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)

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