Calculating original signal from Discerete Fourier Transform
At first glance your first paragraph seemed wildly confusing to me. (The notion of "calculating" an equation makes no sense.) But now I see what your professor is asking of you. He wants you to determine the coefficients of an equation. He gave you a continuous $f(t)$ equation of a signal containing a DC component (1cos(0$\pi$$t$/4) and three sinusoids. And the 2nd and 3rd sinusoids have frequencies that are integer multiples of the 1st sinusoid. 

So, your prof wants you to select the values and length ($N$) of a $t$ time sequence that will make those sinusoids reside exactly at an $N$-point DFT's bin centers. When that happens the bin centers' frequencies will give you the frequencies of the three sinusoids in your $f(t)$ equation. The peak values of the non-zero DFT spectral components (DC plus the three sinusoids) will enable you to establish the [1, 0.75, 0.2, 0.5] peak amplitude coefficients in your $f(t)$ equation.  Your prof wants you to learn how a sinewave whose peak amplitude is $A$ produces a DFT spectral component magnitude value of $M$ given the $N$ length of your DFT.  Your job is to see how $A$, $M$, and $N$ are related to each other. This is a good homework problem. 

Your $t$ = [0,1,2,3] sequence's period is one which means your sample rate is 1 sample/second. And as Ollie said, that's too slow. Set your $t$ sequence to [0, 0.5, 1, 1.5, 2, ..., 6.5, 7, 7.5], compute a 16-point DFT, compute the spectral magnitudes, and see what happens
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