Lesson 17 ~25 min Unit 3 · Trigonometry +85 XP

True Bearings

Three-digit bearings measured clockwise from North. N=000°, E=090°, S=180°, W=270°. Convert between compass and true; find reverse bearings.

Today's hook: True bearing 135° — which direction is that? And what's the reverse bearing for the return journey?

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Think First

warm-up

True bearings ALWAYS use three digits. Why might 045° be written instead of just 45°? (Hint: ambiguity and convention.)

Record your answer in your workbook.

The Big Idea

+5 XP

True bearings are a more modern, computer-friendly way to write directions. Always three digits, measured CLOCKWISE from North, ranging 000° to 360°.

N = 000° or 360°; E = 090°; S = 180°; W = 270°. To find a true bearing from a compass bearing $N\theta E$: bearing = $\theta$ (e.g., $N40°E = 040°$). For $N\theta W$: bearing = $360° - \theta$. For $S\theta E$: bearing = $180° - \theta$. For $S\theta W$: bearing = $180° + \theta$.

True bearings: three digits, clockwise from N, 000-360

Three digits

045° not 45°. 005° not 5°.

Clockwise from N

Always rotate clockwise (like a clock's hands) starting from north.

360° = 000°

Full circle returns to north.

What You'll Master

objectives

Know

True bearings are three digits clockwise from N
N=000°, E=090°, S=180°, W=270°
Reverse bearing = original $\pm$ 180°

Understand

Why true bearings are used in navigation (no ambiguity)
How to convert between compass and true bearings
Why reverse bearings differ by exactly 180°

Can Do

Convert compass ↔ true bearings
Find reverse bearings
Identify direction from a true bearing

Words You Need

vocabulary

True bearingA direction written as three digits, measured clockwise from North.

Three-digit formatBearings are written with three digits even if small. 045°, not 45°.

Reverse bearingThe bearing of the return journey. Differs from the original by exactly 180°.

Clockwise from NRotate in the direction a clock's hands move, starting at north (top).

Compass ↔ trueConversion formulas link compass form (N40°E) and true (040°).

Bearing 360°Same as 000° — due north after a full rotation.

Spot the Trap

heads-up

Wrong: Writing 45° instead of 045° — true bearings need three digits.

Right: Always pad to 3 digits: 045°, 005°, 270°.

Wrong: Measuring counter-clockwise (the wrong direction).

Right: True bearings ALWAYS go clockwise from N.

Compass ↔ True Conversion

+5 XP

To switch between the two systems, use the quadrant rules:

Compass	True
$N\theta E$	$\theta$
$S\theta E$	$180° - \theta$
$S\theta W$	$180° + \theta$
$N\theta W$	$360° - \theta$

Compass → true: use quadrant formula above

Pick the quadrant

Once you know NE/SE/SW/NW, the formula is set.

Sketch helps

A small compass diagram clears any doubt.

Verify with N/E/S/W

E = 090°, S = 180°, W = 270° — check your conversion against these.

Reverse Bearings

+5 XP

The reverse bearing is the direction looking BACK along the path you came — differing from your original bearing by exactly 180°.

If the original bearing is $\theta$, the reverse is $\theta + 180°$ when $\theta < 180°$, and $\theta - 180°$ when $\theta \ge 180°$. (Always keep the answer between 000° and 360°.)

Reverse = original $\pm 180°$, in $[0, 360)$

Add or subtract 180

Pick whichever keeps you in 000-360.

Common pairs

045° ↔ 225°; 090° ↔ 270°; 130° ↔ 310°.

Same line, opposite

Going up the path and coming back — same line, opposite directions.

Watch Me Solve It · 3 examples

Watch Me Solve It · Compass to true

+15 XP per step

PROBLEM

Convert the compass bearing $S60°W$ to a true bearing.

1

Find quadrant

SW → true bearing is between 180° and 270°.
2

Apply formula

$180° + \theta = 180° + 60° = 240°$
3

Format

True bearing: 240°

Answer240°

Watch Me Solve It · True to compass

+15 XP per step

PROBLEM

Express the true bearing 135° as a compass bearing.

1

Find quadrant

135° is between 090° and 180° → SE quadrant.
2

Apply formula

$180° - \theta_{compass} = 135°$, so $\theta_{compass} = 45°$.
3

Write

Compass: $S45°E$

Answer$S45°E$

Watch Me Solve It · Reverse bearing

+15 XP per step

PROBLEM

A plane flies on bearing 070°. What's the reverse bearing for the return trip?

