GCSE Maths: Angles of elevation and depression, and bearings (Foundation)
Part of the GCSE Maths pythagoras and trigonometry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Angles of elevation and depression, and bearings
Angle of elevation. When you look up at something, the angle between the horizontal line and your line of sight is the angle of elevation. Angle of depression. When you look down at something, the angle between the horizontal line and your line of sight is the angle of depression. Both angles are always measured from the horizontal, never from the vertical.
The horizontal line from the top of the cliff and the ground are parallel. So the angle of depression at the top and the angle of elevation of the top from the boat are alternate angles (Chapter 17), and they are equal. This is why we can use the angle at the bottom in the triangle, even though the question gives the angle at the top.
Bearings. A bearing is a direction measured clockwise from north, written with three digits. North is 000°, east is 090°, south is 180° and west is 270°. To go back, the back bearing is the original bearing plus or minus 180°. For example the bearing of B from A is 070°, so the bearing of A from B is 250°. Chapter 22 covers bearings in more detail. Here we use right-angled triangles to solve bearing problems.
When a ship sails on a bearing, draw a north line at the start, mark the bearing angle clockwise from it, and then make a right-angled triangle with east-west and north-south lines. The north line is the side next to the bearing angle, and the east line is opposite it. So for a distance d on a bearing b:
distance east = d sin b distance north = d cos b
These two formulas work for every bearing. If the answer is negative, it means west (for east) or south (for north).
Key ideas
- Elevation: looking up from the horizontal. Depression: looking down from the horizontal.
- The angle of depression from the top equals the angle of elevation from the bottom (alternate angles).
- Bearings: clockwise from north, three digits. Back bearing = bearing ± 180°.
- Draw the right-angled triangle and use SOH CAH TOA or Pythagoras.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw the right-angled triangle. The tower is vertical and the ground is horizontal, so they make a right angle. The angle of elevation is at the observer, between the ground and the line of sight.Opposite the angle: tower height 45 m. Adjacent to the angle: 60 m along the ground.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.45 m is the opposite.
60 m is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 4560 = 0.75
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(4560) = tan−1(0.75) = 36.86989…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 36.9°
Check. Work forwards: tan 36.9° = 0.75 and the ratio was 0.75. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The angle of elevation is 32° at the observer. The height of the tree is vertical, so it is opposite the angle. The ground distance is adjacent.Adjacent = 50 m, opposite = h.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.50 m is the adjacent.
h is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 32° = h50
- Multiply both sides by 50. h is being divided by 50. Multiplying both sides by 50 gets h on its own.h = 50 × tan 32°
- Use the calculator (degree mode). First find tan 32°, then finish the calculation.tan 32° = 0.62486…
h = 50 × 0.62486… = 31.24346… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.h = 31.2 m
Check. The other acute angle is 90° − 32° = 58°. From that angle the opposite and adjacent sides swap, so we use tan: h = 50 ÷ tan 58° = 31.24346…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The cliff is vertical, so the cliff and the sea level make a right angle. The angle of depression 25° at the top equals the angle at the boat (alternate angles).At the boat the angle is 25°. The cliff height 80 m is opposite that angle. The distance x from the cliff is adjacent.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.80 m is the opposite.
x is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 25° = 80x
- Multiply both sides by x. x is on the bottom of the fraction. Multiplying by x brings it to the top.x × tan 25° = 80
- Divide both sides by tan 25°. This leaves x on its own.x = 80tan 25° = 80 ÷ tan 25°
- Use the calculator (degree mode). First find tan 25°, then finish the calculation.tan 25° = 0.46630…
x = 80 ÷ 0.46630… = 171.56055… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.x = 171.6 m
Check. The other acute angle is 90° − 25° = 65°. From that angle the opposite and adjacent sides swap, so we use tan: x = 80 × tan 65° = 171.56055…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. The east and south journeys are at right angles, so use Pythagoras for the distance and tan for the angle. Then use the back bearing rule.
- Draw the journey. Draw a north line at A. B is due east of A (bearing 090°), and C is due south of B (bearing 180° from B). The angle at B is 90°.AB = 12 km, BC = 5 km.
