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16 Trigonometry (3) Case Study 16.1 Applications in Two-dimensional Problems 16.2 Basic Terminology in Three-dimensional Figures 16.3 Applications in Three-dimensional Problems Chapter Summary Case Study How can we walk up the hill in a relatively more comfortable way? It is more comfortable to walk up an inclined road along a zigzag path, let me explain it to you. In order to explain the above question, we can use a prism to illustrate the situation such that AB is the top of the inclined road and PQ is the horizontal ground level. As shown in the figure, M and N are the mid-points of BQ and DQ respectively. Although PMA is longer than PA, PM and MA are less steep than PA, and hence they are more comfortable to walk up. P. 2 16.1 Applications in Two-dimensional Problems A. Angle of Elevation and Angle of Depression When we observe an object above us, the angle q between our line of sight and the horizontal is called the angle of elevation. When we observe an object below us, the angle f between the line of sight and the horizontal is called the angle of depression. These two angles are important in solving practical trigonometric problems. P. 3 16.1 Applications in Two-dimensional Problems A. Angle of Elevation and Angle of Depression Example 16.1T In the figure, TB is a flag. The angles of elevation from a point A to the top T and the base B of the flag are 35 and 20 respectively. If the flag is 3 m long, find the distance between A and T. Solution: TAB 35 20 15 TBA 90 20 (ext. of ) 110 By sine formula, AT 3m sin 110 sin15 3 sin 110 AT m sin 15 10.9 m (cor. to 3 sig. fig.) P. 4 16.1 Applications in Two-dimensional Problems A. Angle of Elevation and Angle of Depression Example 16.2T Fanny looks down from a platform at one end of a swimming pool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6 and 14 respectively. (a) Find the height of the platform if Fanny’s eyes are 1.7 m above the platform. Solution: ACB 8 CAB 6 (alt. s, // lines) CBD 14 (alt. s, // lines) (a) By sine formula, BC 25 m sin 6 sin 8 BC 18.7767 m In CBD, CD sin 14 18.7767 m CD 4.5425 m The height of the platform (4.5425 1.7) m 3 m (cor. to the nearest m) P. 5 16.1 Applications in Two-dimensional Problems A. Angle of Elevation and Angle of Depression Example 16.2T Fanny looks down from a platform at one end of a swimming pool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6 and 14 respectively. (a) Find the height of the platform if Fanny’s eyes are 1.7 m above the platform. (b) How far is boy B from the near end of the pool? (Give the answers correct to the nearest m.) Solution: CBD 14 (alt. s, // lines), BC 18.7767 m (b) In CBD, BD BD 18.2190 m cos14 18.7767 m 18 m (cor. to the nearest m) Boy B is 18 m from the near end of the pool. P. 6 16.1 Applications in Two-dimensional Problems B. Bearing In junior forms, we learnt how to use a compass bearing or a true bearing to indicate the direction of an object from a given point. Compass bearing is also known as reduced bearing, and true bearing is also known as whole circle bearing. When using a compass bearing, directions are measured from the north (N) or the south (S), thus the bearing is represented in the form: Nq E, Nq W, Sq E or Sq W, where 0 q 90. Notes: If q 0 or 90, we simply write it as N, E, S or W. When using a true bearing, all directions are measured from the north in a clockwise direction. The bearing is expressed in the form q, where 0 q 360 and written in three digits such as 007, 056 or 198. P. 7 16.1 Applications in Two-dimensional Problems B. Bearing Example 16.3T Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (Give the answers correct to 1 decimal place.) Solution: (a) PE (12 60 5) m 3600 m PF (10 60 5) m 3000 m FPE 220 150 70 By cosine formula, 3000 m FE 36002 30002 2(3600)(3000) cos 70 m 14 572 364.9 m 3817.3767 m 3817 .4 m (cor. to 1 d. p.) P. 8 3817.3767 m 3600 m 16.1 Applications in Two-dimensional Problems B. Bearing Example 16.3T Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (b) Find the true bearing of Frank from Eric. (Give the answers correct to 1 decimal place.) Solution: (b) a 180 150 30, b a 30 (alt. s, // lines) By sine formula, 3000 m PF PE sin E sin P 3817.3767 m 3000sin 70 0.7385 sin E True bearing 3817.3767 360 47.6026 30 E 47.6026 282.4 (cor. to 1 d. p.) P. 9 3600 m 16.1 Applications in Two-dimensional Problems B. Bearing Example 16.3T Eric and Frank are cycling away from P. Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is cycling in the direction 220 with a speed of 10 m/s. After five minutes, they stop and take a rest. (a) What is the distance between them now? (b) Find the true bearing of Frank from Eric. (c) After having a rest, if they cycle towards each other at the same speed as before, how long does it take for them to meet? (Give the answers correct to 1 decimal place.) Solution: 3000 m 3817.3767 m (12 10) s 2 min 53.5 s (or 2.9 min ) (cor. to 1 d. p.) (c) Time taken P. 10 3817.3767 m 3600 m 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions 1. Angle between Two Straight Lines The figure shows two intersecting straight lines lying on the same plane. The acute angle q is called the angle between the two straight lines AB and CD. In 3-D Figures, we can also identify the angle between two straight lines. For example, the angle between BH and FH is BHF. P. 11 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions 2. Angle between a Straight Line and a Plane When a line is inclined on a plane, we can get a projection of the line on the plane. For example, when a javelin TP hits the ground, the line AP is the projection of TP on the ground. In three-dimensional space, the angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane. For example, the projection of the line AG on AEHD is AH. the angles between the line AG and AEHD is GAH. Remark: If the line is perpendicular to the plane, then the projection of the line on the plane is only a point. P. 12 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions 3. Angle between Two Planes Consider the following two situations. (a) A wooden door is opened. (b) A greeting card is standing on a table. In the above two cases, we observe that there are two planes intersecting with each other. P. 13 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions When two planes intersect, they meet at a straight line which is called the line of intersection. In the figure, a and b are two planes while AQB is the line of intersection. PS and RT are lines on the planes a and b respectively such that PQ ^ AB and RQ ^ AB. The angle q between the lines PQ and RQ is called the angle between planes a and b. Remarks: 1. Actually, the angle between two intersecting planes can be acute or obtuse. 2. Usually, we do not consider the reflex angle as the angle between two intersecting planes. P. 14 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions The figure shows a rectangular block. For planes ABFE and BCHE: Line of intersection: _____________ BC Angle between 2 planes: _____________ ABE / DCH For planes CDEF and EFGH: Line of intersection: _____________ EF Angle between 2 planes: _____________ CFG / DEH The figure shows a right pyramid with a square base. For planes VCD and ABCD: Line of intersection: _____________ CD Angle between 2 planes: _____________ VPQ For planes VBC and VCD: Line of intersection: _____________ VC Angle between 2 planes: _____________ BND P. 15 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions 4. Distance between a Point and a Straight Line Consider a rectangular pyramid. The distance between the point B and the line VC is the perpendicular distance between B and VC, that is, the length of BE. 5. Distance between a Point and a Plane The distance between a point and a plane is the distance between the point and its projection on the plane, that is, the perpendicular distance between the point and the plane. As shown in the figure, PQ is the distance between point P and the plane. P. 16 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.4T The figure shows a cuboid. AB 3 cm, AD 6 cm and BF 4 cm. (a) Find the length of AG and express the answer in surd form. (b) Find the angle between the lines AG and AF. (Give the answer correct to 3 significant figures.) Solution: (a) In EFG, EG 2 EF 2 FG 2 EG 32 6 2 cm 45 cm In AEG, AG 2 AE 2 EG 2 (Pyth. theorem) (Pyth. theorem) AG 4 2 ( 45) 2 cm 61 cm (b) FAG is the angle between the lines AG and AF. In AFG, 6 FG sin FAG 61 AG FAG 50.2 (cor. to 3 sig. fig.) The angle between the lines AG and AF is 50.2. P. 17 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.5T The figure shows a regular rectangular pyramid with base 12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary. Solution: (a) In VPD, VD 2 VP 2 PD 2 15 2 VP 2 5 2 VP 200 cm 10 2 cm (Pyth. theorem) In BCP, BP 2 BC 2 CP 2 (Pyth. theorem) BP 122 52 cm 169 cm 13 cm P. 18 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.5T The figure shows a regular rectangular pyramid with base 12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary. (b) Find the angle between (i) lines VB and VD, (ii) lines VP and BP. (Give the answers correct to 3 significant figures.) Solution: (b) (i) BVD is the angle between lines VB and VD. In BCD, In BVD, by cosine formula, BD 2 BC 2 CD 2 152 152 ( 244) 2 cos BVD (Pyth. theorem) 2(15)(15) 206 BD 122 102 cm 450 244 cm BVD 62.8 (cor. to 3 sig. fig.) The angle between lines VB and VD is 62.8. P. 19 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.5T The figure shows a