Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 18 / a / ii

With inner product $⟨u,v⟩={\int }_{-\pi }^{\pi }u(x)v(x)dx$, orthogonality meansThe constant function has squared norm $2\pi$, and each sine or cosine has squared norm $\pi$. Thus the displayed family is orthogonal but not normalized; these different lengths determine the coefficient factors.