PRANAY RAJPUT

X - B

HEIGHTS AND DISTANCES

How the following diagram allows us to determine the height of the Eiffel Tower without actually having to climb it or the distance between the person and Eiffel Tower without actually walking .

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?What you’re going to do

next?

Heights and Distances

In this situation , the distance or the heights can be founded by using mathematical techniques, which comes under a branch of ‘trigonometry’. The word ‘ trigonometry’ is derived from the Greek word ‘tri’ meaning three , ‘gon’ meaning sides and ‘metron’ meaning measures. Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.

Early Beginning uses of trigonometry for determining heights and distances

Trigonometry (Three-angle-measure)

THE GREAT PYRAMID (CHEOPS) AT GIZA, NEAR CAIRO, ONE OF THE 7 WONDERS OF THE ANCIENT WORD. (THE ONLY ONE STILL SURVIVING).THIS IS THE ONE OF THE EARLIEST USE OF TRIGONOMETRY. PEOPLE USE TRIGONOMETRY FOR DETERMINING HEIGHT OF THIS PYRAMID.

Sun’s rays casting shadows mid-afternoon

Sun’s rays casting shadows late afternoon

An early application of trigonometry was made by Thales on a visit to Egypt. He was surprised that no one could tell him the height of the 2000 year old Cheops pyramid. He used his knowledge of the relationship between the heights of objects and the length of their shadows to calculate the height for them. (This will later become the Tangent ratio.) Can you see what this relationship is, based on the drawings below?

Thales of Miletus 640 – 546 B.C. The

first Greek Mathematician. He predicted the Solar Eclipse of 585 BC.

Trigonometry

Similar Triangles

Similar Triangles

Thales may not have used similar triangles directly to solve the problem but he knew that the ratio of the vertical to horizontal sides of each triangle was constant and unchanging for different heights of the sun. Can you use the measurements shown above to find the height of Cheops?

6 ft

9 ft

720 ft

h

6720 9h

480 ft

(Egyptian f eet of course)46 720

980 f t

xh

Later, during the Golden Age of Athens (5 BC.), the philosophers and mathematicians were not particularly interested in the practical side of mathematics so trigonometry was not further developed. It was another 250 years or so, when the centre of learning had switched to Alexandria (current day Egypt) that the ideas behind trigonometry were more fully explored. The astronomer and mathematician, Hipparchus was the first person to construct tables of trigonometric ratios. Amongst his many notable achievements was his determination of the distance to the moon with an error of only 5%. He used the diameter of the Earth (previously calculated by Eratosthenes) together with angular measurements that had been taken during the total solar eclipse of March 190 BC.

Hipparchus of Rhodes 190-120 BC

Eratosthenes275 – 194 BCThe library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts.

h

20o25o

x d

Early Applications of Trigonometry

Finding the height of a mountain/hill.

Finding the distance to the moon.

Constructing sundials to estimate the time from the sun’s shadow.

Historically trigonometry was developed for work in Astronomy and Geography. Today it is used extensively in mathematics and many other areas of the sciences.

• Surveying

• Navigation

• Physics

• Engineering

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Angle of elevation

Line of s

ight

A

C

B

In this figure, the line AC drawn from the eye of the student to the top of the tower is called the line of sight. The person is looking at the top of the tower. The angle BAC, so formed by line of sight with horizontal is called angle of elevation.

Tow

er

Horizontal level

Angles. of Elevation and Depression

45o Line of sight

Moun

tain

Angle of depression

A

B

C

Object

Horizontal level

In this figure, the person standing on the top of the mountain is looking down at a flower pot. In this case , the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression.

45oAngle of elevation

Line of s

ight

A

C

B

Tow

er

Horizontal level

Method of finding the heights or the distances

Let us refer to figure of tower again. If you want to find the height of the tower i.e. BC without actually measuring it, what information do you need ?

We would need to know the following: i. The distance AB which is the distance between

tower and the person .ii. The angle of elevation angle BAC .Assuming that the above two conditions are given then how can we determine the height of the height of the tower ? In ∆ABC, the side BC is the opposite side in relation to the known angle A. Now, which of the trigonometric ratios can we use ? Which one of them has the two values that we have and the one we need to determine ? Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. Therefore, tan A = BC/AB or cot A = AB/BC, which on solving would give us BC i.e., the height of the tower.

Some Applications of trigonometry based on finding

heights and distance

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28.5m

A

B

C

Here we have to find the height of the school.

Here BC = 28.5 m and AC i.e., the height of the school = tan 45 = AC/BC i.e., 1 = AC/28.5Therefore , AC = 28.5m So the height of the school is 28.5 m.

60o

3.7m

B

A C

Here we have to find the length of the ladder in the below figure and also how far is the foot of the ladder from the house ? (here take √3 = 1.73m)

Now, can you think trigonometric ratios should we consider ? It should be sin 60So, BC/AB = sin 60 or 3.7/AB = √3/2Therefore BC = 3.7 x 2/√3 Hence length of the ladder is 4.28m Now BC/AC = cot 60 = 1/√3i.e., AC = 3.7/√3 = 2.14m (approx) Therefore the foot of the ladder from the house is 2.14m.

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P

D

A

B

Here we need to find the height of the lighthouse above the mountain . Given that AB = 10 m. (here take √3 =1.732).

10 m

Since we know the height of the mountain is AB so we consider the right ∆PAB. We have tan 30 = AB/AP i.e., 1/√3 = 10/AP therefore AP = 10√3m so the distance of the building = 10√3m = 17.32m

Let us suppose DB = (10+x)m now in right ∆PAD tan 45 = AD/AP = 10+x/10√3 therefore 1 = 10+x/10√3 i.e., x = 10(√3-1) =7.32. So, the length of the flagstaff is 7.32m

Summary The line of sight is the line drawn from the eye

of the observer to the point in the object viewed by the observer.

The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object.

The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level , i.e., the case when we lower our the head to look at the object.

The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

Thank you