Chapter 9: Some Applications of Trigonometry

MCQs, Fill in the Blanks, True/False & Short Questions

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A. Multiple Choice Questions (MCQs)

1. The line drawn from the eye of an observer to the point being viewed is called the:

A. Horizontal line
B. Line of sight
C. Vertical line
D. Base line

2. When an observer looks at an object above the horizontal level, the angle formed is called:

A. Angle of depression
B. Right angle
C. Angle of elevation
D. Complementary angle

3. When an observer looks at an object below the horizontal level, the angle formed is called:

A. Angle of elevation
B. Angle of depression
C. Angle of inclination
D. Reflex angle

4. A tower is observed from a point on the ground. If its height and the horizontal distance from the observer are involved, which ratio is generally most convenient?

A. sin θ
B. cos θ
C. tan θ
D. sec θ

5. If the height of an object above the observer’s eye level is $h$ and the horizontal distance is $d$, then:

A. $\tan\theta=\frac{d}{h}$
B. $\tan\theta=\frac{h}{d}$
C. $\sin\theta=\frac{h}{d}$
D. $\cos\theta=\frac{h}{d}$

6. A ladder leaning against a vertical wall forms a right triangle. The ladder represents the:

A. Base
B. Perpendicular
C. Hypotenuse
D. Height

7. If a ladder makes an angle $\theta$ with the horizontal and reaches a height $h$, its length $L$ satisfies:

A. $L=h\sin\theta$
B. $L=\frac{h}{\sin\theta}$
C. $L=h\tan\theta$
D. $L=\frac{h}{\tan\theta}$

8. The shadow of a vertical object generally becomes longer when the Sun’s altitude:

A. Increases
B. Decreases
C. Becomes $90^\circ$
D. Remains unchanged

9. If the angle of elevation of the top of a tower is $45^\circ$, then:

A. Height = twice the distance
B. Height = distance
C. Height = half the distance
D. Height = distance × $\sqrt3$

10. If the angle of elevation of the top of a tower is $60^\circ$, then:$$\frac{\text{height}}{\text{distance}}=$$

A. $1$
B. $\frac1{\sqrt3}$
C. $\sqrt3$
D. $2$

11. A tower is 30 m away from an observation point and its angle of elevation is $30^\circ$. Its height is:

A. $10\sqrt3$ m
B. $30\sqrt3$ m
C. $15\sqrt3$ m
D. 30 m

12. A kite is 60 m above the ground and its string makes $60^\circ$ with the ground. The string length is:

A. 30 m
B. $30\sqrt3$ m
C. $40\sqrt3$ m
D. $60\sqrt3$ m

13. A pole is 5 m high. An electrician needs to reach a point 1.3 m below its top. The height she needs to reach is:

A. 1.3 m
B. 3.7 m
C. 5 m
D. 6.3 m

14. If an observer is 1.5 m tall, the height calculated from the observer’s eye level must be:

A. Multiplied by 1.5
B. Divided by 1.5
C. Increased by 1.5 m
D. Decreased by 1.5 m

15. In a building-and-flagstaff problem, if the building height is $H$ and flagstaff height is $x$, the total height is:

A. $H-x$
B. $Hx$
C. $H+x$
D. $H/x$

16. If two horizontal lines are parallel, the angle of depression from the upper line is equal to the corresponding:

A. Angle of elevation
B. Angle of reflection
C. Right angle
D. Exterior angle

17. For a river-width problem, if the bridge point is above both banks, the total width can be found by:

A. Subtracting the two horizontal distances
B. Adding the two horizontal distances
C. Adding the vertical heights
D. Multiplying the distances

18. In a right triangle, $\tan\theta$ is:

A. Hypotenuse/Base
B. Base/Hypotenuse
C. Perpendicular/Base
D. Perpendicular/Hypotenuse

19. In a right triangle:$$\cot\theta=$$

A. Perpendicular/Base
B. Base/Perpendicular
C. Hypotenuse/Base
D. Base/Hypotenuse

20. If the Sun’s altitude changes from $60^\circ$ to $30^\circ$, the shadow of a tower:

A. Becomes shorter
B. Becomes longer
C. Remains equal
D. Becomes zero

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B. Fill in the Blanks

21. The line joining the observer’s eye to the object being viewed is called the __________.

22. The angle formed by the line of sight with the horizontal when the object is above the observer is called the angle of __________.

