In the given diagram, we need to determine the measure of angle $\angle AC^{\circ}$.

From the problem:

1. Circle $O$ has a radius of 4.
2. Segment $BC$ is 3 units.
3. Line $AD$ is tangent to the circle at point $A$.

Since $AD$ is tangent to the circle at $A$, and $OA$ is a radius, we know that $OA$ is perpendicular to $AD$. This makes $\angle OAD$ a right angle, or $90^\circ$.

If we consider triangle $OAC$, we see that $OC$ is a line segment extending from the center of the circle to the line $BC$ (which is not necessarily tangent). Thus, $OC$ is not necessarily perpendicular to $BC$ or any specific known angle.

However, since $BC$ is not tangent and no additional information is given about $\angle BOC$ or $\angle BAC$, it is difficult to directly calculate $\angle AC$ using typical circle or triangle properties without more details about the relationships or positioning of these segments and points.

The problem seems to require an assumption about the relative positioning of these elements which isn't directly supported by the given information. Thus, calculating $\angle AC$ to an exact degree would likely need either additional information or an assumption that might not be validated by the problem's statement.

If you can provide any further details about the positioning or relationships in the diagram, or if there is additional information not shown here, it would be helpful in solving the problem more accurately.