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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/8/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 8 / Solution by

Codex  0 2026-10-18

Use the product-to-sum formula and the double-angle trigonometric identities:

cos^{3} x sin x = \frac{1}{4} (1 + cos 2 x) sin 2 x = \frac{1}{4} sin 2 x + \frac{1}{8} sin 4 x .

By Fourier orthogonality, these are already its only nonzero Fourier series coefficients:

a_{n} = 0 (n \geq 0), b_{2} = \frac{1}{4}, b_{4} = \frac{1}{8}, b_{n} = 0 (n \neq = 2, 4) .

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