Angle of Elevation
The Angle of Elevation is a fundamental concept in the application of trigonometry to real-world problems. It is covered in CBSE Class 10 Mathematics, Chapter 9 (Some Applications of Trigonometry).
When a person looks upward at an object (such as the top of a building, a flying kite, or a mountain peak), the angle formed between the horizontal line of sight and the line of sight to the object is called the angle of elevation.
This concept is central to solving height and distance problems, which appear regularly in CBSE board examinations. The angle of elevation is always measured from the horizontal, and it increases as the object is higher or closer.
What is Angle of Elevation - Definition, Formula & Solved Examples?
Definition: The angle of elevation of an object as seen by an observer is the angle formed between the horizontal line of sight (at the observer's eye level) and the line of sight directed upward to the object.
Key elements:
- Observer — the person looking at the object.
- Object — the point being observed (above the horizontal).
- Horizontal line — an imaginary line from the observer's eye, parallel to the ground.
- Line of sight — the straight line from the observer's eye to the object.
- Angle of elevation (θ) — the angle between the horizontal line and the line of sight, measured upward.
Important:
- The angle of elevation is always between 0° and 90° (exclusive).
- It is measured at the observer's position, NOT at the object.
- If the observer moves closer to the base of the object, the angle of elevation increases.
- If the observer moves farther from the base, the angle of elevation decreases.
Angle of Elevation Formula
Trigonometric Formula for Angle of Elevation:
In a right-angled triangle formed by the observer, the base of the object, and the top of the object:
tan θ = Opposite / Adjacent = Height of Object / Horizontal Distance
Where:
- θ = angle of elevation
- Opposite side = height of the object above the observer's eye level
- Adjacent side = horizontal distance from the observer to the base of the object
Other useful forms:
sin θ = Height / Line of Sight
cos θ = Horizontal Distance / Line of Sight
When to use which ratio:
- Use tan θ when height and distance are involved (most common).
- Use sin θ when height and line-of-sight length are involved.
- Use cos θ when distance and line-of-sight length are involved.
Derivation and Proof
Setting Up the Right Triangle:
- Let the observer be at point A, standing on level ground.
- Let the object (top of a tower/building) be at point C, directly above point B on the ground.
- AB = horizontal distance (d), BC = height of object (h), AC = line of sight.
- ∠CAB = angle of elevation = θ.
- ∠ABC = 90° (the object is vertical).
Applying trigonometric ratios in △ABC:
- tan θ = BC/AB = h/d → h = d × tan θ
- sin θ = BC/AC = h/(line of sight)
- cos θ = AB/AC = d/(line of sight)
Height of Object = Distance × tan(angle of elevation)
Observer's Eye Height:
In practice, the observer's eye is at some height above the ground (say 1.5 m). The height calculated using tan θ gives the height above the observer's eye. To get the total height of the object:
Total Height = h + Observer's Eye Height
Types and Properties
Problems involving angle of elevation in Class 10 fall into these categories:
Type 1: Finding Height from Angle and Distance
- Given: angle of elevation and horizontal distance.
- Find: height of the object.
- Formula: h = d × tan θ.
Type 2: Finding Distance from Angle and Height
- Given: angle of elevation and height of the object.
- Find: horizontal distance from the observer to the base.
- Formula: d = h / tan θ = h × cot θ.
Type 3: Finding Angle from Height and Distance
- Given: height and horizontal distance.
- Find: angle of elevation.
- Formula: tan θ = h/d, then identify θ from standard values.
Type 4: Two Angles of Elevation (Moving Observer)
- An observer at two positions observes the same object at different angles of elevation.
- Two equations are formed using tan, and solved simultaneously.
Type 5: Two Objects from Same Point
- From one point, angles of elevation to two different objects are given.
- Find heights or distances of the objects.
Type 6: Object on a Hill/Building
- An object (flag, antenna) is placed on top of a building. Angles of elevation to the top and bottom of the object are given.
Methods
Step-by-step method for solving angle of elevation problems:
- Draw the figure — sketch the right-angled triangle with the observer, base, and object clearly marked.
- Mark the known values — label the angle of elevation, height, and/or distance.
- Identify the right triangle — confirm where the 90° angle is (usually at the base of the vertical object).
- Choose the correct trigonometric ratio — based on which sides are given/required: use tan θ (most common), sin θ, or cos θ.
- Set up the equation — write the ratio in terms of the known and unknown quantities.
- Solve for the unknown — use standard trigonometric values (30°, 45°, 60°).
