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CHAPTER 4.4 CHAPTER 4 COMPLEX NUMBERS PART 4 –DeMoivre’s Theorem TRIGONOMETRY MATHEMATICS CONTENT STANDARDS: 17.0 - Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form. 18.0 - Students know DeMoivre’s theorem and can give nth roots of a complex number given in polar form OBJECTIVE(S): Students will learn DeMoivre’s Theorem and how to use to find a power of a complex number. Students will learn how to find the nth roots of a complex number. Powers of Complex Numbers The trigonometric form of a complex number is used to raise a complex number to a power. To accomplish this, consider repeated use of the multiplication rule. z = r cos i sin z2 = r cos i sin r cos i sin = r 2 cos 2 i sin 2 z3 = r 2 cos 2 i sin 2 r cos i sin = r 3 cos 3 i sin 3 z4 = r 4 cos 4 i sin 4 z5 = r 5 cos 5 i sin 5 This pattern leads to DeMoivre’s Theorem, which is named after the French mathematician Abraham DeMoivre (1667-1754). DeMoivre’s Theorem If z r cos i sin is a complex number and n is a positive integer, then z n r cos i sin n = r n cos n i sin n CHAPTER 4.4 EXAMPLE 1: Finding Powers of a Complex Number 12 Use DeMoivre’s Theorem to find 1 3i . First convert to trigonometric form using r = r a 2 b2 = arctan and b a So, the trigonometric form is 1 3i = Then, by DeMoivre’s Theorem, you have = = = = = Roots of Complex Numbers Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial equation of degree n has n solutions in the complex number system. So, the equation x 6 1 has six solutions, and in this particular case you can find the six solutions by factoring and using the Quadratic Formula. x6 1 = = Consequently, the solutions are x = _______, x = ____________________, and x = ____________________ CHAPTER 4.4 Each of these numbers is sixth root of 1. In general, the nth root of a complex number is defined as follows. Definition of an nth Root of a Complex Number The complex number u a bi is an nth root of the complex number z n if z u n a bi . To find a formula for an nth root of a complex number, let u be an nth root of z, where u= and z= By DeMoivre’s Theorem and the fact that u n z , you have s n cos n i sin n r cos i sin . Taking the absolute value of each side of this equation, it follows that ____________. Substituting back into the previous equation and dividing by ____, you get cos n i sin n cos i sin . So, it follows that = and = Because both sine and cosine have a period of _____, these last two equations have solutions if and only if the angles differ by a multiple of _____. Consequently, there must exist an integer k such that n = = By substituting this value of into the trigonometric form of u, you get the following: CHAPTER 4.4 Finding nth Roots of a Complex Number For a positive integer n, the complex number z r cos i sin has exactly n distinct nth roots given by n 2k 2k r cos i sin n n where k = 0, 1, 2, …, n - 1 When k exceeds n – 1, the roots begin to repeat. EXAMPLE 2: Finding the nth Roots of a Real Number Find all the sixth roots of 1. First write 1 in the trigonometric form 1 = _______________________________. Then, by the nth root formula, with n = ____ and r = _____, the roots have the form 6 0 2k 0 2k 1 cos i sin = 6 6 So, for k = 0, 1, 2, 3, 4, and 5, the sixth roots are as follows. k=0 cos 0 i sin 0 = k=1 cos k=2 cos k=3 cos i sin = k=4 cos 4 4 i sin = 3 3 k=5 cos 5 5 i sin = 3 3 k=6 cos 2 i sin 2 = cos 0 i sin 0 Begins to repeat 3 i sin 3 = 2 2 i sin = 3 3 The n distinct nth roots of 1 are called the nth roots of unity. CHAPTER 4.4 EXAMPLE 3: Finding the nth Roots of a Complex Number Find the three cube roots of z 2 2i . Because z lies in Quadrant _____, the trigonometric form of z is r cos i sin z 2 2i r = tan = = = = z = By the formula for nth roots, the cube roots have the form Finally you obtain the roots k=0 = = k=1 = k=2 = DAY