The Phase Information of the DFT
If you assume that for a given DFT length $N$ your signal is given by

$$x[n]=A\cos(2\pi n/N+\phi),\quad n=0,1,\ldots,N-1$$

i.e. its period equals $N$, then the first DFT coefficient (starting to count from $0$) is given by

$$X[1]=A\frac{N}{2}e^{j\phi}$$