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Trig Series

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October 24, 2005
trigonometryabsolute valuecomplex numbers

Problem Statement

For any complex number w=a+biw = a + bi, ∣w∣|w| is defined to be the real number a2+b2\sqrt{a^2 + b^2}. If w=cos⁡40∘+isin⁡40∘w = \cos{40^\circ} + i\sin{40^\circ}, then ∣w+2w2+3w3+⋯+9w9∣−1 |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} equals
<spanclass=′latex−bold′>(A)</span> 19sin⁡40∘<spanclass=′latex−bold′>(B)</span> 29sin⁡20∘<spanclass=′latex−bold′>(C)</span> 19cos⁡40∘<spanclass=′latex−bold′>(D)</span> 118cos⁡20∘<spanclass=′latex−bold′>(E)</span> none of these<span class='latex-bold'>(A)</span>\ \frac{1}{9}\sin{40^\circ} \qquad <span class='latex-bold'>(B)</span>\ \frac{2}{9}\sin{20^\circ} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{9}\cos{40^\circ} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{18}\cos{20^\circ} \qquad <span class='latex-bold'>(E)</span>\text{ none of these}
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