Bearings
From directions to a triangle
- In 5 of 20 recent exams
- 1% of the marks
- Paper 2, calculator
- 1 to 4 marks
How it is set in the exam
Paper 2 (calculator) only, 1–4 marks a part, as a Section A question or inside a longer navigation or rescue question. Typical (5 of 20 exam versions, 15 parts).
Exam phrases to spot (4)
“sails on a bearing of 062°”
“find the bearing of B from A”
“a three-figure bearing”
“which one arrives first?”
In disguise (2)
- an angle between two directions found in an earlier part of the question (sometimes in radians, from a sector: Radians) and then used in the sine rule
- a time asked for when the distance is not known yet: the triangle comes first, and time = distance ÷ speed at the end
Variants (5)
- the interior angle from two bearings, then the sine rule
- the angle between two bearings from one point, then the cosine rule
- a back bearing, or the bearing of one point from another once the triangle is solved
- time = distance ÷ speed, in minutes; two times compared, with a conclusion
- mixed units in one question: metres and kilometres, degrees and radians
Looks like it, but isn't (2)
- a height seen from a boat or from a cliff: Angles of elevation and depression
- a course that changes by a stated angle, with no north lines in the question: Find a side or an angle: choose the rule
The interior angle from two bearings, then the sine rule
A ship leaves port P and sails 14 km on a bearing of 062° to Q. It then turns and sails on a bearing of 145° until it reaches R, which is due east of P.
(a) Show that angle PQR=97°. [2]
(b) Find QR. [3]
(c) Find PR. [2]
Try it yourself first, then compare.
Now try one like it
Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.
- 1
Draw a north line at every point
where a direction is given, and mark each bearing clockwise from its own north line.
- 2
Find the interior angles
from the bearings: differences of bearings at one point, back bearings (±180°), co-interior angles between north lines, the angle sum. Mark them on the diagram.
- 3
Solve the triangle:
a complete pair means the sine rule; two sides and the angle between them mean the cosine rule (Find a side or an angle: choose the rule).
- 4
Answer in the form asked:
a distance; an area, 21absinC with the interior angle between the two sides (The area of a triangle); a bearing, measured at the right point: a known bearing there plus or minus the interior angle (a sketch decides which), written with three figures; a time, as distance ÷ speed, converted to minutes if asked; a comparison, with both values and a sentence.
Measuring anticlockwise, or from east
a bearing is clockwise from north.
Using a bearing as an interior angle
the interior angle is a difference of bearings, or comes from co-interior angles. Draw the north lines.
The back bearing
the bearing of A from B differs from the bearing of B from A by exactly 180°.
More mistakes (6)
The bearing measured at the wrong point
"the bearing of B from A" is measured at A.
The intermediate angle not shown
markschemes want it labelled or marked on a diagram.
Radian mode with degree bearings
it keeps the method marks and loses the answer (check the settings).
Time conversions
0.0712 hours is 4.27 minutes, not 7.12 minutes.
A comparison with one value, or no conclusion
the last mark needs both times, in the same unit, and a sentence saying which arrives first.
Degrees and radians in one rule
an angle from an earlier sector part (radians) and an angle from bearings (degrees) must be in the same unit before the sine rule. Convert one (×π180) and set the mode to match; either unit scores if it is used consistently.