GCSE Maths: Bearings problems: journeys, Pythagoras and trigonometry (Higher)
Part of the GCSE Maths loci, bearings and vectors chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Bearings problems: journeys, Pythagoras and trigonometry
Many bearings questions describe a journey in two parts: a ship sails from A to B on one bearing, then from B to C on another. You are then asked how far C is from A, or what the bearing of C from A is. To solve these, link the bearings to triangles and use the tools you already know: angles in parallel lines, Pythagoras’ theorem and trigonometry (Chapter 20).
- Sketch the journey. Draw the points and the lines, and put a north line at every point where a bearing is given.
- Find the angles you need. Use parallel north lines (co-interior or alternate angles), the back bearing (± 180°), and angles in a triangle.
- Use a triangle. If there is a right angle, use Pythagoras for a length and sin, cos or tan for an angle. In Higher problems with no right angle, use the sine rule and cosine rule.
- Turn the angle back into a bearing. Add it to, or take it away from, a known bearing. Check your answer against the sketch: is it in the right quarter of the compass?
The example below follows these steps. A ship sails 8 km on a bearing of 070° from A to B, then 6 km on a bearing of 160° from B to C. The two bearings differ by 90°, so the path turns through a right angle at B, and triangle ABC has a right angle at B.
Using coordinates. If a place is a number of kilometres east and north of another, make a right-angled triangle from those steps. The angle from the north-south line is found with tan, then you convert it to a bearing: for the north-east quarter the bearing is the angle itself; for the south-east quarter it is 180° minus the angle; for the south-west quarter it is 180° plus the angle; for the north-west quarter it is 360° minus the angle.
Isosceles triangles. If two distances are equal (a ship sails the same distance on two legs), the triangle is isosceles. The base angles are equal, and the angle between the equal sides is the difference between the two bearings.
Higher tier: no right angle. If the path does not turn through exactly 90°, the angle at B in the triangle is 180° minus the turn. Use the cosine rule to find the third side from two sides and the angle between them, then the sine rule to find another angle.
Key ideas
- Draw a north line at every point. North lines are parallel.
- Back bearing: ± 180°. Angles between bearings from the same point are found by subtracting.
- Right angle: Pythagoras for lengths, tan, sin, cos for angles. No right angle (Higher): cosine rule for a length, sine rule for an angle.
- Turn an angle into a bearing by adding or subtracting from a known bearing. Check the quarter of the compass.
- Give bearings to the nearest degree with three figures.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 8 km on a bearing of 070°
B to C: 6 km on a bearing of 160° - Find angle ABC. The ship changes direction from 070° to 160°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.160° − 070° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 82 + 62 = 64 + 36 = 100
AC = √100 = 10.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 6) and the side next to it (AB = 8). Opposite and adjacent means tan.tan(angle BAC) = 6 ÷ 8
angle BAC = tan−1(6 ÷ 8) = 36.9° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 070°.70° + 36.9° = 106.9°
Bearing of C from A = 107° (to the nearest degree) - Find the back bearing. The bearing of A from C is the opposite direction to the bearing of C from A, so add or subtract 180°.107° + 180° = 287°
Bearing of A from C = 287°
Check. Using coordinates from A (east, north): B = (7.52, 2.74) and C = (9.57, −2.90), so C is 10.0 km from A on a bearing of 107°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.P to B: 9 km on a bearing of 310°
B to C: 5 km on a bearing of 220° - Find angle ABC. The ship changes direction from 310° to 220°. The bearings differ by 90°, so the ship turns left through a quarter turn and the path makes a right angle at B.310° − 220° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 92 + 52 = 81 + 25 = 106
AC = √106 = 10.3 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 5) and the side next to it (AB = 9). Opposite and adjacent means tan.tan(angle BAC) = 5 ÷ 9
angle BAC = tan−1(5 ÷ 9) = 29.1° - Take away from the bearing of B from A. C is anticlockwise from B as seen from A, so subtract the angle from 310°.310° − 29.1° = 280.9°
Bearing of C from A = 281° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (−6.89, 5.79) and C = (−10.11, 1.95), so C is 10.3 km from A on a bearing of 281°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the east and north steps as a right-angled triangle, find the angle with tan, then turn it into a bearing measured clockwise from north.