- Find the distance AC. AC is the hypotenuse of the right-angled triangle ABC.AC2 = 122 + 52 = 144 + 25 = 169, so AC = 13 km
- Find angle BAC. At A: the opposite side is BC = 5 and the adjacent is AB = 12 (TOA).tan BAC = 512 = 0.41666…
BAC = tan−1(0.41666…) = 22.61986…° - Find the bearing of C from A. Bearings are measured clockwise from north. East is 090°, and C is further round (towards south) by angle BAC.Bearing = 90° + 22.6° = 112.6°
- Find the bearing of A from C. The back bearing is 180° more (or less).112.6° + 180° = 292.6°, which is 293° to the nearest degree.
Check. Another way: at C the north line points towards B, so angle ACB = 90° − 22.6° = 67.4°. A is to the west of that line, so the bearing is 360° − 67.4° = 292.6°. ✓
Watch out. Draw a north line at each point you measure a bearing from. The bearing at C uses the north line at C.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. The path, the north line and the east line make a right-angled triangle. The bearing angle is between the north line and the path.
- Draw the right-angled triangle. Draw a north line at P. The path to Q makes 60° with it. Go east and north to complete the right angle.Hypotenuse = 20 km. The angle at P (between north and the path) = 60°.
- Label from the 60° angle. The north side is next to the 60° angle (adjacent). The east side is across from it (opposite).North = A, East = O, path = H
- Find the distance east. H and O make SOH.East = 20 × sin 60° = 20 × 0.86602… = 17.32050… km = 17.3 km
- Find the distance north. H and A make CAH.North = 20 × cos 60° = 20 × 0.5 = 10 km = 10.0 km
Check. Pythagoras: 17.32 + 10.02 = 399.3, which is 202 = 400. ✓
Watch out. The bearing is measured from north, so the north side is the adjacent, not the opposite.
Watch out
Bearings start from north and go clockwise, and need three figures (so 40° is written 040°). Draw a north line at every point where you measure a bearing. For the angle of depression, the angle is below the horizontal at the top, not the angle between the line of sight and the cliff.Exercise 20H Elevation, depression and bearings
Calculator allowedKey words. Elevation is an angle upwards from the horizontal. Depression is an angle downwards from the horizontal.
Method. Decide whether the line of sight goes up or down from the horizontal.
- Find the horizontal line. The dashed line along the ground is the horizontal at the observer.The horizontal is at the observer’s feet.
- See where the line of sight goes. The line of sight goes up to the top of the tower.Looking up from the horizontal means an angle of elevation.
Check. If you look up, the angle is of elevation. ✓
Watch out. Elevation: up. Depression: down. Both are measured from the horizontal.
Key words. Elevation is an angle upwards from the horizontal. Depression is an angle downwards from the horizontal.
Method. Decide whether the line of sight goes up or down from the horizontal.
- Find the horizontal line. The dashed line from the observer on top of the cliff is the horizontal.The horizontal is level with the observer’s eyes.
- See where the line of sight goes. The line of sight goes down to the boat.Looking down from the horizontal means an angle of depression.
Check. If you look down, the angle is of depression. ✓
Watch out. The angle of depression is not the angle with the vertical cliff.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The flagpole is vertical and the ground is horizontal. The angle of elevation 40° is at the observer, so the height is opposite it and the 30 m on the ground is adjacent.Adjacent = 30 m, opposite = h.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.30 m is the adjacent.
h is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the adjacent and the opposite. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 40° = h30
- Multiply both sides by 30. h is being divided by 30. Multiplying both sides by 30 gets h on its own.h = 30 × tan 40°
- Use the calculator (degree mode). First find tan 40°, then finish the calculation.tan 40° = 0.83909…
h = 30 × 0.83909… = 25.17298… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.h = 25.2 m
Check. The other acute angle is 90° − 40° = 50°. From that angle the opposite and adjacent sides swap, so we use tan: h = 30 ÷ tan 50° = 25.17298…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio as a fraction, then use the inverse function to turn the ratio into an angle.
- Draw the right-angled triangle. The tree is vertical, so the tree and the ground make a right angle. The angle of elevation x is at the observer: the tree height 12 m is opposite it and the ground distance 20 m is adjacent.Opposite = 12 m, adjacent = 20 m.
