Question

Find the range of $12\sin \theta -9{\sin }^2\theta$

• A. $[4,21]$
• B. $[-21,4]$
  
• C. $[-4,21]$
• D. $[0,\infty )$

Answer 1

The correct option is B

$[-21,4]$

$y=12\sin \theta -9{\sin }^2\theta =4-4+12\sin \theta -9{\sin }^2\theta =4-(4-12\sin \theta +9{\sin }^2\theta )=4-{(2-3\sin \theta )}^2$
As $-1\le \sin \theta \le 1$
$\therefore -21\le y\le 4$