- Add observer's height if needed — if the observer's eye level is given, add it to the calculated height.
Standard Values to Remember:
| θ | 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 |
Solved Examples
Example 1: Finding Height of a Tower
Problem: A person standing 40 m away from the base of a tower observes the top of the tower at an angle of elevation of 60°. Find the height of the tower.
Solution:
Given:
- Horizontal distance (d) = 40 m
- Angle of elevation (θ) = 60°
Using tan θ = h/d:
- tan 60° = h/40
- √3 = h/40
- h = 40√3
- h = 40 × 1.732 = 69.28 m
Answer: The height of the tower is 40√3 m ≈ 69.28 m.
Example 2: Finding Horizontal Distance
Problem: A kite is flying at a height of 30 m. The string makes an angle of 60° with the horizontal ground. Find the horizontal distance of the kite from the person holding the string. (Assume the string is straight.)
Solution:
Given:
- Height (h) = 30 m
- Angle of elevation (θ) = 60°
Using tan θ = h/d:
- tan 60° = 30/d
- √3 = 30/d
- d = 30/√3 = 30√3/3 = 10√3
- d = 10 × 1.732 = 17.32 m
Answer: The horizontal distance is 10√3 m ≈ 17.32 m.
Example 3: Finding Length of Line of Sight
Problem: From a point 20 m away from the foot of a building, the angle of elevation of the top is 30°. Find the length of the line of sight (i.e., the distance from the observer to the top of the building).
Solution:
Given:
- Distance from base (d) = 20 m
- Angle of elevation (θ) = 30°
Using cos θ = d/Line of Sight:
- cos 30° = 20/L
- √3/2 = 20/L
- L = 40/√3 = 40√3/3
- L ≈ 23.09 m
Answer: The line of sight is 40√3/3 m ≈ 23.09 m.
Example 4: Observer's Eye Height Included
Problem: A man whose eye level is 1.5 m above the ground observes the top of a tower at an angle of elevation of 45°. He is standing 50 m from the base of the tower. Find the total height of the tower.
Solution:
Given:
- Eye height = 1.5 m
- Horizontal distance (d) = 50 m
- Angle of elevation (θ) = 45°
Using tan θ = h/d:
- tan 45° = h/50
- 1 = h/50
- h = 50 m (height above eye level)
Total height:
- Total height = h + eye height = 50 + 1.5 = 51.5 m
Answer: The total height of the tower is 51.5 m.
Example 5: Two Angles of Elevation from Different Points
Problem: From a point on the ground, the angle of elevation of the top of a tower is 30°. On walking 20 m towards the tower, the angle of elevation becomes 60°. Find the height of the tower.
Solution:
Given:
- Initial angle = 30°, Final angle = 60°
- Distance walked towards tower = 20 m
Let the height = h and the initial distance from base = d.
From initial position:
- tan 30° = h/d → 1/√3 = h/d → d = h√3 … (i)
From new position (20 m closer):
- tan 60° = h/(d − 20) → √3 = h/(d − 20) → d − 20 = h/√3 … (ii)
Substituting (i) in (ii):
- h√3 − 20 = h/√3
- h√3 − h/√3 = 20
- h(√3 − 1/√3) = 20
- h × (3 − 1)/√3 = 20
- h × 2/√3 = 20
- h = 10√3 ≈ 17.32 m
Answer: The height of the tower is 10√3 m ≈ 17.32 m.
Example 6: Flag on Top of a Building
Problem: From a point on the ground 24 m away from the base of a building, the angle of elevation of the top of the building is 30° and the angle of elevation of the top of a flag on the building is 45°. Find the height of the flag.
Solution:
Given:
- Distance from base = 24 m
- Angle to top of building = 30°
- Angle to top of flag = 45°
Height of building (h₁):
- tan 30° = h₁/24
- 1/√3 = h₁/24
- h₁ = 24/√3 = 8√3 m
Total height — building + flag (h₂):
- tan 45° = h₂/24
- 1 = h₂/24
- h₂ = 24 m
Height of flag:
- Flag height = h₂ − h₁ = 24 − 8√3 = 24 − 13.86 = 10.14 m
Answer: The height of the flag is (24 − 8√3) m ≈ 10.14 m.
Example 7: Shadow Problem
Problem: The shadow of a vertical pole is √3 times its height. Determine the angle of elevation of the sun.
Solution:
Given:
- Let height of pole = h
- Length of shadow = √3 × h
Using tan θ = height/shadow:
- tan θ = h/(√3 h) = 1/√3
- θ = 30°
Answer: The angle of elevation of the sun is 30°.
Example 8: Finding Distance Between Two Objects
Problem: From the top of a 45 m high lighthouse, the angles of elevation and depression of the top and bottom of a cliff are 30° and 60° respectively. Find the height of the cliff.
Solution:
Given:
- Height of lighthouse = 45 m
- Angle of depression to bottom of cliff = 60°
- Angle of elevation to top of cliff = 30°
Finding the horizontal distance (d):
- tan 60° = 45/d → √3 = 45/d → d = 45/√3 = 15√3 m
Finding height of cliff above lighthouse level:
- Let h = height above lighthouse
- tan 30° = h/d → 1/√3 = h/(15√3) → h = 15√3/√3 = 15 m
Total height of cliff:
- = 45 + 15 = 60 m
Answer: The height of the cliff is 60 m.
Real-World Applications
Surveying and Civil Engineering:
- Engineers use theodolites to measure angles of elevation to determine heights of buildings, towers, and terrain features without physically climbing them.
Aviation:
- Pilots use the angle of elevation to calculate the required climb rate to clear obstacles during takeoff.
Astronomy:
- The altitude of a star or planet above the horizon is an angle of elevation, used to determine the object's position.
Navigation:
- Sailors determine the distance to a lighthouse by measuring its angle of elevation and knowing its height.
Architecture:
- Architects calculate shadow lengths, sunlight entry angles, and building heights using angles of elevation.
Everyday Life:
- Estimating the height of a tree, pole, or building using a clinometer (angle-measuring device) and known distance.
Key Points to Remember
- The angle of elevation is the angle between the horizontal line of sight and the line of sight directed upward to an object.
- It is always measured at the observer's position, from the horizontal, going upward.
- The angle of elevation is between 0° and 90° (exclusive).
- The most commonly used ratio for elevation problems is tan θ = height/distance.
- As the observer moves closer to the base, the angle of elevation increases.
- As the observer moves farther from the base, the angle of elevation decreases.
- If the observer's eye height is given, add it to the calculated height.
- The angle of elevation from point A to point B equals the angle of depression from point B to point A (alternate interior angles).
- Always draw a clear figure before attempting the problem.
- Use standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
Practice Problems
- From a point 30 m away from the base of a tower, the angle of elevation of the top is 45°. Find the height of the tower.
- A pole casts a shadow of length 20√3 m when the angle of elevation of the sun is 30°. Find the height of the pole.
- The angle of elevation of the top of a 50 m high building from a point on the ground is 60°. Find the distance of the point from the base.
- From two points on the same side of a tower, the angles of elevation of the top are 30° and 60°. If the points are 50 m apart, find the height of the tower.
- A 1.5 m tall boy sees the top of a building at an angle of elevation of 30° from a distance of 100 m. Find the height of the building.
- The angle of elevation of the top of a tower from two points 50 m and 100 m from its base are complementary. Find the height of the tower.
Frequently Asked Questions
Q1. What is an angle of elevation?
The angle of elevation is the angle formed between the horizontal line of sight and the line directed upward to an object above the observer's eye level. It is measured at the observer's position.
Q2. How is the angle of elevation different from the angle of depression?
The angle of elevation is measured upward from the horizontal (observer looks up), while the angle of depression is measured downward from the horizontal (observer looks down). They are equal when the observer and object exchange roles (alternate interior angles).
Q3. Which trigonometric ratio is most used in elevation problems?
tan θ is the most commonly used ratio because most problems involve height (opposite) and horizontal distance (adjacent). sin θ and cos θ are used when the line of sight (hypotenuse) is involved.
Q4. What happens to the angle of elevation when you move closer to the object?
The angle of elevation increases as you move closer to the base of the object. At the base itself, the angle would approach 90°.
Q5. Does the observer's height matter?
Yes. If the observer's eye level is at height h₀ above the ground, the height calculated using tan θ gives the height above the observer's eye. The total height of the object = calculated height + h₀.
Q6. Can the angle of elevation be 0° or 90°?
At 0°, the observer is looking exactly horizontally (no elevation). At 90°, the observer is looking straight up. Both are limiting cases and not practical in height-and-distance problems.
Q7. How do you solve problems with two angles of elevation?
Form two right triangles sharing the same height. Write tan equations for each angle. Solve the two equations simultaneously to find the unknown height or distance.
Q8. What is a clinometer?
A clinometer is a device used to measure angles of elevation and depression. It consists of a protractor with a plumb line. By sighting the object and reading the angle, you can calculate heights using trigonometric ratios.
