Mathematics • Year 9 • Unit 3 • Lesson 17
True Bearings
Build fluency with three-digit bearings measured clockwise from North. Convert between compass form (N40°E) and true (040°), and find reverse bearings using ± 180°.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Convert the compass bearing S60°W to a true bearing, then find its reverse bearing.
Step 1 — Identify the quadrant.
S60°W lands in the SW quadrant.
Reason: third letter W means the rotation is from S toward W — the SW quadrant.
Step 2 — Pick the right conversion formula.
SW quadrant: true bearing = 180° + θ where θ is the compass angle.
Reason: clockwise from N — past E (090°) past S (180°), then rotate θ further toward W.
Step 3 — Compute the true bearing.
True = 180 + 60 = 240°.
Reason: already three digits, so no padding needed.
Step 4 — Reverse bearing = original ± 180°.
240 is ≥ 180, so subtract: 240 − 180 = 060°. (If we had added: 240 + 180 = 420 → 420 − 360 = 060° — same answer.)
Reason: the reverse bearing must stay in 000°-360°; pad to 3 digits.
Answer: True bearing 240°; reverse 060°.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Convert the true bearing 310° to a compass bearing, then find its reverse bearing.
Step 1 — Identify the quadrant: 310° lies between __________ ° and 360° → the __________ quadrant.
Step 2 — Pick the formula: NW quadrant uses true = __________ − θ, so θ = __________ − true.
Step 3 — Compute θ:
θ = __________ − 310 = __________ °
Step 4 — Write the compass bearing: N __________ °W
Step 5 — Reverse: 310° is ≥ 180, so subtract 180:
Reverse = 310 − 180 = __________ °
Answer: Compass = N __________ °W; Reverse = __________ °.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single conversion). The middle two are standard (conversion + format). The last two are extension (reverse + double check).
Foundation — single conversion
3.1 Convert N30°E to a true bearing. 1 mark
3.2 Convert S45°E to a true bearing. 1 mark
3.3 Convert the true bearing 270° to compass form. 1 mark
3.4 Write each true bearing using correct three-digit format: (a) 8 (b) 75 (c) 358. 1 mark
Standard — combine two ideas
3.5 Convert N15°W to a true bearing (state the formula you used and pad to 3 digits). 2 marks
3.6 Convert true bearing 218° to a compass bearing (state the quadrant and the formula). 2 marks
Extension — reverse and check
3.7 Find the reverse bearing of (a) 075° and (b) 250°. For each, verify that adding the reverse to the original (then reducing modulo 360 if necessary) gives 180°. 3 marks
3.8 A walker heads on true bearing 100° for the outward leg. (a) Convert this bearing to compass form. (b) State the true bearing they should follow to return to start. (c) Convert the return bearing to compass form. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded 310° → compass; reverse)
Step 1: 310° between 270° and 360° → NW quadrant.
Step 2: NW formula: true = 360 − θ, so θ = 360 − true.
Step 3: θ = 360 − 310 = 50°.
Step 4: Compass = N50°W.
Step 5: Reverse = 310 − 180 = 130°.
Answer: N50°W; reverse 130°.
3.1 — N30°E
NE quadrant: true = θ = 30°. Pad: 030°.
3.2 — S45°E
SE quadrant: true = 180 − 45 = 135°.
3.3 — true 270°
270° is due west (a boundary). Compass: W (or N90°W / S90°W as boundary forms).
3.4 — Three-digit format
(a) 8 → 008°; (b) 75 → 075°; (c) 358 → 358° (already 3 digits).
3.5 — N15°W
NW quadrant: formula true = 360 − θ = 360 − 15 = 345°. (Already 3 digits.)
3.6 — true 218°
218 is between 180 and 270 → SW quadrant. Formula true = 180 + θ, so θ = 218 − 180 = 38. Compass: S38°W.
3.7 — Reverse bearings
(a) 075 < 180, so add 180: reverse = 075 + 180 = 255°. Check: 075 + 255 = 330, but we wanted the difference to be 180 — and 255 − 75 = 180 ✓.
(b) 250 ≥ 180, so subtract 180: reverse = 250 − 180 = 070°. Check: 250 − 70 = 180 ✓.
The "difference is 180°" check (rather than the sum) is the clean way to verify.
3.8 — Walker on true 100°
(a) 100 is between 090 and 180 → SE quadrant. θ = 180 − 100 = 80. Compass: S80°E.
(b) Return = 100 + 180 = 280°.
(c) 280 is between 270 and 360 → NW quadrant. θ = 360 − 280 = 80. Compass: N80°W.
Notice the angle 80° recurs — outward and return are mirror directions across the centre.