Initialization

SetOptions[Plot,PlotStyle->{{Black,AbsoluteThickness[2]},{Red,AbsoluteThickness[2]},{Blue,AbsoluteThickness[2]}},AxesLabel->{"t"},ImageSize->500,AxesStyle->Directive[FontFamily->"Arial",FontSize->18,Black,AbsoluteThickness[0.5],Arrowheads[0.04]]];

Work

Guessing the Fourier Expansion of a Function

Look at the function f(t) plotted below.  It is a linear combination of sin(nx) for different values of n, i.e.

f(t) = b1sin(t) + b2sin(2t) + b3sin(3t)+...

Your job is to guess the values of the coefficients.  For simplicity, there are 3 nonzero terms, all of the coefficients are positive or negative integers, and there are no terms for n > 5.

The function is defined below, but I've disguised it so you won't be able to tell what the coefficients are by inspection!

Clear[f,t]

f[t_]:=Sin[t]*4*(1-Cos[t])-4*Sin[t]^3

Plot[f[t],{t,0,4*Pi}]

You might want to start by plotting sin(nx) for various values of n if you don't remember what the sin(nx) function looks like:

Plot[Sin[2*t],{t,0,2*Pi}]

Guess the most important term.  Plot your guess and the function f(t) on the same graph. 
Keep guessing more and more terms until the graph for your guess and the graph for f(t) overlap.  For example:

myguess:=1*Sin[t]+2*Sin[2*t]

The unknown function is BLACK and your guess is RED, and the difference between the two is BLUE (which you want to drive to zero).

Plot[{f[t],myguess,f[t]-myguess},{t,0,2*Pi}]