math project some applications of trigonometry
DESCRIPTION
some applications of trigonometryTRANSCRIPT
Adarsh
PandeyX-B
SOME APPLICATION OF TRIGNOMETRY
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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
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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.
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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).
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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
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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
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4560
12
h
d
d toequals also is 39.16732.0
12
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;732.112h
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)2(732.1312
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h
hhhh
d
hTan
ord
hTan
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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
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hdFrom
d
hTan
d
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9.129732.1*75
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)150(3
)2(
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)2(150
360
)1(3
130
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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
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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.
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Early Beginning uses of trigonometry for determining heights and distances
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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
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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.
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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.
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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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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
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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.
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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 ?
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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.
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Thank you
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