 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/10/solution

sin x cos x = \frac{1}{2} sin 2 x

, and the complementary-angle trigonometric identity gives

cos (π /2 - x) = sin x

. Thus the Fourier series is already the finite expression

sin x + \frac{1}{2} sin 2 x .

The only nonzero Fourier coefficients are

b_{1} = 1

and

b_{2} = 1/2

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