## Angle of Depression in a Triangle

Angle of depression plays an important role in solving triangular and right triangle problems. We can calculate any unknown angle or a side of a triangle by using the angle of depression. In this section, we will see how the angle of depression helps to solve any triangular problem.

## How to find the angle of depression in a triangle?

The angle of depression is the angle between the line of sight of the observer and the horizontal distance. In the case of a triangle, these are two sides i.e. perpendicular and bae. So, the angle of depression can be calculated by using the trigonometric ratios sine, cosine and tangent. See the following examples.

### Example#1

A tree is 18m long at point A from building B. The distance between them is about 20m. If a man on the tree is looking down making angle θ Then a triangle ABC is formed. Find the angle between the man and the building.

Given that in triangle ∆ABC,

AC = h = 18m

AB = d = 20m

We can find the unknown angle by using angle of depression between the man and building B. So,

θ = tan-1 (h / d)

θ = tan-1 (18 / 20)

θ = tan-1 (0.9)

θ = 42

The above example is of a right triangle because the angle between the tree and point B is 90 degrees.

### Example#2

An observer at height 19m makes the angle of depression of 73o with the boat at an unknown distance. Find the distance x. Here we have,

θ = 73

h = 19m

And

x = ?

Using the formula of angle of depression,

$$θ\;=\;tan^{-1}\;(\frac{h}{x})$$

We can write it as

x = h / tantan θ

Substituting the value of h and

x = 19 / tantan 73

x = 20.65m

### Example#3

From a hot air balloon that is flying at a height of 100ft, the angle of depression of a person on the ground is 30 degrees. Find the length of the rope that is tied from the person on the ground to the balloon.

To find the value of x, we have to use the formula of sine as

sinsin θ = 100ft / x

x = 100ft / sinsinθ

x = 100ft / sinsin 30

x = 100ft / (1/2)

x = 200ft

## FAQ’s

### Can an angle of depression be more than 90 degrees?

No, the angle of depression cannot be more than 90 degrees. If it is more than 90 degrees, it will not be the angle of depression.

### What is the angle of depression?

The angle of depression is the measure of angle between an observer and an object with the line of sight at a lower level to the observer.

### What is the maximum angle of depression?

The maximum angle of depression is 90 degrees.