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Angle of Elevation

Introduction

In mathematics and physics, trigonometry has its applications. Mathematical concepts like height, distances, angle of elevation, and angle of depression are examples of applications of trigonometry

Let us imagine; for example, an observer is looking at an aircraft—an angle of elevation formed towards the aircraft from the observer’s view. The angle of elevation helps us find distances using trigonometric ratios like sine, cosine, and tangent.

This article will discuss the angle of elevation, the related terms, and how to calculate the angle of elevation.

Angle of Elevation

Definition

The angle generated by the line from an object to the observer’s eye and the horizontal line or plane is known as the angle of elevation. The line of sight is the line the observer’s eye is on. As the word says, the angle of elevation is thus formed so that it is above the observer’s eye.

For example, an observer is looking at an object, as shown in the figure below. The angle of elevation is represented by θ. 

Here, a right-angle triangle is created if we connect the object vertically and the horizontal line. Thus, we may utilize trigonometry to compute the observer’s distance from the object. The horizontal line will be regarded as the adjacent side of the formed triangle, and the height of the object will be looked at as the opposite side of θ.

Terms Related to Angle of Elevation

These are the terms we must be familiar with when dealing with the angle of elevation.

Angle

When two straight rays or lines meet at a single point, an angle is created. Degrees are used to express angles.

Line of Sight

The line of sight is the line drawn from the observer’s eye to the object being viewed. It is an oblique line which is therefore not horizontal nor vertical. The line of sight, together with the horizontal line, forms the angle of elevation. 

Horizontal Line

The horizontal line is a straight line on a flat surface. In a coordinate system, a horizontal line has points with the same y-coordinate. The line of sight, together with the horizontal line, form the angle of elevation. 

Observer’s Eye

The observer’s eye is the position where the line of sight and the horizontal line meet.

The Angle of Elevation Formulas

There are formulas that will help us calculate the angle of elevation. 

Let us refer to the figure below to show how we use the formulas. Recall that a right-angle triangle is formed if an imaginary line is drawn connecting the object and the horizontal line’s end. As a result, we may use the trigonometry concept to determine how distant the observer is from the object. The horizontal line will be regarded as the adjacent side of the constructed triangle, and the height of the object will be looked at as the opposite side of θ. 

In the figure, side a is the opposite side of , side b is the adjacent side, and side c is the length of the line of sight.

To determine the angle of elevation, we can use the formulas below, provided we know the angles’ two sides. 

sin sinθ =$\frac{opposite}{hypotenuse}$=$\frac{a}{c}$θ = ($\frac{a}{c}$)
cos cosθ=$\frac{adjacent}{hypotenuse}$=$\frac{b}{c}$θ = ($\frac{b}{c}$)
tan tanθ=$\frac{opposite}{adjacent}$=$\frac{a}{b}$θ = ($\frac{a}{b}$)

If we must calculate the angle of elevation given the line of sight and the height of the object from the horizontal line, we can use the sine formula θ = ($\frac{a}{c}$). 

If we must calculate the angle of elevation given the line of sight and the adjacent side of , we can use the cosine formula θ = ($\frac{b}{c}$).

If we must calculate the angle of elevation given the opposite side and the adjacent side of , we can use the tangent formula θ = ($\frac{a}{b}$).

The Angle of Elevation vs. Angle of Depression

Angles of elevation and depression are opposites of one another. The object is positioned above the observer at an angle of elevation, whereas it is positioned below the observer at an angle of depression. The angle of elevation is generated if you are standing and looking at an object on the rooftop. Whereas, if you were to look at the plants on the ground, an angle of depression is formed. In both cases, we can use trigonometry concepts to find the heights and distances. 

Angle of Elevation
Angle of Depression

The table below shows the difference between the angle of elevation and depression.

Angle of ElevationAngle of Depression
When an object is positioned above the observer’s eye level, the angle of elevation is formed.When an object is positioned below the observer’s eye level, an angle of depression is formed.
In terms of the angle of elevation, the horizontal line is below the object.In terms of the angle of depression, the horizontal line is above the object.

