Question

Use DeMoivre's Theorem to write the complex number in standard form.

(√2(cos(20°) + isin(20°))^6


A) 4√3 + 4i

B) - 4 + 4√3 i

C) 4 + 4√3 i

D) - 4√3 + 4i

Answer 1

Hello,

Answer B

(√2(cos(20°) + isin(20°))^6 =8*(cos 120°+isin120°)
=8*(-1/2+i√3/2)
=4(-1+i√3)

Answer 2

Option B is correct

`-4+ 4√(3)i`

Explanation:

Given the complex number:

`(√(2)(\cos(20^(\circ))+i\sin(20^(\circ))))^6`

Using D-Moivre's theorem:

if p is the rational; number:

`(\cos \theta +i\sin \theta)^p = \cos p\theta +i\sin\theta`

then;

Using D-Moivre's theorem:

`(√(2))^6(\cos(6 \cdot 20^(\circ))+i\sin(6 \cdot 20^(\circ)))`

`8((\cos(120^(\circ))+i\sin(120^(\circ)))`

`8(-(1)/(2)+i (√(3))/(2)) = -4+ 4√(3)i`

Therefore, the given complex in the standard form is
`-4+ 4√(3)i`