How to get some law parameters from its FFT figure?
If you feed an FFT a finite set of samples from a function of theoretically infinite range, you have implicitly windowed your function with a rectangular window before the FFT.  Thus, the FFT result will be the convolution of the spectrum of your function and the transform of a rectangular window (a periodic Sinc).  This convolution will be an identity operation only if the samples happen to span exactly an integer multiple of periods of a purely periodic function.  So how much your FFT result will change or differ will depend on the offset and length of your FFT aperture/window with respect to some reference point on and the scale of your function.

In the case of a single sinusoid in zero noise, I reference 2 papers by Clay Turner on solving for the 3 unknowns using only 3 or 4 non-aliased samples, somewhere on my web page: http://www.nicholson.com/rhn/dsp.html   No FFT needed.