Find Angles with Trig Ratios: Right Triangle Quiz

1) A right triangle has opposite = 9 and adjacent = 12. Find θ to the nearest degree.

36°

41°

48°

53°

θ = arctan(9/12) ≈ 36.87° → 36°.

Explanation

θ = arctan(9/12) ≈ 36.87° → 36°.

2) A ladder 10 ft long leans against a wall. The base is 6 ft from the wall. Find the angle of elevation to the nearest degree.

31°

37°

53°

59°

cosθ = 6/10 → θ = arccos(0.6) ≈ 53.13° → 53°.

Explanation

cosθ = 6/10 → θ = arccos(0.6) ≈ 53.13° → 53°.

3) A right triangle has legs 7 and 24. Find the hypotenuse.

23

24

25

26

Hypotenuse = √(7² + 24²) = 25.

Explanation

Hypotenuse = √(7² + 24²) = 25.

4) A ramp rises 2 m over a run of 5 m. Find the angle of elevation to the nearest degree.

21°

23°

25°

27°

θ = arctan(2/5) ≈ 21.80° → 21°.

Explanation

θ = arctan(2/5) ≈ 21.80° → 21°.

5) A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

11

12

13

14

Other leg = √(13² − 5²) = √144 = 12.

Explanation

Other leg = √(13² − 5²) = √144 = 12.

6) A building casts a 20 m shadow. If the sun’s angle of elevation is 40°, find the height of the building.

16.8 m

17.3 m

18.0 m

19.1 m

Height = 20·tan40° ≈ 16.78 m → 16.8 m.

Explanation

Height = 20·tan40° ≈ 16.78 m → 16.8 m.

7) A right triangle has opposite = 8, hypotenuse = 17. Find θ to the nearest degree.

25°

28°

29°

31°

sinθ = 8/17 → θ ≈ 28.1° → 28°.

Explanation

sinθ = 8/17 → θ ≈ 28.1° → 28°.

8) A ladder 15 ft long leans so it makes a 60° angle with the ground. How high does it reach?

11.0 ft

12.5 ft

13.0 ft

13√3 ft

Height = 15·sin60° ≈ 12.99 ft → 13.0 ft.

Explanation

Height = 15·sin60° ≈ 12.99 ft → 13.0 ft.

9) A right triangle has adjacent = 7 and hypotenuse = 25. Find θ to the nearest degree.

74°

74.5°

75°

76°

cosθ = 7/25 → θ ≈ 73.74° → 74°.

Explanation

cosθ = 7/25 → θ ≈ 73.74° → 74°.

10) A 12 m tall pole casts a shadow of 5 m. Find the angle of elevation of the sun to the nearest degree.

65°

67°

68°

70°

θ = arctan(12/5) ≈ 67.38° → 67°.

Explanation

θ = arctan(12/5) ≈ 67.38° → 67°.

11) A right triangle has legs 9 and 40. Find the hypotenuse.

40

41

42

43

Using Pythagoras: √(9² + 40²) = 41. Correct answer is 41.

Explanation

Using Pythagoras: √(9² + 40²) = 41. Correct answer is 41.

12) A 20 m ladder makes a 75° angle with the ground. How high up the wall does it reach?

18.9 m

19.3 m

19.5 m

19.7 m

Height = 20 × sin(75°) = 19.3 m. Correct answer is 19.3 m.

Explanation

Height = 20 × sin(75°) = 19.3 m. Correct answer is 19.3 m.

13) A right triangle has opposite = 15, adjacent = 20. Find θ to the nearest degree.

35°

36°

37°

38°

θ = tan⁻¹(15/20) = 36.9°, approximately 37°.

Explanation

θ = tan⁻¹(15/20) = 36.9°, approximately 37°.

14) A right triangle has hypotenuse 29 and one leg 21. Find the other leg.

19

20

21

22

Other leg = √(29² - 21²) = 20. Correct answer is 20.

Explanation

Other leg = √(29² - 21²) = 20. Correct answer is 20.

15) A tree casts a 30 ft shadow. If the sun’s angle of elevation is 55°, find the height of the tree.

40.0 ft

41.5 ft

42.8 ft

43.2 ft

Height = 30 × tan(55°) = 42.8 ft. Correct answer is 42.8 ft.

Explanation

Height = 30 × tan(55°) = 42.8 ft. Correct answer is 42.8 ft.

16) A right triangle has opposite = 5, hypotenuse = 13. Find θ to the nearest degree.

22°

23°

24°

25°

θ = sin⁻¹(5/13) = 22.6°, approximately 23°.

Explanation

θ = sin⁻¹(5/13) = 22.6°, approximately 23°.

17) A ladder 25 ft long leans so it reaches 24 ft up a wall. Find the angle with the ground to the nearest degree.

72°

73°

74°

75°

θ = sin⁻¹(24/25) = 73.7°, approximately 74°.

Explanation

θ = sin⁻¹(24/25) = 73.7°, approximately 74°.

18) A right triangle has adjacent = 9, hypotenuse = 15. Find θ to the nearest degree.

52°

50°

54°

53°

θ = cos⁻¹(9/15) = 53.1°, approximately 53°.

Explanation

θ = cos⁻¹(9/15) = 53.1°, approximately 53°.

19) A right triangle has opposite = 10, adjacent = 10. Find θ.

40°

55°

50°

45°

θ = tan⁻¹(10/10) = 45.0°. Correct answer is 45°.

Explanation

θ = tan⁻¹(10/10) = 45.0°. Correct answer is 45°.

20) A ramp is 12 ft long and rises 7 ft. Find the angle of elevation to the nearest degree.

34°

35°

36°

37°

θ = sin⁻¹(7/12) = 35.7°, approximately 35°.

Explanation

θ = sin⁻¹(7/12) = 35.7°, approximately 35°.

Finding Angles with Trigonometric Ratios