Introduction Of Trigonometry of Class 10

Power Point Presentation on
Introduction of Trigonometry
By:-
Ram Chandra Gupta

Level:- 10th
Subject:- Math’s
Chapter:- Introduction of Trigonometry

Introductory Activity
Let’s take some examples from our surroundings
where right triangle can be imagined to form:-
If a student is looking at the top of the tower , a
right triangle can be imagined to be made

Half Slice of bread can be imagined to be made a
right triangle

 In all the situation distances or heights can
be found by using some mathematical
techniques which comes under a branch of
mathematics called trigonometry.
 Trigonometry is derived from Greek words
Tri + gon + metron
(three) (sides) (measure)
In fact Trigonometry is the study of relation
between the sides and angles of a triangle.

Historical Background
 The history of trigonometry dates back to the early age
of Egypt and Babylon.
Angles were
then measured in degree.
It was then advanced by
the Greek astronomer
Hipperchus in the second
century of B.C who compiled
a trigonometric table that
measured the length of chord
subtending a various angles in a circle of a fixed
radius r.

Hipparchus is considered the greatest
astronomical observer, and by
some the greatest astronomer
of antiquity. He was the first
Greek to develop quantitative
and accurate models for the
motion of the Sun and Moon.
With his solar and lunar
theories and his numerical
trigonometry, he was probably
the first to
develop a reliable method to predict solar
eclipses.

Key Concepts
 Trigonometric ratios
 Trigonometric ratios at specific angles
(0˚, 30˚, 45˚, 60˚, 90˚)
 Trigonometric ratios at complementary angles
 Trigonometric Identities
 Angle of Elevation
 Angle of Depression

Methodology
 Previous knowledge testing
 Demonstration Method
 Explanation by giving examples
 Relating to daily life
 Concrete to abstract thinking
 Simple to complex
 Learning by doing

B A
C
TRIGONOMETRIC RATIOS
Sin /
Cosec

P
(pandit
)
H
(har)
Cos /
Sec 
B
(badri)
H
(har)
Tan /
Cot 
P
(prasad
)
B
(bole)
This
is
pretty
easy!
BASE (B)
HYPOTENUSE (H)
PERPENDICULAR
(P)


TRIGONOMETRIC VALUES OF SOME COMMON ANGLES
 A 0  30  45  60  90 
Sin A 0 1
Cos A 1 0
Tan A 0 1 Not
Defined
Cosec A Not
Defined
2 1
Sec A 1 2 Not
Defined
Cot A Not
Defined
1 0

Trigonometric Ratio of complementary
Angles
sin( 90° - ) = cos
cos( 90°- ) = sin
tan( 90° - ) = cot
cot( 90° - ) = tan
sec( 90° - ) = cosec
cosec( 90° - ) = sec

13
TRIGONOMETRIC IDENTITIES
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


Vocabulary / Terminology Used
 Right Triangle
 Base, perpendicular and hypotenuse
 Sine ,Cosine, Tangent, Cotangent, Secant and
Cosecant
 Elevation
 Depression

Life Skills Integrated
 Identify and problem solving Attitudes
 It enhance Scientific Skill which improves critical
thinking.

WHAT CAN YOU DO WITH TRIGONOMETRY?
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.

17
Applications
 Measuring inaccessible lengths
 Height of a building (tree, tower, etc.)
 Width of a river (canyon, etc.)
 Angle of elevation and depression

18
 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
HORIZONTAL LEVEL
LINE OF SIGHT
ANGLE OF ELEVATION

19
 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.
 A
L
I
N
E
O
F
S
I
G
H
T
HORIZONTAL LEVEL
ANGLE OF DEPRESSION

20
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 30 is formed
when looking at the top of
the building.

30

120 ft
h = ?
H = 69.28 ft

21
Application: Height
 An observer on top of a hill
measures an angle of
depression of 60 when
looking at a truck parked in the
valley below.
 If the truck is 55 ft from the
base of the hill, how high is the
hill?

