Step by step
The question
Question: A is 9 km south and 12 km west of B: find the distance and the bearing of A from B.
1 Distance
9 km south, then 12 km west: the two short sides of a right triangle.
Pythagoras: 9 squared plus 12 squared is 225.
The square root is 15 km.
BA = √(9² + 12²) = √225 = 15 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 53.1°.
angle = tan⁻¹(129) ≈ 53.1°
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: 233.1°.
bearing = 180° + 53.1° = 233.1°
4 Check
Let's check: the distance times cos of the angle gives back about 9 km south.
So it is right!
15 × cos 53.1° ≈ 9.0 = 9 km south
Let's look back.
First: distance.
Then: angle from the north-south line.
Then: bearing.
Last: check.
Answer
bearing = 180° + 53.1° = 233.1°