Angle of Elevation
The angle of elevation is the angle between the horizontal line and the line of sight which is above the horizontal line. It is formed when an observer looks upwards. Suppose you are standing at the terrace of a building and looking upwards at the sky or at the sun or moon. The angle thus formed between your height from the ground level and the line of sight formed is called the angle of elevation.
| 1. | Angle of Elevation Definition |
| 2. | Angle of Elevation Formula |
| 3. | Angle of Elevation Vs Angle of Depression |
| 4. | FAQS on Angle of Elevation |
Angle of Elevation Definition
The angle of elevation in math is "the angle formed between the horizontal line and the line of sight when an observer looks upwards is known as an angle of elevation". It is always at a height that is greater than the height of the observer. The opposite of the angle of elevation is the angle of depression which is formed when an observer looks downwards. It is important to learn about the angle of elevation and depression while studying heights and distances in trigonometry. The three general words associated with the angle of elevation are angles, horizontal lines, and line of sight.
Let the observer at 'O' is observing an object that is at 'A'. Then the horizontal line is OB and the line of sight is OA. Then the angle formed between OA and OB which is angle AOB is the angle of elevation.
Angle of Elevation Formula
The angle of elevation formula is no different from the formulae of trigonometric ratios. With the help of the formulae given below, we can find the angle of elevation depending on which two sides of the triangle are known. For example, if we have to find the angle of elevation when the height of the object from the horizontal line and the length of the line of sight are known, we can use the sine formula.
For example, to calculate the angle of elevation for an object at the distance of 10 units from the horizontal line (y=10) and 12 units from the observer w.r.t. the horizontal line (x=12), we write, tan θ = 10/12, which can be reduced to tan θ = 5/6. Therefore, the value of θ obtained is tan-1 (5/6). This is the required angle of elevation.
Angle of Elevation Vs Angle of Depression
The angle of elevation and angle of depression are opposites of each other. In an angle of elevation, the object is placed above the observer, while in the case of the angle of depression, it is placed below the observer. If you are standing at your terrace and looking at the sun, then the angle of elevation will be formed. On the other hand, if you will look at the dog standing on the road from your terrace, then the angle of depression will be formed. In both cases, we use trigonometry angles to find the heights and distances. Let us understand the difference between the angle of elevation and depression from the table given below.
| Angle of Elevation | Angle of Depression |
|---|---|
| It is formed when an object is placed above the observer. | It is formed when an object is placed below the observer's eye level. |
| Also known as upwards angle. | Also known as downwards angle. |
| The horizontal line is below the object. | The horizontal line is above the object. |
In the above diagram, θ is the angle of elevation and α is the angle of depression.
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Angle of Elevation Examples
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Example 1: If a girl is standing at point P, which is 8 units away from a building, making an angle of elevation of 45° with point Q, find the height of the building.
Solution: Given that PR=8 units, and ∠QPR=45°. To find the height of the building (QR), we can use the angle of elevation formula tan θ=QR/PR.
tan 45° = QR/8
We know that tan 45° is 1, so,
1 = QR/8
QR=8 units
Answer: Therefore, the height of the building is 8 units.
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Example 2: Find the value of x in the given figure.
Solution: In this figure, there are two angles of elevation given, one is 30° and the other one is 45°. In △POQ, ∠PQO = 30 degrees and OQ=27 feet. Apply the angle of elevation formula tan θ = PO/OQ, we get tan 30 = h/27. The value of tan 30 is 1/√3.
1/√3 = h/27
h = 27/√3
h = 3 √3/√3
h = 3 ft.
Now, apply the same formula in △POR, we get tanθ = PO/OR.
tan 45 = 3/x
The value of tan 45 is 1, and PO = 3 ft.
⇒ 1 = 3/x
x = 3 ft
Answer: Therefore, the value of x is 3 ft.
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Example 3: Ryan is flying a kite that makes an angle of elevation of 30° with the ground. Determine how
high the kite is above the ground when she has let out 100 m of string.
Solution:
Let 'h' be the height of the kite from the ground. Then
Applying sine for the figure,
sin 30 = h/100
We know that sin 30 = 1/2.
1/2 = h/100
h = 100/2 = 50 m
Answer: The kite is 50 m above the ground.
FAQS on Angle of Elevation
What is Meant by Angle of Elevation?
An angle formed when an observer looks at an object placed above its height with respect to the eye level or the horizontal line is known as the angle of elevation. For example, the angle formed between the line of sight and the horizontal line when the sun is observed by a man on the earth is an angle of elevation.
How do you Find the Angle of Elevation?
The angle of elevation can be found when we are given with any two sides of the right triangle formed between the observer and the object. We can use the trigonometric formulae to find the angle of elevation.
How are Angle of Elevation and Depression Related?
The angle of elevation and angle of depression are both measured with respect to the horizontal axis or the horizontal line. The only difference is that when an observer looks upwards at an object, then the angle of elevation is formed, while when she/he looks downwards at an object, the angle of depression is formed.
What is the Angle of Elevation of the Sun?
The angle of elevation of the sun is the angle formed between the horizontal line and your line of sight when you look at the sun. It will keep on changing because the position of the sun keeps on changing. So, even if you stand at the same point in the morning and in the afternoon, the angle of elevation will differ.
Can an Angle of Elevation be More Than 90?
No, an angle of elevation cannot be more than 90 degrees. It always forms a right-angled triangle with the object and the horizontal line. In the right triangle, one angle is 90 degrees which is the angle opposite to the line of sight. So, it is obvious that the other two angles will be less than 90 degrees in order to satisfy the angle sum property of a triangle. Therefore, an angle of elevation cannot be more than 90 degrees.
What is the Formula for Angle of Elevation?
There are three formulae that can be used to find the angle of elevation. The angle of elevation formulas are given below:
How to Find Height with Angle of Elevation?
To find the height with the angle of elevation, we need to use the trigonometric formulas mentioned above based on which two sides of the right triangle are given.
How Do You Find Angle of Elevation and Depression?
Angle of elevation is the angle between a horizontal line of sight and the object when a person is looking up at an object. Whereas the angle of depression is the angle between the horizontal line of sight and the object when a person is looking down at an object.
What is the Importance of Angle of Elevation and Depression?
One of the main aspects of using the angle of elevation and depression is that it is used mostly in word problems in trigonometry when there is a line of sight involved. These angles are used in solving trigonometric problems such as sine, cosine, and tangent along with inverse trigonometric functions.
What is the Altitude of Sun in Angle of Elevation Problems?
The altitude of the sun means the angle of elevation of the sun from the observer. For finding this, we need to use sin, cos, or tan according to the given information.