Lesson 6: Curves in the Complex Plane
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Lesson 6: Curves in the Complex Plane Classwork Opening Exercise a.
b.
Consider the complex number π§π§ = ππ + ππππ. i.
Write π§π§ in polar form. What do the variables represent?
ii.
If ππ = 3 and ππ = 90Β°, where would π§π§ be plotted in the complex plane?
iii.
Use the conditions in part (ii) to write π§π§ in rectangular form. Explain how this representation corresponds to the location of π§π§ that you found in part (ii).
Recall the set of points defined by π§π§ = 3(cos(ππ) + ππ sin (ππ)) for 0 β€ ππ < 360Β°, where ππ is measured in degrees. i.
What does π§π§ represent graphically? Why?
ii.
What does π§π§ represent geometrically?
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.37 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
c.
Consider the set of points defined by π§π§ = 5 cos ππ + 3ππβsin ππ. i.
Plot π§π§ for ππ = 0Β°, 90Β°, 180Β°, 270Β°, 360Β°. Based on your plot, form a conjecture about the graph of the set of complex numbers.
ii.
Compare this graph to the graph of π§π§ = 3(cos(ππ) + ππ sin (ππ)). Form a conjecture about what accounts for the differences between the graphs.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.38 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Example 1 Consider again the set of complex numbers represented by π§π§ = 3(cos(ππ) + ππ sin (ππ)) for 0 β€ ππ < 360Β° . 47T
π½π½ 0
ππππππππ(π½π½)
ππππππππ(π½π½)
(ππππππππ(π½π½), ππππ π¬π¬π¬π¬π¬π¬(π½π½))
ππ 4 ππ 2 3ππ 4 ππ 5ππ 4 3ππ 2 7ππ 4 2ππ
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.39 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
a.
Use an ordered pair to write a representation for the points defined by π§π§ as they would be represented in the coordinate plane.
b.
Write an equation that is true for all the points represented by the ordered pair you wrote in part (a).
c.
What does the graph of this equation look like in the coordinate plane?
Exercises 1β2 1.
Recall the set of points defined by π§π§ = 5 cos(ππ) + 3ππsin (ππ) . 47T
a.
Use an ordered pair to write a representation for the points defined by π§π§ as they would be represented in the coordinate plane.
b.
Write an equation in the coordinate plane that is true for all the points represented by the ordered pair you wrote in part (a).
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.40 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
2.
Find an algebraic equation for all the points in the coordinate plane traced by the complex numbers π§π§ = β2 cos(ππ) + ππ sin (ππ).
Example 2 The equation of an ellipse is given by
π₯π₯ 2
16
+
π¦π¦ 2 4
= 1.
a.
Sketch the graph of the ellipse.
b.
Rewrite the equation in complex form.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.41 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
Exercise 3 3.
The equation of an ellipse is given by
π₯π₯ 2 9
+
π¦π¦ 2
26
= 1.
a.
Sketch the graph of the ellipse.
b.
Rewrite the equation of the ellipse in complex form.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.42 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
Example 3 A set of points in the complex plane can be represented in the complex plane as π§π§ = 2 + ππ + 7cos(ππ) + ππ sin(ππ) as ππ varies. a.
Find an algebraic equation for the points described.
b.
Sketch the graph of the ellipse.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.43 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Problem Set 1.
Write the real form of each complex equation. a. b. c. d.
2.
b. c. d.
π§π§ = β5 cos(ππ) + β10ππ sin (ππ)
π§π§ = 5 β 2ππ + 4 cos(ππ) + 7ππ sin( ππ) π§π§ = 3 cos(ππ) + ππ sin (ππ)
π§π§ = β2 + 3ππ + 4 cos(ππ) + ππ sin (ππ)
(π₯π₯β1)2 9
(π₯π₯β2)2 3
+ +
π¦π¦ 2 25
π¦π¦ 2 15
=1 =1
Write the complex form of each equation. a. b. c. d.
4.
