Corrigé
9.3 Right-angle Trigonometry
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | In a right-angled triangle, the angle $\theta$ has opposite side 3 cm and hypotenuse 5 cm. Write down $\sin \theta$ as a fraction in its simplest form. | $\sin \theta = 3/5$ |
| 2 | Identify the opposite, adjacent, and hypotenuse for angle 30° in a right-angled triangle with hypotenuse 10 cm. | Opp = $10 \sin 30° = 5$; adj = $10 \cos 30° = 5\sqrt{3}$; hyp = 10 |
| 3 | Find sin 60° in surd form. | $\sqrt{3}/2$ |
| 4 | A right triangle has opposite 5 and adjacent 12. Find tan θ. | $\tan \theta = 5/12$ |
| 5 | In a right triangle, $\sin \theta = 0.6$ and hypotenuse = 10. Find the opposite side. | 6 |
| 6 | Use $\sin 30° = 0.5$ to find the opposite side when angle = 30° and hyp = 8 cm. | 4 cm |
| 7 | Find $\tan 45°$ exactly. | 1 |
| 8 | A right triangle has hypotenuse 10 and an angle 30°. Find the side opposite the 30° angle. | 5 |
| 9 | Find tan 30° exactly. | $1/\sqrt{3}$ or $\sqrt{3}/3$ |
| 10 | A right triangle has hyp 13 and one leg 12. Find the third side, and then $\sin \theta$ for the angle opposite the 5-side. | Third side = 5; $\sin \theta = 5/13$ |
| 11 | In a right triangle, the side opposite $\theta$ is 5 cm and hyp = 12 cm. Find $\theta$ to 1 d.p. | $\theta \approx 24.6°$ |
| 12 | A right triangle has hyp 10 cm and angle 35°. Find the opposite side to 2 d.p. | 5.74 cm |
| 13 | A right triangle has adjacent 8 cm and angle 30°. Find the opposite side to 2 d.p. | 4.62 cm |
| 14 | A right triangle has adjacent 6 and opposite 8. Find (a) the hypotenuse, (b) the angle θ to 1 d.p. | (a) 10 (b) ≈ 53.1° |
| 15 | Find the angle of elevation: a tree casts a 12 m shadow when the tree is 7.5 m tall. | ≈ 32° |
| 16 | A right triangle has hyp 10 and angle 30°. Find the adjacent (to 2 d.p.). | 8.66 |
| 17 | A ladder of length 5 m leans against a wall, making a 60° angle with the ground. Find the height up the wall. | $5 \sin 60° \approx 4.33$ m |
| 18 | A right triangle has opposite 8 and adjacent 15. Find (a) the angle θ to 1 d.p., (b) the hypotenuse. | (a) 28.1° (b) 17 |
| 19 | A 25 m vertical building. From the top, the angle of depression to a car is 20°. Find the horizontal distance. | ≈ 68.7 m |
| 20 | Find the angle in a right triangle with hyp 13 and opposite 5. | $\sin^{-1}(5/13) \approx 22.6°$ |
| 21 | A vertical tree casts a shadow of 12 m on horizontal ground. The angle of elevation from the tip of the shadow to the top of the tree is 32°. Find the tree height to 2 d.p. | 7.50 m |
| 22 | A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base. | ≈ 21.8° |
| 23 | A right triangle has angle 30° and hyp 8 cm. Use exact values to find the opposite and adjacent. | Opp 4, adj $4\sqrt{3}$ |
| 24 | An aircraft's angle of approach to a runway is 3°. How high should it be when 7 km away horizontally? Give in m. | ≈ 367 m |
| 25 | A right triangle has hyp 10 and one acute angle 35°. (a) Find the opposite and adjacent. (b) Find the area. | (a) Opp ≈ 5.74, adj ≈ 8.19 (b) ≈ 23.5 |
| 26 | Find $x$ in: angle in a right triangle is $(x + 10)°$ and the opposite/adjacent ratio is 1. | $x = 35$ |
| 27 | A right triangle has one acute angle of 25° and hypotenuse 20 cm. Find the perimeter to 2 d.p. | ≈ 46.58 cm |
| 28 | Two right-angled triangles share a common leg. The first has opposite 3 and adjacent 4. The second has opposite 5 and the same adjacent 4. Find the difference in their hypotenuses. | ≈ 1.40 cm |
| 29 | A right triangle has hyp 25 cm and an angle 60° at one acute vertex. Find the lengths of the two legs. | Legs $25\sqrt{3}/2 \approx 21.65$ and $12.5$ |
| 30 | A boat sails 10 km on bearing 030°. (a) How far north? (b) How far east? | (a) 8.66 km (b) 5 km |
