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9.1 Chapter 9 Applications of Trigonometry Chapter 9 Applications of Trigonometry WARM-UP EXERCISE 1. Find the unknown in each of the following. (Leave your answers in surd form.) (a) A (b) E x 4 D 7 I (c) 12 4 y J 5 K 18 F B z C 2. Simplify the following algebraic fractions. (a) 3x 6y x 2y x 2y (b) 7 4 x y yx 3. Make the variable inside each pair of square brackets as the subject of the formula. 1 y [y] 2b b [b] (a) x 4 (b) a 4. According to each of the following triangles, find the values of cos , sin and tan . (Correct your answers to 2 decimal places if necessary.) (a) D 17 E 15 F (c) D 4.8 E 7.3 8 E D (b) 9.8 6.3 12.2 F F 7.9 9.2 New Trend Mathematics S3B — Junior Form Supplementary Exercises BUILD-UP EXERCISE [ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Set and Advanced Set. You may choose to complete any ONE set according to your need. ] Exercise 9A [ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ] Elementary Set Level 1 1. Find the value of each of the following. 4 (a) tan 45 (c) sin 60 tan 30 (b) cos 2 60 (d) tan 30 sin 30 2. Find the value of each of the following. (a) sin 45 cos 45 (b) 2 cos 30 tan 60 (c) tan 30 cos 60 (d) 2 sin 60 3 cos 30 3. Find the value of each of the following. Ex.9A Elementary Set (a) sin 60 3 tan 45 (b) cos 30 sin 30 tan 30 (c) tan 45 cos 60 sin 30 (d) cos 60 2 sin 30 sin 60 tan 45 4. Find in each of the following. (a) 2 sin 1 (b) 1 sin (c) 3 tan 3 (d) 5. Find in each of the following. tan 45 (a) 2 cos 3 (c) 4 3 4 tan 2 1 2 2 cos 1 0 (b) 2 sin 2 3 0 (d) 3 tan 3 1 0 Chapter 9 Applications of Trigonometry 9.3 Level 2 6. Find the unknown in each of the following figures. (a) D (b) A (c) G H 12 60 b 4.8 a 8 c 45 B E C (d) (e) J I F (f) M P Q f d 5 60 K Ex.9A Elementary Set 30 2 3 e 60 30 N O 15 R L Advanced Set Level 1 1. Find the value of each of the following. (b) cos 2 30 cos 30 (d) sin 60 (f) sin 2 30 tan 2 60 2. Find the value of each of the following. (a) sin 30 cos 60 (c) sin 60 tan 30 cos 60 (b) 2 cos 30 tan 60 (d) 3 tan 30 2 cos 30 sin 45 cos 45 3. Find in each of the following. (a) 2 sin 1 (c) cos 2 2 2 (b) 4 cos 2 3 (d) 2 3 tan 4 6 4. Find in each of the following. (a) 3 tan 2 3 (c) 3 1 tan 5 1 3 1 (d) cos 2 4 (b) 2 sin Ex.9A Advanced Set (a) sin 2 45 1 (c) sin 30 tan 60 (e) sin 30 9.4 New Trend Mathematics S3B — Junior Form Supplementary Exercises Level 2 5. Find the unknown in each of the following figures. (a) A P (b) (c) X Y 60 4 45 3 b R B a c 2 30 Q C Z Ex.9A Advanced Set A (d) A (e) 5 2 45 D E a D A B 60 b y 45 30 4 (f) C BCD is a straight line. 3 B C 45 30 C B AEC and BED are straight lines. D P 6. In the figure, QCDR is a straight line, PQR is an equilateral triangle, ABCD is a rectangle. If A and B are the mid-points of PR and PQ respectively, and each side of PQR is 6 cm long, A (a) find the length and width of rectangle ABCD. (b) find the area of rectangle ABCD. R B D C Q Exercise 9B [ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ] Ex.9B Elementary Set Elementary Set Level 1 1. Simplify each of the following. 1 (a) tan cos (c) 2 tan sin (b) 3 sin cos tan Chapter 9 Applications of Trigonometry 9.5 2. Simplify each of the following. (a) 3sin 2 3cos2 (c) 1 cos 2 sin sin (b) 4 cos2 4 sin 2 (d) tan 2 cos2 cos2 3. Find in each of the following. 4. Simplify each of the following. 3 cos (a) sin( 90 ) (c) sin( 90 ) tan cos(90 ) (b) cos sin 37 1 (d) tan 58 tan (b) cos(90 ) tan(90 ) (d) tan 1 cos(90 ) sin( 90 ) 5. Find the value of each of the following. (a) sin 2 27 sin 2 63 (b) tan 52 (c) tan 18 tan 72 (d) 1 tan 38 cos 69 sin 21 6. Find the value of each of the following. (a) 5 3 tan 53 tan 37 (c) cos2 48 cos2 42 (b) sin 34 cos 56 sin 56 cos 34 sin 18 (d) cos 72 tan18 tan 72 Level 2 3 , find the values of cos and tan . 