A man walks 5 km due north and then 4 km due west. What is his bearing from the starting point?

Let point A be the starting point.
The man walks $5 \text{ km}$ due north from A to an intermediate point, let's call it B.
From B, he walks $4 \text{ km}$ due west to point C.
This movement forms a right-angled triangle $\triangle ABC$, with the right angle at B.
The side AB represents the Northward movement ($5 \text{ km}$).
The side BC represents the Westward movement ($4 \text{ km}$).

We need to find the angle $\angle BAC$, which is the angle measured from the North line (AB) towards the West.
In the right-angled triangle $\triangle ABC$:

Point C is located North-West of point A. Bearings are measured clockwise from the North line ($0^\circ$).
The angle $\angle BAC$ is the angle measured from the North line towards the West.
To find the bearing, we subtract this angle from $360^\circ$ (since it's in the North-West quadrant).

Rounding to one decimal place, the bearing is $321.3^\circ$.