The second paragraph of the Wikipedia page for Fourier Transform says,
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency.
I'm unclear on what,
and whose argument is the phase offset of the basic sinusoid in that frequency.
...actually means in the following context (I have seen no proof anywhere of that claim on Wikipedia by the way): Suppose you have the FT,
$$ℱ_x[sin(x)](ω)=\int_{-\infty}^∞ sin(x) e^{-2πiωx}dx=i π (δ(-2 π ω - 1) - δ(1 - 2 π ω))$$
...and here I can see that a wave of frequency $-\frac{1}{2π}$ and another of frequency $\frac{1}{2π}$ are detected, with no detections of any other frequency. The argument of the FT in each of these cases is $\frac{π}{2}$ and $-\frac{π}{2}$ respectively and so it should follow that I can use these values with a sinusoid to reconstruct $sin(x)$. However, I can't see any way to do this, and I don't want to use the IFT! Thank you for any help you can offer.