Navigate like a pilot. True bearings, compass directions and back bearings connect maps to trigonometry. Solve real-world navigation problems across Australian landscapes.
Today's hook: A ship sails from Sydney on a bearing of 060° for 100 km. If the captain wants to turn around and sail straight back, what heading should they set? The answer is not as simple as "the opposite direction."
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A ship sails from Sydney on a bearing of $060°$ for 100 km. Sketch the journey. What happens if the ship then turns around and sails back to Sydney? What bearing would it follow?
Record your answer in your workbook.
1
The Big Idea
+5 XP to read
A true bearing is an angle measured clockwise from North, written with three digits from $000°$ to $360°$. A compass bearing measures from North or South toward East or West. Every journey has a back bearing -- the direction home -- found by adding or subtracting $180°$.
Navigation problems hide right-angled triangles inside journeys. Break any trip into North-South and East-West components using sine and cosine. Then use Pythagoras or trigonometry to find distances and directions you did not start with.
$\text{back bearing} = \text{bearing} \pm 180°$
Clockwise from North
True bearings always measure clockwise starting at North.
Three digits always
045° not 45°. 090° not 90°. Three digits every time.
Back bearing rule
Less than 180°? Add 180°. More than 180°? Subtract 180°.
2
What You'll Master
objectives
Know
True bearings are measured clockwise from North ($000°$ to $360°$)
Compass directions (N, NE, E, SE, S, SW, W, NW)
Back bearings are found by adding or subtracting $180°$
Understand
How bearings connect to right-angled triangles in navigation
Why back bearings differ by $180°$
Can Do
Convert between compass directions and true bearings
Calculate back bearings
Solve navigation problems using bearings and trigonometry
3
Words You Need
vocabulary
True bearingAn angle measured clockwise from North, written as a three-digit number ($000°$ to $360°$).
Compass bearingA direction written as an angle measured from North or South toward East or West (e.g., N $30°$ E).
Back bearingThe bearing from the destination back to the starting point. Found by adding or subtracting $180°$.
Three-digit notationBearings always written with three digits, e.g., $045°$ not $45°$.
North componentThe distance travelled in the North direction: $d \times \cos(\text{bearing})$.
East componentThe distance travelled in the East direction: $d \times \sin(\text{bearing})$.
4
Spot the Trap
heads-up
Wrong: Writing bearings without three digits. $45°$ instead of $045°$.
Right: Always use three digits: $045°$, $090°$, $180°$, $270°$.
Wrong: Adding $180°$ to a back bearing that is already over $180°$, giving an answer over $360°$.
Right: If bearing $> 180°$, subtract $180°$ for the back bearing. If bearing $< 180°$, add $180°$.
5
Parts of the Whole
+5 XP
Every bearing problem has the same four ingredients: a reference direction (North), a measurement direction (clockwise), an angle, and a distance. Navigation problems add a fifth: the right-angled triangle hidden inside the journey.
Before calculating anything, identify the starting point, the bearing, the distance, and what you need to find. Draw the North line first. Everything else hangs from it.
$\text{North} = d \times \cos \theta$
North line first
Every bearing diagram starts with a vertical North arrow.
Clockwise from North
Measure the angle turning to the right from the North line.
Distance is hypotenuse
The journey distance is the hypotenuse of the triangle.
6
Read the Compass
+5 XP
Compass bearings and true bearings describe the same direction in different languages. Learn to translate between them fluently.
True bearing: always clockwise from North, three digits. Compass bearing: start at North or South, turn toward East or West. N $40°$ E means start at North, turn $40°$ toward East -- true bearing $040°$.
$\text{N } 40° \text{ E} = 040°$
Start at N or S
Compass bearings always start from North or South.
The bearing from A to B and the bearing from B to A always differ by exactly $180°$. This is the back bearing -- the direction you need to travel to retrace your steps.
The rule is simple: if the bearing is less than $180°$, add $180°$. If the bearing is more than $180°$, subtract $180°$. If it equals $180°$, the back bearing is $000°$. The answer is always between $000°$ and $360°$.
$085° \rightarrow 265°$
Less than 180? Add
Bearing 060° → back bearing = 060 + 180 = 240°.
More than 180? Subtract
Bearing 310° → back bearing = 310 - 180 = 130°.
Check: sum to 360?
Bearing + back bearing should always equal 360°.
8
Break the Journey
+5 XP
Navigation problems combine bearings with distances to form right-angled triangles. Break any journey into its North-South and East-West components, then solve each piece separately.
For any bearing, the North component is $d \times \cos(\text{bearing})$ and the East component is $d \times \sin(\text{bearing})$. In other quadrants the signs change: South means negative North, West means negative East.
$\text{North} = d \cos \theta$, $\text{East} = d \sin \theta$
Cos for North, sin for East
In the first quadrant, cos gives the horizontal (North) part.
Watch the signs
Bearings 90-270 give negative North = South. 180-360 give negative East = West.