60

h = ?
55 ft
H = 95.26 ft

23
Application : Surveying
?
70 ft
30

D = 40.41 ft

24
h = ?
HORIZONTAL LEVEL
LIN
E
OF
SIGH
T


CLINOMETER
It is an instrument which is used to measure
the height of distant objects using
trigonometric concepts.
Here, the height of the tree using T. concepts,
h = tan  *(x)
‘x’
units

Triangle Jokes
Trigonometry jokes are a sine of the times.
Fake tan: The major threat to trigonometry
Q. What does trigonometry have in common with a
beach?
A: Tan Gents
Trigonometry for farmers: swine and coswine.
When were trigonometry tables used? “B. C.”, Before
Calculators.

26
CONCLUSION
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!

The Pyramids of
Giza
Primitive forms of
trigonometry were used in
the construction of these
wonders of the world.

Architecture
In architecture, trigonometry plays a massive role in the
compilation of building plans.
For example, architects would have to calculate exact angles of
intersection for components of their structure to ensure
stability and safety.
Some instances of trigonometric use in architecture include
arches, domes, support beams, and suspension bridges.
Architecture remains one of the most important sectors of our
society as they plan the design of buildings and ensure that they
are able to withstand pressures from inside.

How do I get involved in Architecture?
Classes to TAke Salary and Benefits
 Physics
 Geometry
 Trigonometry
 Pre-Calculus and Calculus
 Engineering
 3-D Design
 Drawing
 * Art classes will assist you
in being able to
conceptualize objects!
 Most architects start out with a salary
of
Rs 371268 + per year , and through
experience, may earn up to Rs 500000
or above
 Becoming an architect will open you to
many more careers, including interior
design and building design!

Jantar Mantar
observatory
For millenia, trigonometry
has played a major role in
calculating distances
between stellar objects
and their paths.

Astronomy
Astronomy has been studied for millennia by civilizations in all
regions of the world.
In our modern age, being able to apply Astronomy helps us to
calculate distances between stars and learn more about the
universe.
Astronomers use the method of parallax, or the movement of
the star against the background as we orbit the sun, to discover
new information about galaxies.
Menelaus’ Theorem helps astronomers gather information by
providing a backdrop in spherical triangle calculation.

How do I get involved in Astronomy?
Classes to take Salary and Benefits
 Physics
 Electronics
 Advanced Math
 Geometry
 Precalculus and Calculus
 Astrophysics
 The median salary for Astronomers is
salary packets ranging from Rs. 8 lacs
per annum to around Rs. 10 lacs per
annum for those in senior positions.
 As an aside to being an astronomer,
one can also acquire a teaching
position at a research university!

Grand Canyon
Skywalk
Geologists had to measure
the amount of pressure
that surrounding rocks
could withstand before
constructing the skywalk.

Geology
Trigonometry is used in geology to estimate the
true dip of bedding angles. Calculating the true dip
allows geologists to determine the slope stability.
Although not often regarded as an integral
profession, geologists contribute to the safety of
many building foundations.
Any adverse bedding conditions can result in slope
failure and the entire collapse of a structure.

How do I get involved in Geology?
Classes to take Salary and Benefits
 Physics
 Chemistry
 Pre-calculus and Calculus
 Geometry
 Geochemistry
 Seismology
 Median wages for Geologists are
around Rs 500000+ a year. However, if
involved in oil extraction, earnings
could increase to over Rs 700000+ a
year.
 Geologists can be very flexible in what
they decide to do. There are a
multitude of job options ranging from
agriculture to tourism that require the
work of a geologist.

Reference offline/online:-
NCERT text book of mathematics
www.britanica.com
www.mathsfun.com
http://math.lotsoflessons.com

Home Assignment
1. Related Exercises from text book and exemplar
book of mathematics.
2. Activity:- to prepare a chart of T-ratios, T-ratios
at specific angles, T-identities.
3. Project:- make clinometer ( in group)

Remedial Plan
 Make awareness regarding the ratio by giving
examples from daily life.
 Group leaders will be assigned to make
correction by guiding them.
 Similar problem assigned to practices related to
each concept will be given.

Introduction Of Trigonometry of Class 10

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