De Moivres Theorem simple applications 9162020 By Chtan

De Moivre’s Theorem & simple applications 9/16/2020 By Chtan FYHS-Kulai 1

In mathematics, de Moivre‘s formula, named after Abraham de Moivre. 9/16/2020 By Chtan FYHS-Kulai 2

The formula is important because it connects complex numbers and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x". 9/16/2020 By Chtan FYHS-Kulai 3

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). 9/16/2020 By Chtan FYHS-Kulai 4

Furthermore, one can use a generalization of this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zn = 1. 9/16/2020 By Chtan FYHS-Kulai 5

De Moivre’s theorem For all values of n, the value, or one of the values in the case where n is fractional, of is 9/16/2020 By Chtan FYHS-Kulai 6

Proofing of De Moivre’s Theorem 9/16/2020 By Chtan FYHS-Kulai 7

Now, let us prove this important theorem in 3 parts. 1. When n is a positive integer 2. When n is a negative integer 3. When n is a fraction 9/16/2020 By Chtan FYHS-Kulai 8

Case 1 : if n is a positive integer 9/16/2020 By Chtan FYHS-Kulai 9

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Continuing this process, when n is a positive integer, 9/16/2020 By Chtan FYHS-Kulai 12

Case 2 : if n is a negative integer Let n=-m where m is positive integer 9/16/2020 By Chtan FYHS-Kulai 13

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Case 3 : if n is a fraction equal to p/q, p and q are integers 9/16/2020 By Chtan FYHS-Kulai 15

Raising the RHS to power q we have, but, 9/16/2020 By Chtan FYHS-Kulai 16

Hence, De Moivre’s Theorem applies when n is a rational fraction. 9/16/2020 By Chtan FYHS-Kulai 17

Proofing by mathematic al induction 9/16/2020 By Chtan FYHS-Kulai 18

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The hypothesis of Mathematical Induction has been satisfied , and we can conclude that 9/16/2020 By Chtan FYHS-Kulai 21

e. g. 1 Let z = 1 − i. Find . Soln: First write z in polar form. 9/16/2020 By Chtan FYHS-Kulai 22

Polar form : Applying de Moivre’s Theorem gives : 9/16/2020 By Chtan FYHS-Kulai 23

It can be verified directly that 9/16/2020 By Chtan FYHS-Kulai 24

Properties of 9/16/2020 By Chtan FYHS-Kulai 25

If then 9/16/2020 By Chtan FYHS-Kulai 26

Hence, 9/16/2020 By Chtan FYHS-Kulai 27

Similarly, if Hence, 9/16/2020 By Chtan FYHS-Kulai 28

We have, Maximum value of cosθ is 1, minimum value is -1. Hence, normally 9/16/2020 By Chtan FYHS-Kulai 29

What happen, if the value of is more than 2 or less than -2 ? 9/16/2020 By Chtan FYHS-Kulai 30

e. g. 2 Given that Prove that 9/16/2020 By Chtan FYHS-Kulai 31

e. g. 3 If 9/16/2020 , find By Chtan FYHS-Kulai 32

Do take note of the following : 9/16/2020 By Chtan FYHS-Kulai 33

e. g. 4 9/16/2020 By Chtan FYHS-Kulai 34

Applications of De Moivre’s theorem 9/16/2020 By Chtan FYHS-Kulai 35

We will consider three applications of De Moivre’s Theorem in this chapter. 1. Expansion of 2. Values of . 3. Expressions for terms of multiple angles. 9/16/2020 . By Chtan FYHS-Kulai in 36

Certain trig identities can be derived using De Moivre’s theorem. In particular, expression such as can be expressed in terms of : 9/16/2020 By Chtan FYHS-Kulai 37

e. g. 5 Use De Moivre’s Thorem to find an identity for in terms of 9/16/2020 . By Chtan FYHS-Kulai 38

e. g. 6 Soln: Find all complex cube roots of 27 i. We are looking for complex number z with the property Strategy : First we write 27 i in polar form : - 9/16/2020 By Chtan FYHS-Kulai 39

Now suppose Satisfies. Then, by De Moivre’s Theorem, 9/16/2020 By Chtan FYHS-Kulai 40

This means : where k is an integer. Possibilities are : k=0, k=1, k=2 9/16/2020 By Chtan FYHS-Kulai 41

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In general : to find the complex nth roots of a non-zero complex number z. 1. Write z in polar form : 9/16/2020 By Chtan FYHS-Kulai 44

2. z will have n different nth roots (i. e. 3 cube roots, 4 fourth roots, etc. ) 3. All these roots will have the same modulus the positive real nth roots of r). 4. They will have different arguments : 9/16/2020 By Chtan FYHS-Kulai 45

5. The complex nth roots of z are given (in polar form) by …etc 9/16/2020 By Chtan FYHS-Kulai 46

e. g. 7 Find all the complex fourth roots of -16. Soln: Modulus = 16 Argument = ∏ 9/16/2020 By Chtan FYHS-Kulai 47

Fourth roots of 16 all have modulus : and possibilities for the arguments are : 9/16/2020 By Chtan FYHS-Kulai 48

Hence, fourth roots of -16 are : 9/16/2020 By Chtan FYHS-Kulai 49

e. g. 8 Given that and find the value of m. 9/16/2020 By Chtan FYHS-Kulai 50

e. g. 9 Solve that 9/16/2020 , hence prove By Chtan FYHS-Kulai 51

e. g. 10 Find the cube roots of -1. show that they can be denoted by and prove that 9/16/2020 By Chtan FYHS-Kulai 52

e. g. 11 Solve the following equations, giving any complex roots in the form 9/16/2020 By Chtan FYHS-Kulai 53

e. g. 12 Prove that Hence find 9/16/2020 By Chtan FYHS-Kulai 54

e. g. 13 Show that Use your result to solve the equation 9/16/2020 By Chtan FYHS-Kulai 55

e. g. 14 Use De Moivre’s Theorem to find 9/16/2020 By Chtan FYHS-Kulai 56

e. g. 15 9/16/2020 By Chtan FYHS-Kulai 57

e. g. 16 9/16/2020 By Chtan FYHS-Kulai 58

e. g. 17 9/16/2020 By Chtan FYHS-Kulai 59

e. g. 18 9/16/2020 By Chtan FYHS-Kulai 60

e. g. 19 Express in terms of multiple angles and hence evaluate 9/16/2020 By Chtan FYHS-Kulai 61

e. g. 20 Express in terms of and hence evaluate terms of. 9/16/2020 By Chtan FYHS-Kulai in 62

The end 9/16/2020 By Chtan FYHS-Kulai 63