The Unit Circle & Complex Numbers
DATE:
1/28
CLASS PERIOD:
PreCalc
UNIT:
U.C.-1
.
LESSON OBJECTIVES:
In this extension lesson, students will extend their knowledge of the unit circle to include complex
numbers. Students will explore Pythagorean Triples and the n complex roots of 𝒙𝒏 − 𝟏 = 𝟎. This lesson
corresponds with Common Core State Standards-The Complex Number System N-CN 5. (+) Represent
addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex
plane; use properties of this representation for computation.
MATERIALS
Worksheet. Warm up.
EVALUATION
Checkpoints between sections. Circulate during
worktime.
REVIEW
Absolute Value; equation of the unit circle; complex
numbers in the plane; first Pythagorean Identity
HOMEWORK ASSIGNMENTS
Worksheet 1-5.
ACTIVITIES TO EXTEND UNDERSTANDING
AND/OR RELATED TOPICS
This lesson is an extension.
Schedule
Warm-up (10’)
Activity (20’)
Worktime (15’)
LITERACY STRATEGIES &
ACCOMMODATIONS
Activities
As they enter the room, hand students a warm-up worksheet.
See attached outline.
Hand out WS 1-5.
I.
II.
Review – Go over warm up.
A. Absolute Value. Review definition but share that Absolute Value can be thought
of as “distance from the origin.”
B. Equation of the Unit Circle is 𝑥 2 + 𝑦 2 = 1.
C. Graphing Complex Numbers in the plane as points or vectors (Argand Diagram).
Lesson
A. Absolute Value. How do we find the absolute value of 1+2i? The absolute value
of a complex number 𝑧 = 𝑎 + 𝑏𝑖 is |𝑧| = 𝑎2 + 𝑏 2 . So |1+2i| is 12 + 22 = √5.
This is literally the distance from the origin.
B. Can you think of another equation for the unit circle? What about |z|=1? Here are
some example points on the unit circle: ±1, ±i. See how they correspond to
(1,0), (-1, 0), (0, 1), & (0, -1)?
C. What other points do you think would be there? What other points have
𝑎2 + 𝑏 2 = 1? Do any of the points on the unit circle fit that equation?
Remember from before √𝑖 =
is
√2
√2
+ 2 𝑖.
2
√2
2
+
√2
𝑖
2
? Isn’t that point on the unit circle? So √𝑖
Note 𝑎2 + 𝑏 2 = 1. Similarly, 𝑐𝑜𝑠 2 + 𝑠𝑖𝑛2 = 1 for the same point
𝜋
√2
√2
(4 ). In fact, all four ± 2 ± 2 𝑖 are on the unit circle.
SKIP THIS NEXT SECTION IF TIME IS SHORT. SAVE FOR TOMORROW.
D. Pythagorean Triples. A “Pythagorean Triple” is a set of three integers that make
up a right triangle. The most famous is 3:4:5. 6:8:10 doesn’t count. Another one
is 5:12:13. There are many more – 15:8:17, 7:24:25, 10:24:26, 21:20:29, 9:40:41,
35:12:27, and 11:60:61. In fact, there are infinitely many of them.
𝑎 2
𝑏 2
3
So we have 𝑎2 + 𝑏 2 = 𝑐 2 . If we divide by 𝑐 2 , we have ( 𝑐 ) + ( 𝑐 ) = 1. So, 5 +
4
5
12
1
√3
40
9
𝑖, 13 + 13 𝑖, and 41 + 41 𝑖 are all on the unit circle. (Verify.)
E. Here’s another application of complex numbers on the unit circle. Let
5
III.
𝑧 = 2 + 2 𝑖. Plot this and other powers of z as shown in this table. (see text, p.
408)
Power of z
Coordinates

𝜋
𝑧
1 √3
+
𝑖
3
2
2
2
2𝜋
𝑧
1 √3
− +
𝑖
3
2
2
3
−1
𝑧

4𝜋
𝑧4
1 √3
− −
𝑖
3
2
2
5
5𝜋
𝑧
1 √3
−
𝑖
3
2
2
6
1
2𝜋
𝑧
Show how this table shows the six complex roots of the equation 𝑥 6 − 1 = 0.
Homework. Worksheet 1-5 attached.
Unit Circle Worksheet 1-5 (Complex Numbers and the Unit Circle)
Name _____________________________________
1. Are the following complex numbers on the unit circle (|z|=1)? Answer Y or N.
a.
√3
2
1
+ 2𝑖
c. √2 + √2𝑖
e.
24
7
+ 25 𝑖
25
b. 1+i
d.
f.
3
4
5
+ 5𝑖
27
12
+ 35 𝑖
35
2. Let 𝑧 = 𝑖. Show 𝑧, 𝑧 2 , 𝑧 3 and 𝑧 4 on the unit circle below. What do 𝑧 5 and 𝑧11 equal?
3. Let 𝑧 =
1
√3
+ 2 𝑖.
2
Calculate 𝑧 2 and 𝑧 3 . Show 𝑧, 𝑧 2 , and 𝑧 3 on the unit circle below.
Unit Circle 1-5
Warm Up
Name _____________________
1. Remember absolute value? Solve the following.
a. |5|
b. |-|
c. -|-7|
d. (|-1|)(x)
2. Plot 1+2i on the coordinates below.
Unit Circle 1-5
3. What is the equation of the unit circle?
Warm Up
Name _____________________
1. Remember absolute value? Solve the following.
a. |5|
b. |-|
c. -|-7|
2. Plot 1+2i on the coordinates below.
d. (|-1|)(x)
3. What is the equation of the unit circle?