regular rectangular pyramid with base 12 cm 10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a) Find VP and BP and give the answers in surd form if necessary. (b) Find the angle between (i) lines VB and VD, (ii) lines VP and BP. (Give the answers correct to 3 significant figures.) Solution: (b) (ii) BPV is the angle between lines VP and BP. In BVP, by cosine formula, 132 (10 2 ) 2 152 cos BPV 2(13)(10 2 ) 144 260 2 BPV 66.9 (cor. to 3 sig. fig.) The angle between lines VP and BP is 66.9. P. 20 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.6T The figure shows a wedge with rectangular planes ABCD, EFDA and EBCF. AB 10 cm, BC 16 cm, DCF 35 and DF ^ CF. (a) Find the lengths of BD and DF. (b) Find the angle between line BD and plane EBCF. (Give the answers correct to 3 significant figures.) Solution: (a) In ABD, In CDF, BD2 AB 2 AD2 (Pyth. theorem) CD AB 10 cm DF BD 102 162 cm sin 35 10 cm 356 cm 18.9 cm (cor. to 3 sig. fig.) DF 5.74 cm (cor. to 3 sig. fig.) (b) Since BF is the projection of DF 10 sin 35 sin DBF BD on plane EBCF, DBF 356 BD is the required angle. DBF 17.7 (cor. to 3 sig. fig.) The angle between line BD and plane EBCF is 17.7. P. 21 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.7T The figure shows a right-angled triangular prism with ABCD, AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF 6 cm, FC 8 cm and AD 15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.) Solution: (a) Since EC is the projection of AC on the plane BCFE, ACE is the required angle. In CEF, In ACE, AE 6 CE 2 EF 2 CF 2 (Pyth. theorem) tan ACE CE 17 CE 152 82 cm ACE 19.4 17 cm (cor. to 3 sig. fig.) The angle between the line AC and the plane BCFE is 19.4. P. 22 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.7T The figure shows a right-angled triangular prism with ABCD, AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF 6 cm, FC 8 cm and AD 15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.) Solution: (b) Since EP is the projection of AP on the plane BCFE, APE is the required angle. In BEP, BP 7.5 cm. In AEP, 6 AE tan APE EP 2 BE 2 BP 2 (Pyth. theorem) 120.25 EP EP 82 7.52 cm APE 28.7 120.25 cm (cor. to 3 sig. fig.) The angle between the line AP and the plane BCFE is 28.7. P. 23 16.2 Basic Terminology in Threedimensional Figures A. Terms and Definitions Example 16.8T The figure shows a pyramid with a right-angled triangular base. AB AC 4 cm, VA 5 cm and VAB VAC 90. (a) Find the length of AM where M is the mid-point of BC in surd form. (b) Find the angle between the planes ABC and VBC. (Give the answer correct to 3 significant figures.) Solution: (a) ABC ACB (base s, isos. ) 180 90 ( sum of ) 2 45 In ABM, AM sin 45 4 cm AM 2 2 cm (b) VMA is the angle between the planes ABC and VBC. 5 VMA 60.5 (cor. to 3 sig. fig.) In AVM, tan VMA 2 2 The angle between the planes ABC and VBC is 60.5. P. 24 16.2 Basic Terminology in Threedimensional Figures B. Lines of Greatest Slope In the figure, the inclined plane ABCD intersects the horizontal plane ABEF at the line AB. Three lines XY, PQ and ST are drawn on the inclined plane with PQ perpendicular to AB. Let a, b and g be the angles that XY, PQ and ST make with the horizontal plane respectively. If we compare the three angles, we find that b > a and b > g. In fact, PQ makes the largest angle with the horizontal plane and it is called the line of greatest slope of the inclined plane. Notes: There are infinitely many lines of greatest slope on a given inclined plane, such as l1, l2, and l3 (that are parallel to the line PQ) in the figure. P. 25 16.2 Basic Terminology in Threedimensional Figures B. Lines of Greatest Slope Example 16.9T The figure shows a right-angled triangular prism with ABCD, BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC 5 cm, DE 2 cm and DA 5DE, find (a) the angle between the BN and plane BCEF, (b) the angle between line NC and plane BCEF, (c) the inclination of the line of greatest slope of plane ABCD. (Give the answers correct to 3 significant figures.) Solution: BF 5 cm and AD FE 10 cm Let P be the mid-point of EF. FP 5 cm and NP 2 cm (a) NBP is the required angle. In BFP, BP 52 52 cm 50 cm (Pyth. theorem) P 2 0.2828 50 NBP 15.79 The required angle is 15.8. tan NBP P. 26 16.2 Basic Terminology in Threedimensional Figures B. Lines of Greatest Slope Example 16.9T The figure shows a right-angled triangular prism with ABCD, BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC 5 cm, DE 2 cm and DA 5DE, find (a) the angle between the BN and plane BCEF, (b) the angle between line NC and plane BCEF, (c) the inclination of the line of greatest slope of plane ABCD. (Give the answers correct to 3 significant figures.) Solution: (b) NCP is the required angle. Since CPN BPN (SAS). NCP NBP 15.8 (cor. to 3 sig. fig.) P (c) Line of greatest slope: CD DCE is the required angle. 