Convention

Reading a compass bearing N $α^{\circ}$ E as a right triangle

Trigonometry • Heights and Distances

Statement

A bearing written N

α^{\circ}

E denotes the direction reached by turning

α^{\circ}

from due north towards due east, so a point on that bearing lying

n

units north of the observer lies

e

units east of it with

tan α^{\circ} = \frac{e}{n}

; N

α^{\circ}

W, S

α^{\circ}

E and S

α^{\circ}

W are read the same way with west, south and the corresponding pair substituted.

How to use it

1Draw the north-south line through the observer first, then swing $α^{\circ}$ towards the second letter; the angle is always measured from the north-south line, never from east or west.
2The leg adjacent to $α^{\circ}$ is the north-south offset and the opposite leg is the east-west offset, so $tan α^{\circ}$ is (east-west offset) $/$ (north-south offset).
3With two observers on one east-west line sighting the same point, the two east-west offsets add up to the distance between them when the bearings lean towards each other.

From a point

O

a lighthouse is seen on the bearing N

6 0^{\circ}

E, and the lighthouse is known to be

50

m north of

O

. How far east of

O

is it?

Step-by-step

1
$tan 6 0^{\circ} = \frac{e}{50}$
Why: The north offset $50$ is the leg adjacent to the bearing angle.
2
$e = 50 tan 6 0^{\circ} = 503$

Answer

503

Watch out

A bearing of the form N

α^{\circ}

E is not the same as a three-figure true bearing such as

06 0^{\circ}

measured clockwise all the way round from north, and it is not measured from the east-west line: taking

tan α^{\circ}

as (north offset)

/

(east offset) reverses the triangle.

Builds on

Also known as

compass bearing, N 30 E bearing, bearing measured from north, direction as an angle from north, navigation bearing