Compass and True Bearings Revision Notes for HSC SSCE Mathematics Standard

Compass and True Bearings

What are bearings?

A bearing describes the directional position of one location relative to another point or observer. Bearings are essential in navigation, surveying, and mapping to communicate directions precisely.

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There are two types of bearings used to express direction:

Compass bearings - based on cardinal directions with angles from north or south
True bearings - measured clockwise from north using three-figure notation

Both systems describe the same directions but use different methods of notation.

Compass bearings

The compass directions

Compass bearings use the four main cardinal directions of the compass:

North (N)
East (E)
South (S)
West (W)

The north-south axis runs vertically, while the east-west axis runs horizontally. Between these four cardinal directions are four additional intercardinal directions:

North-east (NE)
South-east (SE)
South-west (SW)
North-west (NW)

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Each intercardinal direction makes an angle of exactly $45°$ with both the north-south and east-west axes. This creates eight evenly-spaced directions around the compass.

How compass bearings work

A compass bearing expresses direction by stating the angle measured from either north or south towards east or west. This system always uses the vertical north-south line as its reference.

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Format for writing compass bearings:

Start with either N or S (never E or W)
State the angle measured from that direction
End with either E or W (the direction you're turning towards)

Example: A compass bearing of S50°W means:

Start at south
Measure an angle of $50°$
Turn towards the west

Finding compass bearings from a diagram

When finding a compass bearing from a diagram, follow these steps:

Step 1: Identify which quadrant the direction lies in:

North-east quadrant
South-east quadrant
South-west quadrant
North-west quadrant

Step 2: Find the angle the direction makes with the vertical (north-south) line.

Step 3: Write the compass bearing using the format: N or S, then the angle, then E or W.

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Worked Example: Finding compass bearings

Find the compass bearing of point $A$ from $O$ and point $B$ from $O$ .

Solution for point A:

The line $OA$ lies in the north-east quadrant
The angle from the north direction is $30°$
Therefore, the compass bearing of $A$ from $O$ is N30°E

Solution for point B:

The line $OB$ lies in the south-east quadrant
The angle shown from north is $120°$
To find the angle from south: $180° - 120° = 60°$
Therefore, the compass bearing of $B$ from $O$ is S60°E

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Exam tip: Always measure compass bearings from the vertical (north-south) line, never from the horizontal (east-west) line.

True bearings

What are true bearings?

A true bearing measures the angle in a clockwise direction starting from north and continuing around to the required direction. True bearings are written using three figures (digits) followed by the letter T.

True bearings are sometimes called three-figure bearings because they must always be written using three digits.

Example: $120°\text{T}$ represents the direction found by measuring $120°$ clockwise from north. This is the same direction as the compass bearing S60°E.

Range and notation

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Three-figure requirement:

The smallest true bearing is $000°\text{T}$ (north)
The largest true bearing is $360°\text{T}$ (which is also north, completing a full circle)
All true bearings between $000°\text{T}$ $000° T$ and $360°\text{T}$ $360° T$ must use three figures
- For example, write $045°\text{T}$ , not $45°\text{T}$
- Write $090°\text{T}$ , not $90°\text{T}$

Cardinal and intercardinal directions as true bearings

The eight main compass directions have the following true bearing equivalents:

Direction	True Bearing
North (N)	$000°\text{T}$
North-east (NE)	$045°\text{T}$
East (E)	$090°\text{T}$
South-east (SE)	$135°\text{T}$
South (S)	$180°\text{T}$
South-west (SW)	$225°\text{T}$
West (W)	$270°\text{T}$
North-west (NW)	$315°\text{T}$

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Memory aid: Notice the pattern - each intercardinal direction adds $45°$ going clockwise. Think of it like a clock:

East = $090°\text{T}$ (quarter circle)
South = $180°\text{T}$ (half circle)
West = $270°\text{T}$ (three-quarters circle)
Full circle back to North = $360°\text{T}$ = $000°\text{T}$

Finding true bearings from a diagram

To find a true bearing from a diagram:

Step 1: Identify the angle measured clockwise from the north direction to the required direction.

Step 2: Write this angle using three figures.

Step 3: Add the letter T after the degree symbol.

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Worked Example: Finding true bearings

Find the true bearing of point $C$ from $O$ and point $D$ from $O$ in the following diagram.

Solution for point C:

The angle measured clockwise from north to $C$ is $210°$
Therefore, $C$ from $O$ is $210°\text{T}$

Solution for point D:

The true bearing of west is $270°\text{T}$
Point $D$ is $60°$ beyond west (measured clockwise)
Therefore: $270° + 60° = 330°$
So $D$ from $O$ is $330°\text{T}$

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Exam tip: When a point lies between two cardinal directions, find the bearing to the nearest cardinal direction first, then add or subtract the additional angle.

Comparing compass and true bearings

Both bearing systems describe the same directions but use different notation methods.

Key differences

Compass Bearing	True Bearing
States the angle from either north or south	Measures the angle clockwise from north only
Uses format: N/S + angle + E/W	Uses three-figure format with T
Example: S60°E	Example: $120°\text{T}$
Angle is always less than $90°$	Angle ranges from $000°$ to $360°$

Conversion examples

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Worked Example: Converting between bearing systems

Example 1: Compass bearing S10°E

Start at north ( $0°$ )
South is at $180°$
S10°E means going $10°$ back from south towards east
$180° - 10° = 170°$
True bearing: $170°\text{T}$

Example 2: Compass bearing N30°W

From north, measure $30°$ towards west (anticlockwise)
Going clockwise: $360° - 30° = 330°$
True bearing: $330°\text{T}$

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Quick conversion guide from compass to true bearings:

For NE quadrant: the true bearing equals the angle
For SE quadrant: true bearing $= 180° -$ angle
For SW quadrant: true bearing $= 180° +$ angle
For NW quadrant: true bearing $= 360° -$ angle

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Key Points to Remember:

A bearing describes the directional position of one location from another point
Compass bearings state the angle from north or south towards east or west (format: N/S + angle + E/W)
The angle in a compass bearing is always measured from the vertical north-south line
True bearings measure the angle clockwise from north using three figures followed by T (format: $000°\text{T}$ to $360°\text{T}$ )
Cardinal directions as true bearings: N = $000°\text{T}$ , E = $090°\text{T}$ , S = $180°\text{T}$ , W = $270°\text{T}$
Intercardinal directions are at $45°$ intervals: NE = $045°\text{T}$ , SE = $135°\text{T}$ , SW = $225°\text{T}$ , NW = $315°\text{T}$
Both systems describe the same directions using different notation - they are interchangeable

Compass and True Bearings (HSC SSCE Mathematics Standard): Revision Notes