Step by step
The question
Question: A is 5 km north and 12 km east of B: find the distance and the bearing of A from B.
1 Distance
5 km north, then 12 km east: the two short sides of a right triangle.
Pythagoras: 5 squared plus 12 squared is 169.
The square root is 13 km.
BA = √(5² + 12²) = √169 = 13 km
2 Angle from the north-south line
At B, the east side is across from the angle and the north side is next to it: tan.
tan to the minus 1 gives about 67.4°.
angle = tan⁻¹(125) ≈ 67.4°
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
North, then turning clockwise to the east side: the bearing is the angle itself, 067.4°.
bearing = 067.4°
4 Check
Let's check: the distance times cos of the angle gives back about 5 km north.
So it is right!
13 × cos 67.4° ≈ 5.0 = 5 km north
Let's look back.
First: distance.
Then: angle from the north-south line.
Then: bearing.
Last: check.
Answer
bearing = 067.4°