Question

If $f\left(\theta \right)=5\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$, then range of $f\left(\theta \right)$ is-

• A. [-5,11]
• B. [-3,9]
• C. [-2,10]
• D. [-4,10]

Answer 1

The correct option is D [-4,10]

$f\left(\theta \right)=5\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$
$=5\cos \theta +3(\cos \theta .\cos \frac{\pi }{3}-\sin \theta .\sin \frac{\pi }{3})+3$
$=(5+\frac{3}{2})\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
$=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
Now since $-7\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le 7$
Therefore range of $f\left(\theta \right)$ is, $[4,10]$