Angle of Depression

Angle of Depression is one of the two important angles in Trigonometry, the other being the angle of elevation. The angle of Depression is the angle that is formed when the object is at the bottom and the observer is at the top, between the horizontal line and the line of sight of the observer. In this article, we will learn about the Angle of Depression including various examples of the angle of depression and key differences between the angle of elevation and the angle of depression. We will also learn, how to calculate the angle of depression. So let’s start our learning for the topic of “Angle of Depression”.

What is Angle of Depression?

In trigonometry, the angle of depression is referred to as the angle, formed between the horizontal line and the line of sight of the observer when the observer is placed over the object. In other words, an angle of depression is formed when the object is at the bottom and the observer is at the top and the measure of it can be affected by changing the various height and distances between the observer and object.

Terms Related to Angle of Depression

There are various terms, which need to know before studying the concept of the Angle of Depression. These terms are discussed as follows:

• Observer: The person observing the object is called the observer.
• Object: The thing that the observer is observing is called the object.
• Horizontal line: The line starting horizontally from the observer is called a horizontal line.
• Line of Sight: The line from the observer’s eye to the object is called the line of sight.

Example of Angle of Depression

Some of the most common examples of the angle of depression in daily life are:

• A man is watching a car from the top of the tower then the horizontal line and the line of sight from the angle of depression.
• A man in the lighthouse is watching a ship in the sea.
• An electrician at the top of the electric pole is looking downwards.
• Angle of Depression From the Top and Bottom

Angle of Depression Formula

As we can see in the following illustration, the horizontal line, line of sight, and the perpendicular drawn from the object to the horizontal line form a right-angled triangle.

From the above triangle POQ we can conclude using the trigonometric ratio:

• sin θ = p / h,
• cos θ = b / h, and
• tan θ = p / b

As θ here is the angle of depression.

So the angle of depression θ is given by:

θ = sin^-1(p/h)

θ = cos^-1(b/h)

θ = tan^-1(p/b)

How to Find Angle of Depression

To find the Angle of Depression for any given condition, we construct a right-angle triangle by drawing perpendicular, and in that right-angle triangle one of the angles is the angle of depression. To find the angle of depression we use the given sides of the triangle and trigonometric ratios and the following steps.

Step 1: Observe the given sides of the triangle.

Step 2: If the perpendicular and hypotenuse of the triangle are given then we use the trigonometric ratio sin.

Step 3: If the base and hypotenuse of the triangle are given then we use the trigonometric ratio cos.

Step 4: If the base and perpendicular of the triangle are given then we use the trigonometric ratio tan.

Step 5: Now, calculate the angle of depression using one of the appropriate formulas as given below:

• θ = sin^-1(p/h)
• θ = cos^-1(b/h)
• θ = tan^-1(p/b)

Therefore, the resultant value will give the angle of depression.

Angle of Depression and Elevation

The angle of depression is the angle formed between the horizontal line and the line of sight of the observer when the object is at the bottom. The angle of elevation is the angle formed between the horizontal line and the line of sight of the observer when the object is at the top. Both angle of depression and the angle of elevation are opposites of each other and the difference between the angle of elevation and the angle of depression is given as follows:

Angle of Depression	Angle of Elevation
It is the angle formed between the horizontal line and the line of sight when the object is at the bottom and the observer at the top.	It is the angle formed between the horizontal line and the line of sight when the object is at the top and the observer at the bottom.
The object is placed below the observer.	The object is placed above the observer.
It is also called a downward angle.	It is also called an upward angle.
The inclination of the line of sight is downwards.	The inclination of the line of sight is upwards.

Solved Examples of Angle of Depression

Problem 1: Find the angle of depression given the base and perpendicular of the 4 cm and 5 cm respectively.

Solution:

Angle of depression θ = tan^-1(p/b)

θ = tan^-1(5/4)

θ = 51.34°

Problem 2: Find the angle of depression given the base and hypotenuse of the 5 cm and 13 cm respectively.

Solution:

Angle of depression θ = cos^-1(b/h)

θ = cos^-1(5/13)

θ = 67.38°

Problem 3: Given the angle of depression formed when the observer is observing the object from top of a pole is 30°. The distance of the object from the pole is 100 m. Find the height of the pole.

Solution:

Angle of depression = 30°

Distance of the object from the pole = 100 m

tan θ = height / distance

tan 30° = h / 100

0.577 = h / 100

Height of the tower h = 57.7 m

Problem 4: Given the height of the tower is 30 m and the distance of the object from the tower is 10 m. Find the angle of depression when the observer is in the tower and observing the object.

Solution:

Height of the tower = 30 m

Distance of the object from the tower = 10 m

Angle of depression θ = tan^-1(p/b)

θ = tan^-1(30 / 10)

θ = tan^-1(3)

θ = 71.56°

Problem 5: A mountaineer is watching camp from the top of a hill with an angle of depression of 30°. The height of the hill is 20km then find the length of the line of sight of the mountaineer.

Solution:

Length of line of sight = hypotenuse of the triangle

Angle of depression θ = 30°

Height of hill = 20km

sin θ = height of hill / length of line of sight

sin 30° = 20 / length of line of sight

Length of line of sight = 20 / sin 30°

Length of line of sight = 20 / (1/2)

Length of line of sight = 20 × 2

Length of line of sight = 40 km

Problem 6: The angle of depression of two ships from a lighthouse are 30° and 60° respectively. Given the height of the lighthouse is 5000 m and ship1 is 4000 m apart from the lighthouse. Find the distance between ship 1 and ship 2.

Solution:

We have drawn below figure according to the question

From the bigger triangle

tan 30° = 5000 / y

y = 5000 / tan 30°

y = 8660.25 m

From the smaller triangle

tan 60° = 5000 / z

z = 5000 / tan 60°

z = 2886.75 m

Since, from above figure y = x + z

x = y – z

x = 8660.25 – 2886.75

x = 5773.5 m

The distance between two ships (x) = 5773.5 m

FAQs on Angle of Depression

Q1. Define Angle of Depression.

Answer:

The angle of depression is defined as the angle formed by the horizontal line and the line of sight of the observer when the object is at the bottom.

Q2. What are the Factors Affecting the Angle of Depression?

Answer:

The factors affecting the angle of depression are height and distance.

Q3. Write the Formula for Angle of Depression if the Perpendicular and Base of the Triangle are Given.

Answer:

The formula for angle of depression if the perpendicular and base of the triangle is:

Angle of depression θ = tan^-1(perpendicular / base)

Q4. In which Condition the Angle of Depression is Formed?

Answer:

The angle of depression is formed when the object is at the bottom and the observer is at the top.

Q5. Write the Formula for Angle of Depression if the Perpendicular and Hypotenuse of the Triangle are Given.

Answer:

The formula for angle of depression if the perpendicular and base of the triangle is:

Angle of depression θ = sin^-1(perpendicular / hypotenuse)

Q6. What is Angle of Elevation?

Answer:

The angle of elevation is the angle formed between the horizontal line and the line of sight of the observer when object is at the top.