Examples

Example 1

Find the value of the angle of elevation ( θ ) in each of the given figures.

Solution

( a ) In the given triangle ABC, ∠A is the angle of elevation, side AC is the opposite side, while side BC is the adjacent side of the angle θ. Since we know two sides of the triangle, opposite and the adjacent sides of the angle θ, we must use the formula for the tangent. 

AC (opposite side ) = 240 ft

BC ( adjacent side ) = 330 ft

Thus, we have,

tan tanθ=$\frac{opposite\: side}{adjacent\: side}$  

tan tanθ=$\frac{240 ft}{330 ft}$  

tan tanθ=0.72  

Therefore, θ=( 0.72 )

Hence, the value of θ ≈ 36°.

( b ) In the given triangle PRQ, ∠R is the angle of elevation, side PR is the length of the line of sight or the hypotenuse of the triangle, while side QR is the adjacent side of the angle θ. Since we know the lengths of the hypotenuse and the adjacent side of the angle θ, we must use the formula for cosine. 

PR (hypotenuse side ) = 26 m

QR ( adjacent side ) = 10 m

Thus, we have,

cos cosθ=$\frac{adjacent\: side}{hypotenuse\: side}$  

cos cosθ=$\frac{10 m}{26 m}$  

cos cosθ=0.38  

Therefore, θ=( 0.38 )

Hence, the value of θ ≈ 68°.

( c ) In the given triangle LMN, ∠M is the angle of elevation, side LM is the length of the line of sight or the hypotenuse of the triangle, while side LN is the opposite side of the angle θ. Since we know two sides of the triangle, hypotenuse and the opposite side of the angle θ, we must use the formula for sine. 

LM (hypotenuse side ) = 13 ft

LN ( opposite side ) = 5 ft

Thus, we have,

sin sinθ=$\frac{opposite\: side}{hypotenuse\: side}$  

sin sinθ=$\frac{5 ft}{13 ft}$  

sin sinθ=0.38  

Therefore, θ=( 0.38 )

Hence, the value of θ ≈ 22°.

( d ) In the given triangle DEF, ∠E is the angle of elevation, side DF is the opposite side of , while side EF is the adjacent side of the angle θ. Since we know two sides of the triangle, the adjacent and the opposite side of the angle θ, we must use the formula for the tangent. 

DF ( opposite side ) = 5 m

EF ( adjacent side ) = 12 m

Thus, we have,

tan tanθ=$\frac{opposite\: side}{adjacent\: side}$  

tan tanθ=$\frac{5 m}{12 m}$  

tan tanθ≈0.42  

Therefore, θ=( 0.42 )

Hence, the value of θ ≈ 23°.

Example 2

The shadow of a tree is 16 meters in length when the angle of elevation of the sun is 47°. Find the height of the tree.

Solution

The angle of elevation θ is 47°, and the length of the shadow, which is the adjacent side is 16 m. The opposite side of angle θ is the height of the tree. Let us use the formula for tangent since the problem involves the opposite and adjacent sides of the triangle. Thus, we have,

tan tanθ=$\frac{opposite\: side}{adjacent\: side}$  

tan tan47°=$\frac{height\: of\: the\: tree}{16\: m}$  

tan 47° ( 16 m ) = height of the tree

17.16 m ≈ height of the tree

height of the tree ≈ 17.16°

Hence, the height of the tree is approximately 17.16 m.

Example 3

At point M, a man is positioned 10 meters from a building, creating a 30° elevation with point T. Find the building’s height.

Solution

In the given scenario, the angle of elevation is 30° while the distance of the man horizontally from the base of the building is 10 meters. The opposite side of angle θ is the height of the building. Let us use the formula for tangent since the problem involves the opposite and adjacent sides of the triangle. Thus, we have,

tan tanθ=$\frac{opposite\: side}{adjacent\: side}$

tan tan30°=$\frac{height\: of\: the\: building}{10\: m}$  

tan 30° ( 10 m ) = height of the building

5.77 m ≈ height of the building

height of the building ≈ 5.77 m

Hence, the height of the building is approximately 5.77 m.

