I am a student majoring in electrical engineering.
There is three equations about Fourier series.
\begin{align} x(t)&=\sum_{n=-\infty}^{\infty}X_n e^{j2\pi nf_0t} &&&& (1)\\ X_n&=\frac1{T_0}\int_{t_0}^{t_0+T_0}x(t)e^{-j2\pi nf_0t}dt &&&& (2)\\ x(t)&=X_0 + \sum_{n=1}^{\infty} A_n \cos(2\pi nf_0t) + \sum_{n=1}^{\infty}B_n \sin(2\pi nf_0t) &&&& (3)\\ &~~~~~~~~~~~~,where~~~~A_n = \frac2{T_0} \int_{t_0}^{t_0+T_0} x(t) \cos(2\pi nf_0t)dt\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~B_n = \frac2{T_0} \int_{t_0}^{t_0+T_0} x(t) \sin(2\pi nf_0t)dt\\ \end{align}
I know equations $(1)$ and $(2)$ are used when changing domain from time to frequency, and vice versa.
I want to know where the equation $(3)$ is used.
I regard equation $(3)$ is weird because $x(t)$ is expressed as a function of $x(t)$ itself.
Please let me know why equation $(3)$ is also important. Thank you.
From a theoretical standpoint, Equation 3 shows you how to decompose $x(t)$ into orthogonal functions (sines and cosines), which can help you draw conclusions about $x(t)$ which aren't obvious from the original function.
See, for instance, Example 1 from the Wikipedia article on Fourier Series: https://en.wikipedia.org/wiki/Fourier_series#Example_1:_a_simple_Fourier_series
Example 2 may help as well: https://en.wikipedia.org/wiki/Fourier_series#Example_2:_Fourier.27s_motivation