- Find the steps east and north. Draw a right-angled triangle from A to B: one side goes east or west, the other goes north or south.East/west step: 5 km east
North/south step: 3 km south - Find the angle from the north-south line. The east/west step is opposite the angle and the north/south step is next to it, so use tan.tan(angle) = 5 ÷ 3
angle = tan−1(5 ÷ 3) = 59.0° - Turn the angle into a bearing. Bearings are measured clockwise from north. Decide which quarter of the compass the point is in.Bearing = 180° − 59.0° = 121.0°
To the nearest degree, bearing = 121°
Check. 121° points into the east half of the compass, which matches the picture. ✓
Watch out. Measure the angle from the north-south line, not from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The triangle is isosceles, so find the base angles. Combine them with the back bearing of the first leg to get the new bearing.
- Draw a sketch with north lines. The north lines at different points are parallel.Bearing of B from A = 040°, bearing of C from A = 100°
AB = AC = 8 km - Find angle BAC. It is the difference between the two bearings measured from the same point.100° − 40° = 60°
- Find the base angles. AB = AC, so triangle ABC is isosceles. Its base angles are equal and the three angles add up to 180°.(180° − 60°) ÷ 2 = 120° ÷ 2 = 60°
- Find the bearing of A from B. This is the back bearing of the first bearing: add or subtract 180°.40° + 180° = 220°
- Turn from BA towards BC. At B the line BC is 60° anticlockwise from BA (the base angle), so subtract it.220° − 60° = 160°
Bearing of C from B = 160°
Check. With coordinates: B = (5.14, 6.13) and C = (7.88, −1.39). The bearing of C from B is 160°. ✓
Watch out. Base angles of an isosceles triangle are the two angles beside the unequal side, not the angle between the equal sides.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the angle inside the triangle at the turning point from the two bearings. Then use the cosine rule for the missing side and the sine rule for an angle. Finish by turning the angle into a bearing.
- Draw a sketch with a north line at A and B. We measure every bearing from a north line.A to B: 10 km, bearing 035°
B to C: 7 km, bearing 110° - Find the bearing of A from B. This back bearing gives the direction of the line B to A. Add or subtract 180°.35° + 180° = 215°
- Find angle ABC. It is the difference between the bearings of A and C as seen from B (take the smaller angle round).215° − 110° = 105°
Angle ABC = 105° - Use the cosine rule to find AC. We know two sides (10 and 7) and the angle between them (105°), so use a2 = b2 + c2 − 2bc cos A.AC2 = 102 + 72 − 2 × 10 × 7 × cos 105°
AC2 = 185.23
AC = 13.6 km - Use the sine rule to find angle BAC. We now know a side and the angle opposite it (AC and angle B), and we want the angle opposite BC (7 km).sin A ÷ 7 = sin 105° ÷ 13.610
sin A = 7 × sin 105° ÷ 13.610 = 0.4968
A = sin−1(0.4968) = 29.8° - Add to (or subtract from) the bearing AB. C is clockwise from B as seen from A, because the ship turned right at B.35° + 29.8° = 64.8°
Bearing of C from A = 065° to the nearest degree - Find the back bearing. The bearing of A from C is half a turn from the bearing of C from A.65° + 180° = 245°
Bearing of A from C = 245°
Check. Using coordinates from A: B = (5.74, 8.19) and C = (12.31, 5.80), so C is 13.6 km from A on a bearing of 065°. ✓
Watch out. Use the angle inside the triangle in the cosine rule, not the turn angle. They add up to 180°.
