Maths ch9 applications of trigonometry

37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.

Quick notes

The line of sight joins the observer's eye to the object.

Angle of elevation: the angle between the line of sight and the horizontal when the object is above the eye.
Angle of depression: the angle between the line of sight and the horizontal when the object is below the eye.

The angle of depression from A to B equals the angle of elevation from B to A (alternate angles).

Most problems use tan θ = height ÷ distance. Useful values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Part A: Multiple choice 16 questions

1. The angle between the line of sight and the horizontal, when looking up at an object, is the

angle of elevation
angle of depression
right angle
reflex angle

Looking up gives elevation.

2. When looking down at an object, the angle formed with the horizontal is the

angle of elevation
angle of depression
straight angle
complementary angle

Looking down gives depression.

3. A tower is 30 m high. From a point 30 m from its foot, the angle of elevation of its top is

30°
60°
45°
90°

tan θ = 30/30 = 1.

4. A kite string is 60 m long and makes 60° with the ground. The height of the kite is

30 m
60√3 m
20√3 m
30√3 m

h = 60 sin 60° = 30√3 m.

5. If a pole's shadow equals its height, the sun's angle of elevation is

45°
30°
60°
90°

tan θ = 1.

6. A tower of height h casts a shadow √3 h long. The sun's elevation is

60°
30°
45°
15°

tan θ = h/(√3h) = 1/√3.

7. A 20 m rope from the top of a pole to the ground makes 30° with the ground. The pole is

20 m
10√3 m
10 m
40 m

h = 20 sin 30° = 10 m.

8. From a point 15 m from a tower's foot, the elevation of the top is 60°. The height is

15 m
5√3 m
15/√3 m
15√3 m

h = 15 tan 60° = 15√3 m.

9. As you walk towards a tower, the angle of elevation of its top

increases
decreases
stays the same
becomes zero

The distance gets smaller, so tan θ gets larger.

10. From the top of a 75 m tower, the angle of depression of a car is 30°. The car is

75 m away
75√3 m away
25√3 m away
150 m away

tan 30° = 75/d, so d = 75√3 m.

11. The angle of depression of B from A is equal to

90° minus the elevation of A from B
twice the elevation
the angle of elevation of A from B
180°

They are alternate angles between parallel horizontals.

12. Pole : shadow = √3 : 1. The sun's elevation is

30°
45°
90°
60°

tan θ = √3.

13. A kite is 60 m high and the string makes 60° with the ground. The string length is

40√3 m
60 m
120 m
30√3 m

60/sin 60° = 120/√3 = 40√3 m.

14. A tree breaks and its top touches the ground 8 m from the foot, making 30° with the ground. The tree's original height was

8 m
8√3 m
16 m
24 m

Standing part = 8/√3, broken part = 16/√3, total = 24/√3 = 8√3 m.

15. Which trigonometric ratio is used most in height and distance problems?

It links height and horizontal distance directly.

16. A 10 m pole casts a 10√3 m shadow. The sun's elevation is

60°
45°
90°
30°

tan θ = 10/(10√3) = 1/√3.

Part B: Fill in the blanks 8 blanks

17. When an object is above the eye, the angle with the horizontal is the angle of [[elevation]].

18. When an object is below the eye, the angle is the angle of [[depression]].

19. A tower is 50 m tall and the elevation of its top is 45°. The observer is [[50]] m from the foot.

20. If the height of a pole equals its shadow, the sun's altitude is [[45]]°.

21. A 20 m rope making 30° with the ground is tied to the top of a pole [[10]] m high.

22. The line from the observer's eye to the object is the line of [[sight]].

23. A 10 m ladder makes 60° with the ground. Its foot is [[5]] m from the wall.

24. As an observer moves away from a tower, the angle of elevation [[decreases]].

Part C: Questions and answers 9 questions

25. A circus artist climbs a 20 m rope stretched from the top of a pole to the ground. The rope makes 30° with the ground. Find the height of the pole.

sin 30° = h/20 → h = 20 × 1/2 = 10 m.

26. A tree breaks due to a storm and the top touches the ground making 30° with it, 8 m from the foot. Find the original height.

Standing part = 8 tan 30° = 8/√3. Broken part = 8/cos 30° = 16/√3.
Height = 24/√3 = 8√3 m.

27. The angle of elevation of the top of a tower from a point 30 m from its foot is 30°. Find the height.

tan 30° = h/30 → h = 30/√3 = 10√3 m.

28. A kite flies 60 m above the ground. The string makes 60° with the ground. Find the length of the string, assuming no slack.

sin 60° = 60/l → l = 60 × 2/√3 = 40√3 m.

29. A 1.5 m tall boy stands some distance from a 30 m building. The angle of elevation of its top changes from 30° to 60° as he walks towards it. How far did he walk?

Height above eye = 28.5 m. Distances: 28.5√3 and 28.5/√3.
Walked = 28.5(√3 − 1/√3) = 28.5 × 2/√3 = 19√3 m.

30. From the top of a 7 m building, the elevation of the top of a tower is 60° and the depression of its foot is 45°. Find the tower's height.

Depression 45° → distance = 7 m.
Extra height above building = 7 tan 60° = 7√3.
Tower = 7(1 + √3) m.

31. Two poles of equal height stand on either side of an 80 m road. From a point between them on the road, the elevations are 60° and 30°. Find the height and the distances.

h = x√3 and h = (80 − x)/√3 → 3x = 80 − x → x = 20.
Height 20√3 m; the point is 20 m and 60 m from the poles.

32. From the top of a 75 m lighthouse, the angles of depression of two ships on the same side are 30° and 45°. Find the distance between the ships.

Distances: 75/tan 45° = 75 and 75/tan 30° = 75√3.
Distance between = 75(√3 − 1) m.

33. A 1.6 m statue stands on a pedestal. From a point on the ground, the elevation of the top of the statue is 60° and of the top of the pedestal is 45°. Find the pedestal's height.

Let pedestal = h. Distance = h (45°). tan 60° = (h + 1.6)/h → √3h = h + 1.6 → h = 1.6/(√3 − 1).
h = 0.8(√3 + 1) m.