Question

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

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

Answer 1

Answer: (C)

$5cos\theta +3cos\left(\theta +\frac{\pi }{3}\right)+3$
$=5cos\theta +3\left(cos\theta cos\frac{\pi }{3}-sin\theta sin\frac{\pi }{3}\right)+3$
$=5cos\theta +\frac{3}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3$
$=\frac{13}{2}cos\theta -\frac{3\sqrt{3}}{2}sin\theta +3$
We know that, the maximum value of
$c+asin\theta +bcos\theta =c+\sqrt{a^2+b^2}$
$\therefore$ Maximum value of $5cos\theta +3cos\left(\theta +\frac{\pi }{3}\right)+3$
$=3+\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(-\frac{3\sqrt{3}}{2}\right)}^2}$
$=3+\sqrt{\frac{196}{4}}=3+\sqrt{49}$
$=3+7=10$