A mathematician can choose to represent a target function defined from $0$ to $L$ using a Fourier Sine Series or a Fourier Cosine Series at her discretion, by temporarily introducing either an odd extension or an even extension, respectively, regardless of whether the target function is actually even, odd, or neither. I want to know if there is a way to translate one type of series into the other, once the choice has been made.
I am working with the sinesum function in Pure Data as a means of synthesizing waveshapes, which seems to eat numeric Fourier Sine Series coefficients as parameters. However, I need to shift some waves vertically, which is easier to do with a Fourier Cosine Series. There is a cosinesum function which eats Fourier Cosine Series coefficients, but it is not as well documented as sinesum.
Given the numeric Fourier Sine Series coefficients of a target wave, can I convert them into numeric Fourier Cosine Series coefficients mathematically?
If I can reverse engineer the pattern of the numeric coefficients as a mathematical expression, can I then convert that expression into an expression for the Fourier Cosine Series coefficients?
Note that I am not seeking a solution like this, where the answer contains an explicitly shifted series of the other type, but the unshifted series the mathematician would have arrived at if she'd used a different extension of the target function.