Question

The maximum value of $5\cos \theta +3\left(\cos \right(\theta +\frac{\pi }{3}\left)\right)+3$ is

• A. $5$
• B. $10$
• C. $11$
• D. $-1$

Answer 1

The correct option is B $10$

$f\left(\theta \right)=5\cos \left(\theta \right)+3\cos (\theta +\frac{\pi }{3})+3$
$=5\cos \theta +\frac{3}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
$=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
Maximum value of $a\cos \theta +b\sin \theta +c=\sqrt{a^2+b^2}+c$
So, maximum value of $f\left(\theta \right)=10$