5 5 (b) If cos , find the values of sin and tan . 7 3 (c) If tan , find the values of sin and cos . 10 7. (a) If sin 8. If sin 1 24 . , find the value of cos 25 tan Ex.9B Elementary Set (a) sin cos 42 1 (c) tan tan 49 9.6 New Trend Mathematics S3B — Junior Form Supplementary Exercises 9. Simplify each of the following. (a) Ex.9B Elementary Set (c) cos 2 sin sin 1 cos 2 tan 2 (b) tan θ cos3 θ sin 3 θ (d) cos 2 cos 4 tan 10. Find in each of the following. 1 tan 4 cos 4 (a) sin 3 cos 2 (b) (c) sin cos( 36) (d) tan(4 10) Advanced Set Level 1 1. Simplify each of the following. cos (a) sin 1 tan 32 (b) cos tan sin sin cos tan (c) sin 2 cos2 (d) 1 (e) sin θ cos θ tan θ 1 (f) sin cos tan cos2 Ex.9B Advanced Set 2. Find in each of the following. (a) sin cos 20 1 (c) tan tan 40 (b) cos sin 45 1 (d) tan 37 tan 3. Simplify each of the following. sin cos(90 ) (b) sin sin( 90 ) (c) sin( 90 ) tan (d) sin tan cos(90 ) tan(90 ) (b) 1 tan 55 tan 35 (a) 4. Find the value of each of the following. (a) sin 2 50 sin 2 40 (c) cos 60 cos 30 (d) tan 22 tan 68 Chapter 9 Applications of Trigonometry 5. Find the value of each of the following. sin 60 (a) 1 cos 30 (c) sin 2 62 sin 2 28 9.7 (b) sin 40 cos 50 sin 50 cos 40 (d) sin 24 tan 66 cos 66 tan 24 Level 2 6. According to each of the following trigonometric ratios, find the remaining two trigonometric ratios. 7. Simplify each of the following. sin cos (a) cos sin (c) cos3 cos sin 2 cos 8. Simplify each of the following. sin( 90 ) (a) cos(90 ) tan(90 ) cos (c) 7 cos(90 ) tan(90 ) 9. Find in each of the following. 1 (a) tan( 55) tan10 (c) sin 3 cos 63 (b) sin 3 4 (b) 4 3cos2 3sin 2 (d) (b) 9 9 cos 2 2 tan 2 cos(90 ) tan(90 ) cos (b) sin( 18) cos( 32) (d) cos( 10) sin( 2 40) 10. Prove that each of the following is an identity. 1 sin 1 (a) (b) tan 2 1 cos tan cos 2 1 tan tan(90 ) sin cos (c) sin 2 (90 ) 1 sin 2 (d) (e) 1 2 sin cos (cos sin ) 2 (f) sin 2 (1 cos ) 2 2(1 cos ) Ex.9B Advanced Set 6 7 8 (c) tan 5 (a) cos 9.8 New Trend Mathematics S3B — Junior Form Supplementary Exercises Exercise 9C [ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ] Elementary Set Level 1 1. Find the area of each of the following triangles. A (a) P (b) L (c) 60 12 cm 7 cm 30 B N 5 cm Q C 60 R M 2. Find the area of each of the following quadrilaterals. (a) A B (b) P 4 cm (c) Q D 4 3 cm 2 5 cm 120 2 3 cm G E Ex.9C Elementary Set 60 60 D S C R F 3. Find the perimeter of each of the following figures. (a) A 5 cm (b) B D (c) P 8 cm 30 45 D 9 cm C Q E 45 R QRS is a straight line. 6 cm G 60 S F 4. In the figure, ABCD is a trapezium, ADC 60 and BC CD. Find the area of trapezium ABCD. A 12 cm B 10 cm 60 D 5. The length of a diagonal of a square is 18 cm. Find the area of the square. C 9.9 Chapter 9 Applications of Trigonometry Level 2 D 6. In the figure, DEFG, AFC and AGB are straight lines. (a) Find the lengths of AC, AD and CD. (b) Find DCE. (c) Find the length of DE. E C F 1 Ex.9C Elementary Set (d) Hence find the value of sin 75 by considering ADG. 30 45 A 7. In the figure, ABCDEFGH is a regular octagon, where DE 2 cm. PFE and HGP are straight lines and GPF 90. G B A (a) Find . B H C G D (b) Find the length of PF. (c) Find the area of FGP. 2 cm (d) Hence find the area of ABCDEFGH. P Advanced Set Level 1 E F 1. Find the area of each of the following figures. B (a) (b) 3 2 cm D (c) E G 45 5 cm 5 cm 30 A (d) F C 7 cm J K I (e) P 10 cm W (f) Q 30 60 12 cm M 4 cm 10 cm Y 60 L S X R Z Ex.9C Advanced Set 30 H 9.10 New Trend Mathematics S3B — Junior Form Supplementary Exercises 2. Find the perimeter of each of the following figures. A (a) 15 cm D (b) Q P (c) E 30 8 3 cm B E C G 10 3 cm 60 60 H S F R 60 D 4 3 