Below we graph the cycloid with these settings and

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Unformatted text preview: 1. Again, assuming β > 0. 852 Applications of Trigonometry √ √ square roots of 16, while 16 means the principal square root of 16 as in 16 = 4. Suppose we wish to find all complex third (cube) roots of 8. Algebraically, we are trying to solve w3 = 8. We √ know that there is only one real solution to this equation, namely w = 3 8 = 2, but if we take the time to rewrite this equation as w3 − 8 = 0 and factor, we get (w − 2) w2 + 2w + 4 = 0. The √ quadratic factor gives two more cube roots w = −1 ± i 3, for a total of three cube roots of 8. In accordance with Theorem 3.14, since the degree of p(w) = w3 − 8 is three, there are three complex zeros, counting multiplicity. Since we have found three distinct zeros, we know these are all of the zeros, so there are exactly three distinct cube roots of 8. Let us now solve this same problem using the machinery developed in this section. To do so, we express z = 8 in polar form. Since z = 8 lies 8 units away on the positive real axis, we get z = 8cis(0). If we let w = |w|cis(α) be a polar form of w, the equation w3 = 8 becomes w3 = 8 (|w|cis(α))3 = 8cis(0) |w|3 cis(3α) = 8cis(0) DeMoivre...
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