A regular Fourier Series will turn out to contain only sine terms if the target function is odd, only cosine terms if the target function is even, and both sine terms and cosine terms if the target function is neither even nor odd.
By contrast, my understanding is that Fourier Sine Series and Fourier Cosine Series are completely different animals, and that the mathematician can choose to represent their target function using a regular Fourier Series, a Fourier Sine Series, or a Fourier Cosine Series at their discretion by temporarily introducing a regular extension, odd extension, or even extension, respectively, regardless of the target function's actual even or odd status. It is perhaps natural to use regular Fourier Series to represent functions within intervals centered around $0$, and Fourier Sine Series or Fourier Cosine Series to represent functions within intervals whose left boundary is $0$, but it should be possible to simply $u$-substitute any finite interval function into either of these forms, do the Fourier Series/Fourier Sine Series/Fourier Cosine Series in $u$, and then revert the substitution.
On the other hand, this would mean that any function (continuous or piecewise smooth, a.k.a., any function that could be represented by a regular Fourier Series) could be represented by a single family of sinusoids, either sines or cosines, which seems to be at odds with the Linear Algebra terminology often associated with Fourier Series. For example, thinking of the sines and cosines in a Fourier Series as a basis leads me to believe that you should expect to require both to represent an arbitrary function, not just one or the other.
Is it always possible to represent any function that could be represented by a regular Fourier Series by your choice of Fourier Sine Series or Fourier Cosine Series? If so, how is this in concord with the Linear Algebraic explanation of Fourier Series?