The greatest value of 12sinθ − 9si$n^2$θ as θ varies is:

Put $t$ = sinθ, so −1 ≤ $t$ ≤ 1 and the expression is $12t-9t^2=4-9(t-2/3{)}^2$. The peak is at $t=2/3$, which is a legal value of sinθ, and there the value is 4. Assuming the maximum happens at sinθ = 1 gives only 12 − 9 = 3.