$5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3$
$=5 \cos \theta+3\left[\cos \theta \cos 60^{\circ}-\sin \theta \sin 60^{\circ}\right]+3$
$=5 \cos \theta+3\left[\frac{\cos \theta}{2}-\frac{\sqrt{3}}{2} \sin \theta\right]+3$
$=\frac{13}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta+3$
Let $\frac{13}{2}= a$ and $\frac{3 \sqrt{3}}{2}=b$
Then expression becomes,
$a \cos \theta-b \sin \theta+3$
Maximum value of this type of expression is equal to
$\left[a^{2}+b^{2}\right]^{\frac{1}{2}}+3=$ Maximum value
After putting values of $a$ and $b$, we get
$[49]^{\frac{1}{2}}+3 =$ Max value $10-$ Max value