5. Let $f(z)=3 z\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$, where $z$ is a complex number. Let $A$ represent the point $f(1+\sqrt{3} i)$, $B$ represent the point $f(0)$, and $C$ represent the point $f(4 i)$. Then $\angle A B C=$ $\qquad$
5. $\frac{\pi}{6}$.

Let $f_{1}(z)=3 z$,
$f_{2}(z)=z\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$.
$f(z)=f_{1} f_{2}(z) \cdot f_{1}(z)$ is a similarity transformation, $f_{2}(z)$ is a rotation transformation, then $f(z)$ is a transformation that preserves angles. Let $A^{*}$ represent the point $1+\sqrt{3} i$, $B^{*}$ represent the point 0, $C^{*}$ represent the point $4 i$, then
$$
\begin{array}{l}
\angle A B C=\angle A^{\cdot} B^{\cdot} C^{\cdot} \\
=\frac{\pi}{2}-\arg (1+\sqrt{3})=\frac{\pi}{2}-\frac{\pi}{3}=\frac{\pi}{6} .
\end{array}
$$