- Label the sides. Stand at the angle we want. The hypotenuse is opposite the right angle. The opposite is across from the angle and the adjacent is next to it.12 m is the opposite.
20 m is the adjacent. - Choose the ratio. We know the opposite and the adjacent, so we use the ratio that contains both. Use SOH CAH TOA.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Write the sides as a fraction (top = opposite, bottom = adjacent).tan x = 1220 = 0.6
- Use the inverse function tan−1. We know the value of tan x and want the angle. The inverse button (SHIFT or 2ndF, then tan) turns the ratio back into an angle.x = tan−1(1220) = tan−1(0.6) = 30.96375…°
- Round to 1 d.p. Give the angle to the accuracy asked, with the degree sign.x = 31.0°
Check. Work forwards: tan 31.0° = 0.60 and the ratio was 0.60. They agree. ✓
Watch out. To find an angle use the inverse button (tan−1), not tan. Do not round the ratio before pressing the inverse button.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The angle of depression 15° at the top equals the angle at the boat (alternate angles). The lighthouse height 60 m is opposite that angle, and the distance d is adjacent.Opposite = 60 m, adjacent = d.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.60 m is the opposite.
d is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 15° = 60d
- Multiply both sides by d. d is on the bottom of the fraction. Multiplying by d brings it to the top.d × tan 15° = 60
- Divide both sides by tan 15°. This leaves d on its own.d = 60tan 15° = 60 ÷ tan 15°
- Use the calculator (degree mode). First find tan 15°, then finish the calculation.tan 15° = 0.26794…
d = 60 ÷ 0.26794… = 223.92304… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.d = 223.9 m
Check. The other acute angle is 90° − 15° = 75°. From that angle the opposite and adjacent sides swap, so we use tan: d = 60 × tan 75° = 223.92304…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The aircraft is directly above a point on the ground at distance d from the airport. The angle of elevation 22° is at the airport, so the height 3000 m is opposite and d is adjacent.Opposite = 3000 m, adjacent = d.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.3000 m is the opposite.
d is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 22° = 3000d
- Multiply both sides by d. d is on the bottom of the fraction. Multiplying by d brings it to the top.d × tan 22° = 3000
- Divide both sides by tan 22°. This leaves d on its own.d = 3000tan 22° = 3000 ÷ tan 22°
- Use the calculator (degree mode). First find tan 22°, then finish the calculation.tan 22° = 0.40402…
d = 3000 ÷ 0.40402… = 7425.26056… - Round to 0 d.p.. Give the answer to the accuracy asked, with the unit.d = 7425 m
Check. The other acute angle is 90° − 22° = 68°. From that angle the opposite and adjacent sides swap, so we use tan: d = 3000 × tan 68° = 7425.26056…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The window is 12 m above the ground. The angle of depression 35° equals the angle at the car (alternate angles), so 12 m is opposite and the distance d from the wall is adjacent.Opposite = 12 m, adjacent = d.
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.12 m is the opposite.
d is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the opposite and the adjacent. Use SOH CAH TOA to pick it.TOA stands for Tan, Opposite, Adjacent:
tan θ = OA (O = opposite, A = adjacent) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.tan 35° = 12d
- Multiply both sides by d. d is on the bottom of the fraction. Multiplying by d brings it to the top.d × tan 35° = 12
- Divide both sides by tan 35°. This leaves d on its own.d = 12tan 35° = 12 ÷ tan 35°
- Use the calculator (degree mode). First find tan 35°, then finish the calculation.tan 35° = 0.70020…
d = 12 ÷ 0.70020… = 17.13777… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.d = 17.1 m
Check. The other acute angle is 90° − 35° = 55°. From that angle the opposite and adjacent sides swap, so we use tan: d = 12 × tan 55° = 17.13777…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the bottom, so you divide. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. East and north are at right angles, so use Pythagoras for the distance and tan for the bearing.
- Draw the journey. The ship sails 5 km east and then 12 km north. The path and the straight line from the start make a right-angled triangle.East 5 km, north 12 km.