cm 3. A slide is 10 m long. The angle between the slide and the ground is 45. Find the height of the top of the slide from the ground. Level 2 Ex.9C Advanced Set 4. In the figure, APQ is a right-angled triangle. PQR is a straight line. QAP 60, QRB 90, ABR 90 and AB 2 cm. (a) Find the length of RB. P (b) Find the length of PR. Q 60 A 2 cm R B D 5. In the figure, AEC and DCFB are straight lines. BDE BAC 30, CFE FBA 90 and CE 60 cm. Find the length of DE. 30 60 cm C E A F 30 B 6. In the figure, AED and BCD are straight lines. D (a) Express AC, BC and CD in terms of a. E (b) Find CDE. C (c) Hence find the value of sin 15 by considering ACE. 15 30 A a B D 7. In the figure, ABCDEFGHIJKL is a regular 12-sided polygon with the sides 1 cm each, and ACEGIK is a regular hexagon. C E F B (a) Find BAC. (b) Use ABC and ALK to form a rhombus. Then find the area of the rhombus. G A H L (c) Hence find the total area of the shaded regions. Ex.9C Advanced Set 9.11 Chapter 9 Applications of Trigonometry I K J Exercise 9D [ In this exercise, correct your answers to 3 significant figures if necessary. ] Elementary Set Level 1 1. Arrange the following gradients in descending order of their angles of inclination. 1 I. 1 : 10 II. 1 : 5 III. 1 : 9 2 IV. 1 : 0.8 V. 1 : 20 VI. 1 : 1.4 1 : 10 3. The horizontal distance and vertical distance of a slope are 40 m and 8 m respectively. Find the gradient of the slope. 4. A person walks 80 m down a road with the gradient of 1 . Find the vertical distance 3 travelled. Level 2 5. The figure shows a wooden platform ABCD. It is given that AB // DC, CE AB, AE 30 m, DC 14 m, EB 8 m and CBE 50. (a) Find the gradients of AD and CB. Express your answers in the form of 1 : n. (b) Which side of the platform is steeper? D 14 m C 50 A 30 m E 8m B Ex.9D Elementary Set 2. The road sign in the figure shows that the gradient of a road is 1 : 10. If the angle of inclination of the road is , find . 9.12 New Trend Mathematics S3B — Junior Form Supplementary Exercises 6. The figure shows a part of a map with the scale of 1 cm : 0.25 km. The length of XY in the map is 1.8 cm. (a) Find the actual distance of XY. 550 m 600 m 650 m (b) Find the angle of inclination of XY. X Y Scale 1 cm : 0.25 km Ex.9D Elementary Set 7. The figure shows a map with the scale of 1 : 10 000. O denotes the location of a signpost, and OA and OB denote two paths to there. On the map, OA 2.8 cm and OB 1.6 cm. 260 m A 240 m O (a) Find the gradient of OA. (b) Find the gradient of OB. 220 m 200 m (c) Which path is flatter? 180 m B 160 m Scale 1 : 10 000 D 8. The figure shows a slope with three sections where their horizontal distances are 244 m, 206 m and 148 m. The gradients of AB, BC and CD are 1 : 2.2, 1 : 1.9 and 1 : 0.84 respectively. If a man travels from A to D, how far does he travel? C B A 244 m Ex.9D Advanced Set Advanced Set Level 1 206 m 148 m 1. Find the angle of inclination corresponding to each of the following gradients. (a) 1 : 3 (b) 1 : 0.5 (c) 1 : 22.5 2. Find the gradient corresponding to each of the following angles of inclination. Express your answers in the form of 1 : n. (a) 10 (b) 40.5 (c) 67.25 9.13 Chapter 9 Applications of Trigonometry 3. A car travels 3.2 km up a slope with the gradient of 1 : 15. Find the vertical rise of the car. 4. The horizontal and vertical distances of a road are 420 m and 35 m respectively. Find the gradient of the road. 1 5. A man walks 240 m down a path with the gradient of . Find the horizontal and vertical 8 distances he walks. Level 2 6. The figure shows a cross section of a hill. The lengths of two sides AB and BC of the hill are 460 m and 530 m respectively, and the height of the hill is 360 m. Find the gradients of AB and BC, and the horizontal