In the above table, it was remarked that the formula for the argument \(\theta\) of a complex number is not so simple.
This is because it depends on which quadrant the complex number \[
z = x+iy
\] lies.
Case 1. \(x \geq 0, y \geq 0\). Then \[
\tan \theta = \frac{y}{x} \iff \theta = \arctan \left( \frac{y}{x} \right).
\]
Case 2. \(x &lt; 0, y \geq 0\). Then \[
\theta = \pi + \arctan \left( \frac{y}{x} \right),
\] noting that the principal value of \(\arctan \left( \frac{y}{x} \right)\) is in the range \(\left( - \frac{\pi}{2},0 \right)\).
You can go ahead to write down the explicit formula for the argument \(\theta\) of a complex number for the remaining cases.

In the above table, it was remarked that the formula for the argument \(\theta\) of a complex number is not so simple.

This is because it depends on which quadrant the complex number 
\[
z = x+iy
\]
lies.

Case 1.  \(x \geq 0, y \geq 0\).

Then 
\[
\tan \theta = \frac{y}{x} \iff \theta = \arctan \left( \frac{y}{x} \right).
\]

Case 2. \(x < 0, y \geq 0\).

Then
\[
\theta = \pi + \arctan \left( \frac{y}{x} \right),
\]
noting that the principal value of \(\arctan \left( \frac{y}{x} \right)\) is in the range \(\left( - \frac{\pi}{2},0 \right)\).

You can go ahead to write down the explicit formula for the argument \(\theta\) of a complex number for the remaining cases.

Formula for argument

Formula for argument