Math project some applications of trigonometry

Adarsh

PandeyX-B

SOME APPLICATION OF TRIGNOMETRY

BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B

INTRODUCTION Trigonometry is the branch of mathematics

that deals with triangles particularly right triangles. For one thing trigonometry works with all angles and not just triangles. They are behind how sound and light move and are also involved in our perceptions of beauty and other facets on how our mind works. So trigonometry turns out to be the fundamental to pretty much everything


BASIC FUNDAMENTALS Angle of Elevation: In the picture below,

an observer is standing at the top of a building is looking straight ahead (horizontal line). The observer must raise his eyes to see the airplane (slanting line). This is known as the angle of elevation.


Angle of Depression: The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel).


600

300

600

a

2a2a

If θ is an angleThe 90-θ is it’s complimentary angle

11

145

2

1 45

2

145

Tan

Cos

Sine


A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent.

Let us look at this triangle…

ac

bө A

B

C

Given the assigned letters to the sides and angles, we can determine the following

trigonometric functions.

The Cosecant is the inversion of the sine, the secant is the inversion of

the cosine, the cotangent is the inversion of the tangent.

With this, we can find the sine of the value of angle A by dividing side a by side c. In order to find the angle itself, we must take the sine of the angle and invert it (in other words, find the cosecant of the sine of the angle).

Sinθ=

Cos θ=

Tan θ=

Side Opposite

Side Adjacent

Side AdjacentSide Opposite

Hypothenuse

Hypothenuse

=

=

= a

bca

b

c


4560

12

h

d

d toequals also is 39.16732.0

12

732.0732.112

;732.112h

(2)(1)..in ngSubstituti

)2(732.1312

60

(1)-------hd ;..145

h

hhhh

d

hTan

ord

hTan


The angle of elevation of the top of a tower from a point At the foot of the tower is 300 . And after advancing 150mtrs Towards the foot of the tower, the angle of elevation becomes 600 .Find the height of the tower

150

h

d

30 60

mh

h

hh

hh

hh

dofvaluethengSubstituti

hd

From

hdFrom

d

hTan

d

hTan

9.129732.1*75

31502

31503

31503

)1503(3

..........

)150(3

)2(

3)1(

)2(150

360

)1(3

130


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 .

?45o

?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.BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B

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

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


45o

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.


Thank you


Date post:	19-Jun-2015
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Math project some applications of trigonometry

Education