Pythagoras for direct
Once you have N and E components, direct distance is $\sqrt{N^2 + E^2}$.
Watch Me Solve It · 3 examples
Watch Me Solve It · Converting bearings
+15 XP per step
Q1
PROBLEM
Convert the following compass bearings to true bearings: (a) N $40°$ E (b) S $30°$ W
1
Part (a) -- N 40° E
Start at North, turn $40°$ toward East (clockwise)
East is clockwise from North, so we add the angle to 000°.
2
Answer (a)
True bearing = $040°$
3
Part (b) -- S 30° W
Start at South ($180°$), turn $30°$ toward West (clockwise)
West is clockwise from South. South is 180°, then add 30° more.
4
Answer (b)
True bearing = $180° + 30° = 210°$
Nice work -- XP earned
Answer(a) $040°$ (b) $210°$
Watch Me Solve It · Back bearing
+15 XP per step
Q2
PROBLEM
The bearing from Sydney to Melbourne is $230°$. Find the bearing from Melbourne to Sydney.
1
Identify the given bearing
Bearing Sydney → Melbourne = $230°$
We need the reverse direction: Melbourne → Sydney.
2
Apply the back bearing rule
$230° > 180°$, so subtract $180°$
When the bearing is greater than 180°, subtract 180° to find the back bearing.
In SE quadrant, we measure from South toward East.
3
Calculate South component
South = $50 \times \cos 60° = 50 \times 0.5 = 25$ km
Cos gives the adjacent side (South direction) from the reference angle.
4
Calculate East component
East = $50 \times \sin 60° = 50 \times 0.866 \approx 43.3$ km
Sin gives the opposite side (East direction) from the reference angle.
Nice work -- XP earned
AnswerSouth = 25 km, East = 43.3 km
9
Common Pitfalls
heads-up
Forgetting three-digit notation
Writing 45° instead of 045°. In navigation, 45° and 045° are not the same -- the three-digit format prevents confusion and is standard across aviation and maritime.
Fix: always write three digits, even if the first one is zero.
Adding 180° to bearings already over 180°
For a bearing of 310°, adding 180° gives 490° which is not a valid bearing. The correct operation is to subtract 180° to get 130°.
Fix: ask "is this bearing less than 180?" If yes, add. If no, subtract.
Using the wrong reference angle for components
For a bearing of 120°, using cos(120°) directly gives a negative value. In component problems, it is often easier to find the reference angle from the nearest cardinal direction (South for 120°) and use that.
Fix: for bearings 90-180, subtract from 180 to get the reference angle from South. For 180-270, subtract 180 to get the reference angle from South.
Copy Into Your Books
True Bearings
Clockwise from North
Three digits: 000° to 360°
N=000, E=090, S=180, W=270
Back Bearings
< 180°: add 180°
> 180°: subtract 180°
= 180°: back bearing = 000°
Components
North = d × cos(θ)
East = d × sin(θ)
Direct distance = √(N² + E²)
Compass to True
N x° E = 000 + x
S x° E = 180 - x
S x° W = 180 + x
N x° W = 360 - x
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Mixed
4 problems
Four bearing problems. Work each one, then reveal the answer.
1 Convert N $40°$ E to a true bearing.
Start at North, turn 40° toward East (clockwise).$= 040°$
2 Find the back bearing of $310°$.
$310° > 180°$, so subtract 180°.$= 130°$
3 A ship sails 50 km on a bearing of $120°$. How far south and how far east has it travelled?
Reference angle = $180° - 120° = 60°$. South = $50 \times \cos 60° = 25$ km. East = $50 \times \sin 60° \approx 43.3$ km.South = 25 km, East = 43.3 km
4 A plane flies 200 km on a bearing of $150°$. How far south has it travelled?
Reference angle = $180° - 150° = 30°$. South = $200 \times \cos 30° = 200 \times 0.866$.$\approx 173$ km
Complete in your workbook.
Multiple Choice · 5 questions
MC1
True bearing of South-West
+10 XP
The true bearing of South-West is:
Correct -- South-West is halfway between South (180°) and West (270°).
South-West is between South and West. Think about the compass rose.
Explanation: South is $180°$ and West is $270°$. South-West is exactly halfway: $180° + 45° = 225°$.
Check the rule: if bearing is less than 180°, add 180°.
Explanation: Since $085° < 180°$, the back bearing = $085° + 180° = 265°$. Verify: $265° - 180° = 085°$ ✓
MC3
Southward displacement
+10 XP
A plane flies 200 km on a bearing of $150°$. Its southward displacement is closest to:
Correct -- South = $200 \times \cos 30° \approx 173$ km.
Bearing 150° is in the SE quadrant. The reference angle from South is 30°.
Explanation: Bearing $150°$ is $30°$ from South ($180° - 150° = 30°$). Southward displacement = $200 \times \cos 30° \approx 200 \times 0.866 = 173.2$ km.
MC4
Compass to true bearing
+10 XP
The compass bearing N $25°$ W is equivalent to the true bearing:
Correct -- N 25° W = $360° - 25° = 335°$.