| 31 | PQRS is a square base 8 cm; pyramid apex V is 12 cm above the centre. M = midpoint of QR. Find (a) GM (G = centre), (b) VM, (c) angle of face VQR with the base. | (a) 4 (b) $4\sqrt{10}$ (c) ≈ 71.6° |
| 32 | In a right triangle, the angle at A is 30° and the hypotenuse is 8 cm. Use exact values to find the legs. | 4 and $4\sqrt{3}$ cm |
| 33 | A cliff of height $h$ stands on level ground. From point A, 28 m from the base, line of sight to the top is 52 m. From B further, sight is 60 m. Find $h$ and the distance from B to the base. | $h \approx 43.82$ m; B ≈ 40.99 m |
| 34 | A flagpole is observed from two points 50 m apart. From the closer point, elevation = 35°. From the further, 22°. Find the height of the flagpole. | ≈ 42.5 m |
| 35 | A ladder of length 6 m leans against a wall. The angle with the ground is initially 70°; it slips to 60°. By how much does the top descend? | ≈ 0.43 m |
| 36 | Find $\sin 75°$ exactly using the identity $\sin(45° + 30°) = \sin 45 \cos 30 + \cos 45 \sin 30$. | $\sin 75° = (\sqrt{6} + \sqrt{2})/4$ |
| 37 | A triangular prism has a right-angled triangular cross-section with legs 6 and 8 cm. The hypotenuse is the base of the prism, lying on a table. Find the angle the prism's slanted face makes with the table. | ≈ 53.1° |
| 38 | A square pyramid has slant edge from base vertex to apex = 13 cm and base diagonal half-length = 5 cm. Find the apex height above the centre. | 12 cm |
| 39 | A skier descends a 100 m slope inclined at 15° to horizontal. Find (a) horizontal distance covered, (b) vertical drop. | (a) ≈ 96.6 m (b) ≈ 25.9 m |
| 40 | A 3D Pythagoras-with-trig problem: a square pyramid has base side 10 cm and slant edge 13 cm. Find the angle between a slant edge and the base. | ≈ 60° |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | In a right-angled triangle, the angle $\theta$ has opposite side 8 cm and hypotenuse 17 cm. Write down $\sin \theta$ as a fraction in its simplest form. | $\sin \theta = 8/17$ |
| 2 | Identify the opposite, adjacent, and hypotenuse for angle 30° in a right-angled triangle with hypotenuse 10 cm. | Same scaled by 12/10. |
| 3 | Find sin 60° in surd form. | $\sqrt{3}/2$ |
| 4 | A right triangle has opposite 5 and adjacent 12. Find tan θ. | $\tan \theta = 7/24$ |
| 5 | In a right triangle, $\sin \theta = 0.6$ and hypotenuse = 10. Find the opposite side. | 12 |
| 6 | Use $\sin 30° = 0.5$ to find the opposite side when angle = 30° and hyp = 8 cm. | 6 cm |
| 7 | Find $\tan 45°$ exactly. | $\sqrt{2}/2$ |
| 8 | A right triangle has hypotenuse 10 and an angle 30°. Find the side opposite the 30° angle. | $10\sqrt{3}$ ≈ 17.32 |
| 9 | Find tan 30° exactly. | $\sqrt{3}$ |
| 10 | A right triangle has hyp 13 and one leg 12. Find the third side, and then $\sin \theta$ for the angle opposite the 5-side. | Third = 15; $\sin \theta = 15/17$ |
| 11 | In a right triangle, the side opposite $\theta$ is 7 cm and hyp = 15 cm. Find $\theta$ to 1 d.p. | $\theta \approx 27.8°$ |
| 12 | A right triangle has hyp 12 cm and angle 50°. Find the opposite side to 2 d.p. | 9.19 cm |
| 13 | A right triangle has adjacent 5 cm and angle 45°. Find the opposite side to 2 d.p. | 5 cm |
| 14 | A right triangle has adjacent 6 and opposite 8. Find (a) the hypotenuse, (b) the angle θ to 1 d.p. | (a) 13 (b) ≈ 67.4° |
| 15 | Find the angle of elevation: a tree casts a 12 m shadow when the tree is 7.5 m tall. | ≈ 55° |
| 16 | A right triangle has hyp 10 and angle 30°. Find the adjacent (to 2 d.p.). | 6 |
| 17 | A ladder of length 5 m leans against a wall, making a 60° angle with the ground. Find the height up the wall. | $8 \sin 45° \approx 5.66$ m |
