1. Find the Value of Each of the Following

[Complete this exercise without using a calculator. Leave your answers in surd form if necessary.]

1. Find the value of each of the following.

(a) (b) (c)

2. Find the value of each of the following.

(a) (b) (c)

3. Find the value of each of the following.

(a) (b) (c)

4. Find the unknown in each of the following figures.

(a) (b) (c)

5. Find the unknown in each of the following figures.

(a) (b) (c)

6. Find the value of each of the following.

(a) (b) (c)

7. Find q in each of the following.

(a) (b) (c)

8. Find q in each of the following.

(a)

(b)

(c)

9. Find the area of each of the following figures.

(a) (b) (c)

[In each of the following, rationalize the denominator of the answer if necessary. (10-12)]

10. Find the unknown in each of the following figures.

(a) (b) (c)

11. In the figure, find the area of trapezium PQRS.

12. A cuboid brick is placed as shown in the figure, where PQRS is a rectangle cross-section. Find the height of point R above the ground.

13. 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 DPQR is 6cm long,

(a) find the lengths of AD and AB.

(b) find the area of rectangle ABCD.

14. Find q in each of the following.

(a)

(b)

(c)

15. In the figure, OABC is a quarter of a circle with the centre at O. The length of radius OA is and OB is the angle bisector of ÐAOC.

(a) Find the lengths of OB and BD.

(b) Find ÐOAB and ÐDAB.

(c) Using the results of (a) and (b), find tan22.5°.

[In each of the following, rationalize the denominator of the answer if necessary. (16-17)]

16. In the figure, DEFG, AFC and AGB are straight lines.

(a) Find the lengths of AC, AD and CD.

(b) Find ÐDCE.

(d) Hence find the value of sin75° by considering DADG.

17. In the figure, AED and BCD are straight lines.

(a) Find the lengths of AC, BC and CD.

(b) Find ÐCDE.

[Complete this exercise without using a calculator. Leave your answers in surd form if necessary.]

18. Simplify the following.

(a) (b) (c)

19. Simplify the following.

(a) (b) (c)

20. Simplify the following.

(a) (b) (c)

21. If, find the values of cosq and tanq.

22. If, find the values of sinq and cosq.

23. If, find the values of sinq and tanq.

24. Simplify the following.

(a) (b)

25. Simplify the following.

(a) (b)

26. (a) Simplify.

(b) If, find the value of .

27. If, find the value of.

[Complete this exercise without using a calculator.]

28. Simplify the following.

(a) (b)

(c)

29. Find the values of the following.

(a) (b) (c)

30. Find q in each of the following.

(a) (b) (c)

31. Simplify the following.

(a) (b)

(c)

32. Find the values of the following.

(a) (b)

33. Find q in each of the following.

(a) (b)

(c)

34. Prove that each of the following is an identity.

(a) (b)

(c)

35. Prove that each of the following is an identity.

(a)

(b)

(c)

(d)

36. (a) Simplify.

(b) If, find the value of.

[In this exercise, unless otherwise stated, give your answers correct to 3 significant figures if necessary.]

37. Find the gradients corresponding to the following angles of inclination.

(a) 12° (b) 27°

38. Find the gradients corresponding to the following angles of inclination. (Express your answers in the form of, where n is correct to the nearest integer.)

(a) 2° (b) 9°

39. Find the angles of inclination corresponding to the following gradients. (Give your answers correct to the nearest degree.)

(a) 0.321 (b) 2.15

(e) (f)

40. Wayne runs up a slope with the gradient of and rises 30m vertically. Find his horizontal run and actual distance moved.

41. Vincent rides a bike 650m down a slope and he falls 70m vertically. Find the gradient of the slope. (Express your answer in the form of , where n is correct to the nearest integer.)

42. The figure shows a cross-section of a hill. The lengths of two sides AB and BC of the hill are 460m and 530m respectively, and the height of the hill is 360m.

(a) Find the horizontal distance of AC.

(b) Find the gradient of AB.

43. A man walks 450m up a slope with the angle of inclination of 19°, and then 640m further up another slope with the angle of inclination of 26°. What is the vertical rise of his final position from his starting point?

44. The figure shows a map in the scale of 1cm:0.25km. It is given thaton the map.

(a) Find the angle of inclination of XY.

(b) Find the actual distance of XY.

45. The figure shows a map in the scale of 1:10000. O denotes the location of a signpost, and OA and OB denote two paths to there. It is given thatandon the map.

(a) Find the angle of inclination of OA.

(b) Find the angle of inclination of OB.

46. The figure shows a wooden platform ABCD. It is given that AB//DC,and .

(a) Find the gradients of AD and CB. Express your answers in the form of 1:n, where n is correct to the nearest integer.

(b) Which side of the platform is steeper?

47. The figure shows a slope with three sections where their horizontal runs are 250m, 240m and 160m respectively. The gradients of AB, BC and CD are 2:5, 1:2 and 5:4 respectively. If Steven travels from A to D, how far does he travel?

