A bearing is provided by a magnetic compass when trying to navigate your way (across land, sea or air). In the context of Middle School Maths, Year 8-11, this starts with recognising a back bearing as being in the opposite direction. Imagine you are trying to travel along a course from A to B and have turned around while between points so as to face back towards A. Then the compass reading will change by 180º. Bearings are always given as three digits, even if this requires a leading zero.

Exercise: find the back bearing to each of 025º,125º, 225º, 325º [205º, 305º, 045º, 145º]

The problems associated with this topic are good examples, especially from Year 10 onwards, of the use of words to obscure the maths attached.

Here are four problems that assume knowledge of GCSE Higher trigonometry, including the sine rule: in each case, drawing a diagram is sensible, as is using other parts of the answer to check that your solution is * consistent*. I tried to put “nice, round numbers” in answers where I could, especially in the angles, which you should round to the nearest degree. I’ve used denser English than Lower School are used to (denser; more meaning per word, fewer words for the same meaning).

1 This might be a walking problem. Points A, B&C are such that the bearing A to C (meaning 'of C from A') is 107º; bearing C to B is 315º, bearing B to A is 235º and the distance AB is 12km. Draw a diagram. Find bearings B to C and C to A; hence find angle BCA. Then calculate angles ABC and BAC and find both lengths AC and BC to the nearest 100m.

2 This might be a sailing or shipping problem. Points P, Q & R are so that the bearing R to P (meaning 'the bearing of P from R') is 280º, from Q to P is 250º and the distances QR=32km and PQ=49km. Find the bearing from P to R and P to Q, hence find angle QPR and calculate QRP (I think it is a nice round number, even to 1d.p.). Thence (from there) find angle PQR, bearing Q to R and calculate length PR.

3 This could be a flying problem. Z to X is a bearing of 250º, Y to X is 210º, Z to Y is 300º and XY is 766km. It is thought that XZ is a round number to three (and?) or four sig.fig, but that YZ is not. Find angles YZX, XYZ and lengths XZ and YZ.

4 This could be a surveying problem. G is north of F by 283m and the bearings of F and G are known from H, to be 285º and 315º respectively. Find angle FGH and lengths FH and GH. It is thought that, to 3 sig.fig., FH is a round number.

DJS 20130220

A possibly silly question: A man leaves his tent to go for a walk. Frightened of the local wildlife, he takes his rifle and walks, according to his compass, a couple of miles east, then he turns and goes a couple of miles north and turns again to go west. The visibility is not brilliant but, being flattish land, he can see adequately maybe a kilometre. A good shot might hit the target of choice at 400m but he isn’t that good. After a little while, maybe most of two miles, he sees what he thinks is a bear approaching his tent (the man’s tent; the bear hasn’t got one yet) and he is thinking of shooting.

What colour is the bear? Oh, you weren’t expecting that, were you?

And, for extension work, where is the tent, to within 50 miles?

Might he be mistaken and be seeing a large bird instead? If it is indeed a bird, what sort of bird is it?

Q1 internal angles 100, 52, 28(=73-45)

Q2 internal angles 30, 50, 100

Q3 YZ = 642.75, XYZ = 90º

Q4 GFH=105º, GH = 546.7