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: