1. Chapter 40 De Moivre’s Theorem & simple applications 12/24/2017
By Chtan FYHS-Kulai

2. In mathematics, de Moivre‘s formula, named after Abraham de Moivre.
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3. The formula is important because it connects complex numbers and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x".
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4. 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).
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5. 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.
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6. 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
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7. Proofing of De Moivre’s Theorem
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8. When n is a positive integer When n is a negative integer
Now, let us prove this important theorem in 3 parts.
When n is a positive integer
When n is a negative integer
When n is a fraction
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9. Case 1 : if n is a positive integer
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10. 12/24/2017
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12. Continuing this process, when n is a positive integer,
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13. Case 2 : if n is a negative integer
Let n=-m where m is positive integer
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14. 12/24/2017
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15. Case 3 : if n is a fraction equal to p/q, p and q are integers
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16. Raising the RHS to power q we have,
but,
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17. Hence, De Moivre’s Theorem applies when n is a rational fraction.
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18. Proofing by mathematical induction
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19. 12/24/2017
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20. 12/24/2017
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21. The hypothesis of Mathematical Induction has been satisfied , and we can conclude that
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22. e.g. 1 Let z = 1 − i. Find . Soln: First write z in polar form.
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23. Polar form : Applying de Moivre’s Theorem gives : 12/24/2017
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24. It can be verified directly that
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25. Properties of
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26. If
then
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27. Hence,
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28. Similarly, if
Hence,
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29. We have,
Maximum value of cosθ is 1, minimum value is -1.
Hence, normally
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30. is more than 2 or less than -2 ?
What happen, if
the value of
is more than 2 or less than -2 ?
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31. e.g. 2
Given that
Prove that
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32. e.g. 3
If , find
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33. Do take note of the following :
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34. e.g. 4
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35. Applications of De Moivre’s theorem
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36. We will consider three applications of De Moivre’s Theorem in this chapter.
1. Expansion of
2. Values of
3. Expressions for in terms of multiple angles.
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37. can be expressed in terms of :
Certain trig identities can be derived using De Moivre’s theorem. In particular, expression such as
can be expressed in terms of :
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38. e.g. 5 Use De Moivre’s Thorem to find an identity for in terms of .
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39. e.g. 6 Soln: Find all complex cube roots of 27i.
We are looking for complex number z with the property
Strategy :
First we write 27i in polar form :-
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40. Satisfies . Then, by De Moivre’s Theorem,
Now suppose
Satisfies Then, by De Moivre’s Theorem,
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41. Possibilities are : k=0, k=1, k=2
This means :
where k is an integer.
Possibilities are : k=0, k=1, k=2
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42. 12/24/2017
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43. 12/24/2017
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44. In general : to find the complex nth roots of a non-zero complex number z.
1. Write z in polar form :
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45. 4. They will have different arguments :
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 :
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46. …etc 5. The complex nth roots of z are given (in polar form) by
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47. Find all the complex fourth roots of -16.
e.g. 7
Find all the complex fourth roots of -16.
Soln:
Modulus = 16
Argument = ∏
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48. Fourth roots of 16 all have modulus :
and possibilities for the arguments are :
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49. Hence, fourth roots of -16 are :
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50. e.g. 8 Given that and find the value of m. 12/24/2017
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51. e.g. 9
Solve , hence prove that
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52. e.g. 10
Find the cube roots of -1. show that they can be denoted by and prove that
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53. e.g. 11
Solve the following equations, giving any complex roots in the form
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54. e.g. 12
Prove that
Hence find
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55. e.g. 13 Show that Use your result to solve the equation 12/24/2017
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56. e.g. 14
Use De Moivre’s Theorem to find
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57. e.g. 15
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58. e.g. 16
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59. e.g. 17
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60. e.g. 18
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61. Express in terms of multiple angles and hence evaluate
e.g. 19
Express in terms of multiple angles and hence evaluate
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62. Express in terms of and hence evaluate in terms of .
e.g. 20
Express in terms of and hence evaluate in terms of
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63. The end
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