Angle Of Elevation: Definition, Formula, Questions and Answers 

Exploring the concept of the angle of elevation opens a window to understanding how sightlines and distances intertwine. This fundamental trigonometric notion plays a pivotal role in various competitive exams. By grasping the angle of elevation's significance, one can precisely decipher heights, distances, and spatial relationships.

Defining Angle Of Elevation

The angle of elevation in trigonometry describes the angle between the horizontal line (ground) and the line of sight when looking up at an object. When calculating the angle of elevation, you can use trigonometric functions like tangent.

For instance, if you know the height and distance from the object, you can determine the angle using the formula: tan(angle) = height/distance.

Key Terms Used In Angle Of Elevation

Angle: The angle refers to the measurement between two lines intersecting at a point. By recognizing the angle, one can gauge the inclination of an object from the observer's viewpoint.

Horizontal Line: A horizontal line runs parallel to the horizon, aiding in determining height variations. The horizontal line acts as a reference for measuring the object's height above ground level accurately.

Line Of Sight: The line of sight is the straight path along which an observer views an object. The line of sight assists in visualizing how high or low an object is positioned relative to the observer.

Interconnected Nature Of These Terms

These terms are intricately connected when determining the angle of elevation. The angle dictates the steepness of elevation, while the horizontal line provides a baseline for measuring vertical distances.

Simultaneously, the line of sight aligns with the angle to establish a clear perspective on an object's position in relation to its top and base.

Angle of Elevation Formula with Example

Let us now delve into the formula for calculating the angle of elevation:

Tangent

The function of tangent is essential in calculating the angle of elevation.

In a right triangle with angle θ, where θ is one of the acute angles, and with sides labelled as follows:

• Opposite represents the side opposite to the angle θ.

• Adjacent represents the side adjacent to the angle θ.

Then, the tangent of angle θ is given by:

tan(θ)= Adjacent/Opposite​

If you're standing a certain distance from a tall building and you look up at the top of the building, the angle between your line of sight (looking up) and the horizontal line (your eye level) is the angle of elevation.

Trigonometric Relation

If you know the height of the object (h) and the horizontal distance from the object (d), you can use trigonometry to find the angle of elevation (θ):

tan⁡(θ)=h/d

Solving for θ:

θ=arctan⁡(h/d)

Visual Representation

Imagine a right triangle where:

• The horizontal leg represents the distance from you to the base of the building.
• The vertical leg represents the height of the building.
• The line of sight forms the hypotenuse of the triangle.

The angle of elevation is the angle at the observer's position between the horizontal leg and the line of sight.

Click here to enhance and upskill your quantitative aptitude related to the angle of elevation (trigonometry) right away!

Angle of Elevation vs. Angle of Depression

Let's look at the key differences between the angle of elevation and the angle of depression:

Angle of Elevation MCQs with Answers

Provided below are some selected questions with answers for you to practice and enhance your quantitative aptitude skill:

Question 1: What is the angle of elevation if you look up at a kite flying at a 45-degree angle above you?

a) 45 degrees

b) 90 degrees

c) 135 degrees

d) None of the above

Solution: a) 45 degree

Explanation: Since the angle is 45 degrees and the tangent of 45 degrees is 1, we have:

tan(45°) = Opposite side / Adjacent side

1 = Opposite side / Adjacent side

Since the opposite side (height of the kite) is equal to the adjacent side (distance between you and the kite) when you are directly below the kite, the angle of elevation is indeed 45 degrees.

Therefore, the angle of elevation when you look up at a kite flying at a 45-degree angle above you is 45 degrees.

Question 2: If the angle of elevation is 30 degrees, how high does a ladder reach if it's 10 meters from a wall?

a) 5 meters

b) 10 meters

c) 15 meters

d) 20 meters

Solution: c) 5 meters

Explanation: Using trigonometry, when the angle of elevation is 30 degrees and the ladder is 10 meters from the wall, we can use the sine function because we have the opposite side (height) and the hypotenuse (length of the ladder). The sine of 30 degrees is 1/2.

So, if the length of the ladder (hypotenuse) is 10 meters, the height it reaches (opposite side) would be (1/2) * 10 meters, which equals 5 meters.

Question 3: What is the height of the tower if the angle of elevation from a point 20 meters away from its base is 60 degrees?

a) 20√3​ meters

b) 10√3 meters

c) 20 meters

d) 10 meters

Solution: a) 20√3 meters

Explanation: To find the height of the tower, we can use trigonometry:

As per the given question, we have the angle of elevation = 60 degrees and distance from the tower's base = 20 meters

We can use the tangent function because we have the opposite side (height) and the adjacent side (distance from the tower's base). The tangent of 60 degrees is √3.

So, if the adjacent side (distance from the tower's base) is 20 meters, the height of the tower (opposite side) would be (√3) * 20 meters. Height of the tower = √3 * 20 ≈ 20√3 meters.

Therefore, the height of the tower is approximately 20√3 meters.

Question 4: If the angle of elevation to the top of a building is 45 degrees and you are standing 20 meters away, how tall is the building?
a) 10 meters
b) 20 meters
c) 20√2​ meters
d) 40 meters

Solution: c) 20 meters

Explanation: As per the given question, we have the angle of elevation = 45 degrees and distance from the building = 20 meters

We can use the tangent function because we have the opposite side (height) and the adjacent side (distance from the building). The tangent of 45 degrees is 1.

