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1
This project has been made with the aim of providing basic understanding on the
subject – MATHEMATICS , in order to cover a part of NCERT syllabus as prescribed
by CBSE. This presentation is titled as - TRIGONOMETRY. Sincere efforts have been
made to make the presentation a unique experience to the viewer. Stress has been
laid on the appearance, neatness and quality of the presentation. No effort has
been spared to make the reading and understanding of the presentation complete
and interesting. I have tried to do my best and hope that the project of mine would
be appreciated by all.
One fine day I was relaxing on the beach where I saw a light house. I started
wondering, what the height of the light house would be? I asked my mathematics
teacher for suggestions and she taught me about the concept of trigonometry in
finding the heights and distances of distant objects without actually measuring it.
This provoked me to gather more information on this topic and to make a beautiful
presentation on it.
Supervised by – *******
Created by – ********
2
One cannot succeed alone no matter how great one’s abilities are, without the
cooperation of others. This project, too, is a result of efforts of many. I would
like to thank all those who helped me in making this project a success.
I would like to express my deep sense of gratitude to my Maths Teacher, Mrs.
******* who was taking keen interest in our lab activities and discussed various
methods which could be employed towards this effect, and I really appreciate
and acknowledge her pain taking efforts in this endeavour.
Supervised by – *******
Created by – *******

3
4
Introduction to
Trigonometry
Right Triangle Trigonometry

5
Q: What is trigonometry?
A: Trigonometry is the study of how the sides and angles
of a triangle are related to each other.
Q: WHAT? That's all?
A: Yes, that's all. It's all about triangles, and you can't get
much simpler than that.
Q: You mean trigonometry isn't some big, ugly monster
that makes students turn green, scream, and die?
6

A: No. It's just triangles.
Some historians say that trigonometry was invented by
Hipparchus, a Greek mathematician. He also introduced the
division of a circle into 360 degrees into Greece.
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.

7
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.
8
C

PERPENDICULAR (P)

B

Sin /
Cosec

Cos /
Sec

Tan /
Cot

P
(pandit)

B
(badri)

P
(prasad)

H
(har)

H
(har)

A

BASE (B)

B
(bole)

This is
pretty
easy!

9
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

10
Sin2

+ Cos2

=1

• 1 – Cos2 = Sin2
• 1 – Sin2 = Cos2

Tan2

+ 1 = Sec2

• Sec2
• Sec2

- Tan2 = 1
- 1 = Tan2

Cot2

+ 1 = Cosec2

• Cosec2
• Cosec2

- Cot2 = 1
- 1 = Cot2
11


Measuring inaccessible lengths
◦ Height of a building (tree, tower, etc.)
◦ Width of a river (canyon, etc.)

12


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

13


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

14
Example 1 of 4

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

15
Example 2 of 4

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.
 If the truck is 55 ft from the base
of the hill, how high is the hill?

h=?

55 ft
H = 136.1 ft

16
17
Example 3 of 4

?

37

70 ft

D = 52.7 ft
18
Example 4 of 4



Road has a grade of 5.5%.
◦ Convert this to an angle expressed in
degrees.

5.5 ft
?

100 ft
A = 3.1
19
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)

h=?

HORIZONTAL LEVEL

‘x’ units
20
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!

21
THANK YOU !
Name

: ********

22

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Trigonometry

  • 1. 1
  • 2. This project has been made with the aim of providing basic understanding on the subject – MATHEMATICS , in order to cover a part of NCERT syllabus as prescribed by CBSE. This presentation is titled as - TRIGONOMETRY. Sincere efforts have been made to make the presentation a unique experience to the viewer. Stress has been laid on the appearance, neatness and quality of the presentation. No effort has been spared to make the reading and understanding of the presentation complete and interesting. I have tried to do my best and hope that the project of mine would be appreciated by all. One fine day I was relaxing on the beach where I saw a light house. I started wondering, what the height of the light house would be? I asked my mathematics teacher for suggestions and she taught me about the concept of trigonometry in finding the heights and distances of distant objects without actually measuring it. This provoked me to gather more information on this topic and to make a beautiful presentation on it. Supervised by – ******* Created by – ******** 2
  • 3. One cannot succeed alone no matter how great one’s abilities are, without the cooperation of others. This project, too, is a result of efforts of many. I would like to thank all those who helped me in making this project a success. I would like to express my deep sense of gratitude to my Maths Teacher, Mrs. ******* who was taking keen interest in our lab activities and discussed various methods which could be employed towards this effect, and I really appreciate and acknowledge her pain taking efforts in this endeavour. Supervised by – ******* Created by – ******* 3
  • 4. 4
  • 6. Q: What is trigonometry? A: Trigonometry is the study of how the sides and angles of a triangle are related to each other. Q: WHAT? That's all? A: Yes, that's all. It's all about triangles, and you can't get much simpler than that. Q: You mean trigonometry isn't some big, ugly monster that makes students turn green, scream, and die? 6 A: No. It's just triangles.
  • 7. Some historians say that trigonometry was invented by Hipparchus, a Greek mathematician. He also introduced the division of a circle into 360 degrees into Greece. 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. 7
  • 8. 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. 8
  • 9. C PERPENDICULAR (P) B Sin / Cosec Cos / Sec Tan / Cot P (pandit) B (badri) P (prasad) H (har) H (har) A BASE (B) B (bole) This is pretty easy! 9
  • 10. 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 10
  • 11. Sin2 + Cos2 =1 • 1 – Cos2 = Sin2 • 1 – Sin2 = Cos2 Tan2 + 1 = Sec2 • Sec2 • Sec2 - Tan2 = 1 - 1 = Tan2 Cot2 + 1 = Cosec2 • Cosec2 • Cosec2 - Cot2 = 1 - 1 = Cot2 11
  • 12.  Measuring inaccessible lengths ◦ Height of a building (tree, tower, etc.) ◦ Width of a river (canyon, etc.) 12
  • 13.  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 13
  • 14.  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 14
  • 15. Example 1 of 4 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 15
  • 16. Example 2 of 4 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.  If the truck is 55 ft from the base of the hill, how high is the hill? h=? 55 ft H = 136.1 ft 16
  • 17. 17
  • 18. Example 3 of 4 ? 37 70 ft D = 52.7 ft 18
  • 19. Example 4 of 4  Road has a grade of 5.5%. ◦ Convert this to an angle expressed in degrees. 5.5 ft ? 100 ft A = 3.1 19
  • 20. 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) h=? HORIZONTAL LEVEL ‘x’ units 20
  • 21. 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! 21
  • 22. THANK YOU ! Name : ******** 22