π§π§ = 6 cos(ππ) + ππ sin (ππ)
Sketch the graphs of each equation. a.
3.
π§π§ = 4 cos(ππ) + 9ππ sin (ππ)
π₯π₯ 2
16
π₯π₯ 2
+
400 π₯π₯ 2
19
π¦π¦ 2 36
+
+
169
π¦π¦ 2 2
(π₯π₯β3)2 100
=1
π¦π¦ 2
=1
=1
+
(π¦π¦+5)2 16
=1
Carrie converted the equation π§π§ = 7cos(ππ) + 4ππ sin(ππ) to the real form
π₯π₯ 2 7
+
π¦π¦ 2 4
= 1. Her partner Ginger said that
the ellipse must pass through the point οΏ½7cos(0), 4sin(0)οΏ½ = (7,0) and this point does not satisfy Carrieβs equation, so the equation must be wrong. Who made the mistake, and what was the error? Explain how you know. 5.
Cody says that the center of the ellipse with complex equation π§π§ = 4 β 5ππ + 2cos(ππ) + 3ππ sin(ππ) is (4, β5), while his partner Jarrett says that the center of this ellipse is (β4,5). Which student is correct? Explain how you know.
Extension: 6.
Any equation of the form πππ₯π₯ 2 + ππππ + πππ¦π¦ 2 + ππππ + ππ = 0 with ππ > 0 and ππ > 0 might represent an ellipse. The equation 4π₯π₯ 2 + 8π₯π₯ + 3π¦π¦ 2 + 12π¦π¦ + 4 = 0 is such an equation of an ellipse. (π₯π₯ββ)2
(π¦π¦βππ)2
a.
Rewrite the equation
b.
Describe the graph of the ellipse, and then sketch the graph.
c.
Write the complex form of the equation for this ellipse.
Lesson 6: Date:
ππ2
+
ππ 2
= 1 in standard form to locate the center of the ellipse (β, ππ).
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.44 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Lesson 6: Curves in the Complex Plane Classwork Opening Exercise a.
b.
Consider the complex number π§π§ = ππ + ππππ. i.
Write π§π§ in polar form. What do the variables represent?
ii.
If ππ = 3 and ππ = 90Β°, where would π§π§ be plotted in the complex plane?
iii.
Use the conditions in part (ii) to write π§π§ in rectangular form. Explain how this representation corresponds to the location of π§π§ that you found in part (ii).
Recall the set of points defined by π§π§ = 3(cos(ππ) + ππ sin (ππ)) for 0 β€ ππ < 360Β°, where ππ is measured in degrees. i.
What does π§π§ represent graphically? Why?
ii.
What does π§π§ represent geometrically?
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.37 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
c.
Consider the set of points defined by π§π§ = 5 cos ππ + 3ππβsin ππ. i.
Plot π§π§ for ππ = 0Β°, 90Β°, 180Β°, 270Β°, 360Β°. Based on your plot, form a conjecture about the graph of the set of complex numbers.
ii.
Compare this graph to the graph of π§π§ = 3(cos(ππ) + ππ sin (ππ)). Form a conjecture about what accounts for the differences between the graphs.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.38 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Example 1 Consider again the set of complex numbers represented by π§π§ = 3(cos(ππ) + ππ sin (ππ)) for 0 β€ ππ < 360Β° . 47T
π½π½ 0
ππππππππ(π½π½)
ππππππππ(π½π½)
(ππππππππ(π½π½), ππππ π¬π¬π¬π¬π¬π¬(π½π½))
ππ 4 ππ 2 3ππ 4 ππ 5ππ 4 3ππ 2 7ππ 4 2ππ
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.39 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
a.
Use an ordered pair to write a representation for the points defined by π§π§ as they would be represented in the coordinate plane.
b.
Write an equation that is true for all the points represented by the ordered pair you wrote in part (a).
c.
What does the graph of this equation look like in the coordinate plane?
Exercises 1β2 1.
Recall the set of points defined by π§π§ = 5 cos(ππ) + 3ππsin (ππ) . 47T
a.
