Math project some applications of trigonometry

SOME APPLICATION OF
TRIGNOMETRY
Adarsh
Pandey
X-B
BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B

 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

 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).

2a 2a
600
300
600
a
If θ is an angle
The 90-θ is it’s complimentary angle
1
1
1

1
1
Sine
45
2
45
2
45

 
Cos
Tan

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…
a
c
b
ө 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
Hypothenuse
Side Adjacent
Hypothenuse
Side Opposite
Side Adjacent
=
=
a
c
b
c
= a
b

60 45
12
h
d
h
45   1;.. d 
h - - - - - - - (1)
Substituting (1)..in (2)
1.732;
h 12


12  1.732  
0.732
16.39 is also equals to d
12
0.732
3 1.732 (2)
12
60
 
      


h
h h h
h
d
h
Tan
or
d
Tan

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
30 60
d
Tan
Tan
1
     
From d h
(1) 3
From
h
d
d
(2)

d h
h
3( 150)
 
Substituting the value of d
.. .. .. .. ..
h h
3( 3 150)
 
h h
3 150 3
 
h h
3 150 3
h
 
2 
150 3
h 75*1.732 129.9
m
(2)
150
60 3
(1)
3
30
 
   

 

Heights and Distances
?
45o
?
What you’re
going to do
next?

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.

Trigonometry
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.
Similar
Triangles
h
Similar
Triangles
6 ft
720 ft
9 ft
Sun’s rays casting shadows mid-afternoon Sun’s rays casting shadows late afternoon
Thales may not have used similar triangles directly to solve the problem but
he knew h
that 6

the ratio of the vertical to horizontal sides of each triangle was
constant 720 and 9
unchanging for different heights of the sun. Can you use the
measurements shown above to find the height of Cheops?
480 ft
(Egyptian 4 feet of course)
6 720
9
80 ft
x
h  

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
Eratosthenes
275 – 194 BC

Early Applications of Trigonometry
h
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
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.
Tower
Horizontal level BY-ADARSH PANDEY X-B BY- ADARSH PANDEY X-B

45o
Mountain
Angle of
depression
A
B
Object
C
Horizontal level

Method of finding the heights or the distances
45o
Angle of elevation
A
C
B
Tower
Horizontal level
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 ΔABC, 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

Math project some applications of trigonometry

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