- Find the distance. The distance from the start is the hypotenuse.52 + 122 = 25 + 144 = 169, so distance = 13 km
- Find the bearing. The bearing is measured clockwise from north. Draw a north line at the start. The angle x between north and the line to the ship has opposite = 5 (east) and adjacent = 12 (north). TOA.tan x = 512, x = tan−1(0.41666…) = 22.61986…°
- Write the bearing with three figures. Round to the nearest degree and write it as three digits.Bearing = 023°
Check. The ship is mostly north and a bit east, so the bearing should be just over 000°. 023° fits. ✓
Watch out. The bearing is the angle from north, so with north 12 and east 5 the angle uses tan−1(5 ÷ 12), not (12 ÷ 5).
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The bearing 040° is the angle between the north line and the path. The east distance is opposite that angle.Hypotenuse = 150 km, angle at the start = 40°, east = E (opposite).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.150 km is the hypotenuse.
E is the opposite. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the opposite. Use SOH CAH TOA to pick it.SOH stands for Sin, Opposite, Hypotenuse:
sin θ = OH (O = opposite, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.sin 40° = E150
- Multiply both sides by 150. E is being divided by 150. Multiplying both sides by 150 gets E on its own.E = 150 × sin 40°
- Use the calculator (degree mode). First find sin 40°, then finish the calculation.sin 40° = 0.64278…
E = 150 × 0.64278… = 96.41814… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.E = 96.4 km
Check. The other acute angle is 90° − 40° = 50°. From that angle the opposite and adjacent sides swap, so we use cos: E = 150 × cos 50° = 96.41814…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Label the sides, pick SOH, CAH or TOA, write the ratio with the numbers, then rearrange to get the unknown side.
- Draw the right-angled triangle. The bearing 040° is the angle between the north line and the path. The north distance is adjacent to that angle.Hypotenuse = 150 km, angle at the start = 40°, north = N (adjacent).
- Label the sides. Stand at the angle θ. The hypotenuse is opposite the right angle. The opposite is across from θ and the adjacent is next to θ.150 km is the hypotenuse.
N is the adjacent. - Choose the ratio. We use the ratio that contains the two sides involved: the hypotenuse and the adjacent. Use SOH CAH TOA to pick it.CAH stands for Cos, Adjacent, Hypotenuse:
cos θ = AH (A = adjacent, H = hypotenuse) - Put the numbers into the ratio. Replace θ with the angle and the sides with the lengths.cos 40° = N150
- Multiply both sides by 150. N is being divided by 150. Multiplying both sides by 150 gets N on its own.N = 150 × cos 40°
- Use the calculator (degree mode). First find cos 40°, then finish the calculation.cos 40° = 0.76604…
N = 150 × 0.76604… = 114.90666… - Round to 1 d.p.. Give the answer to the accuracy asked, with the unit.N = 114.9 km
Check. The other acute angle is 90° − 40° = 50°. From that angle the opposite and adjacent sides swap, so we use sin: N = 150 × sin 50° = 114.90666…, the same answer. ✓
Watch out. Check your calculator is in degree mode (DEG). The unknown is on the top, so you multiply. A quick check: the hypotenuse must be the longest side.
Key words. In a right-angled triangle the hypotenuse is the longest side (opposite the right angle). Measured from the angle θ: the opposite side is directly across from θ, and the adjacent side is next to θ (it is not the hypotenuse). A trigonometric ratio compares two of the sides.
Method. Use Pythagoras for the distance, then tan for an angle, then fit the angle into the bearing.
- Sketch the positions. Take the port P as the origin. A is 8 km north of P and B is 6 km east of P. Triangle PAB has a right angle at P.PA = 8 km, PB = 6 km.
- Find AB. AB is the hypotenuse.AB2 = 82 + 62 = 64 + 36 = 100, so AB = 10 km
- Find angle PAB. At A: opposite = PB = 6, adjacent = PA = 8. TOA.tan PAB = 68 = 0.75
PAB = tan−1(0.75) = 36.86989…° - Find the bearing of B from A. At A the north line points away from P, and P is due south of A (180°). B is round from south towards the east by angle PAB, so we take it away from 180°.Bearing = 180° − 36.9° = 143.1°
To the nearest degree: 143°
Check. B is south-east of A, so the bearing should be between 090° and 180°. 143° fits. ✓
Watch out. Draw the north line at A, not at P.
