Question

Write the maximum value of 12 sin θ − 9 sin² θ.

Answer 1

$$\begin{gathered} \mathrm{Let}f\left(\theta \right)=12\sin \theta -9{\sin }^2\theta \\ =-\left(9{\sin }^2\theta -12\sin \theta \right) \\ =-\left[{\left(3\sin \theta \right)}^2-2.3\sin \theta .2+2^2-4\right] \\ =-\left[{\left(3\sin \theta -2\right)}^2-4\right] \\ =4-{\left(3\sin \theta -2\right)}^2 \\ \mathrm{Minimum}\mathrm{value}\mathrm{of}{\left(3\sin \theta -2\right)}^2\mathrm{is}0. \\ \mathrm{Therefore},\mathrm{maximum}\mathrm{value}\mathrm{of}4-{\left(3\sin \theta -2\right)}^2\mathrm{would}\mathrm{be}4. \end{gathered}$$