Navigation and surveying questions are ordinary trigonometry wrapped in a convention. Learn the convention, draw the diagram, and the mathematics is what you already know.
Today's hook, A ship reports a bearing of 145 degrees. That single number fixes a direction exactly, with no ambiguity anywhere in the world, and it is the reason bearings are always written with three digits.
0/5QUESTS
1
You’re here
Recall, your gut answer first
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
01
Recall, your gut answer first
+5 XP warm-up
North is $000^\circ$ and east is $090^\circ$. What is the true bearing of south-west? And if you look up at a plane at $30^\circ$ above the horizontal, what angle does the pilot look down at to see you?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
auto-saved
2
You’re here
Two conventions, one diagram
Work through the core explanation before applying it.
02
Two conventions, one diagram
+5 XP to read
A true bearing is measured clockwise from north and always written with three digits, so north-east is $045^\circ$. An angle of elevation is measured up from the horizontal, and an angle of depression down from it. Elevation and depression between the same two points are equal, because they are alternate angles between parallel horizontals.
True bearing: clockwise from north, three digits Compass bearing: e.g. N$40^\circ$E Elevation $=$ depression (alternate angles)
Always three digits
Write $045^\circ$, not $45^\circ$. The leading zero is part of the convention and marks are lost without it.
Draw north at every vertex
Bearings are measured from north at the point you are standing. A second leg of a journey needs its own north line.
Elevation equals depression
The angle you look up at equals the angle looked down at from the other end. Mark both on the diagram and the triangle usually solves itself.
3
You’re here
What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
03
What you'll master
Know
Key facts
A true bearing is measured clockwise from north and written with three digits.
A compass bearing such as N$40^\circ$E measures from north or south towards east or west.
An angle of elevation is measured up from the horizontal, and depression down from it.
Elevation and depression between the same two points are equal, being alternate angles.
Understand
Concepts
Why bearings are measured from north rather than from the direction of travel.
Why a new north line is needed at each vertex of a multi-leg journey.
Why the angle of elevation equals the corresponding angle of depression.
Can do
Skills
Convert between true and compass bearings.
Solve right-angled triangle problems set in elevation and depression contexts.
Solve two-leg journey problems by combining bearings with the sine or cosine rule.
04
Key terms
True bearingA direction measured clockwise from north, written with three digits. Like this: due east is $090^\circ$ and south-west is $225^\circ$.
Compass bearingA direction written as an angle east or west of north or south. Like this: N$40^\circ$E means start facing north and turn $40^\circ$ towards the east, which is the true bearing $040^\circ$.
Angle of elevationThe angle you look up through, from the horizontal to the object. Like this: standing 50 m from a tower and looking up at its top at $32^\circ$ above horizontal.
Angle of depressionThe angle you look down through, from the horizontal to the object. Like this: from a cliff top looking down at a boat at $18^\circ$ below horizontal.
Alternate anglesEqual angles formed on opposite sides of a line crossing two parallel lines, which is why elevation equals depression. Like this: the horizontal at the cliff top and the horizontal at the boat are parallel, so both angles are $18^\circ$.
Leg of a journeyOne straight section of a multi-part trip, each with its own bearing. Like this: sail $040^\circ$ for 12 km, then $130^\circ$ for 9 km, which is a two-leg journey.
4
You’re here
True and compass bearings
Work through the core explanation before applying it.
05
True and compass bearings
core concept
A true bearing is measured clockwise from north, starting at $000^\circ$, and is always written with three digits: north is $000^\circ$, east $090^\circ$, south $180^\circ$, west $270^\circ$.
A compass bearing names a starting direction and turns towards east or west, such as N$40^\circ$E or S$25^\circ$W. To convert N$40^\circ$E to a true bearing, measure clockwise from north to get $040^\circ$.
For a southerly compass bearing, work from $180^\circ$. S$25^\circ$W is $180 + 25 = 205^\circ$, and S$25^\circ$E is $180 - 25 = 155^\circ$.