48. In the figure, the scale of the map is 1:100000, where and on the map. It is given that the difference in the heights of every two consecutive contour lines are equal and the gradient of AB is 1:10.

(a) Find the difference in the heights of every two consecutive contour lines.

(b) Find the angle of inclination of BC.

[In this exercise, give your answers correct to 3 significant figures if necessary.]

49. Referring to the figure,

(a) find the angle of elevation of B from C.

(b) find the angle of depression of A from B.

50. 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 of Q from S.

(b) Find the angle of elevation of S from R.

51. Celia is at point A which is 120m away from a building. The angle of elevation of point C at the top of the building from A is 32°. Find the height of the building.

52. In the figure, the height of a lamp-post is 8.2m. If the distance between point A and point B at the base of the lamp-post is 20m, find the angle of elevation of point C at the top of the lamp-post from point A.

53. In the figure, Dennis is standing on a boat. The angle of elevation of point A at the top of a lighthouse from him is 19.5°. If his eye level and point A are 2.4m and 20m above the sea level respectively, find the horizontal distance between Dennis and point A.

54. In the figure, the heights of a building and a church are 82m and 48m respectively. If the building and the church are on the same horizontal level and the distance between them is 245m, find the angle of depression of point B at the top of the church from point A at the top of the building.

55. In the figure, point X, point Y and the base of a lighthouse are on the same horizontal line. The angles of elevation of the top of the lighthouse from point X and point Y are 24° and 32° respectively. If the horizontal distance between point X and the lighthouse is 148m, find the horizontal distance between point Y and the lighthouse.

56. In the figure, the angle of depression of point X from point P at the top of building A is 50°. It is known that the bases of the two buildings and point X are on the same horizontal line, and the distance between the two buildings is 50m. Find the angle of depression of point X from point Q at the top of building B.

57. The angles of elevation of the top and the base of basketball backboard from Bella are 18.3° and 12° respectively. If her eye level is 1.6m above the ground, and she stands 6.2m in front of the backboard, find the height AB of the backboard.

58. In the figure, the angles of depression of the top and base of flagpole A from the top of flagpole B are 22° and 33° respectively. If the height of flagpole B is 7.2m, find the height of flagpole A.

59. Simon stands in front of a bookshelf. The angle of elevation of the top of the bookshelf from him is 31° and the angle of depression of the base of the bookshelf from him is 39°. If his eye level is 1.76m above the ground, find the height of the bookshelf.

60. In the figure, the angles of elevation of a helicopter from point A and point B are 69° and 52° respectively. It is known that the distance between A and B is 200m, and A, B and the projection of the helicopter lie on the same horizontal line. Find the height of the helicopter above the ground.

61. The angles of depression of point C on the ground from point A and point B of a building are 42° and 30° respectively. If A is 14.2m above B, find the height of A from the ground.

62. In the figure, the angles of elevation of point B at the top of a shorter flagpole and point D at the top of a taller flagpole from point E are 62° and 58° respectively. It is known that the distance between the two flagpoles is 30m,, and A, E and C lie on the same horizontal line.

(a) Find the heights of the two flagpoles AB and CD.

(b) Point B and point D are connected by a rope. Find the length of the rope.

63. In the figure, Ivan and Iris stand on a ramp with an angle of inclination of 10° and they are 2.5m apart on the ramp. The eye levels of Ivan and Iris are 1.8m and 1.5m above the ramp respectively.

(a) Find the vertical distance between their positions at the ramp.

(b) Is the angle of Iris’s eyes from Ivan’s an angle of elevation?

64. Rewrite the following true bearings in compass bearings.

(a) 12° (b) 99°

65. Rewrite the following compass bearings in true bearings.

(a) N7°E (b) N23°W

66. For the figure,

(a) find the true bearing of Y from X.

(b) find the compass bearing of X from Y.

67. For the figure,

(a) find the compass bearing of B from A.

(b) find the compass bearing of A from C.

(d) find the true bearing of O from B.

68. A missile P is launched from A at a bearing of 217° to B. If a missile Q is launched from B to hit missile P, which direction should missile Q be launched?

69. In the figure, FG=FH. Find the true bearing of G from F.

70. Alan and Benny start at A and run along the direction N16°W and S62°W for 5 hours to point B and point C respectively. If the speeds of them are both 8km/h, find the compass bearing of B from C.

71. The figure shows a regular pentagon ABCDE.

(a) Find the true bearing of B from A.

(b) Find the true bearing of B from C.

(d) Find the compass bearing of D from A.

72. A car travels 30km along S15°W to B from A, then it travels another 30km NW to C. If it travels 30km from C along N75°E, where is its final position?

[In this exercise, give your answers correct to 3 significant figures if necessary.]

73. P and Q are 8km apart and the compass bearing of P from Q is S27°W. R is due east of P and due south of Q. Find the distance between P and R.