1

Original less than 180°

Add 180°: $70 + 180 = 250°$
2

Format

Return bearing: 250°
3

Verify

Add 180° to 250° → 430° → 430 - 360 = 70° ✓

Answer250°

Common Pitfalls

heads-up

Two-digit bearings

Writing 45° instead of 045°.

Fix: Pad to three digits always.

Wrong rotation direction

Treating bearings as counter-clockwise.

Fix: True bearings are CLOCKWISE from N.

Reverse off by 360

Getting a negative or above-360 answer.

Fix: Keep the answer in 000-360. Add or subtract 360 if needed.

Copy Into Your Books

True bearings

Three digits
Clockwise from N
000-360

Cardinal

N = 000°
E = 090°
S = 180°
W = 270°

Conversion

$N\theta E \rightarrow \theta$
$S\theta E \rightarrow 180-\theta$
$S\theta W \rightarrow 180+\theta$
$N\theta W \rightarrow 360-\theta$

Reverse

$\pm 180°$
Keep in 0-360
Same line, opposite

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Bearings Both Ways

4 problems

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

1 $N20°E$ as true bearing.

NE quadrant: $\theta = 20°$.020°
2 $S40°W$ as true bearing.

SW: $180 + 40 = 220$.220°
3 True 290° as compass.

NW: $\theta = 360 - 290 = 70$.$N70°W$
4 Reverse of 080°.

$80 + 180 = 260$.260°

Complete in your workbook.

Quick Check · 5 questions

True bearing of due east is:

+10 XP

$N50°W$ as a true bearing:

+10 XP

True bearing 200° as a compass bearing:

+10 XP

Reverse of bearing 195°:

+10 XP

Why do true bearings use three digits?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy2 MARKS

Q6. Convert (a) $S70°E$ to a true bearing, (b) the true bearing 315° to a compass bearing.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. A plane flies from town A to town B on true bearing 125°. (a) Express this as a compass bearing. (b) State the reverse bearing for the return flight. (c) Express the reverse as a compass bearing.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A walker is at point P. They want to travel to a hut on a true bearing of 070°, 3 km away. (a) Find their northward and eastward displacement. (b) From the hut, they walk back on the reverse bearing for 1 km. Find their new location relative to P (in N/E components).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — 090°.

2. D — 310°.

3. A — $S20°W$.

4. C — 015°.

5. B — Avoids ambiguity.

Show Your Working Model Answers

Q6 (2 marks): (a) SE quadrant: true bearing $= 180° - 70° = 110°$ [1]. (b) NW quadrant: $\theta = 360° - 315° = 45°$, so compass = $N45°W$ [1].

Q7 (3 marks): (a) SE: $S(180-125)°E = S55°E$ [1]. (b) Reverse: $125 + 180 = 305°$ [1]. (c) 305° is in NW: $\theta = 360-305 = 55$, so $N55°W$ [1].

Q8 (4 marks): (a) North = $3\cos 70° \approx 1.026$ km, East = $3\sin 70° \approx 2.819$ km [1]. (b) Reverse bearing = 070 + 180 = 250°. Walking back: South = $1\cos 70° \approx 0.342$ km, West = $1\sin 70° \approx 0.940$ km [1]. Net from P: North $= 1.026 - 0.342 = 0.684$ km, East $= 2.819 - 0.940 = 1.879$ km [1]. So 0.68 km N and 1.88 km E of P [1].

Stretch Challenge · +25 XP, +10 coins

Three towns triangle

Town A is at $N40°E$ from town B. Town C is due south of town A. What's the true bearing of town C from B if A is 5 km from B and C is 4 km south of A?

Reveal solution

Place B at origin. A = $(5\sin 40°, 5\cos 40°) \approx (3.21, 3.83)$. C = $(3.21, 3.83 - 4) = (3.21, -0.17)$. Bearing of C from B: nearly due east, slightly south. $\tan^{-1}(0.17/3.21) \approx 3°$. Bearing $\approx 090° + 3° = 093°$.

Quick Review

Three digits

045°, not 45°

Clockwise from N

Always

Cardinal

E=090, S=180, W=270

Conversion

Quadrant formulas

Reverse

$\pm 180°$, in 0-360

Pad zeros

005°, 045°

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