Step by step
The question
Question: A is 2 km south and 5 km west of B: find the distance and the bearing of A from B.
1 Distance
2 km south, then 5 km west: the two short sides of a right triangle.
Pythagoras: 2 squared plus 5 squared is 29.
The square root is about 5.4 km.
BA = √(2² + 5²) = √29 ≈ 5.4 km
2 Angle from the north-south line
At B, the west side is across from the angle and the south side is next to it: tan.
tan to the minus 1 gives about 68.2°.
angle = tan⁻¹(52) ≈ 68.2°
Careful!
The angle we found is from the south line.
A bearing is measured from north, turning clockwise.
Careful: a bearing is always from NORTH, clockwise
3 Bearing
From north, clockwise: 248.2°.
bearing = 180° + 68.2° = 248.2°
4 Check
Let's check: the distance times cos of the angle gives back about 2 km south.
So it is right!
5.4 × cos 68.2° ≈ 2.0 = 2 km south
Let's look back.
First: distance.
Then: angle from the north-south line.
Then: bearing.
Last: check.
Answer
bearing = 180° + 68.2° = 248.2°