Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 3 } + 8 i = 0$$
A) $\left\{ 2 \operatorname { cis } 270 ^ { \circ } , 2 \operatorname { cis } 210 ^ { \circ } , 2 \right.$ cis $\left. 150 ^ { \circ } \right\}$
B) $\left\{ 2 \operatorname { cis } 90 ^ { \circ } , 2 \operatorname { cis } 150 ^ { \circ } , 2 \operatorname { cis } 30 ^ { \circ } \right\}$
C) $\left\{ 2 \operatorname { cis } 90 ^ { \circ } , 2 \operatorname { cis } 210 ^ { \circ } , 2 \operatorname { cis } 330 ^ { \circ } \right\}$
D) $\left\{ 2 \operatorname { cis } 270 ^ { \circ } , 2 \operatorname { cis } 30 ^ { \circ } , 2 \operatorname { cis } 150 ^ { \circ } \right\}$

Question

Quizplus · Accepted Answer

First, rewrite the equation as $x^3 = -8i$, which can be expressed in polar form as $8(\cos(270^\circ) + i\sin(270^\circ))$, since $-i$ corresponds to $270^\circ$ on the unit circle. The cube root of $8$ is $2$, and the angles are found by dividing $270^\circ$ by $3$ and adding $120^\circ$ for each subsequent root (due to the $360^\circ/3$ division for a cube root). This results in angles of $90^\circ$, $210^\circ$, and $330^\circ$, all with a radius of $2$.

Find All Solutions of the Equation x3+8i=0x ^ { 3 } + 8 i = 0x3+8i=0

Find All Solutions of the Equation $x ^ { 3 } + 8 i = 0$