Question

$5\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$

Answer 1

$5\cos \theta +3\cos (\theta +\frac{\pi }{3})+3$

$=5\cos \theta +3(\cos \theta \cos \frac{\pi }{3}-\sin \theta \sin \frac{\pi }{3})+3$

$=5\cos \theta +3\ast \cos \theta \ast \frac{1}{2}-3\ast \sin \theta \ast \sqrt{\frac{3}{2}}+3$

$=\frac{13}{2}cos\theta -3\sqrt{\frac{3}{2}}\sin \theta +3$

The range of $a\cos \theta +b\sin \theta +c$ is

$c\pm \sqrt{(a^2+b^2)}$

Hence, the range of given function is

$3\pm \sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}$

$=-4$ & $10$