Watch out
The angle you find with trigonometry is an angle in the triangle, not the bearing. To get the bearing you must add it to a known bearing (or subtract it). Also, make sure you use the right back bearing: the bearing of A from C is not the same as the bearing of C from A.Exercise 22H Bearings and right-angled triangles
Calculator allowedKey words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 12 km on a bearing of 000°
B to C: 5 km on a bearing of 090° - Find angle ABC. The ship changes direction from 000° to 090°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.090° − 000° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 122 + 52 = 144 + 25 = 169
AC = √169 = 13.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 5) and the side next to it (AB = 12). Opposite and adjacent means tan.tan(angle BAC) = 5 ÷ 12
angle BAC = tan−1(5 ÷ 12) = 22.6° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 000°.0° + 22.6° = 22.6°
Bearing of C from A = 023° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (0.00, 12.00) and C = (5.00, 12.00), so C is 13.0 km from A on a bearing of 023°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 12 km on a bearing of 000°
B to C: 5 km on a bearing of 090° - Find angle ABC. The ship changes direction from 000° to 090°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.090° − 000° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 122 + 52 = 144 + 25 = 169
AC = √169 = 13.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 5) and the side next to it (AB = 12). Opposite and adjacent means tan.tan(angle BAC) = 5 ÷ 12
angle BAC = tan−1(5 ÷ 12) = 22.6° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 000°.0° + 22.6° = 22.6°
Bearing of C from A = 023° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (0.00, 12.00) and C = (5.00, 12.00), so C is 13.0 km from A on a bearing of 023°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 20 km on a bearing of 090°
B to C: 15 km on a bearing of 180° - Find angle ABC. The ship changes direction from 090° to 180°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.180° − 090° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 202 + 152 = 400 + 225 = 625
AC = √625 = 25.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 15) and the side next to it (AB = 20). Opposite and adjacent means tan.tan(angle BAC) = 15 ÷ 20
angle BAC = tan−1(15 ÷ 20) = 36.9° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 090°.90° + 36.9° = 126.9°
Bearing of C from A = 127° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (20.00, 0.00) and C = (20.00, −15.00), so C is 25.0 km from A on a bearing of 127°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 8 km on a bearing of 180°
B to C: 6 km on a bearing of 270° - Find angle ABC. The ship changes direction from 180° to 270°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.270° − 180° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 82 + 62 = 64 + 36 = 100
AC = √100 = 10.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 6) and the side next to it (AB = 8). Opposite and adjacent means tan.tan(angle BAC) = 6 ÷ 8
angle BAC = tan−1(6 ÷ 8) = 36.9° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 180°.180° + 36.9° = 216.9°
Bearing of C from A = 217° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (0.00, −8.00) and C = (−6.00, −8.00), so C is 10.0 km from A on a bearing of 217°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 24 km on a bearing of 270°
B to C: 10 km on a bearing of 000° - Find angle ABC. The ship changes direction from 270° to 000°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.000° − 270° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 242 + 102 = 576 + 100 = 676
AC = √676 = 26.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 10) and the side next to it (AB = 24). Opposite and adjacent means tan.tan(angle BAC) = 10 ÷ 24
angle BAC = tan−1(10 ÷ 24) = 22.6° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 270°.270° + 22.6° = 292.6°
Bearing of C from A = 293° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (−24.00, −0.00) and C = (−24.00, 10.00), so C is 26.0 km from A on a bearing of 293°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 9 km on a bearing of 000°
B to C: 12 km on a bearing of 270° - Find angle ABC. The ship changes direction from 000° to 270°. The bearings differ by 90°, so the ship turns left through a quarter turn and the path makes a right angle at B.000° − 270° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 92 + 122 = 81 + 144 = 225
AC = √225 = 15.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 12) and the side next to it (AB = 9). Opposite and adjacent means tan.tan(angle BAC) = 12 ÷ 9
angle BAC = tan−1(12 ÷ 9) = 53.1° - Take away from the bearing of B from A. C is anticlockwise from B as seen from A, so subtract the angle from 000°.0° − 53.1° = 306.9°
Bearing of C from A = 307° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (0.00, 9.00) and C = (−12.00, 9.00), so C is 15.0 km from A on a bearing of 307°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 15 km on a bearing of 180°
B to C: 8 km on a bearing of 090° - Find angle ABC. The ship changes direction from 180° to 090°. The bearings differ by 90°, so the ship turns left through a quarter turn and the path makes a right angle at B.180° − 090° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 152 + 82 = 225 + 64 = 289
