Trigonometry Questions - std 10

 

Trigonometry Questions

1. The angle of elevation to the top of a 17-meter tower is 50°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 17 ÷ tan(50°).
3. Find tan(50°) ≈ 1.1918.
4. Divide: distance ≈ 17 ÷ 1.1918.
5. The distance from the observer to the base of the tower is approximately 14.26 meters.

2. An observer is standing 111 meters away from a building. The angle of elevation to the top of the building is 70°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 111 × tan(70°).
3. Find tan(70°) ≈ 2.7475.
4. Multiply: height ≈ 111 × 2.7475.
5. The height of the building is approximately 304.97 meters.

3. An observer is standing 83 meters away from a building. The angle of elevation to the top of the building is 20°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 83 × tan(20°).
3. Find tan(20°) ≈ 0.3640.
4. Multiply: height ≈ 83 × 0.3640.
5. The height of the building is approximately 30.21 meters.

4. The angle of elevation to the top of a 73-meter tower is 15°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 73 ÷ tan(15°).
3. Find tan(15°) ≈ 0.2679.
4. Divide: distance ≈ 73 ÷ 0.2679.
5. The distance from the observer to the base of the tower is approximately 272.44 meters.

5. An observer is standing 78 meters away from a building. The angle of elevation to the top of the building is 20°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 78 × tan(20°).
3. Find tan(20°) ≈ 0.3640.
4. Multiply: height ≈ 78 × 0.3640.
5. The height of the building is approximately 28.39 meters.

6. An observer is standing 37 meters away from a building. The angle of elevation to the top of the building is 40°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 37 × tan(40°).
3. Find tan(40°) ≈ 0.8391.
4. Multiply: height ≈ 37 × 0.8391.
5. The height of the building is approximately 31.05 meters.

7. The angle of elevation to the top of a 84-meter tower is 35°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 84 ÷ tan(35°).
3. Find tan(35°) ≈ 0.7002.
4. Divide: distance ≈ 84 ÷ 0.7002.
5. The distance from the observer to the base of the tower is approximately 119.96 meters.

8. An observer is standing 70 meters away from a building. The angle of elevation to the top of the building is 65°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 70 × tan(65°).
3. Find tan(65°) ≈ 2.1445.
4. Multiply: height ≈ 70 × 2.1445.
5. The height of the building is approximately 150.12 meters.

9. An observer is standing 71 meters away from a building. The angle of elevation to the top of the building is 40°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 71 × tan(40°).
3. Find tan(40°) ≈ 0.8391.
4. Multiply: height ≈ 71 × 0.8391.
5. The height of the building is approximately 59.58 meters.

10. The angle of elevation to the top of a 47-meter tower is 55°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 47 ÷ tan(55°).
3. Find tan(55°) ≈ 1.4281.
4. Divide: distance ≈ 47 ÷ 1.4281.
5. The distance from the observer to the base of the tower is approximately 32.91 meters.

11. An observer is standing 48 meters away from a building. The angle of elevation to the top of the building is 20°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 48 × tan(20°).
3. Find tan(20°) ≈ 0.3640.
4. Multiply: height ≈ 48 × 0.3640.
5. The height of the building is approximately 17.47 meters.

12. An observer is standing 34 meters away from a building. The angle of elevation to the top of the building is 80°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 34 × tan(80°).
3. Find tan(80°) ≈ 5.6713.
4. Multiply: height ≈ 34 × 5.6713.
5. The height of the building is approximately 192.82 meters.

13. The angle of elevation to the top of a 79-meter tower is 70°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 79 ÷ tan(70°).
3. Find tan(70°) ≈ 2.7475.
4. Divide: distance ≈ 79 ÷ 2.7475.
5. The distance from the observer to the base of the tower is approximately 28.75 meters.

14. An observer is standing 23 meters away from a building. The angle of elevation to the top of the building is 55°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 23 × tan(55°).
3. Find tan(55°) ≈ 1.4281.
4. Multiply: height ≈ 23 × 1.4281.
5. The height of the building is approximately 32.85 meters.

15. An observer is standing 150 meters away from a building. The angle of elevation to the top of the building is 50°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 150 × tan(50°).
3. Find tan(50°) ≈ 1.1918.
4. Multiply: height ≈ 150 × 1.1918.
5. The height of the building is approximately 178.76 meters.

16. The angle of elevation to the top of a 7-meter tower is 40°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 7 ÷ tan(40°).
3. Find tan(40°) ≈ 0.8391.
4. Divide: distance ≈ 7 ÷ 0.8391.
5. The distance from the observer to the base of the tower is approximately 8.34 meters.

17. An observer is standing 63 meters away from a building. The angle of elevation to the top of the building is 20°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 63 × tan(20°).
3. Find tan(20°) ≈ 0.3640.
4. Multiply: height ≈ 63 × 0.3640.
5. The height of the building is approximately 22.93 meters.

18. An observer is standing 132 meters away from a building. The angle of elevation to the top of the building is 25°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 132 × tan(25°).
3. Find tan(25°) ≈ 0.4663.
4. Multiply: height ≈ 132 × 0.4663.
5. The height of the building is approximately 61.55 meters.

19. The angle of elevation to the top of a 77-meter tower is 25°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 77 ÷ tan(25°).
3. Find tan(25°) ≈ 0.4663.
4. Divide: distance ≈ 77 ÷ 0.4663.
5. The distance from the observer to the base of the tower is approximately 165.13 meters.

20. The angle of elevation to the top of a 27-meter tower is 50°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 27 ÷ tan(50°).
3. Find tan(50°) ≈ 1.1918.
4. Divide: distance ≈ 27 ÷ 1.1918.
5. The distance from the observer to the base of the tower is approximately 22.66 meters.

21. An observer is standing 60 meters away from a building. The angle of elevation to the top of the building is 80°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 60 × tan(80°).
3. Find tan(80°) ≈ 5.6713.
4. Multiply: height ≈ 60 × 5.6713.
5. The height of the building is approximately 340.28 meters.

22. The angle of elevation to the top of a 54-meter tower is 75°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 54 ÷ tan(75°).
3. Find tan(75°) ≈ 3.7321.
4. Divide: distance ≈ 54 ÷ 3.7321.
5. The distance from the observer to the base of the tower is approximately 14.47 meters.

23. An observer is standing 108 meters away from a building. The angle of elevation to the top of the building is 70°. Find the height of the building.

Solution:
1. Use the formula: height = distance × tan(θ).
2. Substitute the values: height = 108 × tan(70°).
3. Find tan(70°) ≈ 2.7475.
4. Multiply: height ≈ 108 × 2.7475.
5. The height of the building is approximately 296.73 meters.

24. The angle of elevation to the top of a 54-meter tower is 70°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 54 ÷ tan(70°).
3. Find tan(70°) ≈ 2.7475.
4. Divide: distance ≈ 54 ÷ 2.7475.
5. The distance from the observer to the base of the tower is approximately 19.65 meters.

25. The angle of elevation to the top of a 22-meter tower is 40°. Find the distance from the observer to the base of the tower.

Solution:
1. Use the formula: distance = height ÷ tan(θ).
2. Substitute the values: distance = 22 ÷ tan(40°).
3. Find tan(40°) ≈ 0.8391.
4. Divide: distance ≈ 22 ÷ 0.8391.
5. The distance from the observer to the base of the tower is approximately 26.22 meters.