The reverse bearing. To return along a leg, add $180^\circ$ (or subtract, if that would exceed $360^\circ$). The reverse of $040^\circ$ is $220^\circ$.
Quick check: what is the true bearing of south-west?
True bearings run clockwise from north with three digits: N $000^\circ$, E $090^\circ$, S $180^\circ$, W $270^\circ$. Compass bearings turn from north or south towards east or west. Reverse a bearing by adding or subtracting $180^\circ$.
Pause, copy the four cardinal true bearings, the conversion of N$40^\circ$E and S$25^\circ$W, and the reverse-bearing rule, into your book.
06
Elevation and depression
core concept
We just saw how a bearing fixes a direction in the horizontal plane. That raises a question: how do you describe looking up at a tower or down at a boat? This card answers it → angles of elevation and depression, both measured from the horizontal.
An angle of elevation is measured upward from the horizontal to your line of sight. An angle of depression is measured downward from the horizontal.
The two horizontals, at your eye and at the object, are parallel. The line of sight cuts both, so the angle of elevation from the lower point equals the angle of depression from the higher point. They are alternate angles.
Almost every question of this kind is a right-angled triangle once drawn, so SOH CAH TOA is usually enough. Mark the right angle and label which side you know.
Depression is measured from the horizontal, not the vertical. A depression of $20^\circ$ means $20^\circ$ below horizontal, which is $70^\circ$ from vertical. Reading it from the vertical is the most common error in these questions.
Fill the blank: if the angle of elevation from a boat to a lighthouse top is $18^\circ$, the angle of depression from the lighthouse to the boat is $^\circ$.
Elevation is measured up from the horizontal, depression down from it. They are equal between the same two points because the horizontals are parallel and the line of sight is a transversal, making them alternate angles. Most such problems are right-angled triangles.
Pause, copy the definitions, the parallel-horizontals reason that elevation equals depression, and the warning that depression is measured from the horizontal, into your book.
07
Two-leg journeys, where bearings meet the sine and cosine rules
core concept
We just saw that single elevation and depression problems are right-angled. That raises a question: what happens on a journey that changes direction, where no right angle appears? This card answers it → the bearings give you the angle inside the triangle, and the sine or cosine rule finishes it.
Draw the first leg from the starting point, then draw a **new north line** at the turning point and measure the second bearing from that.
The angle inside the triangle at the turning point is found from the two bearings and the parallel north lines. Often it is $180^\circ$ minus the difference of the bearings, but derive it from the diagram rather than memorising a formula.
Once you have two sides and that included angle, the cosine rule gives the direct distance home. The sine rule then gives the bearing back.
The new north line is the whole trick. Students who draw only one north line get the interior angle wrong and everything after it follows the error. Draw north at every vertex where a bearing is quoted.
Which is NOT needed to find the direct distance across a two-leg journey?
Draw a new north line at every turning point, derive the interior angle from the diagram rather than a formula, then use the cosine rule for the direct distance and the sine rule for the return bearing.
Pause, copy the method (new north line, derive the interior angle, cosine rule then sine rule) and one fully labelled two-leg diagram, into your book.
Worked examples · 3 in a row, reveal as you go
5
You’re here
Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · ELEVATION
From a point $O$ on level ground, the angle of elevation to the top of a cliff is $35^\circ$. The horizontal distance from $O$ to the base of the cliff is 60 m. Find the height of the cliff to the nearest metre.
1
Draw the right-angled triangle: horizontal 60 m, angle $35^\circ$ at $O$
The cliff is vertical, so the triangle is right-angled at the base.
2
$\tan 35^\circ = \dfrac{h}{60}$
Opposite over adjacent, so use tangent.
3
$h = 60\tan 35^\circ \approx 42$ m
Evaluate and round.
PROBLEM 2 · CONVERTING AND REVERSING A BEARING
A yacht sails on a compass bearing of S$32^\circ$W. Write this as a true bearing, and give the true bearing of the return journey.
1
S$32^\circ$W means start facing south, turn $32^\circ$ towards west
South is $180^\circ$ and west is clockwise-further, so add.
