Consider a positive number T. We will focus on the interval [0,T ). If we want to have sines a cosines fit well within this interval, we have to scale them in horizontal direction so that their periods have length commeasurable with T. It is easy to see that the scaling factor is exactly the number ω = 2π/T. Indeed, then we have the following picture.

When we introduce the integer k into the argument, we scale the two functions again, but since the scaling factor is an integer now, we do not spoil the nice fit.

When we work with functions as above, we usually immediately apply substitution to scale them back to the natural interval [0,2π]. For instance, one useful fact about these functions is proved like this:

This substitution goes both ways and without any trouble transforms the setting [0,T ) into the natural setting [0,2π] and back. Thus it is actually enough to know how to work with 2π-periodic functions and other cases can be transformed to this basic one by substitution. For this reason, some authors develop Fourier series only in this natural setting. Since it is not much extra work, we decided to do it in general in order to spare the reader this transforming business.