Chapter 9: Some Applications of Trigonometry
Key Concepts:
• Angle of elevation: angle above horizontal when looking up
• Angle of depression: angle below horizontal when looking down
• Line of sight: line from observer to object
• Use sin, cos, tan ratios in right triangles
Exercise 9.1
Q1. A circus artist climbs a 20m rope from top of a vertical pole to ground. Rope makes 30° angle with ground. Find height of pole.
Solutionsin 30° = h/20
1/2 = h/20
h = 20/2 = 10 m
Q2. A tower stands vertically. From 30m away, angle of elevation of top is 30°. Find height of tower.
Solutiontan 30° = h/30
1/√3 = h/30
h = 30/√3 = 30√3/3 = 10√3 m
Q3. Kite is at 60m height. String makes 60° with ground. Find length of string.
Solutionsin 60° = 60/l
√3/2 = 60/l
l = 120/√3 = 120√3/3 = 40√3 m
Q4. From 12m away from building, angle of elevation of top is 30°. Find height.
Solutiontan 30° = h/12
h = 12/√3 = 4√3 m
Q5. From top of 7m building, angle of elevation of cable tower top is 60°, depression of foot is 45°. Find tower height.
SolutionLet tower height=H, distance=d
tan 45°=7/d → d=7m
tan 60°=(H-7)/7
√3=(H-7)/7
H-7=7√3
H=7(1+√3)=7(1+√3) m
Q6. 1.5m tall boy, angle of elevation to top of 30m building goes from 30° to 60° as he walks towards building. Find distance walked.
SolutionHeight above boy = 30-1.5 = 28.5m
At 60°: tan 60°=28.5/d₁ → d₁=28.5/√3
At 30°: tan 30°=28.5/d₂ → d₂=28.5√3
Distance walked = 28.5√3 - 28.5√3/3
= 28.5√3 × 2/3 = 19√3 m
Q7. From ground, angles of elevation of bottom and top of tower on 20m building are 45° and 60°. Find tower height.
Solutiontan 45°=20/d → d=20m
tan 60°=(20+h)/20
√3=(20+h)/20
h=20√3-20=20(√3-1) m
Q10. Two poles of equal height on either side of 80m road. Angles of elevation 60° and 30°. Find height and distances.
SolutionLet height=h, distance from 60° pole=x
tan 60°=h/x → h=x√3
tan 30°=h/(80-x)
x√3=(80-x)/√3
3x=80-x → 4x=80 → x=20
h=20√3 m
∴ Height=20√3 m, distances=20m and 60m