I know that Fourier transform returns frequency spectrum and it contains both Sine & Cosine terms as "imaginary and real terms". However, we also have individual Sine and Cosine transforms.
- How does one know when to use which?
- What is the strength, weakness of each?
- Are there things that one can do better than the other?
- Are there things that one can do that the others cannot?
Fourier originally used cosine and sine transforms, not the Fourier transform. The nice thing about sine and cosine transforms is that they allow one to simplify the integral when the function is either even or odd, which you cannot do with the standard FT. They are still used in statistics and especially in signal processing.
For example, if the function is even, the sine transform is zero. It it's odd, the cosine transform is zero. This sometimes makes it easier to evaluate integrals.