Angle of Elevation

Angle of Elevation is defined as the upward angle measured from a horizontal line or plane to a point situated above it. It is an important geometrical concept used in trigonometry to find the height and distance of the object. This elevation angle is commonly used to describe the viewing perspective of an object and plays a significant role in activities such as navigation and surveying. The angle of elevation refers to the upward angle formed between an observer's line of sight and a horizontal plane.

The concept of angle of elevation holds fundamental significance in trigonometry and has diverse practical applications across different fields. In this article, we will discuss the 'Angle of Elevation' in detail. We will also look at solved examples and practice problems on the concept of angle of elevation.

What is Angle of Elevation?

Angle of Elevation refers to the angle formed between a horizontal line and the line of sight when an observer looks upward. The angle of elevation is an important geometrical concept indicating the vertical inclination of an observer's line of sight to an object or point positioned above a horizontal reference plane. It is commonly used in trigonometry and surveying to measure the height or distance of an object.

In the world of trigonometry, the angle of elevation is typically measured in degrees and is closely associated with the tangent function. When an individual looks upward at an object, the angle formed between the horizontal line of sight and the observer's line of sight is referred to as the angle of elevation. This measurement is valuable for determining distances, heights and the overall geometric characteristics of the observed object.

Read More Trigonometry and Height and Distance.

Angle of Elevation Definition

Angle of Elevation defines the inclination between a horizontal reference line and the observer's line of sight when gazing upwards

Terms Related to Angle of Elevation

Below listed are the major terms related to Angle of Elevation:

• 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.

Angle of Elevation Example

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: Consider standing on a hill and looking at a distant peak; the angle formed between the hill's slope and your line of sight is the angle of elevation.

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

How to Find Angle of Elevation?

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

Also, Check

• Trigonometric Formulas
• Trigonometric Table
• Trigonometric Identities

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 degrees. If the distance between the two buildings is 50 meters, find by what is extra height of the taller building.

Solution:

Distance between two buildings = 50 meters

⇒ tan(45°) = Extra height of the taller building/ Distance between two buildings

⇒ Extra height of the taller building = tan(45°) × 50

⇒ Extra height of the taller building = 50 meters.

Practice Problems on Angle of Elevation

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?