1. Complex Numbers and DeMoivre’s Theorem
LESSON 9–5
Complex Numbers and DeMoivre’s Theorem

2. Five-Minute Check (over Lesson 9-4) Then/Now New Vocabulary
Key Concept: Absolute Value of a Complex Number
Example 1: Graphs and Absolute Values of Complex Numbers
Key Concept: Polar Form of a Complex Number
Example 2: Complex Numbers in Polar Form
Example 3: Graph and Convert the Polar Form of a Complex Number
Key Concept: Product and Quotient of Complex Numbers in Polar Form
Example 4: Product of Complex Numbers in Polar Form
Example 5: Real-World Example: Quotient of Complex Numbers in Polar Form
Key Concept: De Moivre’s Theorem
Example 6: De Moivre’s Theorem
Key Concept: Distinct Roots
Example 7: pth Roots of a Complex Number
Example 8: The pth Roots of Unity
Lesson Menu

3. Determine the eccentricity, type of conic, and equation of the directrix for the polar equation .
A. e = 4; ellipse; x = 3
B. e = 2; ellipse; y = –3
C. e = 4; hyperbola; x = 12
D. e = 2; hyperbola; x = –3
5–Minute Check 1

4. Determine the eccentricity, type of conic, and equation of the directrix for the polar equation .
A. e = 4; hyperbola; y = 8
B. e = 4; ellipse; x = 2
C. e = 1; parabola; y = 2
D. e = 1; parabola; y = –2
5–Minute Check 2

5. A. Write a polar equation and directrix for the conic with e = 1 and directrix y = −2.B. C. D.
5–Minute Check 3

6. B. Graph a polar equation and directrix for the conic with e = 1 and directrix y = −2.
5–Minute Check 3

7. Write in rectangular form.B.
C. r + x – 3 = 0
D. x 2 = 9 – 6y
5–Minute Check 4

8. What type of conic is given by
A. parabola
B. ellipse
C. hyperbola
D. circle
5–Minute Check 5

9. Convert complex numbers from rectangular to polar form and vice versa.
You performed operations with complex numbers written in rectangular form. (Lesson 0-6)
Convert complex numbers from rectangular to polar form and vice versa.
Find products, quotients, powers, and roots of complex numbers in polar form.
Then/Now

10. absolute value of a complex number polar form trigonometric form
complex plane
real axis
imaginary axis
Argand plane
absolute value of a complex number
polar form
trigonometric form
modulus
argument
pth roots of unity
Vocabulary

11. Key Concept 1

12. A. Graph z = 2 + 3i in the complex plane and find its absolute value.
Graphs and Absolute Values of Complex Numbers
A. Graph z = 2 + 3i in the complex plane and find its absolute value.
(a, b) = (2, 3)
Example 1

13. Absolute value formula
Graphs and Absolute Values of Complex Numbers
Absolute value formula
a = 2 and b = 3
Simplify.
The absolute value of 2 + 3i is
Answer:
Example 1

14. B. Graph z = –3 + i in the complex plane and find its absolute value.
Graphs and Absolute Values of Complex Numbers
B. Graph z = –3 + i in the complex plane and find its absolute value.
(a, b) = (–3, 1)
Example 1

15. Absolute value formula
Graphs and Absolute Values of Complex Numbers
Absolute value formula
a = –3 and b = 1
Simplify.
The absolute value of –3 + i is
Answer:
Example 1

16. Graph 3 – 4i in the complex plane and find its absolute value.
Example 1

17. Key Concept 2

18. A. Express the complex number –2 + 5i in polar form.
Complex Numbers in Polar Form
A. Express the complex number –2 + 5i in polar form.
Find the modulus r and argument .
Conversion formula
a = –2 and b = 5
Simplify.
The polar form of –2 + 5i is about 5.39(cos i sin 1.95).
Answer: 5.39(cos i sin 1.95)
Example 2

19. B. Express the complex number 6 + 2i in polar form.
Complex Numbers in Polar Form
B. Express the complex number 6 + 2i in polar form.
Find the modulus r and argument .
Conversion formula
a = 6 and b = 2
Simplify.
The polar form of 6 + 2i is about 6.32(cos i sin 0.32).
Answer: 6.32(cos i sin 0.32)
Example 2

20. Express the complex number 4 – 5i in polar form.
A. 20(cos i sin 5.61)
B. 20(cos i sin 0.90)
C. 6.40(cos i sin 4.04)
D. 6.40(cos i sin 5.39)
Example 2

21. Graph on a polar grid. Then express it in rectangular form.
Graph and Convert the Polar Form of a Complex Number
Graph on a polar grid. Then express it in rectangular form.
The value of r is 4, and the value of  is
Plot the polar coordinates
Example 3

22. Evaluate for cosine and sine.
Graph and Convert the Polar Form of a Complex Number
To express the number in rectangular form, evaluate the trigonometric values and simplify.
Polar form
Evaluate for cosine and sine.
Distributive Property
The rectangular form of
Example 3

23. Graph and Convert the Polar Form of a Complex Number
Answer:
Example 3

24. Express in rectangular form.
A. –6 – 6i B. C. D.
Example 3

25. Key Concept 4

26. Find in polar form. Then express the product in rectangular form.
Product of Complex Numbers in Polar Form
Find in polar form. Then express the product in rectangular form.
Original expression
Product Formula
Simplify.
Example 4