2 tan DCE DCE 21.8 (cor. to 3 sig. fig.) 5 P. 27 16.3 Applications in Three-dimensional Problems In this section, we shall further study some applications of trigonometric formulas in three-dimensional figures, together with bearings and angles of elevation and depression. P. 28 16.3 Applications in Three-dimensional Problems Example 16.10T The figure shows the plane of a hillside. E is due west of F and C is due south of F. The inclination of the path CD is 8. DF 10 m and AD 65 m. Eric runs directly from C to A. (a) Find the inclination of his path with the ground EBCF. (b) Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.) Solution: (a) Since EC is the projection of AC on the ground EBCF, ACE is the required angle. In CDF, In ACD, AC 2 AD 2 CD 2 (Pyth. theorem) 2 2 10 m AC 65 71 . 853 m sin 8 96.891 m CD In ACE, 10 10 CD m sin ACE ACE 5.92 sin 8 96.891 (cor. to 3 sig. fig.) 71.853 m The inclination of his path with the ground EBCF is 5.92. P. 29 16.3 Applications in Three-dimensional Problems Example 16.10T The figure shows the plane of a hillside. E is due west of F and C is due south of F. The inclination of the path CD is 8. DF 10 m and AD 65 m. Eric runs directly from C to A. (a) Find the inclination of his path with the ground EBCF. (b) Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.) Solution: (b) The projection of AC on the ground EBCF is EC. In CDF, 10 m tan8 CF 10 CF m tan 8 71.1537 m In CEF, EF AD 65 m. 65 tan ECF 71.1537 ECF 42.4122 42.4 (cor. to 3 sig. fig.) The compass bearing of Eric’s path from C is N42.4W. P. 30 16.3 Applications in Three-dimensional Problems Example 16.11T A lighthouse VA with height 90 m stands on the same plane as two ships P and Q. The bearings of the lighthouse from P and Q are N50 E and N65 W respectively. The angle of elevation of V from P is 15 and the distance between P and Q is 800 m. (a) Find the distance between P and A. (b) Find the distance between Q and A. (Give the answers correct to 3 significant figures.) Solution: (a) In VAP, (b) PAQ 50 65 115 (alt. s, // lines) 90 m By cosine formula, tan15 PA QP2 PA2 QA2 2( PA)(QA) cos PAQ 90 m 8002 335.88462 QA2 PA tan15 2(335.8846)(QA) cos115 335.8846 m QA2 283.9019(QA) 527181.5355 0 336 m QA 598 m or 882 m (rejected) (cor. to 3 sig. fig.) (cor. to 3 sig. fig.) P. 31 16.3 Applications in Three-dimensional Problems Example 16.11T A lighthouse VA with height 90 m stands on the same plane as two ships P and Q. The bearings of the lighthouse from P and Q are N50 E and N65 W respectively. The angle of elevation of V from P is 15 and the distance between P and Q is 800 m. (a) Find the distance between P and A. (b) Find the distance between Q and A. (c) Hence find the angle of elevation of V from Q. (Give the answers correct to 3 significant figures.) Solution: (c) In VAQ, 90 597.8677 VQA 8.5607 8.56 (cor. to 3 sig. fig.) The angle of elevation of V from Q is 8.56. tan VQA P. 32 Chapter Summary 16.1 Applications in Two-dimensional Problems 1. Angles of elevation and depression The angle between the line of sight of an object above us and the horizontal is the angle of elevation. The angle between the line of sight of an object below us and the horizontal is the angle of depression. 2. (a) Compass bearing All directions are measured from the north (N) or the south (S). The bearing is expressed in the form Nq E, Nq W, Sq E or Sq W, where 0 q 90. (b) True bearing All directions are measured from the north in a clockwise direction. The bearing is expressed in the form q, where 0 q 360 and written in three digits. P. 33 Chapter Summary 16.2 Basic Terminology in Three-dimensional Figures 1. Angle between Two Straight Lines The angle between two intersecting straight lines is the acute angle formed by the two straight lines lying on the same plane. 2. Angle between a Straight Line and a Plane The angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane. 3. Angle between Two Planes The angle between two planes is the angle between two perpendiculars on the respective planes to the line of intersection of the two planes. P. 34 Chapter Summary 16.2 Basic Terminology in Three-dimensional Figures 4. Distance between a Point and a Straight Line The distance between a point and a straight line is the perpendicular distance from the point to the line. 5. Distance between a Point and a Plane The distance between a point and a plane is the distance between the point and its projection on the plane. 6. Lines of Greatest Slope If PQ ^ AB, then PQ is called the line of greatest slope of the inclined plane ABCD. P. 35 Chapter Summary 16.3 Applications in Three-dimensional Problems In three-dimensional figures, we can also find (a) angles of elevation and depression, and (b) bearing. P. 36