So, if the adjacent side (distance from the building) is 20 meters, the height of the building (opposite side) would also be 20 meters. Therefore, the height of the building is 20 meters.

Question 5: At a certain time of the day, the sun is seen at the angle of 45 degrees. If the shadow of a pole is 10 meters long, what is the height of the pole?
a) 5√2​ meters
b) 10√2​ meters
c) 5 meters
d) 10 meters

Solution: d) 10 meters

Explanation: As per the question,
the angle of elevation of the sun = 45 degrees
Length of the shadow = 10 meters
tan(45°)= Height​/10
The tangent of 45 degrees is 1.
So, solving for the height:
Height=10×tan(45°)=10×1=10
Therefore, the correct answer is 10 meters

Question 6: A ladder is leaning against a wall. The angle of elevation of the ladder is 60 degrees, while the foot of the ladder is 8 meters away from the wall. How long is the ladder?
a) 8 meters
b) 16 meters
c) 8√3​ meters
d) 16√3​ meters

Solution: b) 16 meters

Explanation: As per the given question, 

• Angle of elevation of the ladder = 60°
• Distance of the foot of the ladder from wall = 8 meters

The sine of the angle of elevation is equal to the ratio of the height of the ladder to its length.

sin⁡(60°)=Height/Length of the ladder​ (sine(60°)= √3/2)

Length of the ladder=Height/sin⁡(60°)

Height=8×sin⁡(60°)= 8×√3/2= 4√3

Now, substituting the height into the formula for the length of the ladder:

Length of the ladder=4√3/√3/2=4√3×2/√3=8×2=16

Hence, the length of the ladder is 16 meters.

Question 7: What is the height of the tower when a tower casts a shadow of 60 meters under the sun at an angle of elevation of 30 degrees?
a) 30 meters
b) 60 meters
c) 90 meters
d) 120 meters

Solution: a) 30 meters

Explanation: As per the given question, we have the length of the tower's shadow= 60 meters and the angle of elevation= 30°

Therefore, height=shadow length×tan(angle of elevation)=60×tan(30°)=60×1/√3= 30 meters.

Question 8: What is the height of the tower if the angle of elevation from a point on the ground to the top of the tower is 60 degrees and the point of observation is 100 meters from the base of the tower?

a) 50√3 meters

b) 173.2 meters

c) 100√3 meters

d) 200 meters

Solution: b) 173.2 meters

Explanation: The tower's height is the tangent of 60 degrees times the distance from the observer (100 meters), which is 100 × tan(60°)= 100×√3 ≈ 173.2 meters.

Question 9: What is the distance of the boat from the base of the cliff if an observer on top of a 100-meter cliff sees the boat at an angle of depression of 45 degrees?

a) 50 meters

b) 100 meters

c) 100√2 meters

d) 200 meters

Solution: b) 100 meters 

Explanation: Distance = opposite over tangent (tan 45° = 1), so distance = 100 meters (the opposite side).

Question 10: What is the distance of the building from the point directly below the plane if an aeroplane flying at an altitude of 5 kilometres has an angle of depression of 20 degrees to a building on the ground?

a) 17.3 kilometres

b) 20.4 kilometres

c) 24.4 kilometres

d) 14.5 kilometres

Solution: d) 14.5 kilometres 

Explanation: Distance can be calculated as the opposite side of the angle of depression, which is the altitude of the plane divided by the tangent of the angle of depression, i.e., distance = altitude/tan (angle of depression) = 5 / tan(20°) ≈ 14.5 kilometres.

Conclusion

The exploration of the angle of elevation has shed light on its significance in various fields. Understanding the angle of elevation, its concept, essential terms, and calculations can enhance one's ability to accurately calculate height and distance using trigonometric ratios.

Exploring further examples and practising calculations will solidify one's grasp of elevation angles. Embracing these insights will not only sharpen your aptitude skills but also foster a deeper appreciation for trigonometric ratios in everyday tasks.

Frequently Asked Questions (FAQs)

1. What is the angle of elevation?

When someone gazes up at something higher than their eyes, they see an angle called the angle of elevation. This angle is made between a flat line and the line of sight going upwards to the object.

2. How is the angle of elevation calculated?

To calculate the angle of elevation, the formula tan(angle) = opposite/adjacent is used. By knowing the height of the object and the distance from the observer, you can determine the angle of elevation.

3. What distinguishes elevation angles from depression angles?

Elevation angles are measured when looking upwards from a horizontal line, while depression angles are measured when looking downwards from a horizontal line. Elevation angles are always positive, whereas depression angles are negative.

4. Can you provide real-world examples where understanding elevation angles is crucial?

Understanding elevation angles is essential in various fields such as engineering, architecture, and surveying. For instance, calculating the angle of elevation is critical in designing ramps for accessibility or determining optimal locations for installing surveillance cameras.

5. Why are horizontal lines important when discussing elevation angles?

Horizontal lines serve as reference points for measuring both elevation and depression angles. They provide a baseline for calculating how high or low an object is relative to the observer's eye level, making it easier to determine accurate angle measurements. 

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