Key words. A bearing is an angle measured clockwise from north, written with three figures. The back bearing is the bearing for the return trip.
Method. A back bearing points the opposite way. Add 180°, or subtract 180° if the bearing is more than 180°.
- Decide whether to add or subtract. Bearings go from 000° to 360°. If the bearing is less than 180° add 180°; if it is more, subtract 180°.70° is less than 180°, so add 180°.
- Work it out. Write the answer with three figures.70° + 180° = 250°
Check. The two bearings differ by exactly 180°: 180. ✓
Watch out. Draw a north line at each point. The two north lines are parallel, so alternate angles show the 180° difference.
Key words. A bearing is an angle measured clockwise from north, written with three figures. The back bearing is the bearing for the return trip.
Method. A back bearing points the opposite way. Add 180°, or subtract 180° if the bearing is more than 180°.
- Decide whether to add or subtract. Bearings go from 000° to 360°. If the bearing is less than 180° add 180°; if it is more, subtract 180°.125° is less than 180°, so add 180°.
- Work it out. Write the answer with three figures.125° + 180° = 305°
Check. The two bearings differ by exactly 180°: 180. ✓
Watch out. Draw a north line at each point. The two north lines are parallel, so alternate angles show the 180° difference.
Key words. A bearing is an angle measured clockwise from north, written with three figures. The back bearing is the bearing for the return trip.
Method. A back bearing points the opposite way. Add 180°, or subtract 180° if the bearing is more than 180°.
- Decide whether to add or subtract. Bearings go from 000° to 360°. If the bearing is less than 180° add 180°; if it is more, subtract 180°.310° is more than 180°, so subtract 180°.
- Work it out. Write the answer with three figures.310° − 180° = 130°
Check. The two bearings differ by exactly 180°: 180. ✓
Watch out. Draw a north line at each point. The two north lines are parallel, so alternate angles show the 180° difference.
Key words. A bearing is an angle measured clockwise from north, written with three figures. The back bearing is the bearing for the return trip.
Method. A back bearing points the opposite way. Add 180°, or subtract 180° if the bearing is more than 180°.
- Decide whether to add or subtract. Bearings go from 000° to 360°. If the bearing is less than 180° add 180°; if it is more, subtract 180°.205° is more than 180°, so subtract 180°.
- Work it out. Write the answer with three figures.205° − 180° = 025°
Check. The two bearings differ by exactly 180°: 180. ✓
Watch out. Draw a north line at each point. The two north lines are parallel, so alternate angles show the 180° difference.
Key words. Alternate angles are equal angles between parallel lines, in a Z shape.
Method. Draw the horizontal line at A and the horizontal line (sea level) at B. They are parallel. The line of sight AB crosses both.
- Mark the angles. The angle of depression is between the horizontal at A and the line AB. The angle of elevation is between the line BA and the horizontal at B.Angle of depression at A = 30° (given).
- Name the angle pair. The two horizontal lines are parallel and AB is a transversal. The two angles are on opposite sides of the transversal and between the parallel lines (Z-shape).They are alternate angles.
- Use the alternate angle rule. Alternate angles are equal.Angle of elevation at B = 30°.
Check. Draw the Z shape: the line from A to B is the diagonal of the Z, and the angles are in the two corners. ✓
Watch out. Name the rule: “alternate angles are equal”.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. Show that the two parts of the walk are at right angles, then use Pythagoras’ theorem.
- Find the angle at B. The first leg is on 035° and the second is on 125°. The directions differ by 125° − 35° = 90°, so the two legs are at right angles.125 − 35 = 90, so angle ABC = 90°.
- Find the distance AC. AC is the hypotenuse of the right-angled triangle ABC.AC2 = 62 + 82 = 36 + 64 = 100, so AC = 10 km
- Say what it shows. 6, 8, 10 is a Pythagorean triple.The hiker is 10 km from the start.
Check. Using coordinates: C = (9.99467…, 0.32630…) and its distance from A is √(99.89352… + 0.10647…) = 10. ✓
Watch out. Two directions that differ by 90° make a right angle.