distance of AC. B A Horizontal ground 7. The figure shows a part of a map. The actual horizontal distance of AB is 270 m. Find the angle of inclination of AB. B 360 m 340 m A 320 m 300 m 8. The figure shows a part of a map with the scale of 1 : 5 000. The line segment AB measures 3.2 cm. 450 m A (a) Find the actual horizontal distance of AB. (b) Find the gradient of AB. 425 m 400 m 375 m B 350 m Scale 1 : 5 000 325 m Ex.9D Advanced Set C 9.14 New Trend Mathematics S3B — Junior Form Supplementary Exercises Exercise 9E [ In this exercise, correct your answers to 3 significant figures if necessary. ] Elementary Set Level 1 1. The figure shows a basketball stand. Two players look at A from B and C. Find the respective angles of elevation from B and C to A. A 68 42 C B 2. According to the figure, find A Ex.9E Elementary Set 20 45 B C D (a) the angle of elevation to A from C. (b) the angle of depression to D from A. 3. The figure shows that a person in a boat looks at point A of a lighthouse at the angle of elevation 19.5. If the distances of the eye level and point A from the sea level are 2.4 m and 20 m respectively, find the horizontal distance between the boat and point A. A 20 m 19.5 2.4 m 9.15 Chapter 9 Applications of Trigonometry 4. A person looks at the top Y of building B from the top X of building A at the angle of elevation 28. If the height of building A is 40 m and these two buildings are 240 m apart, find the height of building B. Y X 28 40 m A B 240 m 5. A person looks at peak B from peak A at the angle of depression 7.4. It is given that the heights of peaks A and B are 627 m and 508 m respectively. Find the horizontal distance between the two peaks. A 7.4 B 627 m 6. A person looks at the foot B of a church from the top A of a building at the angle of depression . Given that the height of the building is 82 m, and it is 245 m away from the church, find . Ex.9E Elementary Set 508 m A 82 m 245 m 7. A person sits at the front A of the upper deck of a bus and looks at the top B of a building. Given that A and B are 3.5 m and 120 m from the ground respectively, and the front of the bus is 164 m away from the building, find the angle of elevation to B from the eyes of the person. B B 120 m 3.5 m 164 m A 9.16 New Trend Mathematics S3B — Junior Form Supplementary Exercises Level 2 8. The figure shows a kite which is 30 m above the ground. The angles of elevation to the kite from A and B on the ground are 45 and 62 respectively. Given that the projection of the kite on the ground forms a horizontal line with A and B, find the distance between A and B. 30 m 45 62 A 9. In the figure, the angles of elevation to the top of a tree from A and B on the ground are 40 and respectively. The horizontal distances of A and B from the bottom C of the tree are 18 m and 12 m respectively. Find . B 40 A Ex.9E Elementary Set 10. In the figure, the shorter flagpole is 5 m high, and its shadow BC is 6 m long. The longer flagpole is 8 m high and its shadow is AC. (a) Find . 18 m C 12 m B 8m (b) Find the distance between the two flagpoles. 5m A C B 6m 11. In the figure, a person looks at point A on the ground from the top C of building X at the angle of depression 70. Another one looks at A from the top D of building Y at the angle of depression 40. Given that the two buildings stand on the same horizontal line as point A, find the distance between the two buildings. 