Unit 20 pplications of trigonometry Important facts 1. Key terms gradient ( 斜率 ), angle of inclination ( 傾斜角 ) angle of elevation ( 仰角 ), angle of depression ( 俯角 ), line of sigt ( 視線 ), orizontal ( 水平線 ) bearing ( 方位角 ), compass bearing / reduced bearing ( 羅盤方位角 ), true bearing ( 真方位角 ), wole-circle bearing ( 全方位角 ), due ort ( 正北面 ) 2. asic knowledge Trigonometric ratios: opp. side a a sin θ = =, a = c sinθ, c = yp. c sinθ adj. side b b cos θ = =, b = c cosθ, c = yp. c cosθ opp.side a a tan θ = =, a = b tanθ, b = adj.side b tanθ Special angles sin cos tan c θ b 30 45 60 1 2 3 2 1 3 1 2 = 2 2 23 1 2 1 = 2 2 2 1 3 a Unit 20 pplications of trigonometry age 201
3. reas of plane figures If te two adjacent sides of a a cm cm parallelogram are a cm and b cm long, θ and teir included acute angle is θ, ten: b cm te eigt = a.sin θ area of te parallelogram = a.b.sin θ If two sides of a triangle are a cm and b cm cm a cm long, and teir included acute angle is θ, θ ten: b cm te eigt = a.sin θ 1 area of te triangle = b a sinθ 2 If eac side of an equilateral triangle is x cm, x x ten: 60 te eigt = x.sin 60 x 1 area of te equilateral = x x sin 60 2 1 3 = 2 3 x = x 2 2 2 4 (d) If te equal side and te vertical angle ( 頂 角 ) or a base angle of an isosceles triangle 40 are given, find te base and te eigt first. 40 Example: M = = 20 2 M = 12.cos 20 M M = 12.sin 20 1 area of = 2.( M M) 2 = 12 cos 20.12 sin 20 = 46.3 cm 2 (3 sig. fig.) 4. Gradients ngle of inclination is te angle between an inclined ( 傾斜 ) plane 12cm θ orizontal distance vertical distance age 202 Unit 20 pplications of trigonometry
(or line) and te orizontal ( 水平線 ). In te figure, θ is te angle of inclination. θ is smaller tan 90. Gradient = tan θ, were θ is te angle of inclination, vertical distance gradient =. orizontal distance In problems involving contour maps ( 等高線地圖 ), if te map distance of a road is given, it refers to te orizontal distance, not te distance of te inclined road. Example: In te figure, te road is 5 cm on te 300m 250m map. 1 cm : 40000 cm = 1 cm : 400 m scale 1:40,000 ' = 5 400 m = 2000 m. ' = 300 200 = 100 m. 100 1 te gradient of = =. ' 2000 20 5. ngles of elevation and depression Wen a person looks at an object above im, te angle between is orizontal line of sigt and te orizontal is angle of y called te angle of elevation. It depression is smaller tan 90. Wen a person looks at an object angle of below im, te angle between is elevation x line of sigt and te orizontal is orizontal called te angle of depression. It is smaller tan 90. Te angle of elevation of from is x, and te angle of depression of from is y. x = y because tey are alternate angles ( 錯角 ) of parallel lines. line of sigt 200m 6. ompass bearings (reduced bearings) ompass bearings are measured eiter from te nort () or from te sout (S). In te given figure: Unit 20 pplications of trigonometry age 203
is 25 E from ; 25 is 60 W from ; 60 is S58 W from ; W E is S19 E from. 58 point may be due nort, east, sout or S 19 west of anoter point. E, W, SE and SW means tat te angle is 45. 7. True bearings (wole-circle bearings) It is measured from te nort in te clockwise direction. In te given figures: 50 is 050 from, tat is, 50 E ; is 310 from, tat is, S50 W. 218 is 218 from, tat is, S38 W ; is 027 from, tat is, 38 E. 153 is 153 from, tat is, S27 E ; is 333 from, tat is, 27 W. 346 is 346 from, tat is, 14 W; is 166 from, tat is, S14 E. In bearing problems, pay attention to te angles of parallel lines. You may need to make use of te corresponding angles ( 同位角 ), alternate angles ( 錯角 ) and interior angles ( 同旁內角 ). a x b y r s a = b (corr. s, // lines) x = y (alt. s, // lines) r + s = 180 (int. s, // lines) age 204 Unit 20 pplications of trigonometry
8. Some special figures In te following figures, te given lengt is not a side of any of te rigt-angled triangles: R =, SR = tan 28 tan 55 28 55 50 S R 50 = tan 28 tan 55 1 1 = 50 ( ) = 42.4 (3 sig. fig.) tan 28 tan 55 =, = tan 75 tan32 60 = + 75 32 tan 75 tan 32 60 1 1 = 60 ( + ) tan 75 tan 32 = 32.1 (3 sig. fig.) However, in tis figure: = 180 58 32 = 90 In, = 60 cos 58 58 32 In, = sin 58 60 = (60 cos 58 ) (sin 58 ) = 26.96 (4 sig. fig.) 9. In bearing problems, it is useful to make use of orizontal distances and vertical distances. Example: a F b E E = F + FE = F + = a cosα+ b cos θ E = E + = F + = a sinα+ b sin θ 2 2 = E + E E tan E = E Unit 20 pplications of trigonometry age 205
