Angle of Depression with Examples and FAQs

What is an Angle of Depression?

The Angle of Depression is the angle between the line of sight of the object and the observer and the horizontal line of the level at which the observer is above the object.

From the diagram above we can see that an Observer is higher than the Object, and the line of sight is also depicted. 

Now, the angle θ formed by the line of sight and the line horizontal to the observer is the angle of depression.

From the diagram, we can see that if a perpendicular is dropped from the object point on the horizontal line a right-angled triangle is formed. Hence, using the trigonometric ratios and properties of the right-angled triangle we can easily find the angle of depression if the distance between the object and observer and the height is given.

The formula for Angle of Depression 

The angle of depression can be easily calculated if the length of any two sides of the above triangle is known to us, by using the inverse of trigonometric functions.

From the above diagram the length of the sides of the triangle are mentioned, from this, we can easily get the angle of depression(θ) with the help of the following inverse trigonometric ratios.

We know, 

\sin \theta=\frac{\text { perpendicular }}{\text { hypotenuse }}

\cos \theta=\frac{\text { base }}{\text { hypotenuse }}

\tan \theta=\frac{\text { perpendicular }}{\text { base }}

From the figure above:

\sin \theta=\frac{b}{a}

\cos \theta=\frac{c}{a}

Hence, the value of θ is:

\theta=\sin ^{-1}\left(\frac{b}{a}\right)

\theta=\cos ^{-1}\left(\frac{c}{a}\right)

\theta=\tan ^{-1}\left(\frac{b}{c}\right)

Thus, when the length of any two sides is known the value of θ can easily be calculated. 

How do you differentiate between the angle of elevation and depression?

If a person(observer) is standing at a distance and at a height above the surface of the object he is looking at, then an angle is formed by the line of sight and the line horizontal to the observer which is the angle of depression. Whereas, if a person(observer) is standing lower than the object and at a distance from the object angle of elevation is formed.

In the figure above the Observer is looking at two objects, that are:

• Object E, which is at a distance from the observer and also at a height above the observer, i.e., elevated. In this case, the angle formed between the line of sight for object E and the horizontal line is the angle of elevation(\theta_{1}).
  

• Object D, which is at a distance from the observer and the observer is at a height above the object. In this case, the angle formed between the line of sight for object D and the horizontal line is the angle of depression(\theta_{2}).

This explains how the angle of elevation and depression are different from each other.

Example

1. A man is standing at the top of a 60 m high building, from where he can see his friend coming. The distance of his friend from the building is 80 m. Find the value of depression angle θ from the figure below. 

Solution:

In the given figure the observer is at a height of 60 m and the distance between the building and his friend is 80 m.

From the figure, a perpendicular is dropped from the position of his friend to the horizontal line of the observer.

The length of this perpendicular is equal to the height of the building, i.e., 60 m.

The point at which the perpendicular is dropped on the horizontal line and the observer is at the same distance as his friend is from the building, i.e., 80 m.

Hence, from the trigonometric ratio we have:

\tan \theta=\frac{\text { perpendicular }}{\text { base }}=\frac{60}{80}=\frac{6}{8}=\frac{3}{4}

\theta=\tan ^{-1}\left(\frac{3}{4}\right)

Hence the depression angle\theta=\tan ^{-1}\left(\frac{3}{4}\right).