This is what Mathematica produces for the Fourier transform of Sin[w0 t]: 
Ignoring the Sqrt(Pi/2) factor, the Fourier transform of Sin[w0 t] is: 
(source: https://www.tutorialspoint.com/fourier-transform-of-the-sine-and-cosine-functions & many others). Notice the sign difference of the i (or j, in the engineering world) in the above expressions. Also, quite disturbingly, the time shifting property of the Fourier transform appears to be wrong in WolframAlpha, but nevertheless consistent with the Mathematica behavior reported above: 
compare with: 
What am I doing wrong? I sure hope this is not a bug.