| 18 | A right triangle has opposite 8 and adjacent 15. Find (a) the angle θ to 1 d.p., (b) the hypotenuse. | (a) 36.9° (b) 5 |
| 19 | A 25 m vertical building. From the top, the angle of depression to a car is 20°. Find the horizontal distance. | ≈ 42.8 m |
| 20 | Find the angle in a right triangle with hyp 13 and opposite 5. | ≈ 16.3° |
| 21 | A vertical tree casts a shadow of 15 m on horizontal ground. The angle of elevation from the tip of the shadow to the top of the tree is 48°. Find the tree height to 2 d.p. | 16.66 m |
| 22 | A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base. | ≈ 21.8° |
| 23 | A right triangle has angle 30° and hyp 8 cm. Use exact values to find the opposite and adjacent. | Opp 6, adj $6\sqrt{3}$ |
| 24 | An aircraft's angle of approach to a runway is 3°. How high should it be when 7 km away horizontally? Give in m. | ≈ 875 m |
| 25 | A right triangle has hyp 10 and one acute angle 35°. (a) Find the opposite and adjacent. (b) Find the area. | (a) Opp ≈ 9.00, adj ≈ 10.72 (b) ≈ 48.2 |
| 26 | Find $x$ in: angle in a right triangle is $(x + 10)°$ and the opposite/adjacent ratio is 1. | $x \approx 31.6$ |
| 27 | A right triangle has one acute angle of 25° and hypotenuse 20 cm. Find the perimeter to 2 d.p. | ≈ 32.71 cm |
| 28 | Two right-angled triangles share a common leg. The first has opposite 3 and adjacent 4. The second has opposite 5 and the same adjacent 4. Find the difference in their hypotenuses. | Same. |
| 29 | A right triangle has hyp 25 cm and an angle 60° at one acute vertex. Find the lengths of the two legs. | Both legs $= 10\sqrt{2} \approx 14.14$ |
| 30 | A boat sails 10 km on bearing 030°. (a) How far north? (b) How far east? | (a) 4 km (b) 6.93 km |
| 31 | PQRS is a square base 8 cm; pyramid apex V is 12 cm above the centre. M = midpoint of QR. Find (a) GM (G = centre), (b) VM, (c) angle of face VQR with the base. | (a) 5 (b) 13 (c) ≈ 67.4° |
| 32 | In a right triangle, the angle at A is 30° and the hypotenuse is 12 cm. Use exact values to find the legs. | 6 and $6\sqrt{3}$ cm |
| 33 | A cliff of height $h$ stands on level ground. From point A, 28 m from the base, line of sight to the top is 52 m. From B further, sight is 60 m. Find $h$ and the distance from B to the base. | $h = 45$; B ≈ 46.9 |
| 34 | A flagpole is observed from two points 50 m apart. From the closer point, elevation = 35°. From the further, 22°. Find the height of the flagpole. | ≈ 67.3 m |
| 35 | A ladder of length 6 m leans against a wall. The angle with the ground is initially 70°; it slips to 60°. By how much does the top descend? | ≈ 0.46 m |
| 36 | Find $\sin 75°$ exactly using the identity $\sin(45° + 30°) = \sin 45 \cos 30 + \cos 45 \sin 30$. | $\cos 75° = (\sqrt{6} - \sqrt{2})/4$ |
| 37 | A triangular prism has a right-angled triangular cross-section with legs 6 and 8 cm. The hypotenuse is the base of the prism, lying on a table. Find the angle the prism's slanted face makes with the table. | Same. |
| 38 | A square pyramid has slant edge from base vertex to apex = 13 cm and base diagonal half-length = 5 cm. Find the apex height above the centre. | 15 cm |
| 39 | A skier descends a 100 m slope inclined at 15° to horizontal. Find (a) horizontal distance covered, (b) vertical drop. | (a) ≈ 187.9 m (b) ≈ 68.4 m |
| 40 | A 3D Pythagoras-with-trig problem: a square pyramid has base side 10 cm and slant edge 13 cm. Find the angle between a slant edge and the base. | ≈ 70.5° |
Problèmes — Solutions détaillées
**Tree height.** A tree casts a 5 m shadow when the angle of elevation of the sun is 40°. (a) Find the tree height. (b) Find the slant distance from the tip of the shadow to the top of the tree.
