Answer
$$\theta=\frac{\pi}{6}.$$
Work Step by Step
\begin{align*}
\cos \theta &=\frac{\langle 5,\sqrt{3}\rangle \cdot \langle \sqrt{3},2\rangle}{\|\langle 5,\sqrt{3}\rangle\|\|\langle \sqrt{3},2\rangle\|}\\
&=\frac{5\sqrt{3}+2\sqrt{3}}{\sqrt{25+3}\sqrt{3+4}}\\
&=\frac{7\sqrt{3}}{\sqrt{28}\sqrt{7}}\\
&=\frac{\sqrt{3}}{2}
\end{align*}
That is $$\theta=\cos^{-1}\frac{\sqrt{3}}{2}=\frac{\pi}{6}.$$