AC = √289 = 17.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 8) and the side next to it (AB = 15). Opposite and adjacent means tan.tan(angle BAC) = 8 ÷ 15
angle BAC = tan−1(8 ÷ 15) = 28.1° - Take away from the bearing of B from A. C is anticlockwise from B as seen from A, so subtract the angle from 180°.180° − 28.1° = 151.9°
Bearing of C from A = 152° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (0.00, −15.00) and C = (8.00, −15.00), so C is 17.0 km from A on a bearing of 152°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 10 km on a bearing of 050°
B to C: 7.5 km on a bearing of 140° - Find angle ABC. The ship changes direction from 050° to 140°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.140° − 050° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 102 + 7.52 = 100 + 56.25 = 156.25
AC = √156.25 = 12.5 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 7.5) and the side next to it (AB = 10). Opposite and adjacent means tan.tan(angle BAC) = 7.5 ÷ 10
angle BAC = tan−1(7.5 ÷ 10) = 36.9° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 050°.50° + 36.9° = 86.9°
Bearing of C from A = 087° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (7.66, 6.43) and C = (12.48, 0.68), so C is 12.5 km from A on a bearing of 087°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 16 km on a bearing of 120°
B to C: 12 km on a bearing of 030° - Find angle ABC. The ship changes direction from 120° to 030°. The bearings differ by 90°, so the ship turns left through a quarter turn and the path makes a right angle at B.120° − 030° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 162 + 122 = 256 + 144 = 400
AC = √400 = 20.0 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 12) and the side next to it (AB = 16). Opposite and adjacent means tan.tan(angle BAC) = 12 ÷ 16
angle BAC = tan−1(12 ÷ 16) = 36.9° - Take away from the bearing of B from A. C is anticlockwise from B as seen from A, so subtract the angle from 120°.120° − 36.9° = 83.1°
Bearing of C from A = 083° (to the nearest degree) - Find the back bearing. The bearing of A from C is the opposite direction to the bearing of C from A, so add or subtract 180°.83° + 180° = 263°
Bearing of A from C = 263°
Check. Using coordinates from A (east, north): B = (13.86, −8.00) and C = (19.86, 2.39), so C is 20.0 km from A on a bearing of 083°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the east and north steps as a right-angled triangle, find the angle with tan, then turn it into a bearing measured clockwise from north.
- Find the steps east and north. Draw a right-angled triangle from A to B: one side goes east or west, the other goes north or south.East/west step: 8 km east
North/south step: 6 km north - Find the angle from the north-south line. The east/west step is opposite the angle and the north/south step is next to it, so use tan.tan(angle) = 8 ÷ 6
angle = tan−1(8 ÷ 6) = 53.1° - Turn the angle into a bearing. Bearings are measured clockwise from north. Decide which quarter of the compass the point is in.Bearing = angle from north = 53.1°
To the nearest degree, bearing = 053°
Check. 053° points into the east half of the compass, which matches the picture. ✓
Watch out. Measure the angle from the north-south line, not from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the east and north steps as a right-angled triangle, find the angle with tan, then turn it into a bearing measured clockwise from north.
- Find the steps east and north. Draw a right-angled triangle from A to B: one side goes east or west, the other goes north or south.East/west step: 7 km east
North/south step: 4 km south - Find the angle from the north-south line. The east/west step is opposite the angle and the north/south step is next to it, so use tan.tan(angle) = 7 ÷ 4
angle = tan−1(7 ÷ 4) = 60.3° - Turn the angle into a bearing. Bearings are measured clockwise from north. Decide which quarter of the compass the point is in.Bearing = 180° − 60.3° = 119.7°
To the nearest degree, bearing = 120°
Check. 120° points into the east half of the compass, which matches the picture. ✓
Watch out. Measure the angle from the north-south line, not from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the east and north steps as a right-angled triangle, find the angle with tan, then turn it into a bearing measured clockwise from north.
- Find the steps east and north. Draw a right-angled triangle from A to B: one side goes east or west, the other goes north or south.East/west step: 5 km west
North/south step: 12 km south - Find the angle from the north-south line. The east/west step is opposite the angle and the north/south step is next to it, so use tan.tan(angle) = 5 ÷ 12
angle = tan−1(5 ÷ 12) = 22.6° - Turn the angle into a bearing. Bearings are measured clockwise from north. Decide which quarter of the compass the point is in.Bearing = 180° + 22.6° = 202.6°
To the nearest degree, bearing = 203°
Check. 203° points into the west half of the compass, which matches the picture. ✓
Watch out. Measure the angle from the north-south line, not from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the east and north steps as a right-angled triangle, find the angle with tan, then turn it into a bearing measured clockwise from north.
