Step by step
The question
Question: A is 3 km south and 4 km east of B: find the distance and the bearing of A from B.
1 Distance
3 km south, then 4 km east: the two short sides of a right triangle.
Pythagoras: 3 squared plus 4 squared is 25.
The square root is 5 km.
BA = √(3² + 4²) = √25 = 5 km
2 Angle from the north-south line
At B, the east 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(43) ≈ 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: 126.9°.
bearing = 180° − 53.1° = 126.9°
4 Check
Let's check: the distance times cos of the angle gives back about 3 km south.
So it is right!
5 × cos 53.1° ≈ 3.0 = 3 km south
Let's look back.
First: distance.
Then: angle from the north-south line.
Then: bearing.
Last: check.
Answer
bearing = 180° − 53.1° = 126.9°