Papy

Bearings - Edexcel GCSE Higher Maths

Typically 3-5 marks per paper, most often on Papers 2 and 3 · Spec 1MA1

A bearing is a direction written as an angle. It is measured from north, it turns clockwise, and it is always written with three figures, so due east is 090∘090^\circ and a quarter turn back from north is 315∘315^\circ. On Edexcel's 1MA1 this sits at spec reference G15, next to measuring lines and angles and interpreting scale drawings, and it is common content, so Foundation candidates meet the same three rules. What lifts it into Higher territory is the company it keeps: as soon as the three points do not make a right angle, a bearings question is a sine rule or cosine rule question (G22) wearing a compass, and G22 is Higher only.

Most of the marks are on Papers 2 and 3, where you have a calculator. The short version gives one bearing and asks for the reverse, or gives two bearings from the same point and asks for the angle between them, for 1-3 marks in the first half of the paper. The long version turns up in the last third: three places, two bearings and two distances, then 5 or 6 marks for a distance followed by a bearing. Paper 1 versions do exist and lean on angle facts alone, since the north lines at two different points are parallel and the working is co-interior angles, alternate angles and angles round a point rather than trigonometry. Any printed figure is labelled 'Diagram NOT accurately drawn', so a protractor earns nothing unless the question is explicitly a scale drawing.

Two small words decide the whole question, and they are 'of' and 'from'. The bearing of B from A is measured at A; the bearing of A from B is measured at B, and the two differ by a half turn. The commonest way to lose the last mark is to do the trigonometry perfectly, find a 34∘34^\circ angle inside the triangle, and then write 034∘034^\circ on the answer line, because an angle in a triangle is not a bearing until you have added it to or subtracted it from a bearing at the same point. Draw a north line at every point you measure from, not only the first one, and check the finished answer has three figures and sits between 000∘000^\circ and 360∘360^\circ.

Worked example

Aylesford (A), Denby (D) and Cranwell (C) are three villages. Denby is 14 km from Aylesford on a bearing of 052∘052^\circ. Cranwell is 9 km from Denby on a bearing of 145∘145^\circ. (a) Work out the distance from Aylesford to Cranwell, correct to 1 decimal place. (b) Work out the bearing of Cranwell from Aylesford, correct to the nearest degree.
[5 marks]
  1. The north lines at A and at D are parallel, so turn the first bearing round: the bearing of Aylesford from Denby is 52∘+180∘=232∘52^\circ + 180^\circ = 232^\circ. Angle ADC is the turn at D from DC to DA, so 232∘−145∘=87∘232^\circ - 145^\circ = 87^\circ.
    M1: angle ADC = 87 degrees, from the back bearing 232 and the given bearing 145
  2. Two sides and the angle between them, so the cosine rule: AC2=142+92−2×14×9×cos⁡87∘AC^2 = 14^2 + 9^2 - 2 \times 14 \times 9 \times \cos 87^\circ.
    M1: correct substitution, with the 87 degree angle enclosed by the 14 km and 9 km sides
  3. AC2=196+81−252×0.05233…=263.81…AC^2 = 196 + 81 - 252 \times 0.05233\ldots = 263.81\ldots, so AC=16.242…AC = 16.242\ldots, which is 16.2 km to 1 decimal place.
    A1: 16.2 km; the square root is the step most often dropped
  4. For the direction, find angle DAC with the sine rule, pairing the 9 km side with the angle it faces: sin⁡DAC9=sin⁡87∘16.242…\frac{\sin DAC}{9} = \frac{\sin 87^\circ}{16.242\ldots}, so sin⁡DAC=0.5533…\sin DAC = 0.5533\ldots and angle DAC=33.59…∘DAC = 33.59\ldots^\circ.
    M1: sine rule with DAC paired with DC, using the unrounded AC
  5. C is clockwise of D as seen from A, so add: 52∘+33.59…∘=85.59…∘52^\circ + 33.59\ldots^\circ = 85.59\ldots^\circ. The bearing of Cranwell from Aylesford is 086∘086^\circ.
    A1: 086 degrees, written with the leading zero; the 33.6 on its own scores nothing