- Find the steps east and north. Draw a right-angled triangle from A to B: one side goes east or west, the other goes north or south.East/west step: 3 km west
North/south step: 9 km north - Find the angle from the north-south line. The east/west step is opposite the angle and the north/south step is next to it, so use tan.tan(angle) = 3 ÷ 9
angle = tan−1(3 ÷ 9) = 18.4° - Turn the angle into a bearing. Bearings are measured clockwise from north. Decide which quarter of the compass the point is in.Bearing = 360° − 18.4° = 341.6°
To the nearest degree, bearing = 342°
Check. 342° points into the west half of the compass, which matches the picture. ✓
Watch out. Measure the angle from the north-south line, not from the east-west line.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 300 km on a bearing of 090°
B to C: 400 km on a bearing of 180° - Find angle ABC. The ship changes direction from 090° to 180°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.180° − 090° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 3002 + 4002 = 90 000 + 160 000 = 250 000
AC = √250 000 = 500 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 400) and the side next to it (AB = 300). Opposite and adjacent means tan.tan(angle BAC) = 400 ÷ 300
angle BAC = tan−1(400 ÷ 300) = 53.1° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 090°.90° + 53.1° = 143.1°
Bearing of C from A = 143° (to the nearest degree) - Find the back bearing. The bearing of A from C is the opposite direction to the bearing of C from A, so add or subtract 180°.143° + 180° = 323°
Bearing of A from C = 323°
Check. Using coordinates from A (east, north): B = (300.00, 0.00) and C = (300.00, −400.00), so C is 500 km from A on a bearing of 143°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The triangle is isosceles, so find the base angles. Combine them with the back bearing of the first leg to get the new bearing.
- Draw a sketch with north lines. The north lines at different points are parallel.Bearing of B from A = 030°, bearing of C from A = 110°
AB = AC = 6 km - Find angle BAC. It is the difference between the two bearings measured from the same point.110° − 30° = 80°
- Find the base angles. AB = AC, so triangle ABC is isosceles. Its base angles are equal and the three angles add up to 180°.(180° − 80°) ÷ 2 = 100° ÷ 2 = 50°
- Find the bearing of A from B. This is the back bearing of the first bearing: add or subtract 180°.30° + 180° = 210°
- Turn from BA towards BC. At B the line BC is 50° anticlockwise from BA (the base angle), so subtract it.210° − 50° = 160°
Bearing of C from B = 160°
Check. With coordinates: B = (3.00, 5.20) and C = (5.64, −2.05). The bearing of C from B is 160°. ✓
Watch out. Base angles of an isosceles triangle are the two angles beside the unequal side, not the angle between the equal sides.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The triangle is isosceles, so find the base angles. Combine them with the back bearing of the first leg to get the new bearing.
- Draw a sketch with north lines. The north lines at different points are parallel.Bearing of B from A = 300°, bearing of C from A = 250°
AB = AC = 5 km - Find angle BAC. It is the difference between the two bearings measured from the same point.300° − 250° = 50°
- Find the base angles. AB = AC, so triangle ABC is isosceles. Its base angles are equal and the three angles add up to 180°.(180° − 50°) ÷ 2 = 130° ÷ 2 = 65°
- Find the bearing of A from B. This is the back bearing of the first bearing: add or subtract 180°.300° − 180° = 120°
- Turn from BA towards BC. At B the line BC is 65° clockwise from BA (the base angle), so add it.120° + 65° = 185°
Bearing of C from B = 185°
Check. With coordinates: B = (−4.33, 2.50) and C = (−4.70, −1.71). The bearing of C from B is 185°. ✓
Watch out. Base angles of an isosceles triangle are the two angles beside the unequal side, not the angle between the equal sides.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The north and east legs are at right angles, so use Pythagoras for the distance and tan for the bearing.
- Find the distance. The straight line back to the start is the hypotenuse.52 + 122 = 25 + 144 = 169
√169 = 13 km - Find the angle from north. The east leg (12) is opposite the angle at the start and the north leg (5) is next to it, so use tan.tan x = 12 ÷ 5 = 2.4
x = tan−1(2.4) = 67.4° - Write the bearing. The ship is in the north-east quarter, so the bearing is the angle from north.67.4° is 067° to the nearest degree, as required.
