the Complex Number 5cis⁡45∘,5cis⁡165∘,5cis⁡325∘5 \operatorname { cis } 45 ^ { \circ } , 5 \operatorname { cis } 165 ^ { \circ } , 5 \operatorname { cis } 325 ^ { \circ }5cis45∘,5cis165∘,5cis325∘

Find All Cube Roots of the Complex Number $5 \operatorname { cis } 45 ^ { \circ } , 5 \operatorname { cis } 165 ^ { \circ } , 5 \operatorname { cis } 325 ^ { \circ }$