Question

Express the complex number in trigonometric form.

-3 + 3 square root of 3i

Answer 1

\[put r \cos theta=-3, r \sin \theta=3\sqrt{3}\]
square and add,
\[r ^{2}\left( \cos ^{2}\theta+\sin ^{2}\theta\right)=9+27=36\]
\[r ^{2}=36,r=6,\]
dividing,
\[\tan \theta=\frac{ 3\sqrt{3} }{ -3}=-\sqrt{3}=-\tan \frac{ \pi }{ 3 }=\tan \left( \pi-\frac{ \pi }{3 } \right)\]
\[\tan \theta=\tan \frac{ 2\pi }{ 3 }\]
\[\theta=\frac{ 2\pi }{3 }\]
\[-3+3\sqrt{3}\iota=6\left( \cos \frac{ 2\pi }{ 3 }+\iota \sin \frac{ 2\pi }{3 } \right)\]