12. In the figure, someone looks at points A and B on the ground from the top X of a building at the angles of depression 45 and 38 respectively. Given that the building stands on the same horizontal line as A and B, find the distance between A and B. C 70 D 40 60 m 45 m X Y A X 38 45 90 m A B 9.17 Chapter 9 Applications of Trigonometry 13. From the top Y of building B, someone looks at the top X and foot Z of building A at the angles of depression 15 and 30 respectively as shown. Find the height of building A. Y 15 X 30 65 m Building B Ex.9E Elementary Set Building A Z 14. In the figure, two people look at a helicopter from points A and B on the ground at the angles of elevation 69 and 52 respectively. The distance between A and B is 200 m. If A, B and the projection of the helicopter lie on the same horizontal line, find the height of the helicopter above the ground. C 69 52 A B 200 m Advanced Set Level 1 1. According to the figure, B (a) find the angle of elevation from C to B. (b) find the angle of depression from B to A. A 2. In the figure, the height SP of a tree and the length of PQ are equal. PQR is a horizontal line. (a) Find the angle of depression from S to Q. 38 Ex.9E Advanced Set 15 C S 20 (b) Find the angle of elevation from R to S. P Q R 9.18 New Trend Mathematics S3B — Junior Form Supplementary Exercises 3. In the figure, the height of a school is 20 m and a spot on the ground is 30 m away from the school. If the angle of elevation to the top of the school from the spot is , find . School 20 m 30 m 4. A person looks at an aeroplane from B at the angle of elevation of 25. Given that the aeroplane is flying at a height of 2 000 m above the ground at that time, find the horizontal distance between the person and the aeroplane. A 2 000 m 25 C B 5. On a horizontal ground, Howard stands 25 m away from a flagpole of 15 m high. What is the angle of depression from the top of the flagpole to the position of Howard? Ex.9E Advanced Set 6. According to the figure, find the value of h. hm 50 1.6 m 10 m Level 2 7. In the figure, a helicopter is 160 m above the ground. The angles of elevation to the helicopter from points A and B on the ground are 48 and 69 respectively. Given that the projection of the helicopter on the ground lies on the same horizontal level as A and B, find the distance between A and B. 160 m 69 B 48 A 9.19 Chapter 9 Applications of Trigonometry 8. In the figure, someone looks at the top of a lighthouse from X and Y on the ground at the angles of elevation 24 and 32 respectively. If the horizontal distance between X and the top of the lighthouse is 148 m, find the horizontal distance between Y and the top of the lighthouse. 32 24 X Y 148 m Q 70 P 50 54 m 38 m X Building A 33 10. In the figure, the angles of depression from the top of flagpole B to the top and foot of flagpole A are 22 and 33 respectively. Given that the height of flagpole B is 7.2 m, find the height of flagpole A. 22 7.2 m A 11. Simon is going to find out the height of a bookshelf. He stands in front of it and measures the angle of elevation of the top as 31 and the angle of depression of the foot as 39. If his eye level is 1.76 m above the ground, find the height of the bookshelf. 12. Henry looks at the top and bottom of a basketball backboard from the playground at the angles of elevation of 18.3 and 12 respectively. If his eye level is 1.6 m above the ground, and the horizontal distance between him and the backboard is 6.2 m, find the height AB of the backboard. Building B B 31 39 1.76 m A 18.3 B 12 1.6 m 6.2 m Ex.9E Advanced Set 9. In the figure, two people look at point X on the ground from the top P of building A and the top Q of building B at the angles of depression 50 and 70 respectively. Given that the two buildings stand on the same horizontal line as X, find the distance between the two buildings. 