Explanation

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