Polar Form of Complex Numbers and Trigonometric Connections

Polar Form of Complex Numbers

Introduction to Complex Numbers

Complex numbers can be represented in both rectangular (Cartesian) and polar forms. Understanding these representations is essential for solving equations and analyzing functions in trigonometry and advanced mathematics.

• Rectangular Form: A complex number is written as z = a + bi, where a is the real part and b is the imaginary part.

• Polar Form: A complex number is expressed as z = r(cos
  2 + i
  0sin
  2), where r is the modulus (magnitude) and
  2 is the argument (angle).

Key Concepts and Formulas

• Modulus (r): The distance from the origin to the point (a, b) in the complex plane. Formula: $r = \sqrt{a^2 + b^2}$

• Argument (
  2): The angle formed with the positive real axis. Formula: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

• Conversion from Rectangular to Polar Form: $z = a + bi = r(\cos\theta + i\sin\theta)$

• Conversion from Polar to Rectangular Form: $a = r\cos\theta$ $b = r\sin\theta$

Graphical Representation

The complex number z = a + bi can be plotted as the point (a, b) in the complex plane. The modulus r is the length of the line from the origin to this point, and the argument
2 is the angle this line makes with the positive real axis.

• Example: For z = 3 + 4i:

  • $r = \sqrt{3^2 + 4^2} = 5$

  • $\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ$

  • Polar form: $z = 5(\cos 53.13^\circ + i\sin 53.13^\circ)$

Multiplication and Division in Polar Form

Multiplying and dividing complex numbers is simplified in polar form:

• Multiplication: If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then: $z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$

• Division: $\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$

Quadrants and Argument Adjustment

The value of \theta depends on the quadrant in which the complex number lies. Adjust the argument accordingly:

Quadrant	Sign of a	Sign of b	Adjustment to \theta
I	+	+	None
II	-	+	Add $180^\circ$ (or $\pi$ radians)
III	-	-	Add $180^\circ$ (or $\pi$ radians)
IV	+	-	Add $360^\circ$ (or $2\pi$ radians) if negative

Summary Table: Rectangular and Polar Forms

Form	Expression
Rectangular	$z = a + bi$
Polar	$z = r(\cos\theta + i\sin\theta)$

Additional info: The notes include handwritten calculations and quadrant analysis, reinforcing the importance of correctly determining the argument based on the signs of a and b.