Argument of z

Argument of z

$OP$ makes an angle $\theta$ with the positive direction of the X-axis. $\theta$ is called the argument or amplitude of the complex number $z=a+ib$, denoted $\arg(z)$.

From the right triangle in Fig. 1.3-1.4, the relationships between the modulus $r$, the argument $\theta$, and the coordinates $a,b$ are:

so that

and, when $a\neq0$,

Example. If $z=2+2i$, then $\arg(z)=\theta=\tan^{-1}\left(\dfrac22\right)=\tan^{-1}(1)=\dfrac{\pi}{4}$.

Note on inverse tangent. If $\tan x=y$, its inverse function is written $x=\tan^{-1}y$ or $x=\arctan y$. For example: since $\tan\dfrac{\pi}{6}=\dfrac{1}{\sqrt3}$, then $\tan^{-1}\left(\dfrac{1}{\sqrt3}\right)=\dfrac{\pi}{6}$; and since $\tan\left(-\dfrac{\pi}{4}\right)=-\tan\dfrac{\pi}{4}=-1$, then $\tan^{-1}(-1)=-\dfrac{\pi}{4}$. …