Find all cube roots of the complex number. Leave answers in trigonometric form.
--8
A)2 cis 30°, 2 cis 150°, 2 cis 270°
B)2 cis 180°, 2 cis 0°
C)2 cis 60°, 2 cis 180°, 2 cis 300°
D)2 cis 120°, 2 cis 180°, 2 cis 300°

Question

Quizplus · Accepted Answer

The complex number -8 can be represented in polar form as $8(\cos 180^\circ + i\sin 180^\circ)$. To find its cube roots, we use the formula for the nth roots of a complex number, $z^{1/n} = r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$, where $n=3$ for cube roots, $r=8$, and $\theta=180^\circ$. Calculating for $k=0, 1, 2$ gives the roots: $2(\cos 60^\circ + i\sin 60^\circ)$, $2(\cos 180^\circ + i\sin 180^\circ)$, and $2(\cos 300^\circ + i\sin 300^\circ)$.

Find All Cube Roots of the Complex Number