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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).
Building Approximations for Sin(x)
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).
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