Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }$
A) $- 3.7588 - 1.3681 \mathrm { i }$
B) $- 15.0351 - 5.4723 \mathrm { i }$
C) $- 2.7784 + 15.7569 \mathrm { i }$
D) $- 5.4723 - 15.0351 \mathrm { i }$

Question

Quizplus · Accepted Answer

To solve $\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }$, we use De Moivre's Theorem, which states that $(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta)$. Here, $n=2$, $r=4$, and $\theta = \frac{5\pi}{9}$. Thus, we get $16 \operatorname{cis} \left(\frac{10\pi}{9}\right)$. Converting to rectangular form, $16(\cos(\frac{10\pi}{9}) + i\sin(\frac{10\pi}{9}))$, and calculating the cosine and sine values, we get approximately $-15.0351 - 5.4723i$.

Perform the Indicated Operation [4cis⁡5π9]2\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }[4cis95π​]2

Perform the Indicated Operation $\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }$