824 CHAPTER 8 Applications of Trigonometry (b) A sketch of 5 - 4i shows that u must be in quadrant IV. See Figure 52. r = 252 + 1-422 = 241 and tan u = - 4 5 Use a calculator to find that one measure of u is -38.66°. In order to express u in the interval 30, 360°2, we find u = 360° - 38.66° = 321.34°. 5 - 4i = 241 cis 321.34° S Now Try Exercises 61 and 65. y –4 x 5 – 4i u = 321.34° 5 0 r = √41 Figure 52 An Application of Complex Numbers to Fractals At its basic level, a fractal is a unique, enchanting geometric figure with an endless self-similarity property. A fractal image repeats itself infinitely with ever-decreasing dimensions. If we look at smaller and smaller portions, we will continue to see the whole — it is much like looking into two parallel mirrors that are facing each other. EXAMPLE 5 DeterminingWhether a Complex Number Is in the Julia Set The fractal called the Julia set is shown in Figure 53. To determine whether a complex number z = a + bi is in this Julia set, perform the following sequence of calculations. z2 - 1, 1z2 - 122 - 1, 31z2 - 122 - 142 - 1, c If the absolute values of any of the resulting complex numbers exceed 2, then the complex number z is not in the Julia set. Otherwise z is part of this set, and the point 1a, b2 should be shaded in the graph. Figure 53 Determine whether each number belongs to the Julia set. (a) z = 0 + 0i (b) z = 1 + 1i SOLUTION (a) Here z = 0 + 0i = 0, z2 - 1 = 0 2 - 1 = -1, 1z2 - 122 - 1 = 1-122 - 1 = 0, 31z2 - 122 - 142 - 1 = 0 2 - 1 = -1, and so on. We see that the calculations repeat as 0, -1, 0, -1, and so on. The absolute values are either 0 or 1, which do not exceed 2, so 0 + 0i is in the Julia set, and the point 10, 02 is part of the graph.
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