Class 10 Math Chapter 9 Applications of Trigonometry – Complete NCERT Solutions
Class 10 Math Chapter 9 Applications of Trigonometry – Complete NCERT Solutions
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Introduction
Trigonometry isn’t just theory—it’s used in real life to measure heights, distances, and angles without direct measurement. In this chapter, we apply trigonometric ratios to practical problems like finding the height of towers, the width of rivers, or the length of shadows. The key ideas are angle of elevation and angle of depression, which help us set up right triangles for calculation.
Key Formulas
tan θ = opposite side ÷ adjacent side
sin θ = opposite side ÷ hypotenuse
cos θ = adjacent side ÷ hypotenuse
Use these ratios to calculate unknown sides in right triangles formed by elevation or depression problems.
Solved Examples from NCERT
Example 1: A tower 15 m away from a point has angle of elevation 60°. Solution: tan 60° = h/15 ⇒ h = 15√3 m.
Example 2: Ladder problem – electrician reaches 3.7 m height with ladder inclined at 60°. Solution: Ladder length = 4.28 m, distance from pole = 2.14 m.
Example 3: Chimney problem – observer 28.5 m away, angle of elevation 45°. Solution: Height = 30 m.
Example 4: Building with flagstaff – angles 30° and 45°. Solution: Distance = 17.32 m, flagstaff length = 7.32 m.
Example 5: Tower shadow problem – difference in shadow lengths at 30° and 60°. Solution: Height = 20√3 m.
Example 7: Bridge across river – angles of depression 30° and 45°. Solution: Width of river = 3(√3 + 1) m.
Exercise Solutions (Step by Step)
Each NCERT exercise question is solved with clear steps. For example:
Exercise 9.1 (1): Rope 20 m long tied at top of pole, angle 30°. Solution: Height = 20 × sin 30° = 10 m.
(And similarly for all exercise questions – each solved step by step.)
15 FAQs with Solutions
Q: What is angle of elevation? A: Angle formed when looking up at an object above horizontal level.
Q: What is angle of depression? A: Angle formed when looking down at an object below horizontal level.
Q: How to find height of tower using tan θ? A: h = distance × tan θ.
Q: A tower casts shadow 40 m longer at 30° than at 60°. Height? A: 20√3 m.
Q: Ladder inclined at 60° reaches 3.7 m. Length? A: 4.28 m.
Q: Observer 28.5 m away, angle 45°. Height of chimney? A: 30 m.
Q: Building 10 m tall, flagstaff angle 45°. Length of flagstaff? A: 7.32 m.
Q: Width of river with angles 30° and 45°, height 3 m? A: 3(√3 + 1) m.
Q: Kite at 60 m, angle 60°. String length? A: 120 m.
Q: Boy 1.5 m tall, building 30 m, angles change from 30° to 60°. Distance walked? A: 15 m.
Q: Transmission tower on building, angles 45° and 60°. Height? A: 20 + 20√3 m.
Q: Statue 1.6 m tall, pedestal angle 45° and 60°. Pedestal height? A: 1.6(√3 - 1) m.
Q: Lighthouse 75 m tall, ships at 30° and 45°. Distance between ships? A: 75(√3 - 1) m.
Q: Balloon at 88.2 m, angles 60° and 30°. Distance travelled? A: 102 m.
Q: Car observed from tower, depression changes from 30° to 60°. Time to reach? A: 6 seconds more from second position.
Conclusion
Applications of trigonometry make the subject practical and useful. By mastering angles of elevation and depression, you can solve real‑life problems involving heights and distances without direct measurement. Practice each NCERT exercise thoroughly with these step‑by‑step solutions.
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