NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 6: Curves in the Complex Plane Student Outcomes . Students convert between the real and complex forms of equations for ellipses. Students write equations of ellipses and represent them graphically. Lesson Notes Initially, students review how to represent numbers in the complex plane using the modulus and argument. They review the characteristics of the graphs of the numbers = 푟(cos(휃) + sin(휃)), recognizing that they represent circles centered at the origin with the radius equal to the modulus 푟. They then explore sets of complex numbers written in the form 푧 = 푎 cos(휃) + 푏 sin(휃), identifying the graphs as ellipses. Students convert between the complex and real forms of equations for ellipses, including those whose center is not the origin. They are also introduced to some of the components of ellipses, such as the vertices, foci, and axes. This prepares them to explore ellipses more formally in Lesson 7, where they derive the equation of an ellipse using its foci. Classwork Opening Exercise (5 minutes) This exercise should be completed in pairs or small groups. After a few minutes, students Scaffolding: should discuss their responses to Exercises 1–2 with another pair or group before completing Exercise 5. If the students are struggling with how to convert between the . For students below grade rectangular and polar form, the exercises could be completed as part of a teacher-led level, consider a concrete discussion. Early finishers could display their conjectures and plots for Problem 3, which approach using could be used in a teacher-led discussion of the characteristics of the graph. 푧 = 3 + 2푖, or provide a graphical representation of 푧 = 푎 + 푏푖. Opening Exercise . Advanced students could a. Consider the complex number 풛 = 풂 + 풃풊. explore the properties of i. Write 풛 in polar form. What do the variables represent? the graph of 풛 = 풓(퐜퐨퐬(휽) + 풊 퐬퐢퐧(휽)), where 풓 is the modulus of the complex number and 푧 = 3 cos(휃) + 5푖 sin(휃) 휽 is the argument. and compare it to the graph of ii. If 풓 = ퟑ and 휽 = ퟗퟎ°, where would 풛 be plotted in the complex plane? 푧 = 5 cos(휃) + 3푖 sin(휃) The point 풛 is located ퟑ units above the origin on the imaginary axis. to form conjectures about the properties of graphs represented by 푧 = 푎 cos(휃) + 푏푖 sin(휃). Lesson 6: Curves in the Complex Plane 83 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M3 PRECALCULUS AND ADVANCED TOPICS iii. Use the conditions in part (ii) to write 풛 in rectangular form. Explain how this representation corresponds to the location of 풛 that you found in part (ii). 풛 = 풂 + 풃풊, where 풂 = 풓 퐜퐨퐬(휽) and 풃 = 풓 퐬퐢퐧(휽) 풂 = ퟑ 퐜퐨퐬(ퟗퟎ°) = ퟎ; 풃 = ퟑ 퐬퐢퐧(ퟗퟎ°) = ퟑ Then 풛 = ퟑ풊, which is located three units above the origin on the imaginary axis. b. Recall the set of points defined by 풛 = ퟑ(퐜퐨퐬(휽) + 풊 퐬퐢퐧(휽)) for ퟎ° ≤ 휽 < ퟑퟔퟎ°, where 휽 is measured in degrees. i. What does 풛 represent graphically? Why? It is the set of points that are ퟑ units from the origin in the complex plane. This is because the modulus is ퟑ, which indicates that for any given value of 휽, 풛 is located a distance of ퟑ units from the origin. ii. What does 풛 represent geometrically? A circle with radius ퟑ units centered at the origin c. Consider the set of points defined by 풛 = ퟓ 퐜퐨퐬(휽) + ퟑ풊 퐬퐢퐧(휽). i. Plot 풛 for 휽 = ퟎ°, ퟗퟎ°, ퟏퟖퟎ°, ퟐퟕퟎ°, ퟑퟔퟎ°. Based on your plot, form a conjecture about the graph of the set of complex numbers. ퟐ ퟎ ퟎ −ퟐ ퟐ −ퟐ For 휽 = ퟎ°, 풛 = ퟓ 퐜퐨퐬(ퟎ°) + ퟑ풊 퐬퐢퐧(ퟎ°) = ퟓ + ퟎ풊 ↔ (ퟓ, ퟎ). For 휽 = ퟗퟎ°, 풛 = ퟓ 퐜퐨퐬(ퟗퟎ°) + ퟑ풊 퐬퐢퐧(ퟗퟎ°) = ퟑ풊 ↔ (ퟎ, ퟑ풊). For 휽 = ퟏퟖퟎ°, 풛 = ퟓ 퐜퐨퐬(ퟏퟖퟎ°) + ퟑ풊 퐬퐢퐧(ퟏퟖퟎ°) = −ퟓ + ퟎ풊 ↔ (−ퟓ, ퟎ). For 휽 = ퟐퟕퟎ°, 풛 = ퟓ 퐜퐨퐬(ퟐퟕퟎ°) + ퟑ풊 퐬퐢퐧(ퟐퟕퟎ°) = −ퟑ풊 ↔ (ퟎ, −ퟑ풊). This