Complex numbers' arguments. Obtuse or Acute?

Question

Are the arguments of complex numbers always acute or can there exist obtuse arguments? Or is only the acute version taken every time an argument exceeds $\frac{\pi}{2} \textbf{rad}$ ?

Answer 1

Recall that by definition the argument of a complex number is the angle between the positive real axis and the line joining the point to the origin. The principal value of $\operatorname{Arg}(z)$ is usually defined in the interval $(-\pi,\pi]$ and the set of all arguments of $z$ is $\arg(z) = \{\operatorname{Arg}(z) + 2\pi n\;|\; n \in \mathbb Z\}$.

What are the arguments of $z=-1$ or of $z=-\frac{\sqrt 2}2+i\frac{\sqrt 2}2$?

Maybe your doubt arises from the fact that for $x\neq 0$

• $\operatorname{Arg}(z)=\arctan{\frac y x}\in (-\pi/2,\pi/2)$

but recall that that expression is valid only for $x>0$ and we need to add $\pm \pi$ to obtain $\operatorname{Arg}(z)$or use the function $\arctan2(y,x)$.