Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles, especially right triangles. It has many applications in fields like astronomy, navigation, engineering, and more. Some key uses of trigonometry include measuring inaccessible heights and distances by using trigonometric functions and properties of triangles formed by angles of elevation or depression. For example, trigonometry can be used to calculate the height of a building or tree by measuring the angle of elevation from a known distance away. It also has applications in measuring distances in astronomy, designing curved architectural structures, and calculating road grades. The document provides examples of various real-world applications of trigonometric concepts.

1. Pholoane
Thabile
(2014)

3. Introduction of Trigonometry
Trigonometry is derived from Greek words
trigonon (three angles) and metron ( measure).
Trigonometry is the branch of mathematics
which deals with triangles, particularly triangles
in a plane where one angle of the triangle is 90
degrees
Triangles on a sphere are also studied, in
spherical trigonometry.
Trigonometry specifically deals with the
relationships between the sides and the angles
of triangles, that is, on the trigonometric
functions, and with calculations based on these
functions.

4. Trigonometry is a branch of Mathematics that deals with the distances or
heights of objects which can be found using some mathematical techniques.
The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three) ,
‘gon’ (meaning sides) and ‘metron’ (meaning measure).
Historically, it was developed for astronomy and geography, but scientists
have been using it for centuries for other purposes, too. Besides other fields
of mathematics, trigonometry is used in physics, engineering, and chemistry.
Within mathematics, trigonometry is used primarily in calculus (which is
perhaps its greatest application), linear algebra, and statistics. Since these
fields are used throughout the natural and social sciences, trigonometry is a
very useful subject to know.

5. History
•
•
•
•
•
The origins of trigonometry can be traced to the civilizations
of ancient Egypt, Mesopotamia and the Indus Valley, more
than 4000 years ago.
Some experts believe that trigonometry was originally
invented to calculate sundials, a traditional exercise in the
oldest books
The first recorded use of trigonometry came from the
Hellenistic mathematician Hipparchus circa 150 BC, who
compiled a trigonometric table using the sine for solving
triangles.
The Sulba Sutras written in India, between 800 BC and 500
BC, correctly compute the sine of π/4 (45°) as 1/√2 in a
procedure for circling the square (the opposite of squaring
the circle).
Many ancient mathematicians like Aryabhata,
Brahmagupta,Ibn Yunus and Al-Kashi made significant
contributions in this field(trigonometry).

6. C
PERPENDICULAR (P)
B
Sin /
Cosec
Cos /
Sec
Tan /
Cot
P
(pandit)
B
(badri)
P
(prasad)
H (har)
A
BASE (B)
B
(bole)
H (har)
This is
pretty
easy!
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7. Three Types Trigonometric Ratios
There are 3 kinds of trigonometric
ratios we will learn.
sine ratio
cosine ratio
tangent ratio

8. Right Triangle
A triangle in which one angle is
equal to 90 is called right
triangle.
The side opposite to the right
angle is known as hypotenuse.
AB is the hypotenuse
The other two sides are known
as legs.
AC and BC are the legs
Trigonometry deals with Right Triangles

9. Pythagoras Theorem
 In any right triangle, the area of the square whose side is the
hypotenuse is equal to the sum of areas of the squares whose sides
are the two legs.
 In the figure
AB2 = BC2 + AC2

10. Trigonometry
C
Hypotenuse
O
x0
Opposite
The ‘Hypotenuse’ is always
opposite the right angle
Adjacent
F
y0
C
Hypotenuse
O
Opposite
The ‘Opposite’ is always
opposite the angle under
investigation.
The ‘Adjacent’ is always
alongside the angle under
investigation.
Adjacent
F

11. sin
opposite side
hypotenuse
cos
adjacent side
hypotenuse
tan
opposite side
adjacent side
Make Sure
that the
triangle is
right-angled

12. Relation between different
Trigonometric Identities
• Sine
• Cosine
• Tangent
• Cosecant
• Secant
• Cotangent

13. Sin2 + Cos2 = 1
• 1 – Cos2 = Sin2
• 1 – Sin2 = Cos2
Tan2 + 1 = Sec2
• Sec2 - Tan2 = 1
• Sec2 - 1 = Tan2
Cot2 + 1 = Cosec2
• Cosec2 - Cot2 = 1
• Cosec2 - 1 = Cot2

14. A
0
30
45
60
90
Sin A
0
1
Cos A
1
0
Tan A
0
Cosec A
Not
Defined
Sec A
Not
Defined
Not
Defined
2
1
1
Cot A
1
2
1
Not
Defined
0
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15. How can trigonometry be used to solve realworld problems?

