Lesson 16 ~25 min Unit 3 · Trigonometry +85 XP

Compass Bearings

Read and write directions in compass bearing format: N40°E. Start at N or S, rotate by the angle toward E or W. Draw the path on a compass diagram.

Today's hook: A ship sails ‘N30°E for 50 km’. Can you draw the exact path on a compass diagram, label the angle correctly, and find the end point?

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Think First

warm-up

Compass bearings always START at N (north) or S (south), never at E or W. Why is that the convention, what does it tell you about angle measurement?

Record your answer in your workbook.

The Big Idea

+5 XP

Compass bearings describe direction using a North-South reference. They are written as Nθ°E, Nθ°W, Sθ°E or Sθ°W, where $\theta$ is the acute angle measured from the N or S axis toward E or W.

A bearing of N40°E means: start facing north, then rotate 40° toward east. Similarly, S25°W means: face south, rotate 25° toward west. The angle in a compass bearing is always between 0° and 90° (acute).

$N\theta°E$: rotate $\theta$ from N toward E (0° $\le \theta \le 90$°)

Start N or S

Bearings begin at the north or south axis, never east or west.

Acute angle

$\theta$ is between 0° and 90°.

Sketch first

Drawing the compass with the direction arrow avoids confusion.

What You'll Master

objectives

Know

Compass bearings are NθE, NθW, SθE, or SθW
The angle is the acute angle from the N-S axis
E and W are 90° on a compass

Understand

How compass bearings differ from true bearings (next lesson)
Why the angle is acute (between 0° and 90°)
How a sketch with N at top makes a bearing easy to draw

Can Do

Read a compass bearing and draw it correctly
Write the compass bearing for a given direction arrow
Calculate end points using compass bearings

Words You Need

vocabulary

Compass bearingA direction written as N40°E (or similar), measured from the N or S axis.

Cardinal directionsN, S, E, W, the four primary compass points, separated by 90°.

N40°EStart facing north, rotate 40° toward east. End direction is 40° east of due north.

Acute bearing angleThe bearing's number value, always between 0° and 90°.

Due north / due eastExactly N (no eastward or westward component) or exactly E.

Compass roseA diagram showing the four cardinal directions; the basis for drawing bearings.

Spot the Trap

heads-up

Wrong: “E40°N” is a compass bearing. NO, bearings start at N or S, never E or W.

Right: Compass bearings have form NθE, NθW, SθE or SθW.

Wrong: Measuring the angle from the E or W axis.

Right: Always measure $\theta$ from the N or S axis, turning toward E or W as the third letter dictates.

Drawing a Compass Bearing

+5 XP

To draw any compass bearing, follow three steps:

1. Draw the compass rose (N up, E right, S down, W left). 2. Look at the first letter (N or S), that's where you start the angle. 3. Rotate by the angle toward the third letter (E or W). The drawn line shows the direction.

Start N or S; rotate by angle toward E or W

N up

Always orient your compass diagram with north at the top of the page.

Angle is acute

Maximum is 90° (which equals due E or due W).

Third letter direction

It tells you which way to rotate from the N-S axis.

Compass Bearings → Position

+5 XP

To find an end-point after travelling a distance on a compass bearing, drop the displacement into N-S and E-W components.

Travel distance $d$ on bearing $N\theta E$: northwards component = $d\cos\theta$; eastwards component = $d\sin\theta$. (Because the angle $\theta$ is measured from N, the N-component uses cos.)

On bearing $N\theta E$, travel $d$: N-component $=d\cos\theta$, E-component $=d\sin\theta$

Components

Always split a bearing displacement into N-S and E-W parts.

From N

With angle measured FROM N, the N-component uses cos.

Pythagoras returns

Total distance + two perpendicular components → right triangle, Pythagoras.

Watch Me Solve It · 3 examples

Watch Me Solve It · Draw a compass bearing

+15 XP per step

PROBLEM

Draw the compass bearing $S55°W$ on a compass rose.

1

Start at S

Face south.

Compass starts at the first letter.
2

Rotate by 55° toward W

Rotate clockwise (looking down from above) when starting at S and rotating to W.
3

Mark the direction

55° from due south, toward west.

AnswerA line in the south-west quadrant, 55° from due south.

Watch Me Solve It · Convert NE to a bearing

+15 XP per step

PROBLEM

A boat sails directly north-east. Write this as a compass bearing.

1

NE means

Exactly half-way between N and E.
2

Angle

Half of 90° = 45°.
3

Bearing

$N45°E$.

Answer$N45°E$

Watch Me Solve It · Distance on a bearing

+15 XP per step

PROBLEM

A ship sails 80 km on bearing $N30°E$. How far north and how far east does it travel (2 d.p.)?

