Bearings questions have a reputation for being confusing, but they are actually one of the most predictable topics in the O-Level E Math syllabus. Once you know the fixed set of moves, most bearings questions become a diagram plus one application of the sine or cosine rule. Here is the exact step-by-step method our tutors used when we sat Paper 2.
What a bearing actually means
Three rules cover almost everything:
- A bearing is always measured from North.
- It is always measured clockwise.
- It is always written with three figures, so 62 degrees is written as 062 degrees, and 8 degrees is written as 008 degrees.
Examiners do award and deduct marks on that third point. If your final answer is a bearing and you write "99.9 degrees" instead of "099.9 degrees", you risk losing the accuracy mark. Train yourself to write the leading zero every single time.
The back bearing shortcut
If the bearing of B from A is less than 180 degrees, then the bearing of A from B is that value plus 180 degrees. If it is more than 180 degrees, subtract 180 degrees. This one fact solves half of all bearings problems, because it lets you find the angle inside your triangle.
The five-step method
Use this order every time and you will rarely get stuck.
- Draw a large diagram. Half a page, not a corner scribble. Mark each point clearly.
- Draw a North line at every point mentioned, not just the first one. These North lines are parallel, which is what lets you use alternate angles and co-interior angles.
- Mark every given bearing as an angle from its North line, going clockwise.
- Work out the interior angle of the triangle using back bearings, angles on a straight line, or co-interior angles adding to 180 degrees.
- Choose your tool. Right-angled triangle means SOH CAH TOA. No right angle means sine rule or cosine rule.
Worked example
A ship sails from A to B on a bearing of 062 degrees for 8 km. It then sails from B to C on a bearing of 155 degrees for 6 km. Find the distance AC and the bearing of C from A.
Step 1 and 2. Sketch A, B and C with a North line at A and a North line at B.
Step 3 and 4. Find angle ABC. The bearing of A from B is the back bearing of 062 degrees, so it is 062 + 180 = 242 degrees. The bearing of C from B is 155 degrees. Both are measured from the same North line at B, so
angle ABC = 242 - 155 = 87 degrees.
Step 5. Two sides and the included angle means cosine rule.
AC squared = 8² + 6² - 2(8)(6) cos 87 degrees AC squared = 64 + 36 - 96(0.052336) AC squared = 94.976 AC = 9.7456
So AC = 9.75 km, correct to 3 significant figures.
Now the bearing. Find angle BAC using the sine rule:
sin(BAC) / 6 = sin 87 degrees / 9.7456 sin(BAC) = 6 × 0.99863 / 9.7456 = 0.61482 angle BAC = 37.9 degrees
The bearing of C from A is measured from the North line at A, and C sits 37.9 degrees clockwise beyond the direction of B:
bearing of C from A = 062 + 37.9 = 099.9 degrees.
Notice the three-figure format and the leading zero. If the question also asked for the bearing of A from C, you would add 180 to get 279.9 degrees.
Common mistakes that cost marks
- Measuring anticlockwise. Bearings never go anticlockwise. If your angle looks like it is on the wrong side, redraw.
- Forgetting extra North lines. Without a North line at B and C, you cannot see the alternate angles and you will guess.
- Rounding too early. Keep the unrounded value of AC in your calculator memory before using it in the sine rule. Round only at the end.
- The ambiguous case. If you use the sine rule to find an angle, check whether the obtuse alternative makes sense against your diagram.
- Dropping units or degrees. Distances need km or m, bearings need the degree symbol.
- Answering the wrong direction. "Bearing of C from A" starts at A. Underline the word "from" in the question.
Practise like the exam
Bearings appear most often in Paper 2 as a 6 to 8 mark structured question, frequently combined with area of a triangle using ½ab sin C, or shortest distance from a point to a line. Do at least five past-year bearings questions in one sitting so the diagram-drawing routine becomes automatic, then check your working against the marking scheme wording rather than just the final number.
Want a tutor to walk through a bearings question with you live and check your diagram habits? Message us on WhatsApp at https://wa.link/dsgbkf to book a free trial class with Dojo.