A true bearing describes a direction measured clockwise from the north direction (0°) around a full circle up to 360°. Bearings are always given using three digits (e.g., 045°, 120°, 270°) to avoid confusion.

• A bearing of 000∘ points directly north.

• 090∘ points east.

• 180∘ points south.

• 270∘ points west.

When working with true bearings, clearly draw a compass rose to visualize directions and measure angles clockwise from the north line.

One very useful fact about bearings is that returning in the direction something came from means adding or subtracting 180° from the original bearing - since it is doing a U-turn.

Whether you add or subtract depends on whether the original bearing is smaller or bigger than 180°, since the resulting bearing must be between 0 and 360°.

Example

A ship leaves port P at bearing of 075° and sails 4km to a lighthouse L. It then turns 55° away from north, and sails 3km to a dolphin watching spot D. Find

• The bearing the ship must take to get back to L.

• The distance P from the port.

• The bearing the ship must take to return to port.

The ship starts at a 075° bearing, and then turns 55° away from north, giving 130°. If the ship were to make a U-turn, it would have a bearing of 130°+180°=310° coming back to L.

Since ∠NPL=75°, ∠NLP=180°−75°=105°. Thus the angle ∠PLN=360°−130°−105°=125°. Using the cosine rule:

Now using the sine rule:

So ∠PDL=sin−1(0.526)=31.8°. Finally, we recall that the bearing from D→L was 310°, so the bearing from D→P is 310°−∠PDL=278°.