Topic 3.13: Trigonometry and Polar Coordinates - Complex Numbers

Fundamentals of Complex Numbers

  • Definition and Standard Form: Previously, complex numbers have been written using the form a+bia + bi. In this expression, aa and bb are defined as constants.
  • The Complex Plane: A complex number can be understood as a specific point within the complex plane. This point is determined by its corresponding rectangular or polar coordinates.
  • Rectangular Representation: When a complex number possesses rectangular coordinates (a,b)(a, b), it is expressed in the form a+bia + bi.     * Direct Example: The complex number 3+2i3 + 2i is represented as the point (3,2)(3, 2) in the rectangular coordinate system.     * Visual Mapping: Points on the plane transition from the origin OO to specific locations defined by real and imaginary axes. For instance, the point (3,2)(3, 2) is located 33 units along the real axis and 22 units along the imaginary axis.

Conversion Identities and Polar Representation

  • Coordinate Identities: Foundational identities used to bridge rectangular and polar systems include:     * x=r×cos(θ)x = r \times \text{cos}(\theta)     * y=r×sin(θ)y = r \times \text{sin}(\theta)
  • Polar Form of a Complex Number: Using these identities, a complex number can be expressed in the polar coordinate system. When a complex number has polar coordinates (r,θ)(r, \theta), it is expressed as:     * (r×cos(θ))+i(r×sin(θ))(r \times \text{cos}(\theta)) + i(r \times \text{sin}(\theta))

Analytical Examples of Coordinate Conversion

  • Example 6: Rectangular to Polar Conversion:     * Scenario: A complex number is represented by a point in the complex plane with rectangular coordinates (−2,−2)(-2, -2).     * Task: Identify one way to express the complex number using its polar coordinates (r,θ)(r, \theta).     * Calculation of Magnitude (rr): Using the Pythagorean relationship: (−2)2+(−2)2=8(-2)^2 + (-2)^2 = 8, therefore r = \text{\sqrt{8}}.     * Calculation of Angle (θ\theta): Using the tangent identity: tan(θ)=−2−2=1\text{tan}(\theta) = \frac{-2}{-2} = 1. Since the point (−2,−2)(-2, -2) is located in Quadrant III (Q3Q3), the reference angle must be adjusted. The angle is determined to be \frac{5\text{\pi}}{4}.     * Final Polar Expression: \text{\sqrt{8}} \times \text{cos}(\frac{5\text{\pi}}{4}) + i(\text{\sqrt{8}} \times \text{sin}(\frac{5\text{\pi}}{4})).     * Verification: Evaluating the expression yields \text{\sqrt{8}} \times (-\frac{\text{\sqrt{2}}}{2}) + i(\text{\sqrt{8}} \times (-\frac{\text{\sqrt{2}}}{2})) = -2 - 2i.

  • Example 7: Polar to Rectangular Conversion:     * Scenario: A complex number is expressed in polar coordinates as 4 \times \text{cos}(\frac{7\text{\pi}}{6}) + i(4 \times \text{sin}(\frac{7\text{\pi}}{6})).     * Task: Express the complex number using its rectangular coordinates (x,y)(x, y).     * Calculation of xx: x = 4 \times \text{cos}(\frac{7\text{\pi}}{6}) = 4 \times (-\frac{\text{\sqrt{3}}}{2}) = -2\text{\sqrt{3}}.     * Calculation of yy: y = 4 \times \text{sin}(\frac{7\text{\pi}}{6}) = 4 \times (-\frac{1}{2}) = -2.     * Final Rectangular Coordinates: (-2\text{\sqrt{3}}, -2).

Worksheet A: Topic 3.13 Problem Sets

  • Problem 17: Converting Rectangular (2, -2) to Polar:     * Given: Rectangular coordinates (2,−2)(2, -2).     * Radius Calculation: r = \text{\sqrt{(2)^2 + (-2)^2}} = \text{\sqrt{8}} = 2\text{\sqrt{2}}.     * Quadrant Identification: The point is in Quadrant IV (Q4Q4).     * Correct Expression: Based on the quadrant and radius, the expression is (2\text{\sqrt{2}} \times \text{cos}(\frac{7\text{\pi}}{4})) + i(2\text{\sqrt{2}} \times \text{sin}(\frac{7\text{\pi}}{4})). This corresponds to Option (C) in the worksheet notation: (2\text{\sqrt{2}} \times \text{cos}(\frac{7\text{\pi}}{4})) + i(2\text{\sqrt{2}} \times \text{sin}(\frac{7\text{\pi}}{4})) (noting the transcript includes a potential sign variation in options, option C specifically uses the angle \frac{7\text{\pi}}{4}).

  • Problem 18: Unit Circle Application:     * Given: Rectangular coordinates (-\frac{\text{\sqrt{3}}}{2}, \frac{1}{2}).     * Radius Identification: Since these coordinates reside on the Unit Circle, r=1r = 1.     * Quadrant Identification: The point is in Quadrant II (Q2Q2).     * Polar Expression: The angle corresponding to this point is \frac{5\text{\pi}}{6}. The expression is (\text{cos}(\frac{5\text{\pi}}{6})) + i(\text{sin}(\frac{5\text{\pi}}{6})), which corresponds to Option (A).

  • Problem 19: Polar to Rectangular in Q4:     * Given: Polar expression 2 \times \text{cos}(-\frac{\text{\pi}}{3}) + i(2 \times \text{sin}(-\frac{\text{\pi}}{3})).     * Task: Find rectangular coordinates (x,y)(x, y).     * Location: Quadrant IV (Q4Q4).     * Calculation of xx: x = r \times \text{cos}(\theta) = 2 \times \text{cos}(-\frac{\text{\pi}}{3}) = 2 \times (\frac{1}{2}) = 1.     * Calculation of yy: y = r \times \text{sin}(\theta) = 2 \times \text{sin}(-\frac{\text{\pi}}{3}) = 2 \times (-\frac{\text{\sqrt{3}}}{2}) = -\text{\sqrt{3}}.     * Result: The coordinates are (1, -\text{\sqrt{3}}). This corresponds to Option (B).

  • Problem 20: Polar to Rectangular in Q3:     * Given: Polar expression 10 \times \text{cos}(\frac{5\text{\pi}}{4}) + i(10 \times \text{sin}(\frac{5\text{\pi}}{4})).     * Task: Find rectangular coordinates (x,y)(x, y).     * Location: Quadrant III (Q3Q3).     * Calculation of xx: x = 10 \times \text{cos}(\frac{5\text{\pi}}{4}) = 10 \times (-\frac{\text{\sqrt{2}}}{2}) = -5\text{\sqrt{2}}.     * Calculation of yy: y = 10 \times \text{sin}(\frac{5\text{\pi}}{4}) = 10 \times (-\frac{\text{\sqrt{2}}}{2}) = -5\text{\sqrt{2}}.     * Result: The coordinates are (-5\text{\sqrt{2}}, -5\text{\sqrt{2}}). This corresponds to Option (A).