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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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