Example 4

Marivic is flying a kite that is 60 degrees above the ground. Find the height of the kite above the ground after she has released 40 meters of string.

Solution

The angle of elevation θ is 60°. The length of the released string, 40 m, will be used as the hypotenuse of the right triangle. The opposite side of angle θ is the height of the kite above the ground. Let us use the formula for sine since the problem involves the opposite and hypotenuse of the triangle. Thus, we have,

sin sinθ=$\frac{opposite\: side}{hypotenuse\: side}$  

sin sin60° = ( height of the kite from the ground ) / (40 m ) 

sin 60° ( 40 m ) = height of the kite from the ground 

20 √3 m = height of the kite from the ground 

height of the kite from the ground ≈ 34.64 m

Thus, the height of the kite from the ground is approximately 34.64 m.

Example 5

A ladder climbs to the top of a vertical wall and makes a 45-degree angle with the ground. Determine the length of the ladder if the ladder’s foot is 5 meters from the wall.

Solution

The angle of elevation θ is 45°, while the ladder length is the right triangle’s hypotenuse. The distance between the foot of the wall is 5 meters, the distance of the adjacent side of the right triangle. Let us use the formula for cosine since the problem involves the adjacent and the hypotenuse of the triangle. Thus, we have,

cos cosθ=$\frac{adjacent\: side}{hypotenuse\: side}$  

cos cos45°=$\frac{adjacent\: side}{length\: of\: the\: ladder}$  

Length of the ladder = $\frac{5\: m}{cos\: cos45°}$

Length of the ladder = 5 √2 m

Length of the ladder ≈ 7 m

Hence, the length of the ladder is approximately 7 m.

Summary

Angle of Elevation

The angle generated by the line from an object to the observer’s eye and the horizontal line or plane is known as the angle of elevation of the object as sighted by the observer. The line of sight is the line that the observer’s eye is on. As the word says, the angle of elevation is thus formed so that it is above the observer’s eye.

Angle of Elevation
Angle of Depression
When an object is positioned above the observer’s eye level, the angle of elevation is formed.When an object is positioned below the observer’s eye level, an angle of depression is formed.
In terms of the angle of elevation, the horizontal line is below the object.In terms of the angle of depression, the horizontal line is above the object.

Terms Related to Angle of Elevation

AngleObserver’s Eye
When two straight rays or lines meet at a single point, an angle is created. Degrees are used to express angles.The observer’s eye is the position where the line of sight and the horizontal line meet.
Line of SightHorizontal Line
The line of sight is the line drawn from the observer’s eye to the object being viewed. It is an oblique line which is therefore not horizontal nor vertical. The line of sight, together with the horizontal line, form the angle of elevation. The horizontal line is a straight line on a flat surface. In a coordinate system, a horizontal line has points with the same y-coordinate. The line of sight, together with the horizontal line, form the angle of elevation. 

The Angle of Elevation Formulas

To determine the angle of elevation, we can use the formulas below, provided we know the angles’ two sides. 

sin sinθ =$\frac{opposite}{hypotenuse}$=$\frac{a}{c}$θ = ($\frac{a}{c}$)
cos cosθ=$\frac{adjacent}{hypotenuse}$=$\frac{b}{c}$θ = ($\frac{b}{c}$)
tan tanθ=$\frac{opposite}{adjacent}$=$\frac{a}{b}$θ = ($\frac{a}{b}$)

Frequently Asked Questions on Angle of Elevation ( FAQs )

What is meant by angle of elevation?

The angle generated by the line from an object to the observer’s eye and the horizontal line or plane is known as the angle of elevation of the object as sighted by the observer. The line of sight is the line that the observer’s eye is on. The angle of elevation is thus created, as the word “elevation” suggests, so it is above the observer’s eye.

What differentiates the angle of elevation from the angle of depression?

Angles of elevation and depression are opposites of one another. The object is positioned above the observer at an angle of elevation, whereas it is positioned below the observer at an angle of depression. The angle of elevation is generated if you are standing and looking at an object on the rooftop. On the other side, if you were to gaze at the plants on the ground, an angle of depression would be generated.