23. The angle formed by the line of sight with the horizontal when the object is below the observer is called the angle of __________.

24. In a right triangle, \tan\theta=\frac{\text{__________}}{\text{base}}

25. In a right triangle, \sin\theta=\frac{\text{perpendicular}}{\text{__________}}

26. In a right triangle, \cos\theta=\frac{\text{__________}}{\text{hypotenuse}}

27. If the horizontal distance from a tower is $d$ and the angle of elevation is $\theta$, the height above eye level is __________.

28. If an observer’s eye is $e$ metres above the ground, the total height of an object is d\tan\theta+__________.

29. A ladder leaning against a wall represents the __________ of the right triangle.

30. When the Sun’s altitude decreases, the shadow of a tower generally becomes __________.

31. For an angle of $45^\circ$, \tan45^\circ=__________.

32. \tan60^\circ=__________.

33. \tan30^\circ=__________.

34. \cot45^\circ=__________.

35. The total height of a building and its flagstaff is equal to the building height __________ the flagstaff height.

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C. True or False

36. The line of sight is always horizontal.

37. An angle of elevation is formed when the observer looks upward.

38. An angle of depression is formed when the observer looks downward.

39. The ladder in a ladder-against-wall problem is usually the hypotenuse.

40. If the observer’s eye is above ground level, that height may need to be added to the calculated height.

41. When the Sun’s altitude increases, the shadow of a tower becomes longer.

42. $\tan45^\circ=1$.

43. $\tan60^\circ=\frac1{\sqrt3}$.

44. In a river-width problem, the distances from the bridge’s vertical projection to the two banks can be added to obtain the river width.

45. Angle of depression and the corresponding angle of elevation can be equal when the horizontal lines are parallel.

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D. Match the Following

Column A	Column B
46. Line of sight	a. $1$
47. Angle of elevation	b. Looks downward
48. Angle of depression	c. Eye to object
49. $\tan45^\circ$	d. Looks upward
50. Ladder	e. Hypotenuse

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E. Very Short Answer Questions

51. What is meant by the line of sight?

52. Define angle of elevation.

53. Define angle of depression.

54. When is the angle of elevation used?

55. When is the angle of depression used?

56. Which trigonometric ratio is usually useful when height and horizontal distance are involved?

57. What side of a right triangle does a ladder represent when it rests against a vertical wall?

58. Why is an observer’s height sometimes added to the calculated height?

59. What happens to the shadow of a tower when the Sun’s altitude decreases?

60. What is the relationship between the angle of depression and the corresponding angle of elevation?

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F. Numerical / Application-Based Questions

These are especially important because the chapter is primarily application-based.

61. A tower is 15 m away from a point on the ground. If its angle of elevation is $60^\circ$, find its height.

62. A ladder reaches a point 3.7 m above the ground and makes an angle of $60^\circ$ with the horizontal. Find the length of the ladder.

63. For the ladder in Q62, find the distance of its foot from the wall.

64. An observer 1.5 m tall stands 28.5 m from a chimney. The angle of elevation of its top from her eyes is $45^\circ$. Find the height of the chimney.

65. A 10 m building is observed from a point where the angle of elevation of its top is $30^\circ$. Find the distance of the point from the building.

66. From the same point, the angle of elevation of the top of a flagstaff fixed on the building is $45^\circ$. If the building is 10 m high, find the flagstaff’s height.

67. The shadow of a tower is 40 m longer when the Sun’s altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower.

68. From the top of a building, the angles of depression of the top and bottom of an 8 m building are $30^\circ$ and $45^\circ$, respectively. Determine the height of the taller building and the distance between the buildings.

69. A bridge is 3 m above the banks of a river. From a point on the bridge, the angles of depression of the opposite banks are $30^\circ$ and $45^\circ$. Find the width of the river.

70. A tower stands at the top of a 20 m building. From a point on the ground, the angles of elevation of the bottom and top of the tower are $45^\circ$ and $60^\circ$, respectively. Find the tower’s height.

71. A statue 1.6 m tall stands on a pedestal. From a point on the ground, the angles of elevation of the top of the statue and the top of the pedestal are $60^\circ$ and $45^\circ$, respectively. Find the pedestal’s height.

72. A lighthouse is 75 m high. Two ships on the same side have angles of depression $30^\circ$ and $45^\circ$. Find the distance between the ships.

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G. Assertion–Reason Questions

For each question, choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

73.

