What does N40 degrees E mean?

N40 degrees E is a compass bearing: start facing north, then turn 40 degrees toward east. As a three-figure bearing it's 040 degrees. The N or S tells you the starting direction; the E or W tells you which way to turn.

What does N40 degrees E mean?

N40 degrees E is a compass bearing: start facing north, then turn 40 degrees toward east. As a three-figure bearing it's 040 degrees. The N or S tells you the starting direction; the E or W tells you which way to turn.

Year 11 General Applications Of Trigonometry

Bearings And Navigation

Q: What is a three-figure bearing?

A three-figure bearing is the angle measured clockwise from north, written with three digits. North is 000 degrees, east 090 degrees, south 180 degrees, west 270 degrees. The leading zeros (for bearings under 100 degrees) make the notation unambiguous.

Q: How do I find the reverse bearing?

Add 180 degrees to the original bearing. If the result exceeds 360 degrees, subtract 360 degrees. For example, the reverse of 072 degrees is 252 degrees; the reverse of 250 degrees is 070 degrees (since 250 plus 180 equals 430, then 430 minus 360 equals 70).

Q: How do I find north and east distances from a bearing?

For a bearing theta measured from north in the first quadrant (between 000 and 090 degrees), the north component is d cos theta and the east component is d sin theta, where d is the distance travelled. For other quadrants, sketch and use the reference angle from the nearest N or S direction.

Q: What's the difference between a bearing and a maths angle?

A bearing starts at north and increases clockwise: N is 0 degrees, E is 90 degrees, S is 180 degrees, W is 270 degrees. A maths angle starts at the positive x-axis (east) and increases anticlockwise. So a bearing of 030 degrees is the same direction as a maths angle of 60 degrees. Always think bearing — from north, clockwise.

10 practice questions 2 video lessons Theory + worked examples

Theory

A bearing is a direction expressed as an angle. The three-figure bearing is measured clockwise from north (e.g.\ $040^{\circ}$ ); the compass bearing is written as N/S then an angle then E/W (e.g.\ $N 40^{\circ} E$ ). Both describe the same direction in different ways.

A bearing is a way of describing a direction using angles. There are two common notations:

Three-figure bearing — the angle measured clockwise from north, always written with three digits. North is $000^{\circ}$ , east is $090^{\circ}$ , south is $180^{\circ}$ , west is $270^{\circ}$ . Examples: $040^{\circ}$ , $215^{\circ}$ , $290^{\circ}$ .
Compass bearing — written as N or S, then an angle, then E or W. Example: $N 40^{\circ} E$ means $40^{\circ}$ east of north.

The reverse bearing rule: if the bearing of $B$ from $A$ is $θ$ , then the bearing of $A$ from $B$ is $θ + 180^{\circ}$ (subtract $180^{\circ}$ if the sum exceeds $360^{\circ}$ ). For example, if $B$ is on bearing $072^{\circ}$ from $A$ , then $A$ is on bearing $252^{\circ}$ from $B$ .

For a journey of distance $d$ on bearing $θ$ (in the first quadrant), the north component is $d \cos θ$ and the east component is $d \sin θ$ .

Three-figure bearing of P from O is $060^{\circ}$

Compass bearings name the quadrant first, then the angle

Converting between bearing notations

Quadrant	Compass	Three-figure
N → E	$N θ^{\circ} E$	$θ^{\circ}$
E → S	$S (90 - θ)^{\circ} E$	$(90 + θ)^{\circ}$
S → W	$S θ^{\circ} W$	$(180 + θ)^{\circ}$
W → N	$N (90 - θ)^{\circ} W$	$(360 - θ)^{\circ}$

Reverse bearing

Reverse bearing = θ + 180^{\circ} (or θ - 180^{\circ} if the sum exceeds 360^{\circ})

reverse = θ + 180 °

Distance components from a bearing

For a journey of distance $d$ on bearing $θ$ measured from north (first-quadrant bearing):

north component = d \cos θ east component = d \sin θ

N = d \cos θ, E = d \sin θ

          Outside the first quadrant: always sketch the situation and use the reference angle from the nearest N or S direction. Then apply SOH-CAH-TOA to the right triangle you've drawn.
        

How to solve a bearings problem

Sketch the compass at every reference point in the problem, with N pointing straight up.
Draw the bearing line(s), measuring clockwise from north each time.
Identify the right triangle formed by the bearing line and the N–S or E–W axes.
Apply SOH-CAH-TOA — or for multi-leg journeys, use the angle between successive bearings as the angle inside your triangle.

Example 1 — Convert notation

Convert

N 40^{\circ} E

to a three-figure bearing, and convert

215^{\circ}

to a compass bearing.

