Find Powers of Complex Numbers in Polar Form
-$\left[ 10 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 3 }$
A) $- 500 \sqrt { 2 } + 500 \sqrt { 2 } \mathrm { i }$
B) $50 \sqrt { 2 } + 50 \sqrt { 2 } \mathrm { i }$
C) $15 \sqrt { 2 } + 15 \sqrt { 2 } \mathrm { i }$
D) $5 \sqrt { 2 } + 5 \sqrt { 2 } \mathrm { i }$

Question

Quizplus · Accepted Answer

Using De Moivre's theorem, $(r(\cos \theta + i\sin \theta))^n = r^n (\cos n\theta + i\sin n\theta)$. Here, $r = 10$, $\theta = \frac{3\pi}{4}$, and $n = 3$. So, $10^3(\cos \frac{9\pi}{4} + i\sin \frac{9\pi}{4}) = 1000(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) = 1000(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) = -500\sqrt{2} + 500\sqrt{2}i$.

Find Powers of Complex Numbers in Polar Form - [10(cos⁡3π4+isin⁡3π4)]3\left[ 10 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 3 }[10(cos43π​+isin43π​)]3

Find Powers of Complex Numbers in Polar Form
- $\left[ 10 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 3 }$