Height and Distance

Height and Distance is a part of trigonometry which is studied by scholars from all over the world. It is used to calculate distances from the Earth to the planets and stars since years. One of the best uses of height and distance was in finding the height of the highest mountain in the world that is Mount Everest named after Sir George Everest. It was the Great Trigonometric Survey conducted by British India in 1852 where the height of Mt. Everest was found out from a distance of 160 km by using the instrument the Giant Theodolites. In recent days, we use the concept of height and distance in the detection of crime, marine biology, navigation, military, calculus, astronomy, aviation, etc. It is also helpful to measure the height of the mountain, pillar, building, monument, etc.

Height and Distance Concept

Height and distance of anything can be found by using Trigonometric formulas. You can find the height of Qutub Minar if you know the distance of your position from Qutub Minar and the angle at which you are able to see the top of the Minar. To find the height we use trigonometry because the surface of the ground, the height of Minar and the line of elevation all together form a right angle triangle with 90 degrees between the Minar and the ground. 


Distance is usually the ‘base’ of the right-angled triangle formed by the height of Minar and line of sight. The length of the horizontal level also tells us the distance, as it also forms the base of the triangle which is at a certain height but parallel to the ground. Line of sight forms the ‘hypotenuse’ of the right-angled triangle. The line perpendicular to the ground forms the ‘height’ of the triangle.


To calculate the angle of elevation or depression we can use the following formula: 

\[\sin \theta  = \frac{{Height}}{{Hypotenuse}}.\cos \theta  = \frac{{Base}}{{Hypotenuse}},\tan \theta = \frac{{Height}}{{Base}}\]


Here, θ is either the angle of elevation or depression.

Terms Related to Height and Distance

Line of Sight: It is the straight line that is drawn from the eye of an observer to the point of an object which is to be viewed.

Horizontal Level: It is the horizontal line drawn from the eye of the viewer.

The angle of elevation: It is the angle formed between the line of sight and horizontal level if the object is above the horizontal level.

The Angle of Depression: It is the angle formed between the line of sight and the horizontal level if the object is below the horizontal level.


Pythagorean Theorem 

Since height and distance involve a right-angled triangle so Pythagoras theorem can be used to find the length of the sides. Pythagoras theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the square of its base and height. 


\[{\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Perpendicular} \right)^2}\]

If the length of the base, perpendicular and hypotenuse of a right-angle triangle is a, b and c respectively. 

Then,  a2+ b2= c2.


Thus, if the length of any two sides is known then the length of the third side can be found by using the Pythagoras theorem which is also called the Pythagorean triple. 

Height and Distance Solved Examples:

  1. An observer 1.5m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45 degrees. What is the height of the chimney?

Solution:

Here, AB is the chimney, CD the observer and angle ADE is the angle of elevation. In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney.

We have

AB = AE + BE = AE + 1.5

and DE = CB = 28.5m


To determine AE, we choose a trigonometry ratio, which involves both AE and DE. Let us choose the tangent of the angle of elevation.

Now,   tan 45o = AE/DE

                    1 = AE/28.5

                   AE = 28.5 

So the height of the chimney (AB) = (28.5 + 1.5)m = 30m.


  1. From a point P on the ground, the angle of elevation of the top of a 10m tall building is 30degree. A flag is hoisted at the top of the building and the angle of elevation of the top of flagstaff from P is 45 degrees. Find the length of the flagstaff and the distance of the building from point P.

Solution:

AB denotes the height of the building. BD the flag and P the given point. Note that there are two right triangles PAB and PAD. We are required to find the length of the flagstaff, i.e., DB and the distance of the building from the point, i.e, PA.


Since we know the height of the building AB, we will first consider the right PAB.


We have tan30o = \[\frac{AB}{AP}\]


i.e,         \[\frac{1}{\sqrt{3}}\] = \frac{10}{AP}\]

Therefore, AP = 10 \[\sqrt{3}\].


i.e, the distance of the building from P is 10 root 3 m = 17.32m.


Next, let us suppose DB = xm. Then AD = (10+x)m.


