Hello SB,
the issue here is that the signal window is not the correct width to give the periodicity that you would like. Right now
t(1) = 0 t(end) = 10 y(1) = 3.1124 y(end) = 3.1124
The first point in y is the same value as the the last point. Periodicity means that time windows can be appended side-by-side to continue the waveform. That means that 3.1124, which is the last point in the first time window, and the first point in the next time window, will be repeated. You no longer have three continuous cosine waves, and instead there is a very small flat part. You can see the effect of that in the fft because the signal, which supposedly only contains three discrete frequencies, shows nonzero amplitude for a lot of other frequencies as well.
If you delete the last point with
t = linspace(0,T,N);
t(end) = [];
and proceed from there, everything works. There are three distinct sharp frequency peaks, no other frequency content, and the phases are correct.