N x° W means start at North and turn x° toward West (anti-clockwise).
Explanation: N $25°$ W means $25°$ West of North. Since true bearings measure clockwise from North, West of North means subtracting from $360°$: $360° - 25° = 335°$.
MC5
Two-leg journey direction
+10 XP
A ship sails from port on a bearing of $070°$ for 30 km, then on $160°$ for 40 km. The second leg is mainly in which direction?
Correct -- $160°$ is between 90° and 180°, which is South-East.
Check which quadrant 160° falls into.
Explanation: A bearing of $160°$ lies between East ($090°$) and South ($180°$), placing it in the South-East quadrant. The ship is travelling mainly South-East on the second leg.
Short Answer · 3 questions
Q6
Hiker navigation
+15 XP
Q6
SHORT ANSWER
A hiker walks 8 km on a bearing of $040°$. (a) How far north of the starting point is the hiker? (b) How far east of the starting point is the hiker? (c) What is the back bearing to return to the start?
Write your working in your book.
(a) North = $8 \times \cos 40° \approx 8 \times 0.766 \approx 6.1$ km.
(b) East = $8 \times \sin 40° \approx 8 \times 0.643 \approx 5.1$ km.
(c) Back bearing = $040° + 180° = 220°$.
Q7
Ship's two-leg journey
+15 XP
Q7
SHORT ANSWER
A ship sails from port A to port B on a bearing of $120°$ for 50 km. It then sails from port B to port C on a bearing of $210°$ for 70 km. (a) Sketch the ship's journey showing the two legs. (b) Show that the angle between the two paths is $90°$. (c) Calculate the direct distance from port A to port C.
Write your working in your book.
(a) Diagram with port A, port B 50 km away on $120°$, port C 70 km from B on $210°$.
(b) Angle between paths = $210° - 120° = 90°$.
(c) Since angle at B is $90°$, use Pythagoras: $AC = \sqrt{50^2 + 70^2} = \sqrt{2500 + 4900} = \sqrt{7400} \approx 86.0$ km.
Q8
Lighthouse triangulation
+15 XP
Q8
SHORT ANSWER
Two lighthouses, A and B, are 20 km apart on a north-south line. Lighthouse A is due north of lighthouse B. A ship is spotted from lighthouse A on a bearing of $110°$ and from lighthouse B on a bearing of $050°$. (a) Draw a diagram showing the positions of A, B and the ship. (b) Calculate the angle at the ship between the two lines of sight. (c) Using the sine rule or otherwise, find the distance from the ship to lighthouse A. (d) Explain why it is important for navigation that lighthouses can be seen from two different bearings.
Write your working in your book.
(a) Diagram with A north of B, ship to the east.
(b) From A: bearing $110°$ → angle from North = $110°$. From B: bearing $050°$ → angle from North = $50°$. Angle at ship = $180° - (110° - 50°) = 180° - 60° = 120°$.
(c) Using sine rule: $\frac{AS}{\sin 50°} = \frac{20}{\sin 60°}$ → $AS = \frac{20 \times \sin 50°}{\sin 60°} \approx \frac{20 \times 0.766}{0.866} \approx 17.7$ km.
(d) Two bearings allow triangulation -- the intersection of two lines of sight pinpoints the exact location. This is the principle behind GPS and coastal navigation.
S
Stretch Challenge · Lighthouse triangulation
+20 XP
S
STRETCH
Two lighthouses, A and B, are 20 km apart on a north-south line. Lighthouse A is due north of lighthouse B. A ship is spotted from lighthouse A on a bearing of $110°$ and from lighthouse B on a bearing of $050°$. Find the distance from the ship to lighthouse A using the sine rule. Show all working including the diagram and angle calculations.
Record in your book -- full marks require clear working.
Draw diagram: A north of B, ship to the east.
At A: bearing $110°$ means angle between North and line AS is $110°$.
At B: bearing $050°$ means angle between North and line BS is $50°$.
Angle at A inside triangle = $180° - 110° = 70°$.
Angle at B inside triangle = $050°$ (alternate angle to bearing).
Angle at ship = $180° - 70° - 50° = 60°$... wait, recheck: angle at A = $180° - 110° = 70°$ is incorrect. The interior angle at A = $110° - 0° = 110°$ measured from South? No.
Correct: bearing from A is $110°$ (clockwise from North). The angle between AB (South) and AS = $180° - 110° = 70°$.
Bearing from B is $050°$ (clockwise from North). The angle between BA (North) and BS = $050°$.
Use the interactive below to plot bearings on a compass rose, convert between compass and true bearings, and test yourself with the quiz. Try to predict the back bearing before checking.
Record two observations about how bearings relate to the compass rose.
Consolidation Game -- Doodle Jump Quiz
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Jump your way to the top by answering questions on bearings, navigation and trigonometry. The higher you climb, the harder the questions.
Lesson Complete
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Mark this lesson as complete to earn your bonus XP.