Step by step
The question
Question: A is 10 km north and 4 km west of B: find the distance and the bearing of A from B.
1 Distance
10 km north, then 4 km west: the two short sides of a right triangle.
Pythagoras: 10 squared plus 4 squared is 116.
The square root is about 10.8 km.
BA = √(10² + 4²) = √116 ≈ 10.8 km
2 Angle from the north-south line
At B, the west side is across from the angle and the north side is next to it: tan.
tan to the minus 1 gives about 21.8°.
angle = tan⁻¹(410) ≈ 21.8°
Careful!
The angle we found is from the north line.
A bearing is measured from north, turning clockwise.
Careful: a bearing is always from NORTH, clockwise
3 Bearing
From north, clockwise: 338.2°.
bearing = 360° − 21.8° = 338.2°
4 Check
Let's check: the distance times cos of the angle gives back about 10 km north.
So it is right!
10.8 × cos 21.8° ≈ 10.0 = 10 km north
Let's look back.
First: distance.
Then: angle from the north-south line.
Then: bearing.
Last: check.
Answer
bearing = 360° − 21.8° = 338.2°