f(x) = 1/(2) g(t) e^{^(i tx)} dt

g(x) = 1/(2) f(t) e^{^(-i tx)} dt

Fourier Sine and Cosine Transforms

f(x) = (2/) g(x) cos(xt) dt (Cosine Transform)
f(x) = (2/) g(x) sin(xt) dt (Sine Transform)

IF f(x) is even, THEN FourierSineTransform( FourierSineTransform(f(x)) ) = f(x)
IF f(x) is odd, THEN FourierCosineTransform( FourierCosineTransform(f(x)) ) = f(x)

Under certain restrictions of continuity.