The angle of depression is an important component used in trigonometric operations. The angle of depression is formed when the observer stands on a ground higher than the object which he/she is trying to observe. When the observer is looking down on the object, then you can draw a line connecting the object and the observer’s eyes (the line of sight) and you can draw another horizontal line from the eye level of the observer. The angle formed between the horizontal line and the line of sight is called the angle of depression. The angle of depression is equal to the angle of elevation.

The angle of elevation is just the opposite of the angle of depression. If you change the setting and place the observer at a lower level than the object so that the observer is looking above at the object, then the angle of elevation is formed. Similar to the angle of depression, a line connects the object and the observer’s eyes (the line of sight) and a horizontal line is drawn from the eye level of the observer. The angle between the horizontal line and the line of sight is known as the angle of elevation. Both the angle of depression and the angle of elevation are important for trigonometric operations.

Now, the basic definition of the depression angle is the angle formed between the horizontal plane of the observer and the line of sight. This line of sight refers to the unobstructed vision or a line that connects both the observer and the object. The angle of depression is a potent tool in solving our everyday problems. It is used in trigonometric applications to calculate the distance between the observer and the object if the height and the angle of depression are known to us. So, in the example below, the angle 2, which is formed by the horizontal and the line of sight, is the depression angle.

In the above examples, you will see that all the settings of the angle of depression form a right-angle triangle.

We have linked another image below. From the reference image below, we will be able to define angle of depression mathematically as:

Tan(angle of depression) = (Height at which the observer is standing at)/(The distance between the object and the observer)

From the example below, ∠FEG is the angle of depression, and ∠DGE is the angle of elevation. If we take a good look at the image we see that by congruency of alternate interior angles, both the angle of depression and angle of elevation are equal. Mathematically it would be represented as,

∠FEG= ∠DGE,

So, Tan(∠FEG) = Tan(∠DGE)

And, Tan(∠DGE) = ED/DG (We already know, Tan θ = Perpendicular/Base)

So, Tan(∠FEG) = Tan(∠DGE) = ED/DG

Example 1:

Paul is standing at the top of the building. From the top of the building, he sees a vehicle parked at a distance of 100 meters. The angle made by the line of sight and the horizontal plane is 60 degrees. From this information, calculate the height of the building.

Solution:

From the problem statement, we can infer and create a triangle PQR given below in the figure.

So, in the figure above, P is the position at which Paul is standing. R is the vehicle's position, and h is the height of the building we have to calculate.

It is given in the problem that,

QR = 100m, ∠SPR is the angle of depression and is equal to 60 degrees.

So, ∠PRQ = ∠SPR = 60 (Congruency of alternate interior angles)

So, in Right triangle PQR,

∠R = 60, ∠P = 180-(90+60) = 180-150 = 30

So, PQ = h =?

Applying the tangent of the angle R:

Tan(r) = h/QR = h/100 -------- (1)

Since, ∠R = 60 degrees. Tan(r) = Tan(60) = √3.

Substituting the value of tan r in equation 1, we get

√3 = h/100 or h = √3 * 100 = 173.20 m

Hence, the building that Paul is standing on is 173.20 meters tall.

FAQ (Frequently Asked Questions)

1. Is it necessary for the angle of depression and angle of elevation to be equal in every case?

Yes, if you draw a simple construction to find out the angle of depression and angle of elevation, you would see that in each case, both the angles would be equal. Additionally, both the angles in question would always be alternate interior angles. Since, the ground on which the observer and the object are situated are parallel to each other. This means we can use the principles of parallel lines and alternate interior angles in this case. By the rules of congruency of alternate interior angles, both of them would always be equal. Look at the diagram below to understand the relationship between the angle of depression and angle of elevation.

2. What would happen if the depression angle is zero?

It is possible for the depression angle to be completely zero as well. The depression angle is formed between the horizontal plane and the observer's line of sight. If the object and the observer are standing on the same level ground, and the observer is looking straight at the object in front of him/her, then during this time, there will not be a slope which indicates an angle. Rather, the depression in such a case will be zero as the object lies on the same level as the observer.

Fun facts and real-world application of angle of depression:

If you are on a trip to the mountainside. You see a huge mountain which is approximately the size of Mount Everest. You can actually find out the distance between you and the mountain by applying the concept of angle of depression and angle of elevation. All you need is the angle of depression from the top of the mountain or angle of elevation from where you are standing. By quick referencing let’s say you found the angle to be 60 degrees.

So, by the definition of angle of depression and angle of elevation,

Tan(60) = height of mount everest/ the distance between you and mountain

So, The distance = height /Tan(60),

in this case height = 8,848m

So, the distance = 8848/Tan(60) = 5108m (as Tan(60) = √3).