Solution

$N 40^{\circ} E$ means $40^{\circ}$ clockwise from north. Three-figure bearing $= 040^{\circ}$ .

$215^{\circ}$ is past south by $215^{\circ} - 180^{\circ} = 35^{\circ}$ , heading toward west:

$215^{\circ} - 180^{\circ}$	$=$	$35^{\circ}$ past south, toward west
Compass	$=$	$S 35^{\circ} W$

N40°E = 040 °, 215 ° = S35°W

Example 2 — North/east components

A ship sails

120

km on a bearing of

043^{\circ}

. How far north of its starting point is it (nearest km)?

Solution

Sketch the right triangle, with north up the page:

The bearing $43^{\circ}$ is measured from north, so the north component uses $\cos$ :

$north$	$=$	$120 \cos 43^{\circ}$
$north$	$\approx$	$88 km$

N = 120 \cos 43 ° \approx 88 km

Example 3 — Reverse bearing

The bearing of

B

from

A

072^{\circ}

. Find the bearing of

A

from

B

Solution

Add $180^{\circ}$ to the original bearing:

$return bearing$	$=$	$072^{\circ} + 180^{\circ}$
	$=$	$252^{\circ}$

reverse = 072 ° + 180 ° = 252 °

Example 4 — Find the bearing

A bushwalker walks

12

km east, then

8

km south. Find the bearing of her finishing point from her starting point (nearest degree).

Solution

Sketch the L-shaped path, then form the right triangle from start to finish:

From the start, the angle $α$ from south (toward east) satisfies $\tan α = \frac{12}{8}$ :

$\tan α$	$=$	$\frac{12}{8} = 1.5$
$α$	$=$	$\tan^{- 1} (1.5) \approx 56^{\circ}$
$bearing$	$=$	$180^{\circ} - 56^{\circ} = 124^{\circ}$

bearing = 180 ° - 56 ° = 124 °

Common pitfalls

Bearings start at north and go clockwise. They are NOT the same as standard maths angles (measured from the positive x-axis, anticlockwise). Forgetting this is the most common source of wrong answers.

Always sketch the compass. East is on the right, west on the left. Sketching avoids accidentally flipping a direction in your head.

Multi-leg journeys: angle BETWEEN bearings. The angle inside the triangle at a turn is the angle between the two bearings (e.g.\ a turn from

060^{\circ}

150^{\circ}

gives an interior angle of

90^{\circ}

at the turn).

Three-figure bearings need three digits. Write

040^{\circ}

, not

40^{\circ}

. Both mean the same thing but the leading zero signals the notation clearly.

Frequently asked questions

What is a three-figure bearing?

A three-figure bearing is the angle measured clockwise from north, written with three digits. North is $000^{\circ}$ , east $090^{\circ}$ , south $180^{\circ}$ , west $270^{\circ}$ . The leading zeros (for bearings under $100^{\circ}$ ) make the notation unambiguous.

What does N40°E mean?

$N 40^{\circ} E$ is a compass bearing: start facing north, then turn $40^{\circ}$ toward east. As a three-figure bearing it's $040^{\circ}$ . The N (or S) tells you the starting direction; the E (or W) tells you which way to turn.

How do I convert between compass and three-figure bearings?

Sketch the compass and place the bearing line. Three-figure goes clockwise from north. Compass uses the quadrant (NE / SE / SW / NW) plus an angle from N or S. For example, $215^{\circ}$ is in the SW quadrant, $35^{\circ}$ past south, so it's $S 35^{\circ} W$ .

How do I find the reverse bearing?

Add $180^{\circ}$ to the original bearing. If the result exceeds $360^{\circ}$ , subtract $360^{\circ}$ . For example, the reverse of $072^{\circ}$ is $252^{\circ}$ ; the reverse of $250^{\circ}$ is $070^{\circ}$ (since $250 + 180 = 430$ , then $430 - 360 = 70$ ).

How do I find north and east distances from a bearing?

For a bearing $θ$ measured from north in the first quadrant (between $000^{\circ}$ and $090^{\circ}$ ), the north component is $d \cos θ$ and the east component is $d \sin θ$ , where $d$ is the distance travelled. For other quadrants, sketch and use the reference angle from the nearest N or S direction.

What's the difference between a bearing and a maths angle?

A bearing starts at north and increases clockwise: N is $0^{\circ}$ , E is $90^{\circ}$ , S is $180^{\circ}$ , W is $270^{\circ}$ . A maths angle starts at the positive x-axis (east) and increases anticlockwise. So a bearing of $030^{\circ}$ is the same direction as a maths angle of $60^{\circ}$ . Always think "bearing — from north, clockwise".