Question

Express the complex number in trigonometric form.

-2+2rad3 i

Answer 1

That's

Answer 2

\(z=a+bi=r\exp(i\theta)\)
where
\[\begin{cases}a=r\cos\theta\\b=r\sin\theta\end{cases}\]
\[\begin{cases}-2=r\cos\theta\\2\sqrt3=r\sin\theta\end{cases}\]
\(r^2=(-2)^2+(2\sqrt3)^2~~\Rightarrow~~r=4\)

Finding \(\theta\) shouldn't be too demanding.

Answer 3

find the inverse tangent of imaginary/real

Answer 4

I always know how to find r, but I can't ever find theta for some reason.

Answer 5

divide imag/real to get - sqr(3)
find atan(-sqr(3))
if you plot -2, 2sqr(3) you see you are in the 2nd quadrant.

Answer 6

\[-2=4\cos\theta~~\Rightarrow~~\cos\theta=-\frac{1}{2}~~\Rightarrow~~\theta=\frac{2\pi}{3}\text{ or }\frac{4\pi}{3}\]
To check which one, use the sine equation:
\[2\sqrt3=4\sin\theta~~\Rightarrow~~\sin\theta=\frac{\sqrt3}{2}~~\Rightarrow~~\theta=\frac{\pi}{3}\text{ or }\frac{2\pi}{3}\]

Answer 7

Ah alright, thanks