We consider the periodic continuation of a 'neat' function \(f(x)\) on the interval \((-\pi,\pi)\) and approximate it with the Fourier series: \[f(x) = a_0+\sum_{n=1}^{\infty}\bigl(a_n\cos(nx) +b_n\sin(n x)\bigr)\] The Fourier coefficients \(a_n\) and \(b_n\) (with \(n\ge 1\) ) both describe a trigonometric contribution with the same frequency \(n\): The only difference is that \(a_n\) is a cosine contribution, whereas \(b_n\) is a sine contribution. This cosine can be regarded as a shifted sine. After all, \(\sin(x)=\cos(x-\tfrac{1}{2}\pi)\). The total contribution of the terms to the frequency is \(a_n\cos(nx)+b_n\sin(nx)\). Also, this sum can be written as a shifted sine: for given \(a_n\) and \(b_n\) there exist numbers \(c_n\) and \(\varphi_n\) such that \[a_n\cos(nx)+b_n\sin(nx)=c_n\sin(nx+\varphi_n)\] The coefficient is called the amplitude of the frequency \(n\) in the function \(f\). The shift is given by the phase angle \(\varphi_n\). The following applies to the amplitude: \[c_n=\sqrt{a_n^2+b_n^2}\] The sequence \(c_1, c_2, c_3, \ldots\) is called an amplitude spectrum. The bar graph of the function \(n\mapsto c_n\) is called the frequency-amplitude spectrum of the function \(f\). In this spectrum, we always represent the coefficients \(c_n\); only in the case of a Fourier sine series (or Fourier cosine series) we have \(c_n=|b_n|\) (or \(c_n=|a_n|\) ).