How to get coefficients for sine/cosine function from complex FFT?
This is a no brainer for anyone who's studied signal processing (Im wondering why its been so long since its been answered) 

The Fourier Transform itself is a function that tells you what sinusoid do you need to add together to create this continuous signal. (Its like the Fourier series for more complex signals) 

the FFT algorithm gives you COMPLEX coefficients (a+bj) (where j is the imaginary number... it looks like and i but backwards....) 

calculate the magnitude of each coefficient A(n) = sqrt(a(n)^2 + b(n)^2) 
calculate the phase of each coefficient phi(n) = inverse_tan(b(n)/a(n)); (remember to use atan2 so that it results in the correct quadrant) 

now remember based on your sampling rate each coeff represents a "bin" of frequencies, however you should think of them as evenly distributed frequencies:

So if you have N bins each frequency_bin they represent is f = Fs/(2*N)

then your signal becomes

$$
f(x) = \Sigma^N_0 A(n)*cos(2*\pi*f(n) + \phi(n))
$$

also: Predicting function: Impossible. that would be asking for a non-causal system.