Check. 067° is a little more than north-east (045°), which matches a longer east leg. ✓
Watch out. The angle from north is found with the east distance over the north distance, not the other way round.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The angle in the triangle is measured from the line AB, not from north. Add it to the bearing of B.
- Say what 36.9° is. It is the angle between AB and AC, inside the triangle. It is not measured from north.Angle BAC = 36.9°
- Find where AC is. The ship turned right, so C is clockwise from B. Add the angle to the bearing of B from A.70° + 36.9° = 106.9°
- Write the bearing. Round and write three figures.Bearing of C from A = 107°
Check. C is south-east of A (between 090° and 180°). 036.9° would be north-east. ✓
Watch out. A bearing is always measured from north.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find how far each ship has sailed. The bearings differ by 90°, so the ships and the port make a right-angled triangle with the right angle at P.
- Find the distances sailed. Distance = speed × time.A: 12 × 2 = 24 km. B: 16 × 2 = 32 km.
- Check for a right angle at P. The bearings are 045° and 135°.135° − 45° = 90°, so angle APB = 90°
- Find the distance AB. AB is the hypotenuse.AB2 = 242 + 322 = 576 + 1024 = 1600
AB = 40 km - Find the angle at A. PB (32) is opposite the angle at A and PA (24) is next to it.tan(angle PAB) = 32 ÷ 24
angle PAB = 53.1° - Find the back bearing of P from A. The bearing of A from P is 045°, so add 180°.45° + 180° = 225°
- Find the bearing of B from A. B is anticlockwise from the line AP as seen from A, so subtract the angle.225° − 53.1° = 171.9° ≈ 172°
Check. Coordinates (east, north): A = (16.97, 16.97), B = (22.63, −22.63). B is 5.66 east and 39.60 south of A, so the bearing is 180° − tan−1(5.66 ÷ 39.60) = 171.9°. ✓
Watch out. Draw the sketch first: the ships sail in two directions at right angles.
Exercise 22I Harder bearings problems
Calculator allowedKey words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw the journey and mark the right angle where the ship turns. Use Pythagoras for a distance and trigonometry for an angle, then turn the angle into a bearing.
- Draw a sketch and put a north line at A and at B. A bearing is always measured from north, so we need a north line at every point we measure from.A to B: 6 km on a bearing of 060°
B to C: 4 km on a bearing of 150° - Find angle ABC. The ship changes direction from 060° to 150°. The bearings differ by 90°, so the ship turns right through a quarter turn and the path makes a right angle at B.150° − 060° = 90°
Angle ABC = 90°. Triangle ABC is right-angled. - Use Pythagoras to find AC. AC is opposite the right angle, so it is the hypotenuse: add the squares, then square root.AC2 = 62 + 42 = 36 + 16 = 52
AC = √52 = 7.2 km - Find angle BAC with trigonometry. In the right-angled triangle we know the side opposite the angle (BC = 4) and the side next to it (AB = 6). Opposite and adjacent means tan.tan(angle BAC) = 4 ÷ 6
angle BAC = tan−1(4 ÷ 6) = 33.7° - Add to the bearing of B from A. C is further round clockwise from north than B is, so add the angle to 060°.60° + 33.7° = 93.7°
Bearing of C from A = 094° (to the nearest degree)
Check. Using coordinates from A (east, north): B = (5.20, 3.00) and C = (7.20, −0.46), so C is 7.2 km from A on a bearing of 094°. ✓
Watch out. The angle found with trigonometry is inside the triangle. You must add it to (or subtract it from) a bearing before you answer.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The triangle is isosceles, so find the base angles. Combine them with the back bearing of the first leg to get the new bearing.
- Draw a sketch with north lines. The north lines at different points are parallel.Bearing of B from A = 020°, bearing of C from A = 120°
AB = AC = 9 km - Find angle BAC. It is the difference between the two bearings measured from the same point.120° − 20° = 100°
- Find the base angles. AB = AC, so triangle ABC is isosceles. Its base angles are equal and the three angles add up to 180°.(180° − 100°) ÷ 2 = 80° ÷ 2 = 40°
- Find the bearing of A from B. This is the back bearing of the first bearing: add or subtract 180°.20° + 180° = 200°
- Turn from BA towards BC. At B the line BC is 40° anticlockwise from BA (the base angle), so subtract it.200° − 40° = 160°
Bearing of C from B = 160°
Check. With coordinates: B = (3.08, 8.46) and C = (7.79, −4.50). The bearing of C from B is 160°. ✓
Watch out. Base angles of an isosceles triangle are the two angles beside the unequal side, not the angle between the equal sides.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Draw a north line at S and at C. Use the back bearing of C from S, and the fact that angle CSL is 90°.