In tis exercise, give te answers to 3 significant figures or 1 decimal place wen appropriate. (I) Warm-up items, o.1-22 (Time: ~60 min) 1. Find te areas of te following figures. 9cm 26cm M 60 R 18cm 14cm (d) 20cm 5cm 2. Find te area of a regular exagon wit side 8 cm. Express te answer in surd form. 3. road as a gradient of 1 : 15. Wat is its angle of inclination? 4. car rises 30 m after travelling 116 m up a road. Find te gradient of te road. R 72 15cm 116m 30m age 206 Unit 20 pplications of trigonometry
5. car travels 15 km up a road of gradient 1 in 10 and ten 20 km up anoter road of gradient 1 in 12. Find te vertical distance te car rises. 6. Te figure sows part of a map wit scale 1 : 25000. straigt road crosses te 350m and 400m contours at and respectively. If = 4cm, wat is te average angle of inclination of te road? 7. Te lengt of Fiona s sadow is 2.14m wen te angle of elevation of te sun is 35. Find te lengt of er sadow wen te angle of elevation of te sun is 60. 450m 500m 400m 350m 300m Scale 1:25000 35 60 2.14m 8. Te angle of depression from te top of a curc to a point due west of it is 47 and to a point due east of it is 63. If te eigt of te curc is 22m, find te distance between and. Unit 20 pplications of trigonometry age 207
9. From te top of a building, te angles of depression of two cars on te same side of te building are 24 and 36 respectively. If te eigt of building is 120 m, find te distance between te cars. 24 36 120m 10. Te angles of elevation from two sips to te top of a cliff are 25 and 40. Te sips are 75m apart. Let m be te eigt of te cliff. Find and in terms of. Find te eigt of te cliff to te nearest m. 25 40 75m 11. Two boats are on opposite sides of a ligtouse. Te angles of elevation of te top of te ligt ouse from te two boats are 38 and22 respectively. If te boats are 120 m apart, find te eigt of te ligtouse correct to te nearest integer. 38 22 120m m age 208 Unit 20 pplications of trigonometry
12. Find te values of a, b, x and y in te given figure. (d) Find te compass bearing of from. Find te true bearing of from. Find te true bearing of from. 13. In te figure, and are te positions of two cars. Find te bearing of from. 14. is S35 W of. If R =, and R = 90, find te compass bearing of from R. 15. In te figure, M is an equilateral triangle. If M is 318 from, find te true bearing of M from. b y 45 40 x a 15 39 42 M R E 35 Unit 20 pplications of trigonometry age 209
16. is 24 km and S 65 W from wile is 30 km and S 25 E from. Find te compass bearing of from. 24km 65 25 17. ar starts travelling from a point at 80 km/ in te direction of 055. t te same time, car starts travelling from at 60 km/ in te direction of 325. 55 Wat is te distance between and after 2 ours? 325 18. Gary walked 1.5 km from to at on bearing of S50 W and ten 3 km from to on a bearing of S50 E. If e walks back to at a constant speed of 50 meter per minute, find te sortest time needed. 50 19. man walks 150 m on a true bearing of 286 and ten 240 m on a true bearing of 210. Find is distance: west of te starting point, sout of te starting point, from te starting point. 50 30km 210 286 150 m 240 m age 210 Unit 20 pplications of trigonometry
20. spere is put inside a sealed rigt cone so tat te igest point of te spere touces te base of te cone. If te eigt of te cone is 8 cm and its vertical angle is 56, wat is te radius of te spere? 21. is a rectangular card wit dimensions 20 cm by 30 cm. It s vertices and touc te edges of a table, and makes an angle of 30 wit te orizontal line E. Find te vertical distances from and to E. Find te orizontal distance from to E. 30cm 22. elicopter is flying orizontally at a certain altitude at a speed of 180 km/. Wen it is at position H, its angle of depression of an airport is 44. fter 15minutes, it comes to position K and te angle of depression of becomes 79. Find te distance between H and K. Find te altitude of te elicopter. 20cm 30 56 79 44 K E H Unit 20 pplications of trigonometry age 211
