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Amiss cis

Source: 2021 Fall AMC 12B #21

November 17, 2021
AMCAMC 12AMC 12 B

Problem Statement

For real numbers xx, let P(x)=1+cos⁡(x)+isin⁡(x)−cos⁡(2x)−isin⁡(2x)+cos⁡(3x)+isin⁡(3x)P(x)=1+\cos (x)+i \sin (x)-\cos (2 x)-i \sin (2 x)+\cos (3 x)+i \sin (3 x) where i=−1i=\sqrt{-1}. For how many values of xx with 0≤x<2π0 \leq x<2 \pi does P(x)=0?P(x)=0 ?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4<span class='latex-bold'>(A)</span>\: 0\qquad<span class='latex-bold'>(B)</span> \: 1\qquad<span class='latex-bold'>(C)</span> \: 2\qquad<span class='latex-bold'>(D)</span> \: 3\qquad<span class='latex-bold'>(E)</span> \: 4
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