I am trying to think about the sine transform of symmetric functions, chiefly the unit rectangle and the unit triangle function. I am poor with LaTeX so I'll do this as best I can
Say I have the Triangle function. Then, its sine transform is defined as
$$\int_{-1}^0(t+1)\sin(2\pi s t) d t + \int_0^1 (1-t)\sin(2\pi s t) d t$$
Now, since the triangle function is an even function, and the sine function is an odd function, the odd function times an even function gives me an odd function. Knowing that the interval of the FT integral will be symmetric, these facts lead me to conclude the Sine transform of the Triangle function is zero.
This also leads me to believe that the cosine transform of the triangle function is equal to the Fourier transform of the triangle function, strictly because the sine transform is zero.
Is my train of thought correct?