Find all specified roots.
-Sixth roots of i.
A) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$

B) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$

C) $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$

D) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }$, cis $\frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$

Question

Find all specified roots.
-Sixth roots of i.
A) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$

B) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$

C) $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$

D) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }$, cis $\frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$

Quizplus · Accepted Answer

The sixth roots of $i$ can be found using the formula for the $n$th roots of a complex number, which in polar form is $r(\cos(\theta+2k\pi/n)+i\sin(\theta+2k\pi/n))$, where $k=0,1,...,n-1$. Since $i = \cos(\pi/2) + i\sin(\pi/2)$, the sixth roots are given by setting $n=6$ and $\theta=\pi/2$, resulting in roots at angles $\pi/12$, $5\pi/12$, $3\pi/4$, $7\pi/4$, $13\pi/12$, and $17\pi/12$, corresponding to the options in D.

Find All Specified Roots B) C) D) cis⁡π12,cis⁡5π12,cis⁡3π4\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }cis12π​,cis125π​,cis43π​

Find All Specified Roots B) C) D) $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }$