📐 SOME APPLICATIONS OF TRIGONOMETRY

Class 10 Mathematics • Chapter 9

Heights & Distances • Line of Sight • Elevation • Depression

THE GOAL • HANDWRITTEN EXAM NOTES
01. What is this Chapter About?
In the previous chapter, we studied trigonometric ratios. Now we use those ratios to solve real-life problems involving heights and distances.
💡 Easy Hindi: Is chapter mein hum trigonometry ko real-life situations mein use karte hain.

Jaise:
• Tower ki height find karna
• Building ki height find karna
• River ya object ki distance find karna
• Kisi object ko dekhne par angle of elevation/depression use karna
Core idea: Almost every problem can be converted into a right-angled triangle. Once the triangle is formed, choose the correct trigonometric ratio.
02. Line of Sight
The straight line joining the eye of the observer to the point being observed is called the line of sight.
Observer Top of object Line of sight Horizontal θ Height
✏️ Example
A student standing on the ground looks at the top of a tower. The line joining the student's eye to the top of the tower is the line of sight.
👉 Simple language mein: Eye se object ke point tak jo imaginary straight line draw hoti hai = Line of Sight.
03. Angle of Elevation ⭐⭐⭐
When an observer looks at an object above the horizontal level of the observer's eye, the angle formed between the horizontal line and the line of sight is called the angle of elevation.
Observer Top θ Angle of Elevation Line of Sight Height Horizontal
💡 Hindi: Agar observer kisi object ke upar wale point ko dekh raha hai, to horizontal line se upar jo angle banta hai use Angle of Elevation kehte hain.
Remember:

Object is above observer's eye level
→ Elevation
✏️ Example
A boy observes the top of a tower at an angle of elevation of 30°. If the distance between the boy and the foot of the tower is 20 m, find the height of the tower.

Here:
θ = 30°
Base = 20 m
Since height and base are involved:
tan 30° = Height / Base
1/√3 = h/20
Therefore:
h = 20/√3 m
✓ Height = 20/√3 m
04. Angle of Depression ⭐⭐⭐
When an observer looks at an object below the horizontal level of the observer's eye, the angle formed between the horizontal line and the line of sight is called the angle of depression.
Observer Horizontal Line of Sight θ Angle of Depression Object Height
💡 Hindi: Agar observer kisi object ke neeche wale point ko dekh raha hai, to horizontal line se neeche jo angle banta hai use Angle of Depression kehte hain.
⚠️ Very Important: Angle of depression observer ke paas banta hai, lekin question solve karte waqt uska relationship with the corresponding angle of elevation at the object use kiya ja sakta hai because the horizontal lines are parallel.
✏️ Example
From the top of a building, a person observes an object on the ground at an angle of depression of 45°. If the horizontal distance is 30 m, find the height.
tan 45° = Height / 30
1 = Height / 30
Height = 30 m
✓ Height = 30 m
05. Angle of Elevation vs Angle of Depression
Point Elevation Depression
Object position Above observer Below observer
Line of sight Upward Downward
Angle Above horizontal Below horizontal
Typical situation Looking at tower top Looking at object below
🧠 Easy trick:
UP → Elevation
DOWN → Depression
06. The Most Important Triangle
Once the situation is understood, convert it into a right-angled triangle.
A B C Base / Distance Height Line of Sight / Hypotenuse θ
Height

Usually perpendicular side.
Distance

Usually horizontal/base side.
Line of Sight

Usually the hypotenuse.
👉 Diagram banane ke baad:

