Oxford University MAT 2021 Solved Papers

The function inside the brackets is $6{\sin }^2\mathit{x}-8\sin \mathit{x}+3$ which is a quadratic for $\sin \mathit{x}$. We could therefore consider the quadratic $6{\mathit{u}}^2-8\mathit{u}+3$ for $-1\le \mathit{u}\le 1$. Complete the square to write this as $6{\left(\mathit{u}-\frac{2}{3}\right)}^2+\frac{1}{3}$. This reaches a minimum value when $\mathit{u}=\frac{2}{3}$. For $\mathit{u}=\sin \mathit{x}$ in the range $0\le \mathit{x}\le {360}^{\circ }$, this happens for two values of $\mathit{x}$ both in . The value there is . Only one of the graphs reaches a negative minimum value twice in that range.