Find Roots of Complex Numbers in Polar Form
-The complex cube roots of $64 \left( \cos 150 ^ { \circ } + \mathrm { i } \sin 150 ^ { \circ } \right)$ (polar form)
A) $4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , 4 \left( \cos 170 ^ { \circ } + i \sin 170 ^ { \circ } \right) , 4 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right)$
B) $4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , 4 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 4 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right)$
C) $- 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , 4 \left( \cos 170 ^ { \circ } + i \sin 170 ^ { \circ } \right) , - 4 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right)$
D) $- 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , 4 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , - 4 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

The roots of a complex number in polar form can be found using De Moivre's theorem. For a complex number $z = r(\cos \theta + i\sin \theta)$ and its nth root, the formula is $z^{1/n} = r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$, where $k = 0, 1, 2, ..., n-1$. For the cube roots ($n=3$) of $64(\cos 150^\circ + i\sin 150^\circ)$, we first find the magnitude of the root, which is $64^{1/3} = 4$, and then apply the formula for $k = 0, 1, 2$ to find the angles. The angles are $50^\circ$, $170^\circ$, and $290^\circ$ respectively, leading to the roots given in option A.

Find Roots of Complex Numbers in Polar Form -The Complex 64(cos⁡150∘+isin⁡150∘)64 \left( \cos 150 ^ { \circ } + \mathrm { i } \sin 150 ^ { \circ } \right)64(cos150∘+isin150∘)

Find Roots of Complex Numbers in Polar Form
-The Complex $64 \left( \cos 150 ^ { \circ } + \mathrm { i } \sin 150 ^ { \circ } \right)$