Height → Perpendicular
Horizontal distance → Base
Line of sight → Hypotenuse

Phir ratio select karo.
07. Which Trigonometric Ratio Should I Use?
Given / Required Best Ratio
Height and Base tan θ
Height and Hypotenuse sin θ
Base and Hypotenuse cos θ
sin θ = Height / Line of Sight
cos θ = Base / Line of Sight
tan θ = Height / Base
💡 Most height-distance questions mein height + horizontal distance diya hota hai. Isliye tan θ bahut commonly useful hota hai. Lekin blindly tan use nahi karna. Question ke given sides ko triangle mein identify karke ratio choose karna hai.
08. Example: Finding Height of a Tower ⭐⭐⭐
Question
A man standing 20 m away from the foot of a tower observes the top of the tower at an angle of elevation of 30°. Find the height of the tower.
30° 20 m h Line of Sight Top
Given:
θ = 30°
Base = 20 m
Required:
Height = h
Use:
tan θ = Height / Base
Therefore:
tan 30° = h / 20
1/√3 = h/20
h = 20/√3 m
✓ Height = 20/√3 m
09. Example: Finding Distance
Question
The angle of elevation of the top of a tower from a point on the ground is 45°. If the tower is 25 m high, find the distance of the point from the foot of the tower.
Let distance = x.
tan 45° = 25/x
1 = 25/x
Therefore:
x = 25 m
✓ Distance = 25 m
👉 Yahan height aur base unknown hai, lekin height given hai. Because: tan θ = perpendicular / base we can directly find the base.
10. Example: Finding Line of Sight
Question
A person observes the top of a tower from a point 20 m away. The angle of elevation is 60°. Find the length of the line of sight.
Let line of sight = l. Here:
Base = 20 m
θ = 60°
We need Hypotenuse. Use:
cos θ = Base / Hypotenuse
Therefore:
cos 60° = 20/l
1/2 = 20/l
l = 40 m
✓ Line of sight = 40 m
11. Observer's Eye Height ⭐⭐⭐
Sometimes the observer's eye is not at ground level. In such questions, the vertical distance from the ground to the observer's eye must be considered separately.
Eye height θ Line of Sight Tower
✏️ Example
A person whose eye is 1.5 m above the ground observes the top of a tower at an angle of elevation of 45°. The distance from the person to the tower is 20 m. Find the height of the tower.
Height above eye level:
h = 20 tan 45°
h = 20 m
But total tower height includes the eye height:
Total Height = 20 + 1.5
= 21.5 m
✓ Height of tower = 21.5 m
⚠️ Important: Agar question mein observer ki eye height given hai, to final height mein usse include karna mat bhoolna.
12. Angle of Depression → Horizontal Distance
Question
From the top of a 40 m high building, the angle of depression of a car on the ground is 45°. Find the distance of the car from the foot of the building.
40 m 45° Line of Sight Car Distance = x
Due to the corresponding angle relationship, the relevant angle in the right triangle is 45°. Therefore:
tan 45° = 40/x
1 = 40/x
x = 40 m
✓ Distance = 40 m
13. Two Angles of Elevation ⭐⭐⭐
Sometimes two different positions are given for observing the same object. Each position forms a different right-angled triangle.
A B Top α β Two observation points
💡 Agar observer object ke closer aata hai, generally angle of elevation increase hota hai.

Is type ke question mein har observation point se alag triangle banao aur equations form karo.
✏️ Basic Example
From point A, the angle of elevation of the top of a tower is 30°. From point B, which is closer to the tower, the angle of elevation is 60°. Let tower height = h. Then:
tan 30° = h / distance from A
and:
tan 60° = h / distance from B
The two equations can be solved together according to the distance information given in the question.
14. Important Standard Values
Angle tan θ sin θ cos θ
30° 1/√3 1/2 √3/2
45° 1 1/√2 1/√2
60° √3 √3/2 1/2
For Heights and Distances, the most frequently used ratios are often:

tan θ → height and horizontal distance
sin θ → height and line of sight
cos θ → horizontal distance and line of sight
15. How to Solve Any Heights & Distances Question
STEP 1: Read the question carefully.

STEP 2: Draw a rough figure.

STEP 3: Mark the given angle.

STEP 4: Identify the height and horizontal distance.

STEP 5: Identify the required side.

STEP 6: Choose sin, cos or tan.

STEP 7: Substitute the standard value.

STEP 8: Solve the equation.

STEP 9: Write the answer with correct unit.
🧠 Golden Rule: Question ko directly formula mein mat ghusao.

Question → Figure → Triangle → Ratio → Calculation

Ye sequence follow karoge to silly mistakes bahut kam hongi.
16. Common Exam Mistakes ⚠️
1. Angle of elevation aur depression ko interchange karna.

2. Line of sight ko horizontal maan lena.

3. Height aur distance ko wrong sides par mark karna.

4. tan ki jagah sin/cos unnecessarily use karna.

5. Observer ki eye height ignore karna.

6. Depression question mein corresponding angle relationship ko correctly identify na karna.

7. Final answer mein metre / metres jaise units bhool jana.

8. Calculator-style decimal answer dena jab exact form required ho.
17. Quick Formula Sheet 🎯
sin θ = Perpendicular / Hypotenuse
cos θ = Base / Hypotenuse
tan θ = Perpendicular / Base
tan θ = Height / Horizontal Distance
Height = Distance × tan θ
Distance = Height / tan θ
sin θ = Height / Line of Sight
cos θ = Distance / Line of Sight
18. One-Minute Concept Revision
Line of Sight

Eye → Point observed
Elevation

Looking upward
Depression

Looking downward
Height

Usually perpendicular side.
Horizontal Distance

Usually base of the right triangle.
Most Important:
tan θ = Height / Horizontal Distance
❤️ Chapter ka actual idea: Trigonometry yahan sirf formula nahi hai. Hum real-life situation ko right-angled triangle mein convert karte hain, phir appropriate trigonometric ratio use karke unknown height ya distance find karte hain.