Now, in right triangle PAD, 

tan 45 degree = \[\frac{{AD}}{{AP}} = \frac{{\left( {10 + x} \right)}}{{\left( {10{\text{ }}root{\text{ }}3} \right)}}\]


Therefore, 1= \[\frac{(10+x)}{10\sqrt{3}}\]


i.e,  x= 10(\[\sqrt{3}\] - 1) = 7.32.

So, the length of the flagstaff is 7.32m.


Summary of the Types of Problems:


Type

Given Information

Find

Type 1

The distance of the object and angle of elevation

Height of the object.

Type 2

The distance of the object and angle of elevation

The hypotenuse of the triangle made with distance (base) and height

Type 3

The angle of elevation and height of the objects

Distance between two objects

Type 4

The distance of the object and angle of depression

Height of the object.

Type 5

The angle of elevation and angle of depression from a certain height.

Height of the object.

Type 6

The angle of depression and height of the observer from the ground.

Distance between two objects

Type 7

The angle of depression and height of the observer from the ground.

Time taken (speed)

Practical Application of Height and Distance:

The concept of Height and Distance are used in various fields like:

To Find the Slope and Height During Construction:

We can find the height of a building or monument if the angle of elevation and the distance of the building is given. Similarly, the determination of angle of elevation, angle of depression and distance of the object can be found out if the other pieces of information are given.


The Use of Trigonometry to Measure the Height of a Mountain or a Building:

Basically, the height of the mountain or building can be easily measured using the trigonometric ratios. Consider the base of the mountain or building as adjacent and the height as opposite, then the third-longest part of the triangle will be hypotenuse as shown above in the figure. Triangle should be a right-angled triangle. Hence, this is the real-life application of trigonometry.                                        


To Calculate the Angle of Elevation or Depression Required to Reach the Target:


Incase of aviation, wind plays a most important role. In aviation, the angle of depression and the angle of elevation is used depending on the case. The angle of depression is when the line of sight to the object is below the horizontal line and the angle of elevation is when the line of sight to the object is above the horizontal line.


The wind of flight direction is considered as the two perpendicular sides of a right-angled triangle in which the speed of the flight and the speed of wind are measured in their direction. 

Explanation: From the above figure, the distance between the plane and the final point is one of the perpendicular distances. The distance of the plane from the ground is another perpendicular distance. Slant distance is the hypotenuse.

For this, trigonometric ratios are used for finding the distance or angle of aviation.

To Calculate the Height of the Criminal or Angle of the Gun:

In criminology, trigonometry can be used to find the shots of bullet or how tall the shooter was. This helps the police in the investigation. For this also trigonometric ratios are used i.e., sine, cosine, tangent. This is also the real-life application of trigonometry.

From the figure, it can be clearly seen that the triangulation is done through which the distance and angle can be found.


To Find the Distance or the Angle of Movement of Marine Organisms:

Marine Biology is the study of marine life. In marine biology, the biologists study marine life like plants, animals and other organisms that live there.

In marine biology, for predicting how far and at what angle the organism or plant is from the observer, trigonometry concepts are used.

  • Here, the water level is the base of the right-angled triangle and 

  • the depth of the object is the height 

  • slant distance is the line of sight 

  • Angle could be an angle of depression or angle of elevation, it depends on the observer who is observing.                  

Hence, trigonometry ratio and formula are used for finding the distance, angle.

Hence the exact location of the underwater living or non- living things can be measured easily with the help of trigonometry ratios sine, cosine and tangent. Hence, this is again a real life application of trigonometry.

To Control the Movement of Vehicles:

In navigation, controlling and monitoring of moving vehicles are studied. In this, the movement of the plane, ships, submarines, etc is studied. Navigation is of many types- marine navigation, aeronautical navigation, space navigation, and land navigation. Navigation is the study of moving objects and the angle or distance is measured using the concept of trigonometry at that point where we have to measure.

To measure the exact location of the moving body, compass, pinpoint poles, trigonometric ratios and formula are used.


To Find the Position and Angle of Satellite:

In satellite, trigonometry is used. Where the satellite is and at what angle this all can be easily measured using the trigonometry. Exact location or position of the satellite can be easily measured using the trigonometric ratios.