Complex Numbers Yr 2

Question 1

What is De Moivre’s Theorem?

Answer 1

Question 2

Prove De Moivre’s Theorem by induction

Answer 2

Question 3

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

Answer 6

Question 7

What is the equation to the roots of z^n =1

Answer 7

Question 8

How are the nth roots of unity shown on a complex number diagram?

Answer 8

Question 9

What is the sum of the roots of unity?

Answer 9

Always = 0

Question 10

Answer 10

Question 11

Answer 11

Question 12

What is the equation of roots of any complex number?

Answer 12

Question 13

Answer 13

Question 14

Derive the equation for the nth roots of unity

Answer 14

Question 15

Derive the formula of equation for complex roots of any complex number

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

Answer 18

Question 19

Answer 19

Question 20

Answer 20

Question 21

What is the form of e^i°?

Answer 21

Question 22

Answer 22

Question 23

What is Euler’s identity?

Answer 23

Question 24

What are the two important identities for sin and cos in terms of e^° and i ?

Answer 24

Question 25

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Question 29

Answer 29

Question 30

Answer 30

Question 31

Answer 31

Question 32

What are the nth roots of unity?

Answer 32

Complex numbers that when raised to the nth power, equal 1. Each root represents a point on the complex plane on the unit circle, creating a regular polygon with n vertices

Question 33

Derive the nth roots of unity formula

Answer 33

Question 34

General nth roots of unity formula

Answer 34

Question 35

Answer 35

.

Question 36

Derive the nth roots for any complex number formula

Answer 36

Question 37

What is the general formula for the nth roots of any complex number ?

Answer 37

Question 38

What is the nth roots for any complex number?

Answer 38

Similar to the nth roots of unity but instead of lying on a unit circle, they lie on a circle at with radius r^1/n and starting at angle theta/n, rather than at 1,0

Question 39

Answer 39

Question 40

Answer 40

Question 41

Answer 41

Question 42

Answer 42

Question 43

Answer 43

Question 44

Answer 44

Question 45

Answer 45

Question 46

Answer 46

Question 47

Answer 47

Question 48

Answer 48

Question 49

Complete the identities

Answer 49

Question 50

Answer 50

Question 51

Using the standard results fill out the following:

Answer 51

Question 52

Answer 52

Question 53

Answer 53

Question 54

Answer 54

Question 55

Answer 55

Question 56

Answer 56

Question 57

Answer 57

Question 58

Answer 58

Question 59

Answer 59

Question 60

Answer 60

Don’t forget the pi and the i !

Question 61

Answer 61

Question 62

Part a

Answer 62

Question 63

Part b

Answer 63

Question 64

Exam practice

Answer 64

Question 65

Part a

Answer 65

Question 66

Answer 66

Question 67

Part A

Answer 67

Question 68

Answer 68