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 $c_n=|a_n|$ ).