Find all specified roots.
-Fifth roots of $1 .$
A) $1 , \operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , \operatorname { cis } \frac { 8 \pi } { 5 }$
B) $- 1 , \operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , \operatorname { cis } \frac { 8 \pi } { 5 }$
C) $1 , \operatorname { cis } \frac { \pi } { 5 } , \operatorname { cis } \frac { 3 \pi } { 5 } , \operatorname { cis } \frac { 5 \pi } { 5 } , \operatorname { cis } \frac { 7 \pi } { 5 }$
D) $1 , \operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , - 1$

Question

Quizplus · Accepted Answer

The fifth roots of 1 can be found using De Moivre's theorem, which states that the nth roots of a complex number $z = r(\cos \theta + i\sin \theta)$ are given by $r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$ for $k = 0, 1, ..., n-1$. For $1 = \cos(0) + i\sin(0)$, the fifth roots are $\cos(\frac{2k\pi}{5}) + i\sin(\frac{2k\pi}{5})$, where $k = 0, 1, 2, 3, 4$, which matches the options in choice A.

Find All Specified Roots 1.1 .1. A) B) C) D) 1,cis⁡2π5,cis⁡4π5,cis⁡6π5,−11 , \operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , - 11,cis52π​,cis54π​,cis56π​,−1

Find All Specified Roots $1 .$ A) B) C) D) $1 , \operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , - 1$