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TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. Q1. The diagram shows four points, A, B, C and D, on a piece of horizontal land. It is given that AB = 45 meters, AD = 25 meters and BD = 28 meters. (a) Calculate angle ADB. (b) Given also that CD = 22 meters and that angle ACD = 33o, calculate angle ADC. (c) The line BD is produced beyond D. Calculate the shortest distance from C to this extended line. (d) D is the foot of a vertical mast, DT. The angle of elevation of the top of the mast, T, from A is 40o. Calculate the angle of elevation of T from B. Ans. (a) 252 282 452 cos ADB 2(25)(28) ADB 116.1o (b) sin 33o 22 25 DAC 28.64o sin DAC DAC 180o 33o 28.64o 118.4o (c) Diagram E is the foot of the perpendicular from C to BD produced. EDC ADC ADE ADC (180o ADB) 118.4o 180o 116.1o 54.5o The shortest distance from C to the extended Line = CE = 22 sin 54.5o = 17.9 m Compiled By : Sir Rashid Qureshi www.levels.org.pk 1 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. (d) DT = 25 tan 40o DT 25 tan 40o tan TDB = DB 28 Angle of elevation of T from B = 36.8o Q2. A vertical flagpole, 18 m high, stands on horizontal ground. Calculate the angle of elevation of the top of the flagpole from a point, on the ground 25 m from its base. Ans. 35.8o Q3. The diagram shows the position of a harbor, H, and three island A, B and C. C is due North of H. ˆ = 128o. The bearing of A from H is 062o and HAB HA = 54 km and AB = 31 km. (a) Calculate the distance HB. (b) Find the bearing of B from A. (c) The bearing of A from C is 133o. Calculate the distance AC. (d) A lightship, L, is positioned due North of H and equidistant from A and H. Calculate the distance HL. (c) Q4. 65.2 km (d) 57.5 km An aircraft waiting to land is flying around a triangular Circuit ABC. A, B and C are vertically above three beacons, X, Y and Z. T is the control tower at the airport, and T, X, Y and Z lie in a horizontal plane. BC = 18 km, CA = 22 km and AB =b24 km. (a) (i) The plane is flying at 200 km/h. Calculate the time, in minutes and seconds, Compiled By : Sir Rashid Qureshi www.levels.org.pk 2 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. that The aircraft takes to complete one circuit. (ii) (b) Calculate the largest angle of triangle ABC. Z is due West of T. The bearing of X from Z is 042o and the bearing of X from T is 338o. (c) (i) Find the angles of triangle TXZ. (ii) Calculate TX. The aircraft is flying at a constant height of 2600 meters. Calculate the angle of depression of the tower, T, from the aircraft when it is at A. Ans. Q5. (a) (i) 19 min 12 sec (b) 64o (c) Angle of depression = 8.4o (ii) C = 73.0o (ii) TX = 17.6 km The diagram shows the positions of A and B. Find the bearing of (a) A from B, (b) B from A. Ans. (a) 300o Q6. Three paths, AB, BC and CA, run along the edges of a horizontal triangular field ABC. (b) 120o BC = 51 m, AC = 72 m and angle ACB = 81o. (a) Calculate the length of AB. (b) Calculate the area of the field ABC. (c) Calculate the shortest distance from C to AB. (d) A vertical tree, CT, has its base at C. The angle of elevation of the top of the tree from A is 15o. (e) John measured the largest angle of elevation of the top of the tree as seen Compiled By : Sir Rashid Qureshi www.levels.org.pk 3 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. form the path AB. Calculate this angle of elevation. Ans. (a) 81.5 m (b) 1810 m2 correct to 3 figures (c) 44.5 m (d) CT = 19.3 m (e) 23.4o A is due North of O. Q7. (a) ˆ = 12o. A ship sailed from O to B, where AOB Write down the bearing of B from O. (b) At B, the ship turned and sailed to C, ˆ = 50o. Where OBC Calculate the bearing of C from B. Ans. (a) 348o (b) 218o Q8. Three points, A, B and C, lie on a horizontal field. Angle BAC = 75o and the bearing of C from A is 217o. AB = 72 m and AC = 60 m. (a) (b) Calculate (i) the bearing of B from A, (ii) BC, (iii) Angle ABC, (iv) the bearing of C from B. A girl standing at B is flying a kite. The kite, K, is vertically above A. The string, BK, attached to the kite is at 24o to the horizontal. Calculate the angle of elevation of the kite when viewed form C. Compiled By : Sir Rashid Qureshi www.levels.org.pk 4 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. Ans. Q9. (a) (i) 292o (ii) (iii) ABC = 45.7o 80.9 m (iv). 