Angle of Depression

What is Angle of Depression? 

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.

What is 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.

Define Angle of Depression:

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.

The Formula of The Angle of Depression: 

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 

Solved Examples

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.