## How to solve problems with angle of depression and elevation?

In our daily life, whenever we look up at something in the sky or look down at some object on the earth, we make an angle of depression and elevation with our eyes. Therefore, we can easily solve our real life problems with the angle depression and elevation. Moreover, these angles also help to find an unknown factor in geometrical problems. For example, the height of the object or the distance between the observer and the object.

When we look up at the top of a building or at a bird flying in the sky, the angle of elevation is formed. Similarly, when we look down from the top of a building, the angle of depression is formed. The knowledge of basic trigonometry is required to solve problems related to these angles. See the following examples.

### Example#1

A man is standing 50m away from a building. The angle of elevation of 70 degrees is formed when he looks at the top of the building of height z. Find the height z of the building and the hypotenuse h.

Since the angle between man and the building is the angle of elevation so,

θ = tan^{-1} (vertical distance / horizontal distance)

Since the vertical distance is unknown so,

70 = tan^{-1} (z / 50)

Or,

tan tan 70 = (z / 50)

So,

z = 60m

Now to find hypotenuse h,

sin sin θ = (z / h)

sin sin 70 = (60 / h)

Simplifying to get,

h = 46.4m

### Example#2

From the top of a 72 m high vertical cliff, a boat has an angle of depression of 32°. How far is the boat from the base of the cliff? Give your answer to an appropriate degree of accuracy. State any assumptions you have made in your calculation.

In this example, the boat makes an angle of depression with the vertical cliff. The height of the cliff is known but the horizontal distance between the cliff and the boat is unknown. Therefore, using the formula of angle of depression.

θ = tan^{-1} (vertical distance / horizontal distance)

Substituting the values,

32 = tan^{-1} (72m / y)

tan tan 32 = (72m / y)

Simplifying to get,

y=115 m

So, the distance between the boat and the cliff is 115 m.

### Example#3

A radio station tower was built in two sections. From a point 87ft from the base of the tower, the angle of elevation of the top of the first section is 25 degrees, and the angle of3elevation of the top of the second section is 40 degrees. To the nearest foot, what is the height of the top section of the tower?

In this problem,

d = 87ft

x = ?

We have to find the height of top section, so the angle of top section is,

θ = 40^{o} - 25^{o} = 15^{o}

Now,

θ = tan^{-1} (x / d)

tan tan θ = (x / d)

tan tan 15 = (x / 87)

x = 23.3ft

So the height of the upper section of the tower is 23.3ft.

### Example#4

A man is standing 3 m from the building and looking at the top of it. Assume the man is 1.5 m tall up to his eyes. The height of the building from the eye level of the man to the top is 5.2 m find the angle of elevation.

In the given problem, the distance between the line of sight of the man and the building is

d = 3m

And the height of the building from the line of sight of the man is

h = 5.2m

Now, using the tangent function

θ = tan^{-1} (vertical distance / horizontal distance)

Substituting the values of height and distance

θ = tan^{-1} (5.2 / 3)

θ = 60^{o}

The angle of elevation is approximately 60 degrees.

## FAQ’s

### Which refers to the angle of depression?

The angle between the vertical distance and horizontally downward distance between the observer’s eye and the object is referred to as the angle of depression.

### How are angles of elevation and depression related?

The angle of depression formed when the observer looks downward and the angle of elevation formed when her looks upward. The angle of elevation and angle of depression are equal but opposite to each other. In other words, they are congruent.

### How do you find the angle of elevation and depression?

The angle of elevation and the angle of depression can be found by using trigonometric ratio inverse tangent.

### Is angle of elevation equal to angle of depression?

Yes, the angle of depression and the angle of elevation are equal and opposite to each other.