Orthogonality Solves a Brutal Trigonometric Integral
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This 2023 MIT Integration Bee Final problem looks brutal: a squared ratio of eight sine functions. The key is to simplify the ratio first using double-angle and product-to-sum identities. The entire ratio expression collapses into a cosine polynomial: R(x)=2 cos(8x)+2 cos(6x)-2 cos(2x)-1 Then cosine orthogonality on ([0,pi]) makes every cross ...
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