Question

Represent the complex numbers in polar form
$-3\sqrt{3}+3i$

Answer 1

Given

$z=-3\sqrt{3}+3i\{x+iy\}$

we have every complex no. can be written in polar form

$\Rightarrow z=r(\cos +\theta +i\sin \theta );$ where $\theta =anglez$

where $r=\sqrt{x^2+y^2}$

$\therefore r=\sqrt{(-3\sqrt{3}{)}^2+(3{)}^3}=\sqrt{27+9}=6$

Now we know that

$r\cos \theta =-3\sqrt{3}\Rightarrow 6\cos \theta =-3\sqrt{3}$

$r\sin \theta =3\Rightarrow 6\sin \theta =3$

$\Rightarrow \cos \theta \frac{-\sqrt{3}}{2};\sin \theta =\frac{1}{2};\theta =\pi /6$

$\therefore \theta$ lies in second quadrant

$\therefore \arg \left(z\right)=\pi -\theta =\pi \frac{-\pi }{6}=\frac{5\pi }{6}$

Polar form of $z=6(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6})$