9.20 New Trend Mathematics S3B — Junior Form Supplementary Exercises Ex.9E Advanced Set 13. In the figure, the angle of elevation to a kite from X on the ground is 64, and the angle of elevation to the kite from Y on the ground is 38. The distance between X and Y is 74 m. If X, Y and the projection of the kite on the ground lie on the same horizontal line, find the height of the kite. 64 38 X 14. Two people look at point C on the ground from A and B of a building at the angles of depression 42 and 30 respectively. If A is 14.2 m above B, find the height of A from the ground. Y 74 m A 42 14.2 m B 30 C Exercise 9F Elementary Set Level 1 1. Complete the following table. Compass bearing True bearing N30E S42W Ex.9F Elementary Set S19E 010 123 270 310 N 2. According to the figure, find the compass bearing and true bearing of B from A. A 30 B 9.21 Chapter 9 Applications of Trigonometry N 3. According to the figure, find (a) the true bearing of Y from X. Y N (b) the true bearing of X from Y. X N Ex.9F Elementary Set 4. According to the figure, find 15 B (a) the compass bearing of C from A. (b) the compass bearing of B from O. 60 A 70 D (c) the compass bearing of B from A. 60 (d) the compass bearing of C from B. 65 O E C Level 2 5. Victor drives a car from city A at a bearing of S47W to city B. If he has to drive from city B back to city A, in what direction should he drive? Advanced Set Level 1 1. If the compass bearing of B from A is N54W, what is the compass bearing of A from B? 2. If the true bearing of Y from X is 230, what is the true bearing of X from Y? N A (b) the compass bearing of A from C. 36 30 (c) the true bearing of B from C. B (d) the true bearing of O from B. 25 30 O C E Level 2 4. A missile P is launched from A at a bearing of 217 to position B. If a missile Q is launched from position B to hit missile P, in what direction should missile Q fly? 5. Patrick runs from location A for 100 m at a bearing of 045 to location B, then he runs for 100 2 m at a bearing of 270 to location C. If he then runs from B for 100 m at bearing of 135, what is his final location? Ex.9F Advanced Set 3. According to the figure, find (a) the compass bearing of B from A. 9.22 New Trend Mathematics S3B — Junior Form Supplementary Exercises Exercise 9G [ In this exercise, correct your answers to 1 decimal place if necessary. ] Elementary Set Level 1 1. Jane walks 500 m due south and then 750 m due west. Find the compass bearing of her final position from the starting point. 2. If Alfred walks 250 m at a bearing of 297 and then walks to the west of the starting point at a bearing of 180, find the distance between the final position and the starting point. N 3. Vivian walks 5 km from A at a bearing of N62W to B, and then walks 3 km at a bearing of S28W to C. Find the true bearing of C from A. B N 5 km 3 km 62 28 A C Ex.9G Elementary Set 4. The figure shows three locations P, Q and R on a map. P is at a bearing of N28W from Q, Q is at a bearing of S62W from R, and P is at a bearing of S95W from R. It is given that PQ 45 km. N 95 R P 62 N 28 (a) Find the distance between P and R. Q (b) Find the distance between Q and R. 5. The bearing of a lighthouse from a ship at A is N72E. The ship then sails eastward to B and the bearing of the lighthouse from B is N29W. If the distance between A and B is 1 500 m, what will be the shortest distance between the ship and the lighthouse during the journey? N N 29 72 A 1 500 m B Level 2 6. In the figure, a ship sails 10 km from A to B. The bearing of B from A is N45W. Then the ship sails 15 km to C from B at the direction N20W. (a) Find the true bearing of C from A. (b) Find the distance of AC. C N 15 km 20 45 N B 10 km A 9.23 Chapter 9 Applications of Trigonometry N C N 40 60 A 10 km Ex.9G Elementary Set 7. The bearing of C from a car at A is N60E. The car travels 10 km eastward to B and the bearing of C from B is N40E. If the car continues to travel