16. Applications of Trigonometry
•
This field of mathematics can be applied in astronomy,navigation,
music theory, acoustics, optics, analysis of financial markets,
electronics, probability theory, statistics, biology, medical imaging
(CAT scans and ultrasound), pharmacy, chemistry, number theory
(and hence cryptology), seismology, meteorology, oceanography,
many physical sciences, land surveying and geodesy, architecture,
phonetics, economics, electrical engineering, mechanical
engineering, civil engineering, computer graphics, cartography,
crystallography and game development.
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17. Applications
• Measuring inaccessible lengths
•
•
Height of a building (tree, tower, etc.)
Width of a river (canyon, etc.)

18. Application: Height
•
•
To establish the height of a
building, a person walks 120 ft
away from the building.
At that point an angle of
elevation of 32 is formed when
looking at the top of the building.
h=?

32
120 ft
H = 74.98 ft
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19. Application: Height

68
•
•
An observer on top of a hill
measures an angle of depression of
68 when looking at a truck parked
in the valley below.
h=?
If the truck is 55 ft from the base of
the hill, how high is the hill?
55 ft
H = 136.1 ft
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20. • Angle of Depression –
It 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 head to look
at the object.
HORIZONTAL LEVEL

A
ANGLE OF DEPRESSION
20

21. • Angle of Elevation –
It 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.
A

ANGLE OF ELEVATION
HORIZONTAL LEVEL
21

22. Angles of Elevation and Depression
Line of sight: The line from our eyes to the
object, we are viewing.
Angle of Elevation:The angle through which
our eyes move upwards to see an object
above us.
Angle of depression:The angle through
which our eyes move downwards to see an
object below us.
22

23. 23

24. ?
37
70 ft
D = 52.7 ft
24

25. • Road has a grade of 5.5%.
•
Convert this to an angle expressed in degrees.
5.5 ft
?
100 ft
A = 3.1
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26. Applications of Trigonometry in
Astronomy
•
•
•
•
Since ancient times trigonometry was used in astronomy.
The technique of triangulation is used to measure the distance to nearby stars.
In 240 B.C., a mathematician named Eratosthenes discovered the radius of the
Earth using trigonometry and geometry.
In 2001, a group of European astronomers did an experiment that started in 1997
about the distance of Venus from the Sun. Venus was about 105,000,000
kilometers away from the Sun .

27. •
•
Application of Trigonometry in
Architecture
Many modern buildings have beautifully curved surfaces.
Making these curves out of steel, stone, concrete or glass is
extremely difficult, if not impossible.
•
One way around to address this problem is to piece the
surface together out of many flat panels, each sitting at an
angle to the one next to it, so that all together they create
what looks like a curved surface.
•
The more regular these shapes, the easier the building
process.
•
Regular flat shapes like squares, pentagons and hexagons,
can be made out of triangles, and so trigonometry plays an
important role in architecture.
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28. Trigonometry begins in the right triangle, but it doesn’t have to be restricted to
triangles. The trigonometric functions carry the ideas of triangle trigonometry
into a broader world of real-valued functions and wave forms.
Trig functions are the relationships amongst various sides in right triangles.
The enormous number of applications of trigonometry include astronomy,
geography, optics, electronics, probability theory, statistics, biology, medical
imaging (CAT scans and ultrasound), pharmacy, seismology, land surveying,
architecture.
I get it!
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29. Reference list
Niona. N. (2011). Trigonometry. (online), available:http://www.slideshare.net/krunittayamath.06 december 2011.
accessed by thabile on 05 march 2014.
Pete013 .( 2013). Trigonometry.(online), available:
http://www.slideshare.net/pete013?utm_campaign=profiletracking&utm_medium=sssite&utm_source=ssslidevie
w . 04 november 2013. Accessed by Thabile on 05 March 2014
Watts. M. (2011). Real world application of trigonometry. (online),
available:http://www.slideshare.net/m42watts?utm_campaign=profiletracking&utm_medium=sssite&utm_source
=ssslideview. 23 june 2011. Accessed by Thabile on 05 march 2014.
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30. THANK YOU !
Name
: thabile
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