Question

Range of $5\cos \Theta +3\cos (\Theta -\frac{\pi }{3})+8$

• A. $[-7,1]$
• B. [1,8]
• C. [1, 15]
• D. [2, 15]

Answer 1

Answer: (C)

[1, 15]

$f\left(\theta \right)=5cos\theta +3cos(\theta -\frac{\pi }{3})+8$
$=5cos\theta +3[cos\theta cos\frac{\pi }{3}-sin\theta sin\frac{\pi }{3}]+8$
$=5cos\theta +\frac{3}{2}[cos\theta -sin\theta ]+8$
$f\left(\theta \right)=\frac{13cos\theta -3sin\theta +16}{2}$
range of $13cos\theta -3sin\theta ϵ[-\sqrt{169+9},\sqrt{169+9}]$
$ϵ[-\sqrt{174},\sqrt{174}]$
$16\le \sqrt{174}\le 17$
$\therefore$ range of $f\left(\theta \right)ϵ[\frac{16-\sqrt{174}}{2},\frac{16+\sqrt{174}}{2}]$
$[1,15]$ is a part of $[\frac{16-\sqrt{174}}{2},\frac{16+\sqrt{174}}{2}]$