27. Now find the rectangular form of the product.
Product of Complex Numbers in Polar Form
Now find the rectangular form of the product.
10(cos π + i sin π) Polar form
= 10(–1 + 0i) Evaluate.
= –10 + 0i Distributive Property
The polar form of the product is 10(cos π + i sin π). The rectangular form of the product is –10 + 0i or –10.
Answer: 10(cos π + i sin π); –10
Example 4

28. Find Express your answer in rectangular form.
A. – i
B. –19.80 – 19.80i
C. – i
D. – i
Example 4

29. Express each number in polar form.
Quotient of Complex Numbers in Polar Form
ELECTRICITY If a circuit has a voltage E of 100 volts and an impedance Z of 4 – 3j ohms, find the current I in the circuit in rectangular form. Use E = I • Z.
Express each number in polar form.
100 = 100(cos 0 +j sin 0)
4 – 3j = 5[cos (–0.64) + jsin (–0.64)]
Example 5

30. Solve for the current I in E = I • Z.
Quotient of Complex Numbers in Polar Form
Solve for the current I in E = I • Z.
I • Z = E Original equation
Divide each side by Z.
E = 100(cos 0 + j sin 0)
Z = 5[cos (–0.64) + j sin (–0.64)]
Example 5

31. Now, convert the current to rectangular form.
Quotient of Complex Numbers in Polar Form
Quotient Formula
Simplify.
Now, convert the current to rectangular form.
I = 20(cos j sin 0.64) Original equation
= 20( j) Evaluate.
= j Distributive Property
The current is about j amps.
Answer: j amps
Example 5

32. ELECTRICITY If a circuit has a voltage of 140 volts and a current of 4 + 3j amps, find the impedance of the circuit in rectangular form.
A j
B – 16.8j
C j
D j
Example 5

33. Key Concept 6

34. Find and express in rectangular form.
De Moivre’s Theorem
Find and express in rectangular form.
First, write in polar form.
Conversion formula
a = 3 and b =
Simplify.
Simplify.
Example 6

35. Now use De Moivre's Theorem to find the fourth power.
The polar form of is
Now use De Moivre's Theorem to find the fourth power.
Original equation
De Moivre's Theorem
Simplify.
Example 6

36. De Moivre’s Theorem
Evaluate.
Simplify.
Therefore,
Answer:
Example 6

37. Find and express in rectangular form.
A. 1728i
B. 1728 C. D.
Example 6

38. Key Concept 7

39. Find the fifth roots of –2 – 2i.
pth Roots of a Complex Number
Find the fifth roots of –2 – 2i.
First, write –2 – 2i in polar form.
–2 – 2i =
Now, write an expression for the fifth roots.
Example 7

40. Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots.
pth Roots of a Complex Number
Simplify.
Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots.
Example 7

41. Let n = 0 First fifth root Let n = 1 pth Roots of a Complex Number
Example 7

42. Second fifth root Let n = 2 Third fifth root
pth Roots of a Complex Number
Second fifth root
Let n = 2
Third fifth root
Example 7

43. Let n = 3 Fourth fifth root Let n = 4 Fifth fifth root
pth Roots of a Complex Number
Let n = 3
Fourth fifth root
Let n = 4
Fifth fifth root
Example 7

44. pth Roots of a Complex Number
The fifth roots of –2 – 2i are approximately i, – i, –1.22 – 0.19i, –0.19 – 1.22i, and 1.10 – 0.56i.
Answer: i, – i, –1.22 – 0.19i, –0.19 – 1.22i, – 0.56i
Fifth fifth root
Example 7

45. Find the cube roots of A. –1.04 + 5.91i, –4.60 – 3.86i, 5.64 – 2.05i
B. 3 – 5.20i, i, 5.64  2.05i
C. – i, –1.39 – 1.17i, 1.71 – 0.62i
D – 1.57i,  i , –1.82
Example 7

46. Find the fifth roots of unity.
The pth Roots of Unity
Find the fifth roots of unity.
First write 1 in polar form.
1 = 1 • (cos 0 + i sin 0)
Now write an expression for the fifth roots.
Simplify.
Example 8

47. Let n = 0 to find the first root of 1.
The pth Roots of Unity
Let n = 0 to find the first root of 1.
Let n = 0
First Roots
Notice that the modulus of each complex number is 1. The arguments are found by , resulting in  increasing by for each successive root. Therefore, we can calculate the remaining roots by adding to each previous .
Example 8

48. cos 0 + i sin 0 or 1 1st root 2nd root 3rd root 4th root 5th root
The pth Roots of Unity
cos 0 + i sin 0 or 1 1st root
2nd root
3rd root
4th root
5th root
Example 8

49. The pth Roots of Unity
The fifth roots of 1 are approximately 1, i, – i, –0.81 – 0.59i, and 0.31 – 0.95i, as shown in the figure.
Answer: 1, i, – i, –0.81 – 0.59i, 0.31 – 0.95i
Example 8

50. Find the fourth roots of unity.A.
B i, – i, –0.92 – 0.38i, – 0.92i C.
D. 1, i, –1, –i
Example 8

51. Complex Numbers and DeMoivre’s Theorem
LESSON 9–5
Complex Numbers and DeMoivre’s Theorem