Use an ordered pair to write a representation for the points defined by π§π§ as they would be represented in the coordinate plane.
b.
Write an equation in the coordinate plane that is true for all the points represented by the ordered pair you wrote in part (a).
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.40 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
2.
Find an algebraic equation for all the points in the coordinate plane traced by the complex numbers π§π§ = β2 cos(ππ) + ππ sin (ππ).
Example 2 The equation of an ellipse is given by
π₯π₯ 2
16
+
π¦π¦ 2 4
= 1.
a.
Sketch the graph of the ellipse.
b.
Rewrite the equation in complex form.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.41 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
Exercise 3 3.
The equation of an ellipse is given by
π₯π₯ 2 9
+
π¦π¦ 2
26
= 1.
a.
Sketch the graph of the ellipse.
b.
Rewrite the equation of the ellipse in complex form.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.42 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M3
PRECALCULUS AND ADVANCED TOPICS
Example 3 A set of points in the complex plane can be represented in the complex plane as π§π§ = 2 + ππ + 7cos(ππ) + ππ sin(ππ) as ππ varies. a.
Find an algebraic equation for the points described.
b.
Sketch the graph of the ellipse.
Lesson 6: Date:
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.43 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Problem Set 1.
Write the real form of each complex equation. a. b. c. d.
2.
b. c. d.
π§π§ = β5 cos(ππ) + β10ππ sin (ππ)
π§π§ = 5 β 2ππ + 4 cos(ππ) + 7ππ sin( ππ) π§π§ = 3 cos(ππ) + ππ sin (ππ)
π§π§ = β2 + 3ππ + 4 cos(ππ) + ππ sin (ππ)
(π₯π₯β1)2 9
(π₯π₯β2)2 3
+ +
π¦π¦ 2 25
π¦π¦ 2 15
=1 =1
Write the complex form of each equation. a. b. c. d.
4.
π§π§ = 6 cos(ππ) + ππ sin (ππ)
Sketch the graphs of each equation. a.
3.
π§π§ = 4 cos(ππ) + 9ππ sin (ππ)
π₯π₯ 2
16
π₯π₯ 2
+
400 π₯π₯ 2
19
π¦π¦ 2 36
+
+
169
π¦π¦ 2 2
(π₯π₯β3)2 100
=1
π¦π¦ 2
=1
=1
+
(π¦π¦+5)2 16
=1
Carrie converted the equation π§π§ = 7cos(ππ) + 4ππ sin(ππ) to the real form
π₯π₯ 2 7
+
π¦π¦ 2 4
= 1. Her partner Ginger said that
the ellipse must pass through the point οΏ½7cos(0), 4sin(0)οΏ½ = (7,0) and this point does not satisfy Carrieβs equation, so the equation must be wrong. Who made the mistake, and what was the error? Explain how you know. 5.
Cody says that the center of the ellipse with complex equation π§π§ = 4 β 5ππ + 2cos(ππ) + 3ππ sin(ππ) is (4, β5), while his partner Jarrett says that the center of this ellipse is (β4,5). Which student is correct? Explain how you know.
Extension: 6.
Any equation of the form πππ₯π₯ 2 + ππππ + πππ¦π¦ 2 + ππππ + ππ = 0 with ππ > 0 and ππ > 0 might represent an ellipse. The equation 4π₯π₯ 2 + 8π₯π₯ + 3π¦π¦ 2 + 12π¦π¦ + 4 = 0 is such an equation of an ellipse. (π₯π₯ββ)2
(π¦π¦βππ)2
a.
Rewrite the equation
b.
Describe the graph of the ellipse, and then sketch the graph.
c.
Write the complex form of the equation for this ellipse.
Lesson 6: Date:
ππ2
+
ππ 2
= 1 in standard form to locate the center of the ellipse (β, ππ).
Curves in the Complex Plane 2/9/15
Β© 2015 Common Core, Inc. Some rights reserved. commoncore.org
S.44 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