1

N-component

$80\cos 30° \approx 80 \times 0.866 \approx 69.28$ km north
2

E-component

$80\sin 30° = 80 \times 0.5 = 40$ km east
3

Position

End point: $\approx 69.28$ km N, 40.00 km E of start.

Answer$\approx 69.28$ km N, 40 km E

Common Pitfalls

heads-up

Starting at E or W

Writing ‘E20°N’ instead of ‘N70°E’.

Fix: Compass bearings start at N or S only. Re-express as the complement.

Obtuse angle

Writing ‘N120°E’, angle exceeds 90°.

Fix: If the angle exceeds 90°, switch to S as the starting axis. E.g. N120°E = S60°E.

Component swap

Using sin for the N-component when angle is from N.

Fix: Angle FROM N: N-component is cos (adjacent), E-component is sin (opposite).

Copy Into Your Books

Compass form

$N\theta E$, $N\theta W$, $S\theta E$, $S\theta W$
Angle 0-90°
Acute only

Read it

First letter: start axis
Number: rotation angle
Third letter: rotate toward

Components

From N: N=cos, E=sin
From S: S=cos, E=sin
Pick the right axis

Draw it

Compass rose first
N up, E right
Then mark angle

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Compass Drills

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 Convert ‘directly east’ to compass bearing.

Due east is 90° from N.$N90°E$ (also written E)
2 Bearing $N50°E$, rotate by how much from N?

By 50° toward E.50°
3 Travel 10 km on $N60°E$. East-component?

$10\sin 60° \approx 8.66$ km.$\approx 8.66$ km E
4 ‘Half-way between S and W’ as compass bearing.

45° from S toward W.$S45°W$

Complete in your workbook.

Quick Check · 5 questions

Compass bearings always:

+10 XP

What direction is $S30°W$?

+10 XP

A ship sails 100 km on bearing $N45°E$. North-component (2 d.p.):

+10 XP

Due east as a compass bearing:

+10 XP

A walker heads $N20°W$. The angle between their direction and due North is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy2 MARKS

Q6. Draw the compass bearing $N60°W$ and write a sentence describing how it is constructed.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. A boat sails 60 km on bearing $N40°E$. (a) How far east does the boat travel? (b) How far north? (c) Express its final position using northward and eastward distances from the start.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A walker leaves base camp and travels 5 km on bearing $S70°E$. (a) Find their southward and eastward components. (b) From this new position, they travel a further 8 km on bearing $S20°W$. Find the new southward and westward components. (c) Find the total southward distance from base camp.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B Start at N or S.

2. C From S toward W.

3. A$100\cos 45° \approx 70.71$.

4. D$N90°E$.

5. A20° from N.

Show Your Working Model Answers

Q6 (2 marks): Draw compass rose with N up [1]. From the centre, mark a line 60° from the north axis, rotated toward the west. Result is in the NW quadrant, closer to W than to N [1].

Q7 (3 marks): (a) East = $60\sin 40° \approx 38.57$ km [1]. (b) North = $60\cos 40° \approx 45.96$ km [1]. (c) Boat is approximately 45.96 km N and 38.57 km E of start [1].

Q8 (4 marks): (a) S = $5\cos 70° \approx 1.71$ km, E = $5\sin 70° \approx 4.70$ km [1]. (b) S = $8\cos 20° \approx 7.52$ km, W = $8\sin 20° \approx 2.74$ km [1]. (c) Total S = $1.71 + 7.52 = 9.23$ km south of base camp [1]. (Note: net E-W = $4.70 - 2.74 = 1.96$ km east) [1].

Stretch Challenge · +25 XP, +10 coins

Returning home

A boat sails 30 km on bearing $N40°E$, then turns and sails 20 km on bearing $N50°W$. (a) What's the boat's position relative to start (km N and E)? (b) What compass bearing should it take to return directly to start, and how far is that?

Reveal solution

Leg 1: N $\approx 22.98$, E $\approx 19.28$. Leg 2: N $\approx 12.86$, W $\approx 15.32$. Total: N $\approx 35.84$, E (net) $\approx 19.28 - 15.32 = 3.96$. Distance from start: $\sqrt{35.84^2 + 3.96^2} \approx 36.06$ km. Return bearing: south + slight west. $\tan^{-1}(3.96/35.84) \approx 6.3°$, so return bearing $\approx S6.3°W$.

Quick Review

Form

$N\theta E$, $N\theta W$, $S\theta E$, $S\theta W$

Acute

$\theta \in [0°,90°]$

Start at N or S

Never E or W

Draw compass rose

N up always

Components

From N: N=cos, E=sin

Sketch first

Saves arithmetic errors

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