Q:

Express the complex number in trigonometric form. -6 + 6 [tex]\sqrt{3}[/tex] i

Accepted Solution

A:
Answer:[tex]\large\boxed{-6+6\sqrt3i=12\left(\cos\dfrac{2\pi}{3}+i\sin\dfrac{2\pi}{3}\right)}[/tex]Step-by-step explanation:Look at the picture.The trigonometric form of a complex number:[tex]z=|z|(\cos\alpha+i\sin\alpha)[/tex]where:[tex]|z|=\sqrt{a^2+b^2}\\\\\cos\alpha=\dfrac{a}{|z|}\\\\\sin\alpha=\dfrac{b}{|z|}[/tex]We have the complex number:[tex]z=-6+6\sqrt3i\to a=-6,\ b=6\sqrt6[/tex]Substitute:[tex]|z|=\sqrt{(-6)^2+(6\sqrt3)^2}=\sqrt{36+108}=\sqrt{144}=12[/tex][tex]\cos\alpha=\dfrac{-6}{12}=-\dfrac{1}{2}\\\\\sin\alpha=\dfrac{6\sqrt3}{12}=\dfrac{\sqrt3}{2}[/tex]Therefore[tex]\alpha=\dfrac{2\pi}{3}[/tex]Finally:[tex]-6+6\sqrt3i=12\left(\cos\dfrac{2\pi}{3}+i\sin\dfrac{2\pi}{3}\right)[/tex]