- Find the bearing of C from S. The bearing of S from C is 060°, which is less than 180°, so add 180°.60° + 180° = 240°
- Use the right angle at S. Angle CSL = 90°. L is anticlockwise from the line SC as seen from S, because L is to the east and C is to the south-west. Subtract 90°.240° − 90° = 150°
- Write the answer. Bearing of L from S = 150°
Check. L is due east of C. From S (north-east of C) L is south-east of S, so 150° is sensible. ✓
Watch out. Add or subtract 90° in the right direction: sketch it.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. The bearings at G differ by 90°, so triangle GHL has a right angle at G. Find HL with Pythagoras and the bearing with trigonometry and the back bearing.
- Find angle HGL. The bearings of H and L from G are 040° and 130°.130° − 40° = 90°
- Find HL. HL is the hypotenuse.HL2 = 32 + 42 = 25
HL = 5 km - Find angle GHL. GL (4) is opposite the angle at H and GH (3) is next to it.tan(GHL) = 4 ÷ 3
GHL = 53.1° - Find the bearing of G from H. This is the back bearing of 040°: add 180°.40° + 180° = 220°
- Find the bearing of L from H. L is anticlockwise from the line HG as seen from H, so subtract.220° − 53.1° = 166.9° ≈ 167°
Check. Coordinates from G: H = (1.93, 2.30), L = (3.06, −2.57). The bearing of L from H is 166.9°. ✓
Watch out. Check with a sketch: L is mostly south of H.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the angle inside the triangle at the turning point from the two bearings. Then use the cosine rule for the missing side and the sine rule for an angle. Finish by turning the angle into a bearing.
- Draw a sketch with a north line at A and B. We measure every bearing from a north line.A to B: 10 km, bearing 035°
B to C: 7 km, bearing 110° - Find the bearing of A from B. This back bearing gives the direction of the line B to A. Add or subtract 180°.35° + 180° = 215°
- Find angle ABC. It is the difference between the bearings of A and C as seen from B (take the smaller angle round).215° − 110° = 105°
Angle ABC = 105° - Use the cosine rule to find AC. We know two sides (10 and 7) and the angle between them (105°), so use a2 = b2 + c2 − 2bc cos A.AC2 = 102 + 72 − 2 × 10 × 7 × cos 105°
AC2 = 185.23
AC = 13.6 km - Use the sine rule to find angle BAC. We now know a side and the angle opposite it (AC and angle B), and we want the angle opposite BC (7 km).sin A ÷ 7 = sin 105° ÷ 13.610
sin A = 7 × sin 105° ÷ 13.610 = 0.4968
A = sin−1(0.4968) = 29.8° - Add to (or subtract from) the bearing AB. C is clockwise from B as seen from A, because the ship turned right at B.35° + 29.8° = 64.8°
Bearing of C from A = 065° to the nearest degree
Check. Using coordinates from A: B = (5.74, 8.19) and C = (12.31, 5.80), so C is 13.6 km from A on a bearing of 065°. ✓
Watch out. Use the angle inside the triangle in the cosine rule, not the turn angle. They add up to 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the angle inside the triangle at the turning point from the two bearings. Then use the cosine rule for the missing side and the sine rule for an angle. Finish by turning the angle into a bearing.