The table below shows the essential difference between the angle of elevation and the angle of depression.

Angle of ElevationAngle of Depression
When an object is positioned above the observer’s eye level, the angle of elevation is formed.When an object is positioned below the observer’s eye level, an angle of depression is formed.
In terms of the angle of elevation, the horizontal line is below the object.In terms of the angle of depression, the horizontal line is above the object.

In a triangle, what is the angle of elevation?

In terms of angle of elevation, the right-angle triangle is formed if an imaginary line is drawn connecting the object and the horizontal line’s end. 

As a result, we may use the trigonometry concept to determine how distant the observer is from the object. The horizontal line will be regarded as the adjacent side of the constructed triangle, and the height of the object will be looked at as the opposite side of θ. 

How do we apply trigonometric ratios to get the angle of elevation?

To determine the angle of elevation, we can use the formulas below, provided we know the angles’ two sides. 

sin sinθ =$\frac{opposite}{hypotenuse}$=$\frac{a}{c}$θ = ($\frac{a}{c}$)
cos cosθ=$\frac{adjacent}{hypotenuse}$=$\frac{b}{c}$θ = ($\frac{b}{c}$)
tan tanθ=$\frac{opposite}{adjacent}$=$\frac{a}{b}$θ = ($\frac{a}{b}$)

If we must calculate the angle of elevation given the line of sight and the height of the object from the horizontal line, we can use the sine formula θ = ($\frac{a}{c}$). 

If we must calculate the angle of elevation given the line of sight and the adjacent side of , we can use the cosine formula θ = ($\frac{b}{c}$).

If we must calculate the angle of elevation given the adjacent side of , we can use the tangent formula θ = ($\frac{a}{b}$).

What are the terms related to the angle of elevation?

These are the terms we must be familiar with when dealing with the angle of elevation.

Angle

When two straight rays or lines meet at a single point, an angle is created. Degrees are used to express angles.

Line of Sight

The line of sight is the line drawn from the observer’s eye to the object being viewed. It is an oblique line which is therefore not horizontal nor vertical. The line of sight, together with the horizontal line, form the angle of elevation. 

Horizontal Line

The horizontal line is a straight line on a flat surface. In a coordinate system, a horizontal line has points with the same y-coordinate. The line of sight, together with the horizontal line, form the angle of elevation. 

Observer’s Eye

The observer’s eye is the position where the line of sight and the horizontal line meet.

What is the angle of elevation, and how do you calculate it?

The angle generated by the line from an object to the observer’s eye and the horizontal line or plane is known as the angle of elevation of the object as sighted by the observer. The line of sight is the line that the observer’s eye is on. As the word says, the angle of elevation is thus formed so that it is above the observer’s eye.

To find the angle of elevation, we can use the formulas below, provided we know the angles’ two sides. 

If we must calculate the angle of elevation given the line of sight and the height of the object from the horizontal line, we can use the sine formula θ = ($\frac{a}{c}$). 

If we must calculate the angle of elevation given the line of sight and the adjacent side of , we can use the cosine formula θ = ($\frac{b}{c}$).

If we must calculate the angle of elevation given the opposite and the adjacent side of , we can use the tangent formula θ = ($\frac{a}{b}$).

For example, let us use the triangle LMN below.

In the given triangle LMN, ∠M is the angle of elevation, side LM is the length of the line of sight or the hypotenuse of the triangle, while side LN is the opposite side of the angle θ. Since we know the two sides of the triangle, the hypotenuse and the opposite side of the angle θ, we must use the formula for sine. 

LM (hypotenuse side ) = 13 ft

LN ( opposite side ) = 5 ft

Thus, we have,

sin sinθ=$\frac{opposite\: side}{hypotenuse\: side}$  

sin sinθ=$\frac{5\: ft}{13\: ft}$  

sin sinθ=0.38  

Therefore, θ=( 0.38 )

Hence, the value of θ ≈ 22°.

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