Argument

Argument:

The value of the angle $\alpha$ receives the name of argument

For a given complex number, the argument supports an infinite set of values, which differ from each other in $2\cdot{k}\cdot{\pi}; k\in{\mathbb{Z}}$

It's called principal value of the argument the one that meets $0\leq\alpha\leq{2}\cdot{\pi}$

It can be calculated by:

$\alpha=Arg(z)=atan2(b, a)=\begin{cases} \arctan(\frac{b}{a}) & \text{if }a > 0 \\ \arctan(\frac{b}{a}) + \pi & \text{if }b \geq 0, a < 0 \\ \arctan(\frac{b}{a}) - \pi & \text{if }b < 0, a < 0 \\ \frac{\pi}{2} & \text{if }b > 0, a = 0 \\ \frac{-\pi}{2} & \text{if }b < 0, a = 0 \\ \text{Undefined} & \text{if }b = 0, a = 0 \end{cases}$

This result is obtained in radians and sometimes it will be useful to convert it to degrees:

$\alpha=\frac{atan2(b, a)\cdot{360}}{2\cdot\pi}$

You can also use the following table that expresses trigonometric reasons:

	$>rad$	$>\sin \alpha$	$>\cos \alpha$	$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$
$>0^{\circ}$	$>0$	$>0$	$>1$	$>0$
$>30^{\circ}$	$>\frac{\pi}{6}$	$>\frac{1}{2}$	$>\frac{\sqrt{3}}{2}$	$>\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}$
$>45^{\circ}$	$>\frac{\pi}{4}$	$>\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$	$>\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$	$>1$
$>60^{\circ}$	$>\frac{\pi}{3}$	$>\frac{\sqrt{3}}{2}$	$>\frac{1}{2}$	$>\sqrt{3}$
$>90^{\circ}$	$>\frac{\pi}{2}$	$>1$	$>0$	$>\text{Undefined}$
$>180^{\circ}$	$>\pi$	$>0$	$>-1$	$>0$
$>270^{\circ}$	$>\frac{3\cdot\pi}{2}$	$>-1$	$>0$	$>\text{Undefined}$

Example of argument

$\begin{cases}z=(-3)+4\cdot{i} \\ \alpha=atan2(4, -3)=2.2143\text{ radians}\end{cases}$

$\alpha=\frac{atan2(4, -3)\cdot{360}}{2\cdot\pi}=\frac{2.2143\cdot{360}}{2\cdot\pi}=\frac{797.148}{2\cdot\pi}=126.8701^{\circ}\approx\frac{2\cdot\pi}{3}$

So we have to:

$\alpha\approx\frac{2\cdot\pi}{3}$