Angle of elevation is the angle formed between the horizontal line and the line of sight when you look upward at an object above your eye level.
In simple words, if you’re standing on the ground and looking up at something—like the top of a building, a tree, or a hill—the angle your eyes make with the ground is called the angle of elevation.
Related Terms
- Horizontal Line: The reference line used in measuring the angle of elevation. It is also called Horizon
- Line of Sight: The direct line of vision from the observer's eye to the point of interest.
- Zenith Line: The imaginary vertical line from the observer's position directly overhead.
- Angular Altitude: The measurement of an object's apparent height in the sky, considering the angle of elevation.
Examples
In practical applications, surveyors use the angle of elevation to calculate the height of structures like buildings or towers, utilizing trigonometric principles for accurate assessments. Similarly, astronomers and geographers apply this concept to gain insights into the positions and dimensions of celestial objects and geographical features.
- Example 1: Imagine standing on level ground and looking up at the top of a tree; the angle formed between the ground and your line of sight is the angle of elevation.
- Example 2: When you look up at a distant peak, the angle between your horizontal line and your line of sight is called the angle of elevation.
Formula
The formula for the angle of elevation (θ) is given by:
- Sine Ratio (sin): The sine of an angle of elevation is the ratio of the perpendicular to the hypotenuse. In mathematical terms, it can be expressed as
sin θ = Perpendicular/Hypotenuse
- Cosine Ratio (cos): The cosine of an angle of elevation is the ratio of the adjacent side to the hypotenuse. Mathematically, it is represented as
cos θ = Base/Hypotenuse
- Tangent Ratio (tan): The tangent of an angle of elevation is the ratio of the perpendicular to the base. In mathematical terms, it can be expressed as
tan θ = Perpendicular / Base
Steps
To find the angle of elevation, measure the height and horizontal distance then use the formula mentioned earlier.
Step 1: Identify the Triangle
Determine the right-angled triangle formed by the horizontal line of sight and the line of sight to the object.
Step 2: Identify Sides
Label the sides of the triangle: the side opposite the angle of elevation, the adjacent side, and the hypotenuse.
Step 3: Apply Trigonometry
Use either the trigonometric ratios such as sine, cosine, tangent function depending on the known sides to find the angle of elevation.
Calculate Angle of Elevation
Depending on the known sides, use either the sine, cosine or tangent function to find the angle of elevation. When examining angles of elevation, trigonometric functions come into play. These functions help us understand the relationship between the angle of elevation and the sides of a right-angled triangle.
To determine the angle of elevation measure the vertical height and the horizontal distance, then use the formula:
Using Tangent (tan θ) = Perpendicular/Base
Using Sine (sin θ) = Perpendicular/Hypotenuse
Using Cosine (cos θ) = Base/Hypotenuse
Example: A boy standing 20 m away from a pole sees the top of the pole of height 20√3 m. Find the angle of elevation.
Solution:
Let the Pole be AB of height 20√3 m and the boy is standing at point C 20 away from the pole. If we join all the three points we will get a right triangle ABC.
In triangle ABC let the angle of elevation i.e. ∠ACB be θ
tan θ = Perpendicular/Base
⇒ tan θ = AB/AC = 20√3/20 = √3
Now since, tan θ = √3
Hence, θ = tan-1(√3) = 60°
Hence, angle of elevation is 60°.
Angle of Elevation and Depression
Angle of Elevation and Angle of Depression both these geometric concept are used in various fields including architecture, navigation and physics.
While the angle of elevation is formed when looking upward, the angle of depression is formed when looking downward from a horizontal line. When observing an object above eye level such as a mountain or a building, the angle of elevation helps determine the angle at which the observer must look to see the top of the object. Similarly, when observing an object below eye level such as a submarine or a water-plants, the angle of depression helps determine the angle at which the observer must look to see the bottom of the object.
Below is the tabular difference between Angle of Elevation and Angle of Depression
Angle of Elevation | Angle of Depression |
|---|---|
The angle formed when looking upward from a horizontal line or plane. | The angle formed when looking downward from a horizontal line or plane. |
The angle at which the observer must look to see the top of the object. | The angle of depression helps determine the angle at which the observer must look to see the bottom of the object. |
Example: Observing an object above eye level such as a mountain or a building | Example: Observing an object below eye level such as a submarine or a water-plants |
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Solved Examples on Angle of elevation
Example 1. A person standing 100 meters away from a tower observes the top of the tower at an angle of elevation of 45 degrees. Determine the height of the tower.
Solution:
Distance of tower from person = 100 meters
⇒ tan(45°) = height of the tower/ distance of tower from person
⇒ height of the tower = 100 × tan(45°)
⇒ Height of the tower = 100 meter
Example 2. An archer aims an arrow at a 30-degree angle of elevation. If the arrow travels a horizontal distance of 200 meters, find the arrow's vertical displacement.
Solution:
Horizontal Distance from archer = 200 meters
⇒ tan(30°) = vertical distance of archer/ horizontal distance of archer
⇒ Vertical Displacement= 200× tan (30°)
⇒ Vertical Displacement= 200/√3
Example 3. From a point on the ground, the angle of elevation to the top of a cliff is 60 degrees. If the cliff is 80 meters distance, calculate the height of the cliff.
Solution:
Distance of cliff from person = 80 meters
⇒ tan(60°) = height of the cliff/ distance of cliff from person
⇒ height of the cliff = 80 × tan(60°)
⇒ Height of the cliff = 80√3 meter
Example 4. A person on a boat sees the top of a lighthouse at a 30 degree angle of elevation. If the lighthouse is 30 meters tall, calculate the distance between the boat and the lighthouse.
Solution:
Height of Lighthouse = 30 meters
⇒ tan(30°) = Height of Lighthouse/ distance between the boat and the lighthouse
⇒ distance between the boat and the lighthouse = Height of Lighthouse/ tan(30°)
⇒ distance between the boat and the lighthouse = 30√3 meter
Example 5. From the top of a building, the angle of elevation to the top of a taller building is 15°. The horizontal distance between the two buildings is 50 m. Find the additional height of the taller building.
Solution:
Distance between buildings = 50 m
tan(15°) = Additional height / 50
Additional height = 50 × tan(15°)
= 50 × 0.268
≈ 13.4 m
Practice Problems
Q1: A person at the beach looks up at a kite with an angle of elevation of 60 degrees. If the kite is 100 meters above the ground, calculate the horizontal distance from the person to the kite.
Q2: An observer on a mountain peak measures the angle of elevation to a lower peak as 45 degrees. If the horizontal distance between the peaks is 500 meters, find the difference in their heights.
Q3: A sniper aims at a target with an angle of elevation of 30 degrees. If the bullet travels a horizontal distance of 500 meters, determine the vertical displacement of the bullet.
Q4: An airplane is flying at an altitude of 10,000 meters. If the angle of elevation from the ground is 20 degrees, determine the horizontal distance from the airplane to the observer.
Q5: A ladder leans against a wall forming a 60-degree angle of elevation. If the base of the ladder is 15 meters away from the wall, find the length of the ladder.
Q6: What is the relationship between the angle of elevation of a cloud, the height of the cloud, and the distance from the observer?