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Angle of Depression is formed when an observer looks at an object such the object is placed at a height lower than the height of the observer.
- It is the angle between the horizontal line originating from the eyes of the observer and the line of sight.
- Here line of sight refers to the line joining the object and observer’s eye.
- The angle of depression can be calculated by using the concept of trigonometry.
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Key Terms: Right-angled triangle, Line of sight, Perpendicular, Base, tan x, Angle of depression, Angle of Elevation, Horizontal line, Angle of depression formula
What is Angle of Depression?
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The Angle of Depression is the angle made by a horizontal line from the eyes of the observer with the line joining the object and the observer’s eye.
- This angle can only be formed when the height of the observer is greater than the height of the object.
- For example, let a person standing on the roof of a building look at an object placed on the ground.
- Then the angle between the horizontal line from the observer’s eye and line of sight is known as the angle of depression.
It generally depends on two factors
- The height of the observer from the ground
- The horizontal distance between the observer and the object.
Angle of Depression
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| Value of Cos 180 | Value of Sin 180 | Sin 30 Degrees |
| Trigonometric Identities | Ratios and Identities in Trigonometry | Trigonometry Table |
Angle of Depression Formula
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The angle of depression can be calculated by using the trigonometric ratios formulae. If any two sides of a right-angled triangle are known, then the formula of the angle of depression is given by
| \(tan \: \theta = \frac {Opposite\: side}{Adjacent \:side}\) \(\theta = tan^{-1} \frac {Opposite \:side}{Adjacent \: side}\) |
Where
- θ is the angle of depression
- The opposite side is equal to the height of the observer from the ground
- Adjacent side is the perpendicular distance between the object and the observer
Angle of Depression and Angle of Elevation Comparison
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The angle of depression and angle of elevation are opposite to each other.
- The angle of depression is formed when the object is placed at a height lesser than the height of the observer.
- The angle of elevation is formed when the object is placed at a height larger than the height of the observer.
- It is the angle made by the horizontal line from the observer’s eye and the line of sight.
Solved Examples
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| Ques. If a person standing at the top of the building of height 50√3 ft is looking at her daughter standing at a distance of 50 ft away from the building, what will be the angle of depression formed? Ans. Given
The angle of depression can be calculated as θ = tan-1 (h/d) ⇒ θ = tan-1 (50√3/50) ⇒ θ = tan-1 (√3) ⇒ θ = 60∘ Hence, the angle of depression is 60∘ Ques. Find the angle of depression given that the base and perpendicular are 4√3 cm and 4 cm, respectively. Ans. Given
The angle of depression can be calculated as θ = tan-1 (p/b) ⇒ θ = tan-1 (4/4√3) ⇒ θ = tan-1 (1/√3) ⇒ θ = 30∘ Hence, the angle of depression is 30∘ |
Also check:
| Chapter Related Links | ||
|---|---|---|
| Geometric Mean | Arithmetic Progression | Value of Log 0 |
| Geometric Progression | Sequence and Series | Sin cos Tan |
Things to Remember
- The angle of depression is the angle between the horizontal line originating from the eyes of the observer and the line of sight.
- Line of sight refers to the line joining the object and the observer’s eye.
- It is formed when an observer standing at a height looks at an object placed on the ground.
- The formula for the angle of depression is given by tan x = tan-1 (Opposite side / Adjacent side).
- The angle of elevation is formed when the object is placed at a height larger than the height of the observer.
Sample Questions
Ques. What is the Angle of Deviation? (1 Mark)
Ans. The angle between the horizontal line from the observer’s eye and the line of sight is known as the angle of depression.
Ques. What is the formula for the Angle of Depression? (1 Mark)
Ans. The formula for the angle of depression is given by
θ = tan-1 (Opposite side/Adjacent side)
Ques. Is the angle of depression and inclination the same? (2 Marks)
Ans. The angle of inclination is the angle formed above the horizontal axis. It is equivalent to the angle of elevation. The angle of depression is the same as the angle of declination since they both form below the horizontal axis and have a downward slope.
Ques. How is the angle of depression related to real life? (1 Mark)
Ans. The angle of depression is used to calculate heights and distances that need long calculations. It has applications in a variety of fields, including architecture, engineering, and science.
Ques. What is the Angle of Elevation? (2 Marks)
Ans. The angle of Elevation refers to that particular angle that forms when the observer usually stands on a platform and observes an object that is kept above the platform.
Ques. If a tower 30 m high casts a shadow 10\(\sqrt{3}\)m long on the ground, then what is the angle of elevation of the sun? (3 Marks)
Ans. Let the required angle be θ.
tan θ = 30/10\(\sqrt{3}\)
⇒ tan θ =\(\sqrt{3}\)
⇒ tan θ = tan 60°
⇒ θ = 60°
Ques. If a person standing at the top of the building of height 70√3 m is looking at her daughter standing at a distance of 70 m away from the building, what will be the angle of depression formed? (3 Marks)
Ans. Given
- The height of the building, h = 70√3 m
- The distance between the person and her daughter, d = 70 m
The angle of depression can be calculated as
θ = tan-1 (h/d)
⇒ θ = tan-1 (70√3/70)
⇒ θ = tan-1 (√3)
⇒ θ = 60∘
Hence, the angle of depression is 60∘
Ques. Find the angle of depression given that the base and perpendicular are 9√3 cm and 9 cm, respectively. (2 Marks)
Ans. Given
- Base, b = 9√3 cm
- Perpendicular, p = 9 cm
The angle of depression can be calculated as
θ = tan-1 (p/b)
⇒ θ = tan-1 (9/9√3)
⇒ θ = tan-1 (1/√3)
⇒ θ = 30∘
Hence, the angle of depression is 30∘
Ques. A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then calculate the height of the wall. (3 Marks)
Ans. We have
∠BAC = 180° – 90° – 60° = 30°
sin 30° = BC/AC
⇒ 1/2 = BC/15
⇒ 2BC = 15
⇒ BC = 15/2 m
Ques. In the figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of the pole. If AD = 2.54 m. Find the length of the ladder. (3 Marks)
Ans. We have
BD = AB – AD = 6 – 2.54 = 3.46 m
In right-angled triangle DBC, sin 60° = BD/DC
⇒ 3/√2 = 3.46/DC
⇒ 3\(\sqrt{DC}\)= 3.46 x 2
∴ Length of the ladder, DC = 6.92 / \(\sqrt{3}\) = 6.92/1.73
⇒ DC = 4 m
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