(a) ≈ 4.20 m (b) ≈ 6.53 m
**Building observation.** A pedestrian 1.7 m tall stands 40 m from a 31 m monument. (a) Find the angle of elevation of the top of the monument from the pedestrian's eyes. (b) The pedestrian moves to 20 m from the monument. Find the new angle.
(a) ≈ 36.3° (b) ≈ 55.6°
**Zipline.** A zipline runs from a 154 m UN building to a 443 m Empire State Building, separated horizontally by 1488 m. (a) Find the zipline length. (b) Find the angle of elevation from the UN to the ESB.
(a) ≈ 1516 m (b) ≈ 11°
**Bearings.** A boat sails 12 km on a bearing of 60°. (a) Find the easterly displacement. (b) Find the northerly displacement. (c) The boat then sails 10 km on bearing 150°. Find the resulting position.
(a) ≈ 10.4 km (b) 6 km (c) east ≈ 15.4, north ≈ -2.7
**Pyramid trigonometry.** A pyramid has square base side 10 cm, apex 12 cm above the centre. (a) Find the half-diagonal of the base. (b) Find the slant edge length. (c) Find the angle the slant edge makes with the base.
(a) $5\sqrt{2} \approx 7.07$ cm (b) $\sqrt{194} \approx 13.93$ cm (c) ≈ 59.5°
**Cuboid diagonal trigonometry.** A cuboid has length 8, width 6, height 4. (a) Find the base diagonal. (b) Find the space diagonal. (c) Find the angle the space diagonal makes with the base.
(a) 10 (b) $\sqrt{116} \approx 10.77$ (c) ≈ 21.8°
**Exact-value triangles.** For a 30-60-90 triangle with shorter leg 1: (a) Find the hypotenuse. (b) Find the longer leg. (c) State the ratios sin, cos, tan for 30° and 60°.
(a) 2 (b) $\sqrt{3}$ (c) See working
**45-45-90 triangle.** A square has side 1. (a) Find the diagonal. (b) Find sin 45°, cos 45°, tan 45°.
(a) $\sqrt{2}$ (b) $\sin 45° = \cos 45° = \sqrt{2}/2$; $\tan 45° = 1$
**Angle of depression problem.** A coastguard 50 m above sea level sees a boat with angle of depression 8°. (a) Find the horizontal distance to the boat. (b) The boat moves so the angle is 12°. Find the new distance. (c) How fast does the boat's distance to the coastguard's vertical change?
(a) ≈ 356 m (b) ≈ 235 m (c) Depends on time — distance decreased by ≈ 121 m
**Find the side or angle.** A right triangle has hyp 20 cm and angle 25°. (a) Find both legs. (b) Find the area. (c) Find the perimeter.
(a) Opp ≈ 8.45, adj ≈ 18.13 (b) ≈ 76.6 cm² (c) ≈ 46.58 cm
**3D trig in a cuboid.** A box is 12 cm × 8 cm × 6 cm. A diagonal is drawn from one corner to the opposite corner. (a) Find the base diagonal. (b) Find the space diagonal. (c) Find the angle between the space diagonal and the longest face.
(a) $\sqrt{208} \approx 14.42$ cm (b) $\sqrt{244} \approx 15.62$ cm (c) See working
**Application: roof slope.** A house has a roof that rises 3 m over a horizontal span of 5 m. (a) Find the angle of the roof above horizontal. (b) Find the slant length of the roof. (c) Find the area of one side of the roof, if the house is 10 m long.
(a) ≈ 31° (b) ≈ 5.83 m (c) ≈ 58.3 m²