The principal angles of the complex arguments are in the range $-\pi \le x\le \pi $.
If so why does the comlex number
$\omega =\frac{\left(-1-\sqrt{3}i\right)}{2}$ is represented by
where $\frac{4\pi }{3}$ is greater than $\pi $, instead of representing it as
Different textbooks and different authors will not follow the exact same conventions. Some authors will choose their angles in the range $0 \le x < 2\pi$, others in the range $-\pi < x \le \pi$. There might be other possibilities too.
The reason for lack of consistency is that the choice of convention for the principal angle is not mathematically very important. What is much more important is simply to recognize that the complex exponential function is periodic, with period $2 \pi i$, and therefore the collection of all Euler forms for $\omega$ can be written as $$\omega = \exp\biggl(\frac{4}{3}\pi i + 2 \pi i \, n\biggr) = \cos\biggl(\frac{4}{3}\pi i + 2 \pi i \, n\biggr) + i \sin\biggl(\frac{4}{3}\pi i + 2 \pi i \, n\biggr) $$ It can also be written as $$\omega = \exp\biggl(-\frac{2}{3}\pi i + 2 \pi i \, n\biggr) = \cos\biggl(-\frac{2}{3}\pi i + 2 \pi i \, n\biggr) + i \sin\biggl(-\frac{2}{3}\pi i + 2 \pi i \, n\biggr) $$ There is no difference of mathematical validity between these two forms. If there is a difference in popularity, perhaps it is because some people avoid minus signs and others prefer smaller numbers.