- Draw a sketch with a north line at A and B. We measure every bearing from a north line.A to B: 15 km, bearing 200°
B to C: 9 km, bearing 260° - Find the bearing of A from B. This back bearing gives the direction of the line B to A. Add or subtract 180°.200° − 180° = 20°
- Find angle ABC. It is the difference between the bearings of A and C as seen from B (take the smaller angle round).260° − 20° = 240°
That is more than 180°, so the angle inside the triangle is 360° − 240° = 120° - Use the cosine rule to find AC. We know two sides (15 and 9) and the angle between them (120°), so use a2 = b2 + c2 − 2bc cos A.AC2 = 152 + 92 − 2 × 15 × 9 × cos 120°
AC2 = 441.00
AC = 21.0 km - Use the sine rule to find angle BAC. We now know a side and the angle opposite it (AC and angle B), and we want the angle opposite BC (9 km).sin A ÷ 9 = sin 120° ÷ 21.000
sin A = 9 × sin 120° ÷ 21.000 = 0.3712
A = sin−1(0.3712) = 21.8° - Add to (or subtract from) the bearing AB. C is clockwise from B as seen from A, because the ship turned right at B.200° + 21.8° = 221.8°
Bearing of C from A = 222° to the nearest degree - Find the back bearing. The bearing of A from C is half a turn from the bearing of C from A.222° − 180° = 42°
Bearing of A from C = 042°
Check. Using coordinates from A: B = (−5.13, −14.10) and C = (−13.99, −15.66), so C is 21.0 km from A on a bearing of 222°. ✓
Watch out. Use the angle inside the triangle in the cosine rule, not the turn angle. They add up to 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the angle inside the triangle at the turning point from the two bearings. Then use the cosine rule for the missing side and the sine rule for an angle. Finish by turning the angle into a bearing.
- Draw a sketch with a north line at A and B. We measure every bearing from a north line.A to B: 12 km, bearing 310°
B to C: 8 km, bearing 020° - Find the bearing of A from B. This back bearing gives the direction of the line B to A. Add or subtract 180°.310° − 180° = 130°
- Find angle ABC. It is the difference between the bearings of A and C as seen from B (take the smaller angle round).130° − 20° = 110°
Angle ABC = 110° - Use the cosine rule to find AC. We know two sides (12 and 8) and the angle between them (110°), so use a2 = b2 + c2 − 2bc cos A.AC2 = 122 + 82 − 2 × 12 × 8 × cos 110°
AC2 = 273.67
AC = 16.5 km - Use the sine rule to find angle BAC. We now know a side and the angle opposite it (AC and angle B), and we want the angle opposite BC (8 km).sin A ÷ 8 = sin 110° ÷ 16.543
sin A = 8 × sin 110° ÷ 16.543 = 0.4544
A = sin−1(0.4544) = 27.0° - Add to (or subtract from) the bearing AB. C is clockwise from B as seen from A, because the ship turned right at B.310° + 27.0° = 337.0°
Bearing of C from A = 337° to the nearest degree
Check. Using coordinates from A: B = (−9.19, 7.71) and C = (−6.46, 15.23), so C is 16.5 km from A on a bearing of 337°. ✓
Watch out. Use the angle inside the triangle in the cosine rule, not the turn angle. They add up to 180°.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. Find the angle ABC inside the triangle from the bearings, then use area = ½ ab sin C.
- Find the back bearing of A from B. 35° is less than 180°, so add 180°.35° + 180° = 215°
- Find angle ABC. It is the difference between the bearings of A and C from B.215° − 110° = 105°
- Use the area formula. The two sides are AB = 10 and BC = 7, and angle B is between them.Area = ½ × 10 × 7 × sin 105° = 35 × 0.9659 = 33.8 km2
Check. Use the base AB = 10 and the perpendicular height from C: 7 × sin 105° = 6.76, so area = ½ × 10 × 6.76 = 33.8. ✓
Watch out. Use the angle between the two known sides.
Key words. A bearing is an angle that gives a direction. It is measured from north, clockwise, and written with three figures (for example 045° or 270°). A back bearing is the bearing of the way back along the same line.
Method. PA = PB = 20 km, so triangle PAB is isosceles with apex angle 60°. That makes it equilateral.
- Find angle APB. The bearings from P are 050° and 110°.110° − 50° = 60°
- Find the other angles. PA = PB so the base angles are equal. The angles add to 180°.(180° − 60°) ÷ 2 = 60°
- Find AB. All three angles are 60°, so the triangle is equilateral.AB = 20 km
- Find the back bearing of P from A. The bearing of A from P is 050°; add 180°.50° + 180° = 230°
- Find the bearing of B from A. B is anticlockwise from the line AP at A, with angle PAB = 60°. Subtract.230° − 60° = 170°
Check. With coordinates: A = (15.32, 12.86), B = (18.79, −6.84). The bearing of B from A is 170.0°. ✓
Watch out. When both distances are equal, look for an isosceles triangle.