Symmetry Considerations of The Fourier Series

We noticed that the Fourier series of Example $ 1 $ consisted only of the sine terms. One may wonder if a method exists whereby one can know in advance that some Fourier coefficients would be zero and avoid the unnecessary work involved in the tedious process of calculating them. Such a method does exist; it is based on recognizing the existence of symmetry. Here we discuss three types of symmetry:
(1) even symmetry,
(2) odd symmetry,
(3) half-wave symmetry.