eastward, what will be its shortest distance from C? B N 8. May is at a bearing of S40E and 150 m away from Winnie. They start travelling at the same time and May travels to the north at 5 m/s. Winnie decides to meet May by taking the shortest path. 5 m/s 40 (a) At which direction should Winnie go? 150 m N (b) How far should May travel to meet Winnie? (c) How far should Winnie travel to meet May? (d) Find the lowest speed taken by Winnie to meet May. Advanced Set Level 1 1. In the figure, the distance between two buoys A and B is 150 m. Boat C is due north of A, and the compass bearing of boat C from B is N72W. Find the distances of C from buoys A and B. N C N 72 A B Ex.9G Advanced Set 150 m N 2. In the figure, two ships B and C sail from A. Ship B sails 23 km to the east and ship C sails 18 km to the south. A 23 km B B (a) Find the distance between the two ships. (b) Find the true bearing of ship C from ship B. 18 km B 3. Sam and Raymond sail for 3 hours on windsurfers from the starting point A to B and C at bearings S28W and S62E respectively. The speed of Sam is 2 km/h, and the speed of Raymond is 3 km/h. Find the true bearing of C from B. C N A 62 28 B C 9.24 New Trend Mathematics S3B — Junior Form Supplementary Exercises N 4. The figure shows the locations of three ships A, B and C. Ship B is due east of ship A, and ships C and B are 240 m apart. The bearings of ship C from ships A and B are S25W and S62W respectively. Find the distance between ships A and C. N A B 62 25 240 m C Level 2 N 5. The figure shows the locations of three lighthouses X, Y and Z. Ben sails 19 km in a boat from lighthouse X to lighthouse Y at a bearing of 203, and then sails 22 km to lighthouse Z at a bearing of 102. (a) Find the distance between lighthouses X and Z. 203 X 19 km N (b) Find the true bearing of X from Z. N 102 Y 22 km Z Ex.9G Advanced Set 6. In the figure, cars A and B travel from O at the same time at bearings N30W and N64E respectively. Car A travels at 82 km/h and car B travels at 100 km/h. (a) Find the distance between cars A and B after 2 hours. (b) Find the compass bearing of car B from car A after 2 hours. 7. The bearing of location B from location A is N54E, and locations A and B are 154 km apart. Helicopters P and Q fly from A and B respectively, and arrive at C at the same time. It is given that helicopter P flies at the speed of 160 km/h at a bearing of S12E for 1.8 hours to reach C. (a) Find the distance between B and C. A B N 30 64 O N 154 km B 54 A 12 (b) Find the speed of helicopter Q. (c) Find the compass bearing of C from B. C 8. A typhoon (at location B ) is observed 300 km away from a city (at location A) at the bearing of S75E. This typhoon moves at a bearing of N40W at 18 km/h. It is known that the area within 200 km from the centre of the typhoon will be affected by the typhoon. Will this city be affected by the typhoon? If it does, how long will it be affected? CHAPTER TEST N N A 75 300 km 40 B (Time allowed: 1 hour) [ In this test, unless otherwise stated, correct your answers to 3 significant figures if necessary. ] Section A (1) [ 3 marks each ] 1. Find the value of the expression tan 30 tan 60 sin 60 cos 30 without using a calculator. 2. Find the value of sin 2 75 sin 2 15 . 3. Lawrence walks 200 m up a slope with the gradient of 1 . What is his vertical rise? 5 4. The angle of depression from the top of a building of 100 m high to a car on the ground is 35. What is the distance between the car and the building? 5. A ship sails 15 km to the east from A to B, and then sails 10 km to the north to C. Find the bearing of A from C. (Correct your answer to 1 decimal place.) 6. If sin 5 , find the values of cos and tan . 