157.7o (b) 28.1o (b) Diagram I represents part of the framework of the ride. The points A, B, C, D, E and F are on the framework. The points H, C, G, E and F lie on a horizontal lien. The lines BH and DG are vertical BC = 80 cm, HC = 60 m, DG = 40 m GE = 35 m and ˆ = 32o DCG Calculate (i) ˆ , HCB (ii) CD, (iii) the angle of depression of E from D. (c) Diagram II shows part of the ride. The carriage that carried the family was 4.6 m long. It was travelling at a constant speed of 15 m/s as it passed the point F. Compiled By : Sir Rashid Qureshi www.levels.org.pk 5 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. (i) Calculate, correct to the nearest hundredth of a second, the time taken for the carriage to pass the point F. Ans. Q10. (ii) Express 15 m/s in kilometers per hour. (b) (i) HCB 41.4o (ii) CD = 75.5 m (c) (i) 0.31 s (ii) 54 km/h (iii) 48.8o In the diagram, A and B are two points on a straight coastline. B is due east of A and AB = 7 km. The position of a boat at different times was noted. (a) At 8 a.m, the boat was at C, where ˆ = 66o and ABC ˆ = 48o ACB Calculate (b) (i) the bearing of B from C, (ii) the distance AC. At 9 a.m, the boat was at D, where ˆ = 41o. AD = 6.3 km and DAB Calculate (i) the area of triangle ADB, (ii) the shortest distance from the boat to the coastline. Compiled By : Sir Rashid Qureshi www.levels.org.pk 6 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. (c) At 11 a.m, the boat was at E, where AE = 9 km and BE = 5 km. Calculate the bearing of E from A. Ans. (a) (i) 138o (ii) 5.69 km (b) (i) 14.5 km2 (ii) 4.13 km [ Q11. A vertical flagpole, BF, stands at the top of a hill. AB is the steepest path up the hill. ˆ = 90o. N lies vertically below B and ANB AN = 100 m and AB = 104 m. (a) Show that BN = 28.6 m. (b) ˆ = 25o. It is given that FAN (i) Write down the size of the angle of depression of A from F. (ii) (c) Calculate the height, BF, of the flagpole. The diagram shows three other straight paths (CB, DB and ACD) on the hill. ˆ NAC ˆ 90o . The path ACD is horizontal and BAC CN and DN are horizontal lines. (i) Given that AC = 60 m, ˆ . Calculate BCN (ii) ˆ 10o , Given that BDN ˆ . Calculate DBA Compiled By : Sir Rashid Qureshi www.levels.org.pk 7 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. Ans. (a) BN 2 AB 2 AN 2 Pythagoras’ 1042 1002 816 m2 BN 816 28.6 m Q12. (b) (i) 25o (ii) BF = 18.1 m (c) (i) BCN 13.8o (ii) DBA 50.8o Diagram I In Diagram I, the point D lies on AC and N is the foot of the perpendicular from C to BD. AB = 61m, AD = 30 m and DC = 45m. Angle BAC = 41o. (a) (b) Calculate BD. Show that, correct to the nearest square meter. The area of triangle BDA is 600 m2. (f) area of BCD 3 . area of BDA 2 (c) Explain why (d) Calculate the area of triangle BCD. (e) Hence calculate CN. Diagram II The same points B, C, D and N lie on a sloping plane. The point E is 15 m vertically below C. The points B, E, D and N lie on a horizontal plane. Diagram II represents this information. Calculate the angle of elevation of C from N. Ans. (a) BD = 43.1 m (b) Consider BDA Compiled By : Sir Rashid Qureshi www.levels.org.pk 8 TOPIC#10 Trigonometry --- Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems. Area 12 6130sin 41 600.29 600 m2 Area of BCD 2 CD h area of BDA 12 AD h 1 (c) CD AD 45 3 30 2 (d) area of BCD 32 600 900m (e) area of BCD 12 BD CN 1 900 43.113 CN 2 CN 41.8m2 (f) angle of elevation = 21.1o Compiled By : Sir Rashid Qureshi www.levels.org.pk 9