Theoretical Background
Let \( Z \) be a complex number given in standard form by:
\[ Z = a + bi \]The modulus \( |Z| \) of the complex number \( Z \) is given by:
\[ |Z| = \sqrt{a^2 + b^2} \]And the argument of the complex number \( Z \) is the angle \( \theta \) in standard position given by:
\[ \tan(\theta) = \dfrac{b}{a} \]
Note:
Since the trigonometric equation has an infinite number of solutions (due to the periodicity of the tangent function), two major conventions are adopted for the range of \( \theta \):
- Convention (1): Defines the argument \( \theta \) in the range \( 0 \le \theta < 2\pi \).
- Convention (2): Defines the argument \( \theta \) in the range \( (-\pi, +\pi] \).
The four quadrants, as defined in trigonometry, are determined by the signs of \( a \) and \( b \). If the terminal side of \( Z \) lies in quadrant I or II, both conventions give the same value of \( \theta \). If it lies in quadrant III or IV, convention 1 gives a positive angle and convention 2 gives a negative angle.