Question

The maximum value of 12 $\sin \theta -9{\sin }^2\theta$ is

• A. 3
• B. 4
• C. 5
• D. none of these

Answer 1

Answer: (B)

$12\sin \theta -9{\sin }^2\theta =-9\left({\sin }^2\theta -\frac{12}{9}\sin \theta \right)$
$=-9\left({\sin }^2\theta -\frac{4}{3}\sin \theta \right)$
$=-9\left[{\sin }^2\theta -\frac{4}{3}\sin \theta +{\left(\frac{2}{3}\right)}^2-{\left(\frac{2}{3}\right)}^2\right]$
$=-9{\left(\sin \theta -\frac{2}{3}\right)}^2+9\times \frac{4}{9}\le 4$