What's the difference between Fourier Cosine and Sine Series besides the periodic function?

Question

I'm aware for Fourier Cosine Series you have an even extension of f(x) and the Sine Series has an odd extension, the former requiring a_o, a_n, and cosine as the periodic function, with the latter containing b_n with sine as the periodic function. However, can't any function be translated to either its sine or cosine series equivalent? Why bother learning both methods? Is there some inherent difference that I'm not recognizing? If anything, shouldn't you always just calculate the sine series for every f(x) since you only have to compute one coefficient (b_n)?

It's really baffling and I can't find any articles online describing the obvious differences or advantages/disadvantages to either method.

Thanks!

Answer 1

However, can't any function be translated to either its sine or cosine series equivalent?

No.

Recall that a linear combination of even functions is even, and a linear combination of any odd functions is odd. That means neither class is sufficient to represent functions that are neither even nor odd.

Now, what is true is that any function can be written as a sum of an even and an odd function, so you can decompose the even parts in terms of cosine, the odd parts in terms of sine, and add them up to get the Fourier series.

Answer 2

The cosine series captures the even part of a function

$$f_{e}(x) = \dfrac{f(x)+f(-x)}{2}$$

The sine series captures the odd part of a function

$$f_{o}(x) = \dfrac{f(x)-f(-x)}{2}$$

And one can encounter functions for which you can never have a purely even or purely odd function, no matter how the function is shifted, e.g.

$$f(x) = cos(x) + sin(x)$$

So you need both to handle any possible function you might encounter.

Answer 3

Consider the function graphed here:

This function is neither odd nor even. Moreover, you cannot translate it to make it odd or to make it even.

In general you need both cosine and sine terms, not just one or the other. The even and odd functions are the exceptional cases.