Heights and Distance

The concept of “Height and Distance” has been employed to measure distances from Earth to the stars and planets for many years. It is fascinating how one can find the height of an object or the distance between two bodies just by using some simple trigonometric formulas. The real-life applications of heights and distances include measuring the heights of a building and the depth of a water body, and it is employed in many sections of biology and physics. Heights and distances can be measured using simple trigonometric ratios, Pythagoras theorem, and other simple formulas.   

Introduction to heights and distances

Trigonometric formulas determine the distance and height of any object. It is possible to determine how tall Qutub Minar is if you are aware of the distance between yourself and the monument and the angle from which you see the monument. There are some important definitions related to “heights and distances.” Some of them are;

1. Line of sight- It is an imaginary line from the eye of the observer to the object. In simpler words, it tells us about the object seen by the observer. It also tells us about the angle of inclination made with the horizontal.

2. Angle of Elevation- The angle of elevation is the angle between a horizontal plane and an oblique line from the eyes of an observer of an object kept above his eyes. 

3. Angle of depression- The angle of depression is the angle between horizontal level and line of sight when the object is below the level of the eye of an observer. 

Formulas of Heights and Distances

The formulas used in heights and distances are nothing new. We use the trigonometric ratios, Pythagoras theorem, and other basic formulas to solve questions in heights and distances. 

For instance, if we take a right-angled triangle ABC, then. 

Sin θ= perpendicular/ hypotenuse

Cos θ= base/ hypotenuse 

Tan θ= perpendicular/base

With the help of an example, let us see how we can use these ratios to solve problems related to heights and distances. 

1. A man standing 30m away from a building is observing it with an angle of 30°. Then how can we find the height of the building? 

To solve this, we have to consider a right angled triangle ABC. ∠ CAB= 30°.Then. 

Tan 30°= perpendicular/ base 

1/ √3= perpendicular/base= p/30 

1/ √3= p/30

p= 1/√3 x 30

p= 10√3. 

Perpendicular= 10√3. Hence the height of the building is 10√3. 

Some illustrative examples

If there is a pole 6m tall and it casts a shadow of 2√3m on the ground, how can you find the elevation of the sun?

Answer:

To find the elevation of the sun, we have to find the angle by using the below equation 

tanӨ= height of pole/ shadow= 6/2√3= 3/√3= √3. 

tanӨ= √3.

Ө= 60°. 

Ө is the elevation of the sun. Hence the elevation of the sun is 60. 

A man observes a building from two different distances and angles of inclination. When he is standing at a distance d from the building, the inclination angle is 60°, and when he moves 30 m away from that point, the inclination angle becomes 30°. Find the height of the building. 

Answer:

Let the height of the building be h. In the first case, when the man is standing at a distance d from the building, them 

Tan 60°= h/d 

√3= h/d 

d= h/ √3

Now the man moves 30m away from initial point.Then

Tan 300= h/d+30

1/√3= h/ d+30

√3h= d+30. 

Putting the value of d in the above equation, we can find that 

h= 15√3 m. 

Hence the height of the building= 15√3m.

You may be provided with one or two of the following details:

• The distance between the object from the observer

• The highest point of the particular object

• The angle from which an observer can see the highest point of the object (angle of elevation)

• The angle from which an observer looks at something when they are at the top of a building/tower (angle of depression)

Some important points

1. The angle of elevation increases if the observer moves towards the object. 

2. If the angle of elevation is mentioned, it indicates that the object is above the observer’s line of sight.

3. If the angle of depression is mentioned, it indicates that the object is below the observer’s line of sight. 

4. The angle of depression decreases if the observer moves away from the object. 

Important uses of heights and distances in daily life

Some of the uses of heights and distances in real-life situations are: 

1. It can be used to determine the height of buildings, trees, lighthouses, etc 

2. It is utilised in sub-marine and marine systems to measure water depth.

3. It is utilised in navigation systems to aid the navigation of vehicles, boats, and other vessels.

4. It is employed in satellite systems to determine the angle and position of space-crafts and satellites.

5. It is used in different fields such as biology, chemistry, and physics.

Conclusion

Measuring the heights and distances is perhaps one of the most useful applications of trigonometry. Astronomers use it to calculate the distance between two celestial bodies and navigators to measure distances in the ocean. The angle of elevation and the angle of depression are the two most important terms used in ‘heights and distances.’ The angle of elevation is the angle between a horizontal plane and an oblique line from the eyes of an observer of an object kept above his eyes. In contrast, the angle of depression is the angle between horizontal level and line of sight when the object is below the level of the eye of an observer.