Question

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

• A. 5
• B. 11
• C. 10
• D. $-1$

Answer 1

Answer: (C)

$5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3$
$=5\cos \theta +3\left(\cos \theta \cos \frac{\pi }{3}-\sin \theta \sin \frac{\pi }{3}\right)+3$
$=5\cos \theta +\frac{3}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
$=\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3$
$\therefore$ Maximum value $=3+\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(-\frac{3\sqrt{3}}{2}\right)}^2}$
$-3\sqrt{\frac{196}{4}}-3+7-10$