838 CHAPTER 8 Applications of Trigonometry 52. Basins of Attraction The fractal shown in the figure is the solution to Cayley’s problem of determining the basins of attraction for the cube roots of unity. The three cube roots of unity are w1 = 1, w2 = - 1 2 + 23 2 i, and w3 = - 1 2 - 23 2 i. A fractal of this type can be generated by repeatedly evaluating the function ƒ1z2 = 2z3 + 1 3z2 , where z is a complex number. We begin by picking z1 = a + bi and successively computing z2 = ƒ1z12, z3 = ƒ1z22, z4 = ƒ1z32, . . . . Suppose that if the resulting values of ƒ1z2 approach w1, we color the pixel at 1a, b2 red. If they approach w2, we color it blue, and if they approach w3, we color it yellow. If this process continues for a large number of different z1, a fractal similar to the figure will appear. Determine the appropriate color of the pixel for each value of z1. (Data from Crownover, R., Introduction to Fractals and Chaos, Jones and Bartlett Publishers.) (a) z1 = i (b) z1 = 2 + i (c) z1 = -1 - i 53. The screens here illustrate how a pentagon can be graphed using a graphing calculator. Note that a pentagon has five sides, and the Tstep is 360 5 = 72. The display at the bottom of the graph screen indicates that one fifth root of 1 is 1 + 0i = 1. Use this technique to find all fifth roots of 1, and express the real and imaginary parts in decimal form. Relating Concepts For individual or collaborative investigation (Exercises 59 – 62) In earlier work we derived identities, or formulas, for cos 2u and sin 2u. These identities can also be derived using De Moivre’s theorem. Work Exercises 59 – 62 in order, to see how this is done. 59. De Moivre’s theorem states that 1cos u + i sin u22 = ________. 60. Expand the left side of the equation in Exercise 59 as a binomial and combine like terms to write the left side in the form a + bi. 61. Use the result of Exercise 60 to obtain the double-angle formula for cosine. 62. Repeat Exercise 61, but find the double-angle formula for sine. 54. Use the method of Exercise 53 to find the first three of the ten 10th roots of 1. Use a calculator to find all solutions of each equation in rectangular form. 55. x2 - 3 + 2i = 0 56. x2 + 2 - i = 0 57. x5 + 2 + 3i = 0 58. x3 + 4 - 5i = 0 −1.2 −1.8 1.2 1.8 The calculator is in parametric, degree, and connected graph modes.
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