Assertion: The angle of elevation is measured when the object is above the observer’s horizontal level.
Reason: In this situation, the observer has to look upward.

74.

Assertion: A ladder leaning against a vertical wall forms a right triangle.
Reason: The wall and the ground are perpendicular.

75.

Assertion: A lower Sun altitude produces a longer shadow.
Reason: For a fixed tower height, shadow length is inversely related to $\tan\theta$.

76.

Assertion: The angle of depression can be equal to the corresponding angle of elevation.
Reason: The horizontal lines involved are parallel.

77.

Assertion: In a tower problem involving height and horizontal distance, $\tan\theta$ is often convenient.
Reason: Tangent relates the perpendicular and base of a right triangle.

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H. Important Conceptual Questions

78. Why is a right-angled triangle formed in most heights-and-distances problems?

79. Why must a diagram be drawn before solving a trigonometric application problem?

80. How can the height of a tall object be determined without directly measuring it?

81. Explain why two different angles may be needed in a building-and-flagstaff problem.

82. How does the position of the Sun affect the shadow of a tower?

83. How can the width of a river be determined using the height of a bridge and angles of depression?

84. In a problem involving a person observing the top of a tower, why is the distance from the person’s eye—not necessarily the person’s feet—used in the trigonometric triangle?

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Answer Key

MCQs

Q	Answer
1	B
2	C
3	B
4	C
5	B
6	C
7	B
8	B
9	B
10	C
11	A
12	C
13	B
14	C
15	C
16	A
17	B
18	C
19	B
20	B

Fill in the Blanks

21. Line of sight
22. Elevation
23. Depression
24. Perpendicular
25. Hypotenuse
26. Base
27. dtan⁡θd\tan\theta
28. ee
29. Hypotenuse
30. Longer
31. 1
32. 3\sqrt3
33. 1/31/\sqrt3
34. 1
35. plus (+)

True/False

36. False
37. True
38. True
39. True
40. True
41. False
42. True
43. False
44. True
45. True

Match the Following

46. c
47. d
48. b
49. a
50. e

Very Short Answers

51. The straight line joining the observer’s eye to the point being viewed.
52. The angle between the horizontal and line of sight when the object is above the observer.
53. The angle between the horizontal and line of sight when the object is below the observer.
54. When the observer looks upward at an object.
55. When the observer looks downward at an object.
56. Tangent.
57. Hypotenuse.
58. Because the trigonometric calculation may give the height only from the observer’s eye level to the object’s top.
59. It becomes longer.
60. They are equal when they are corresponding angles formed by a transversal with parallel horizontal lines.

Numerical Answers

61. $\boxed{15\sqrt3\text{ m}}$
62. $\boxed{4.28\text{ m approximately}}$
63. $\boxed{2.14\text{ m approximately}}$
64. $\boxed{30\text{ m}}$
65. $\boxed{10\sqrt3\text{ m}\approx17.32\text{ m}}$
66. $\boxed{7.32\text{ m approximately}}$
67. $\boxed{20\sqrt3\text{ m}}$
68. $\boxed{\text{Height}=4(\sqrt3+1)+8\text{ m; distance}=4(\sqrt3+1)\text{ m}}$
69. $\boxed{3(1+\sqrt3)\text{ m}}$
70. $\boxed{20(\sqrt3-1)\text{ m}}$
71. $\boxed{1.6(\sqrt3+1)\text{ m}}$
72. $\boxed{75(\sqrt3-1)\text{ m}}$

Assertion–Reason

73. A
74. A
75. A
76. A
77. A

Key conceptual answers

78. Because the horizontal ground and vertical height are perpendicular, producing a right triangle suitable for trigonometric ratios.
79. A diagram makes the height, distance, angle and relevant right triangle clear, reducing the chance of choosing the wrong ratio.
80. Measure a suitable horizontal distance and angle of elevation, then use a trigonometric ratio such as tangent. If necessary, add the observer’s eye height.
81. One angle can determine the building’s height-to-distance relationship, while the second angle gives the relationship for the combined building-plus-flagstaff height.
82. A higher Sun altitude gives a shorter shadow; a lower altitude gives a longer shadow.
83. Split the river width into the two horizontal distances from the foot of the bridge’s vertical height to each bank, calculate each using tangent, then add them.
84. Because the line of sight starts from the observer’s eye, so the vertical side of the relevant triangle represents the height from eye level to the object.