The mınımum value of $5\cos \mathrm{\theta }+3\cos (\mathrm{\theta }+\mathrm{\pi }/3)+3 \mathrm{is}$

$5\cos \mathrm{\theta }+3\cos (\mathrm{\theta }+\mathrm{\pi }/3)+3=5\cos \mathrm{\theta }+3[\cos \mathrm{\theta cos}(\mathrm{\pi }/3)-\sin \mathrm{\theta sin}(\mathrm{\pi }/3\left)\right]+3$
$=\frac{13}{2}\cos \mathrm{\theta }-\frac{3\sqrt{3}}{2}\sin \mathrm{\theta }+3$.

 Minimum value $=3-\sqrt{169/4+27/4}=3-7=-4$