2
True bearing $= 180 + 32 = 212^\circ$
Written with three digits as $212^\circ$.
3
Return: $212 - 180 = 032^\circ$
Subtract $180^\circ$ because adding would exceed $360^\circ$.
PROBLEM 3 · TWO-LEG JOURNEY
A ship sails 12 km on a bearing of $040^\circ$, then 9 km on a bearing of $130^\circ$. Find its distance from the starting point, to one decimal place.
1
Draw a new north line at the turning point, $P$
The second bearing is measured from north there, not from the first leg.
2
The interior angle at the turning point is $180^\circ - (130^\circ - 40^\circ) = 90^\circ$
From the parallel north lines and the two bearings.
3
$d^2 = 12^2 + 9^2 - 2(12)(9)\cos 90^\circ = 225$, so $d = 15.0$ km
Cosine rule; here the angle is a right angle, so it reduces to Pythagoras.
6
You’re here
Quick-fire practice
Work through the core explanation before applying it.
09
Quick-fire practice
+10 XP
Write N$65^\circ$W as a true bearing.
The angle of depression from a $40$ m cliff to a boat is $22^\circ$. How far is the boat from the base?
A plane flies on a bearing of $118^\circ$. What is the bearing of the return flight?
Explain why the angle of elevation from A to B equals the angle of depression from B to A.
auto-saved
7
You’re here
Revisit the plane overhead
Run the quick drill and copy the summary into your book.
10
Revisit the plane overhead
At the start you worked out what angle the pilot looks down at. Name the geometric reason the two angles are equal, and explain why a bearing needs three digits when an elevation angle does not.
auto-saved
1
You’re here
Multiple choice
Answer the drill bank and rate your confidence.
01
Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
2
You’re here
Short answer
Write full responses, then check them against the model answers.
02
Short answer
ApplyBand 43 marks
Q1. From a window 18 m above the ground, the angle of depression to a car is $27^\circ$. Find the horizontal distance from the base of the building to the car, correct to one decimal place. (3 marks)
auto-saved
ApplyBand 54 marks
Q2. A hiker walks 8 km on a bearing of $055^\circ$, then 6 km on a bearing of $145^\circ$. Find how far she is from her starting point, and the true bearing she must walk to return directly. (4 marks)
auto-saved
UnderstandBand 32 marks
Q3. Explain the difference between a true bearing and a compass bearing, and convert S$48^\circ$E to a true bearing. (2 marks)
auto-saved
📖 Comprehensive answers (click to reveal)
Practice 1: $360 - 65 = 295^\circ$. Practice 2: $\tan 22^\circ = \frac{40}{d}$, so $d = \frac{40}{\tan 22^\circ} \approx 99.0$ m. Practice 3: $118 + 180 = 298^\circ$. Practice 4: the horizontals at A and B are parallel and the line of sight is a transversal, so the two angles are alternate angles and therefore equal.
Q1 (3 marks): The angle of elevation from the car equals the depression, $27^\circ$ (alternate angles) [1]. $\tan 27^\circ = \frac{18}{d}$ [1]. $d = \frac{18}{\tan 27^\circ} \approx 35.3$ m [1].
Q2 (4 marks): Interior angle at the turning point $= 180 - (145 - 55) = 90^\circ$ [1]. $d^2 = 8^2 + 6^2 - 2(8)(6)\cos 90^\circ = 100$, so $d = 10$ km [1]. For the return bearing, $\sin\theta = \frac{6}{10} = 0.6$, so $\theta \approx 36.9^\circ$ from the first leg reversed [1]. Reverse of $055^\circ$ is $235^\circ$, so the return bearing is $235 + 36.9 \approx 272^\circ$ [1].
Q3 (2 marks): A true bearing is measured clockwise from north and written with three digits; a compass bearing is measured from north or south towards east or west [1]. S$48^\circ$E $= 180 - 48 = 132^\circ$ [1].
1
You’re here
Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
01
Boss battle · Navigator
earn bronze · silver · gold
Convert bearings, solve elevation and depression triangles, and close two-leg journeys. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.