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 $=3+\sqrt{\left(\frac{13}{2}\right)^2+\left(-\frac{3 \sqrt{3}}{2}\right)^2}$
$-3 \sqrt{\frac{196}{4}}-3+7-10$