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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

Respuesta :

caylus
Hello,

Answer B

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

Answer:

Option B is correct

[tex] -4+ 4\sqrt{3}i[/tex]

Step-by-step explanation:

Given the complex number:  

[tex](\sqrt{2}(\cos(20^{\circ})+i\sin(20^{\circ})))^6[/tex]

Using D-Moivre's theorem:

if p is the rational; number:

[tex](\cos \theta +i\sin \theta)^p = \cos p\theta +i\sin\theta[/tex]

then;

Using D-Moivre's theorem:

[tex](\sqrt{2})^6(\cos(6 \cdot 20^{\circ})+i\sin(6 \cdot 20^{\circ}))[/tex]

[tex]8((\cos(120^{\circ})+i\sin(120^{\circ}))[/tex]

[tex]8(-\frac{1}{2}+i \frac{\sqrt{3}}{2}) = -4+ 4\sqrt{3}i[/tex]

Therefore, the given complex in the standard form is [tex] -4+ 4\sqrt{3}i[/tex]