Class 10 Maths Applications of Trigonometry – NCERT Examples & Solutions

Class 10 Maths Chapter 9: Some Applications of Trigonometry

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Introduction

This chapter teaches the practical use of trigonometry in solving real-world problems like measuring heights, distances, and angles using right triangles.

Basic Rules and Formulas

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent
• cosec θ = 1 / sin θ
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Examples

Example 1: A tree casts a shadow of 10 m when the angle of elevation of the sun is 30°. Find the height of the tree.

tan 30° = height / shadow → 1/√3 = h / 10 → h = 10/√3 ≈ 5.77 m

25 Questions with Solutions

Q1. Find the height of a pole if its shadow is 15 m and angle of elevation is 45°.

tan 45° = h / 15 → h = 15 m

Q2. A tower casts a shadow of 20 m. If the sun's elevation angle is 30°, find the height of the tower.

tan 30° = h / 20 → h = 20 / √3 ≈ 11.55 m

Q3. Find the distance between two points on opposite sides of a river, given angles of elevation from a point are 30° and 60°.

Use tan θ formula and sum distances → d = h(cot 30° + cot 60°)

Q4. From a point on the ground, the angle of elevation of the top of a building is 60°. Distance from the building is 50 m. Find the height.

tan 60° = h / 50 → h = 50√3 ≈ 86.6 m

Q5. A man 1.8 m tall observes the top of a building at an angle of 30° from eye level. If he is 10 m away, find building height.

tan 30° = (H - 1.8) / 10 → H = 10 / √3 + 1.8 ≈ 8.57 m

Q6. Find the angle of elevation if a 10 m pole casts a shadow of 10 m.

tan θ = 10 / 10 = 1 → θ = 45°

Q7. Two poles 6 m and 11 m high are 12 m apart. Find angle of elevation from top of shorter pole to top of taller pole.

θ = tan⁻¹((11 - 6)/12) = tan⁻¹(5/12) ≈ 22.62°

Q8. Find height of a tree using shadow of 7 m and angle 60°.

tan 60° = h / 7 → h = 7√3 ≈ 12.12 m

Q9. Distance between two points on a hill observed at angles 30° and 60°?

Use tan θ formula → d = h(cot30° - cot60°)

Q10. A building's shadow is 25 m. Angle of elevation 45°, find height.

tan 45° = h / 25 → h = 25 m

Q11. A tower 20 m high casts shadow. Angle = 30°. Find shadow length.

tan 30° = 20 / shadow → shadow = 20√3 ≈ 34.64 m

Q12. Find angle of elevation if height 10 m and shadow 10√3 m.

tan θ = 10 / 10√3 = 1/√3 → θ = 30°

Q13. From 40 m away, building top seen at 60°. Find height.

tan 60° = h / 40 → h = 40√3 ≈ 69.28 m

Q14. Two ships observe a lighthouse at angles 30° and 45°, distance between ships 50 m. Find lighthouse height.

Use tan θ formula → h = 50 / (cot30° - cot45°) ≈ 28.87 m

Q15. From top of hill, angle of depression 30°, height 100 m. Find distance from base.

tan 30° = 100 / d → d = 100√3 ≈ 173.2 m

Q16. A pole 8 m high casts shadow 4√3 m. Find sun angle.

tan θ = 8 / 4√3 = 2/√3 → θ ≈ 49.1°

Q17. Height of pole = ?, shadow = ?, angle 60°

h = shadow × tan 60°

Q18. Two hills 500 m apart. Angle elevation from base = 30° & 45°. Find hill height difference.

Use tan θ → h1 = x tan30°, h2 = (x+500)tan45° → difference = h2 - h1

Q19. Angle elevation 60°, shadow 5 m, find height.

h = 5√3 ≈ 8.66 m

Q20. Angle 45°, shadow 10 m, height?

h = 10 m

Q21. Two poles 10 m apart, angles 30° & 60°. Find height difference.

Use tan θ → Δh = h2 - h1 = d(tan60° - tan30°)

Q22. Angle 30°, shadow 20 m, find height.

h = 20 / √3 ≈ 11.55 m

Q23. Height of tree 15 m from 20 m away, angle 37°. Find top distance.

Use Pythagoras → d² = 15² + 20² → d ≈ 25 m

Q24. Angle elevation 45°, distance 50 m, find height.

h = 50 m

Q25. Find height if tan θ = 2, shadow = 5 m.

h = 5 × 2 = 10 m

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15 FAQs on Applications of Trigonometry

Q1. What is application of trigonometry?
Measuring heights, distances, navigation, architecture, engineering.

Q2. Which angles are used for elevation & depression?
Angles above and below horizontal.

Q3. How to find building height?
Use tan θ = height/shadow.

Q4. How to find distance across river?
Use angles from two points and trigonometry formulas.

Q5. Can trigonometry solve real-life problems?
Yes, height, distance, navigation, construction.

Q6. What is angle of depression?
Angle below horizontal.

Q7. What is angle of elevation?
Angle above horizontal.

Q8. Can shadow measure height?
Yes, using tan θ = height/shadow.

Q9. What is cot θ?
Reciprocal of tan θ.

Q10. What is sec θ?
Reciprocal of cos θ.

Q11. What is cosec θ?
Reciprocal of sin θ.

Q12. How to solve two-pole height problem?
Use difference of heights and angles formulas.

Q13. Can angles > 90° be used?
No, only right triangle situations.

Q14. What is practical use of tan θ?
Height / base ratio to calculate unknowns.

Q15. Are trigonometric formulas same in real-life problems?
Yes, identities and ratios remain same.

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