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams, so every position is given in words: sketch it, draw a north arrow at each point you are asked to measure from, and mark the bearings on before you calculate anything. The later questions pair bearings with right-angled trigonometry and with the sine and cosine rules, which is how Edexcel builds the 5 and 6 mark versions on Papers 2 and 3, so work in degree mode and hold the full display value between stages.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    A direction is 40 degrees clockwise from north. Which of these is the correct way to write it as a bearing?
    Answer options for question 1
  2. Question 2 Warm-up [1 mark]
    Write the bearing of due west as a three-figure bearing. Give your answer in degrees.
  3. Question 3 Exam pace [3 marks]
    A ship sails 30 km due east from A to B, then 30 km due north from B to C. Work out the bearing of A from C. Give your answer as a three-figure bearing in degrees.
  4. Question 4 Exam pace [3 marks]
    A helicopter flies 25 km in a straight line from a base H on a bearing of 205 degrees. Work out how far south of H the helicopter finishes, correct to 1 decimal place.
  5. Question 5 Exam pace [3 marks]
    A path leaves camp C and runs in a straight line for 7 km on a bearing of 300 degrees to a hut D. Work out how far west of C the hut D is, correct to 1 decimal place.
  6. Question 6 Stretch [5 marks]
    Two ships leave a port P at the same time. Ship A sails on a bearing of 048 degrees at 18 km/h. Ship B sails on a bearing of 126 degrees at 22 km/h. Work out the distance between the two ships after 2 hours, correct to 1 decimal place.

Common mistakes examiners see

  • Reading the direction backwards. The question says the bearing of B from A is 073∘073^\circ and asks for the bearing of A from B, and the answer given is 073∘073^\circ again, when it should be 253∘253^\circ.

    Put your pencil on the second letter, because that is where the north line goes and where the angle is measured. Say the phrase as an instruction: 'of A, from B' means stand at B and look at A.

  • Dropping the leading zero, so a direction of 47 degrees is written as 47 rather than 047∘047^\circ. On a one-mark answer line that is the whole mark.

    Count the digits before you move on. Fewer than three means pad the front with zeros, never the back: 047∘047^\circ is a bearing, 470∘470^\circ is not a direction at all.

  • Adding 180∘180^\circ to every bearing when finding the reverse, so 245∘245^\circ comes back as 425∘425^\circ.

    Below 180∘180^\circ, add 180∘180^\circ. At 180∘180^\circ or above, subtract 180∘180^\circ. Either way the answer has to land between 000∘000^\circ and 360∘360^\circ, so check that before you write it on the line.

  • Using the co-interior angle as the reverse bearing: a bearing of 080∘080^\circ is turned into 100∘100^\circ instead of 260∘260^\circ.

    The 100∘100^\circ is a genuine angle in the picture, the one between the north line at B and the line BA, but it has been swept anticlockwise. Bearings only ever turn clockwise, so the bearing is 360∘−100∘=260∘360^\circ - 100^\circ = 260^\circ.

  • Handing in the triangle angle as the answer. The sine rule gives angle BAC =34∘= 34^\circ and 034∘034^\circ goes on the answer line, when the bearing of B from A was 052∘052^\circ and the bearing wanted is 086∘086^\circ.

    An angle from the sine or cosine rule is measured between two lines in the triangle, not from north. Combine it with a bearing you already know at that same vertex, adding if the new point lies clockwise of the old one and subtracting if it lies anticlockwise, then sanity-check the compass quadrant.

Frequently asked questions

Are bearings on the Foundation paper as well as Higher?
Yes. Bearings sit under spec reference G15 on the 1MA1 specification, which is common content, so Foundation candidates measure them, draw them and reverse them too. The Higher difference is what they get attached to: once a bearings question needs the sine rule or the cosine rule it is Higher only, because those are spec reference G22.
Why do bearings have to be written with three figures?
So a direction can never be misread. Every bearing is an angle from 000 to 360 measured clockwise from north, and padding the front with zeros keeps them all the same width: 7 degrees is written 007, 65 degrees is written 065, and 240 stays as it is. Dropping the zeros is the quickest way to turn a correct calculation into an answer that does not answer the question.
How do I work out a back bearing?
Add 180 if the bearing is less than 180, and subtract 180 if it is 180 or more. So a bearing of 054 reverses to 234, and a bearing of 288 reverses to 108. The rule works because the north lines at the two points are parallel, which makes the outward and return directions a half turn apart.
How many marks are bearings worth on Edexcel Higher?
A reverse bearing or a bearing read off a scale drawing is 1-2 marks. A bearing built on right-angled trigonometry is usually 3-4, and the version that wants the cosine rule for a distance and then the sine rule for the direction runs to 5 or 6 marks near the end of a calculator paper. Across the three 80-mark papers, 3-5 marks per paper is a fair expectation.

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