13 Section A (2) [ 6 marks each ] 7. Find the unknowns in the figure and leave your answers in surd form. b cm 135 30 10 cm a cm Ex.9G Advanced Set 9.25 Chapter 9 Applications of Trigonometry 9.26 New Trend Mathematics S3B — Junior Form Supplementary Exercises 8. Prove that sin cos (1 tan 2 ) tan is an identity. 9. The figure shows a part of a map with the scale of 1 : 10 000. The length of AB is 4 cm. A (a) Find the actual horizontal distance of AB. (b) Find the average angle of inclination of AB. 250 m 200 m 150 m B 100 m 50 m Scale 1 : 10 000 10. In the figure, the compass bearing of B from A is S38E, the compass bearing of B from C is S52W, and the compass bearing of C from A is N85E. A car travels at the speed of 40 km/h from A to B for 2.5 hours. (a) Find ABC. (b) Find the distance of AC. N N A C 85 52 38 B Section B 11. S3A students are having body height measurements with the ruler on the wall. Alfred is 156 cm tall. But when Patrick stands facing him 60 cm apart, he reads Alfred’s height as 159 cm. (a) According to the figure, explain why there is a difference in Alfred’s height. (3 marks) (b) If the distance between the top of Alfred’s head and the ruler on the wall is 10 cm, what is the angle of elevation from Patrick’s eyes to the top of Alfred’s head? (3 marks) (c) What is the distance between Patrick’s eye level and the ground? (3 marks) (d) If Patrick walks 60 cm backwards, what will he read about Alfred’s height? (Correct your answer to the nearest cm.) (4 marks) 60 cm 10 cm Chapter 9 Applications of Trigonometry 9.27 Multiple Choice Questions [ 3 marks each ] 12. Find the area of the following parallelogram, correct to the nearest 1 cm 2. 12 cm 16. Simplify sin 1 cos 2 . A. 0 B. 1 120 C. sin 9 cm □ D. tan A. 50 cm 2 17. William walks 500 m up a path with the gradient of 1 : 12. Find his vertical rise. B. 72 cm 2 C. 94 cm 2 □ D. 108 cm 2 13. Find the value of 3 A. 4 1 B. 4 C. D. A. 498 m B. 497 m C. 41.7 m cos 30 sin 30 . tan 30 □ D. 41.5 m 18. If the angle of elevation from A to B is 57, then the angle of depression from B to A is 3 2 3 4 A. 33. □ B. 43. C. 57. □ D. 137. 6 14. Given that cos , find the values of 7 sin and tan . 13 6 13 A. sin , tan 7 13 19. What is the bearing of D from A? N A 13 13 B. sin , tan 7 6 34 32 13 13 C. sin , tan 6 6 D. sin 6 13 13 , tan 13 6 64 □ B 42 C A. S34E 15. Given that sin 28.7 cos , find . B. S66E A. 27.8 C. N42E B. 28.7 D. N64E C. 61.3 D. 65.2 D □ □ 9.28 New Trend Mathematics S3B — Junior Form Supplementary Exercises 20. Find the area of the following ABC. 24. The following shows a map of the scale 1 : 5 000. If the length of MN on the map is 3.2 cm, find the average angle of inclination of the path, correct to 1 decimal place. A 8 cm 45 30 B C A. 42 2 cm 2 B. 54 2 cm 2 M N C. (8 3 24) c m 2 □ D. (8 3 8) c m 2 300 m 250 m 200 m 150 m 21. Simplify 1 tan (90 ) 2 tan 2 . Scale 1 : 5 000 A. 43.2 A. 2 cos 2 B. 46.8 B. 2 tan 2 C. 58.4 C. 1 □ D. 0 22. Islands A and B are 350 m apart. The true bearing of B from A is 130. A ship sails from the west of B along a straight line passing through B. What is the shortest distance between the ship and A? (Correct your answer to the nearest m.) A. 225 m 25. Find the angle of elevation from S to P, correct to 1 decimal place. P 2.2 m Q 2m R 1.6 m S B. 268 m 2.8 m 3.6 m 1.5 m C. 392 m □ D. 417 m A. 53.7 B. 51.6 C. 38.4 23. If sin cos 1 , find . 2 □ D. 69.6 2 D. 36.3 □ A. 0 B. 30 C. 60 D. 90 □ 26. The bearing of Wendy at location B from Desmond at location A is 140, and they are 20 m apart. Find the distance, correct to 1 decimal place, that Wendy should walk to the north such that the bearing of Wendy from Desmond is 110. Chapter 9 Applications of Trigonometry N 110 A 140 20 m N B A. 4.7 m B. 7.3 m C. 10.6 m D. 35.3 m □ 9.29