Question

need help with de moivre's theorum? fan+medal

Answer 1

Use DeMoivre's Theorem to write the complex number in trigonometric form.
\[(1-i \sqrt{3})^{3}\]

Answer 2

\[z=x+iy=r\exp(i\theta)\]
\[r=\sqrt{x^2+y^2}~~\Rightarrow~~r=\sqrt{1^2+(-\sqrt3)^2}=2\\
\theta=\tan^{-1}\frac{y}{x}~~\Rightarrow~~\theta=\tan^{-1}\frac{-\sqrt3}{1}=-\frac{\pi}{3}\]
So,
\[1-i\sqrt3=2\exp\left(-i\frac{\pi}{3}\right)\]
By DeMoivre's Theorem,
\[(1-i\sqrt3)^2=\left(2\exp\left(-i\frac{\pi}{3}\right)\right)^2=4\exp\left(-i\frac{2\pi}{3}\right)=4\left(\cos\frac{2\pi}{3}-i\sin\frac{2\pi}{3}\right)\]

Answer 3

thanks :)