Age 16 to 18 Challenge Level:
As a brute force approach you can choose 4 values of x, work out
sin(x) and then solve the 4 simultaneous equations in 4 variables.
This will give a polynomial which matches the values for your
choices of x.
However, you can save time by noting the symmetry properties of
sin(x), in that sin(-x) = - sin(x).