Question

se DeMoivre's Theorem to write the complex number in trigonometric form.

(1 - i√3)^3


A) 8(cosπ + isinπ)

B) 2(cos(5Ï€/3) + isin(5Ï€/3))

C) 8(cos(5Ï€/3) + isin(5Ï€/3))

D) 2(cos(2Ï€/3) + isin(2Ï€/3))

Answer 1

let \[z=r \left( \cos \theta+\iota \sin \theta \right)=1-\iota \sqrt{3}\]
\[r \cos \theta=1,r \sin \theta=-\sqrt{3}\]
square and add
\[r=\sqrt{1^2+\left( -\sqrt{3} \right)^2}=2\]
divide
\[\tan \theta=\frac{ -\sqrt{3} }{ 1 }=-\sqrt{3}=-\tan \frac{ \pi }{ 6 }=\tan \frac{ -\pi }{ 3 }=\tan \left( 2 \pi-\frac{ \pi }{ 3 }\right)=\tan \frac{ 5 \pi }{ 3 }\]
\[\theta=\frac{ 5 \pi }{ 3 }\]
now you can complete.