The given expression is

$5\cos \theta +3\left(\cos \theta \cos {60}^{\mathrm{o}}-\sin \theta \sin {60}^{\mathrm{o}}\right)+3$

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

$=\cos \theta \left(5+\frac{3}{2}\right)+\sin \theta \left(-\frac{3\sqrt{3}}{2}\right)+3$

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

$\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \in \left[-k,k\right]$ where

$k=\sqrt{{\left(\frac{13}{2}\right)}^2+{\left(-\frac{3\sqrt{3}}{2}\right)}^2}=\sqrt{\frac{169}{4}+\frac{27}{4}}=\frac{14}{2}=7$

$\Rightarrow \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \in \left[-\mathrm{7,7}\right]$

$\Rightarrow \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3\in \left[-\mathrm{4,10}\right]$

$\Rightarrow a=-4$ and $b=10$

$\Rightarrow a+b=6$