set of points seems to form an oval shape centered at the origin. ii. Compare this graph to the graph of 풛 = ퟑ(퐜퐨퐬(휽) + 풊 퐬퐢퐧(휽)). Form a conjecture about what accounts for the differences between the graphs. The coefficients of 퐜퐨퐬(휽) and 풊 퐬퐢퐧(휽) are equal for 풛 = ퟑ(퐜퐨퐬(휽) + 풊 퐬퐢퐧(휽)), which results in a circle, which has a constant radius, while the coefficients are different for 풛 = ퟓ 퐜퐨퐬(휽) + ퟑ풊 퐬퐢퐧(휽), which seems to stretch the circle. Lesson 6: Curves in the Complex Plane 84 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M3 PRECALCULUS AND ADVANCED TOPICS Example 1 (5 minutes) The students are led through an example to demonstrate how to convert the equation of a circle from its complex form to real form (i.e., an equation in 푥 and 푦). This prepares them to convert the equations of ellipses from complex form to real form. We have seen that in the complex plane, the graph of the set of complex numbers defined by 푧 = 3(cos(휃) + sin(휃)) for 0° ≤ 휃 < 360° is a circle centered at the origin with radius 3 units. How could we represent each point on the circle using an ordered pair? (3 cos(휃) , 3 sin(휃)) . Let’s say we wanted to represent 푧 in the real coordinate plane. We’d need to represent the points on the circle using an ordered pair (푥, 푦). How can we write any complex number 푧 in terms of 푥 and 푦? 푧 = 푥 + 푦 . So for 푧 = 3(cos(휃) + sin(휃)), which expressions represent 푥 and 푦? 푧 = 3 cos(휃) + 3 sin(휃), so 푥 = 3 cos(휃) and 푦 = 3 sin(휃) . What is the resulting ordered pair? (3 cos(휃) , 3 sin(휃)) . Let’s graph some points and verify that this gives us a circle. Complete the table for the given values of 휃. Example 1 Consider again the set of complex numbers represented by 풛 = ퟑ(퐜퐨퐬(휽) + 풊 퐬퐢퐧(휽)) for ퟎ° ≤ 휽 < ퟑퟔퟎ°. 휽 ퟑ 퐜퐨퐬(휽) ퟑ 퐬퐢퐧(휽) (ퟑ퐜퐨퐬(휽) , ퟑ풊 퐬퐢퐧(휽)) ퟎ ퟑ ퟎ (ퟑ, ퟎ) 흅 ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ ( , ) ퟒ ퟐ ퟐ ퟐ ퟐ 흅 ퟎ ퟑ (ퟎ, ퟑ) ퟐ ퟑ흅 ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ − (− , ) ퟒ ퟐ ퟐ ퟐ ퟐ 흅 −ퟑ ퟎ (−ퟑ, ퟎ) ퟓ흅 ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ − − (− , − ) ퟒ ퟐ ퟐ ퟐ ퟐ ퟑ흅 ퟎ −ퟑ (ퟎ, −ퟑ) ퟐ ퟕ흅 ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ ퟑ√ퟐ − ( , − ) ퟒ ퟐ ퟐ ퟐ ퟐ ퟐ흅 ퟑ ퟎ (ퟑ, ퟎ) Lesson 6: Curves in the Complex Plane 85 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M3 PRECALCULUS AND ADVANCED TOPICS . Plot the points in the table, and determine the type of curve created. The curve is a circle with center (0, 0) and radius 3. If 푥 = 3 cos(휃) and 푦 = 3 sin(휃), write an equation that relates 푥2 and 푦2. 푥2 + 푦2 = (3 cos(휃))2 + (3 sin(휃))2 . Now, simplify the right side of that equation. 푥2 + 푦2 = 9 cos2(휃) + 9 sin2(휃) 푥2 + 푦2 = 9(cos2(휃) + sin2(휃)) . Do you know a trigonometric identity that relates sin2(휃) and cos2(휃)? cos2(휃) + sin2(휃) = 1 . Substitute the identity into the previous equation. 푥2 + 푦2 = 9 . How does the graph of this equation compare with the graph of our equation in complex form? Both graphs are circles centered at the origin with radius 3 units. a. Use an ordered pair to write a representation for the points defined by 풛 as they would be represented in the coordinate plane. (ퟑ 퐜퐨퐬(휽) , ퟑ 퐬퐢퐧(휽)) b. Write an equation that is true for all the points represented by the ordered pair you wrote in part (a). Since 풙 = ퟑ 퐜퐨퐬(휽) and 풚 = ퟑ 퐬퐢퐧(휽): 풙ퟐ + 풚ퟐ = (ퟑ 퐜퐨퐬(휽))ퟐ + (ퟑ 퐬퐢퐧(휽))ퟐ 풙ퟐ + 풚ퟐ = ퟗ(퐜퐨퐬(휽))ퟐ + ퟗ(퐬퐢퐧(휽))ퟐ 풙ퟐ + 풚ퟐ = ퟗ((퐜퐨퐬(휽))ퟐ + (퐬퐢퐧(휽))ퟐ) We know that (퐬퐢퐧(휽))ퟐ + (퐜퐨퐬(휽))ퟐ = ퟏ, so 풙ퟐ + 풚ퟐ = ퟗ. Lesson 6: Curves in the Complex Plane 86 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M3 PRECALCULUS AND ADVANCED TOPICS c. What does the graph of this equation look like in the coordinate plane? The graph is a circle centered at the origin with radius ퟑ units.