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

Question

Quizplus · Accepted Answer

First, rewrite the equation in polar form: $x^3 = -8i = 8(\cos(270^\circ) + i\sin(270^\circ))$. Taking the cube root, we find the solutions are $2(\cos(90^\circ + \frac{360^\circ k}{3}) + i\sin(90^\circ + \frac{360^\circ k}{3}))$ for $k = 0, 1, 2$, resulting in angles of $90^\circ$, $210^\circ$, and $330^\circ$.

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

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