(II) Stimulating items, o. 23-41 23. Te figure sows a map wit scale 1 cm : 400 m. man standing at is looking down at a tree wic is 6 m tall. is 4 cm long on te map, and te man s eyes are 1.6 m above is feet. Tere is a straigt road connecting and. Find te actual lengt of tis road. Find te gradient and te angle of inclination of te road. Find te angle of depression from te man to te tree. 24. From alf-way up a building te angle of depression of a car is 32. Find te angle of depression from te top of te building. 25. Te figure sows two trees standing vertically on a slope of 20. Te eigts of te trees are 2 m and 5 m respectively, and tey are 100 m 5m apart. Find te angle of depression from te top of te 2m taller tree to te top of te sorter 20 tree. 100m 200m 300m 100m 26. From a tower top, te elevation of a building is 58, wile from te tower base it is 68. Find te eigt of te building if te tower is 18 m ig. age 212 Unit 20 pplications of trigonometry
27. and are two buildings of eigts 65 m and m respectively. Te angle of depression of from is 22 and te angle of elevation of from is 39. Find te value of. Find te distance between te two buildings. 28. Te angles of elevation of a big balloon from te top and te bottom of a building are 37 and 65 respectively. Te building is 80 m tall. Find te vertical eigt of te balloon. 29. and E are two buildings of eigts x and y respectively. Te angle of elevation of from and are respectively θ and 45. Find y in terms of x and θ. x 45 37 65 80m y 30. t 11:00 a.m., a typoon is at 680 km S70 W of Hong Kong. It is moving in a direction of 18 E at 160 km/. Wat will be its sortest distance from Hong Kong? t wat time will it be nearest Hong Kong? Give your answer to te nearest minute. θ E Unit 20 pplications of trigonometry age 213
31. In te figure, is due east of. bears 21 W from and is due nort of. is due nort of and bears 35 E from. Find te bearing of from. 32. From a point of a small town, a man observes tat te bearing of a tower is 230. He drives along a road running due west of at a constant speed of 60 km/. fter 20 minutes, e reaces a curc R from were e observes tat te bearing of becomes 140. Find te distance between R and. 35 21 Te man drives a furter 10 km along te same road and reaces a farm S. Find te true bearing of from S. If tere is a straigt road joining S and, ow long will it take im to drive from S to at te same speed? Give te answer to te nearest 0.1 minute. 140 S 10km R 230 33. ligtouse was 40 km and 32 W from ligtouse. t 1:30 p.m., a sip left and moved at a speed of 28 km/ in te direction of S60 E. Wen te sip was nearest to, it canged te course to 20 W. Te sip ten reaced ligtouse R wic is due nort of. age 214 Unit 20 pplications of trigonometry
Find te bearing and distance of te sip from wen te sip was nearest to it. t wat time, correct to te nearest minute, did te sip reac R? 34. Tree corners of a cube are marked, and as sown. Wat is te value of cos? 35. In te figure, is a square and R is an isosceles triangle in wic = R. If = 6 cm, = 52, find. 40km 60 32 6 cm 36. In te figure, is te centre and is a cord of te circle, and is produced to. It is known tat te radius of te circle is 5 cm, = 8 cm and = 56. If passes troug and is te mid-point of, find. Find. Find te area of. 52 R 20 R Unit 20 pplications of trigonometry age 215
37. In te figure, = = 12cm. If = 40 and = 30, find. 38. In te figure, TR and SR are diameters of two semicircles. and TR = SR = 90. If TS = 2, find in terms of θ. 39. Given : is a square, E = E and E EF. rove E FE. Find te value of tan θ. T 2 S 40. Te figure sow te cross-section of a rectangular block resting on te plane T wic inclines at an angle of 28 wit te orizontal. If RS = 6 cm, R = 18 cm and T = 12 cm, find R te eigts of and R above te ground. 28 S T 6cm 18cm 40 30 E R F age 216 Unit 20 pplications of trigonometry
41. Find te value of y in Figure 41a. (d) Figure 41b sows two identical lampposts, one on te orizontal level, anoter on a slope of 20. Te angle of elevation of te sun is 60. Find te values of a and b. Wen te angle of elevation of te sun is 60, te sadow of te lamppost on te orizontal level is 2.3m long. Find te eigt of te lamppost. Find te lengt of te sadow of te lamppost on te slope wen te angle of elevation of te sun is 60. 8 60 75 y Figure 41a 60 2.3m a 20 Figure 41b x b Unit 20 pplications of trigonometry age 217