Write the expression in the standard form a + bi.
-$\left[ 2 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) \right] ^ { 3 }$
A) $3 + 3 i$
B) $4 \sqrt { 2 } + 4 \sqrt { 2 } \mathrm { i }$
C) $3 \sqrt { 2 } + 3 \sqrt { 2 } \mathrm { i }$
D) $4 + 4 \mathrm { i }$

Question

Quizplus · Accepted Answer

This expression can be simplified using De Moivre's theorem, which states that $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i\sin(n\theta)$ for any real number $n$. Applying this theorem, we get $\cos(45^\circ) + i\sin(45^\circ)$, which simplifies to $\sqrt{2}/2 + i\sqrt{2}/2$. Multiplying by 8 (since $2^3 = 8$) gives $4\sqrt{2} + 4\sqrt{2}i$.

Write the Expression in the Standard Form a + Bi [2(cos⁡15∘+isin⁡15∘)]3\left[ 2 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) \right] ^ { 3 }[2(cos15∘+isin15∘)]3

Write the Expression in the Standard Form a + Bi $\left[ 2 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right) \right] ^ { 3 }$