Exploring Trigonometry: A Journey Through Mathematical Relationships

 
Module 1 of 3
 
 
Ratna Rathor
AECS-2
Mumbai
 
1
 
Trigonometric Ratios
 
2
 
The  word  'trigonometry'  is  derived from  the
Greek  words  'tri'  (meaning  three), 'gon'
(meaning  sides)  and 'metron' (meaning
measure).
 
In fact, trigonometry is the study of the
relationship between the sides and angles of a
triangle.
 
‘tri’
  
 
Three
‘gon’
  
 
Sides
Metron
 
 
Measure
 
3
 
OBJECTIVES
 
   
At the end of this  lesson, you will be able to-
State the trigonometric ratios of an acute angle in a
right-angled triangle.
Calculate the trigonometric ratios of an acute
angle using Pythagoras Theorem.
Establish some identities involving these ratios,
called Trigonometric Identities.
 
4
 
Trigonometry is said to be the most important
 mathematical relationship ever discovered.
 
Triangles are one of the most simple forms found in
Nature, but their mathematics has vital importance,
especially where precise distance measurements are
needed.
 
5
 
Applications in Real Life
 
In ancient timed, it was used for astronomy
   in finding the distance of stars.
Finding the radius of the earth
Finding the height of hills, buildings, trees etc.
Navigation – Airplane, Ships etc.
Defence
 
 
 
 
6
 
Historical
 
Background
 
The history of trigonometry dates back to
the early age of Egypt and Babylon. Angles
were then measured in degrees.
 
It was then advanced by  the Greek
astronomer Hipparchus in the second
century  B.C. He compiled  a trigonometric
table that  measured the length of a chord
subtending various angles in a circle of a
fixed  radius r.
 
 
He is known as the father of TRIGONOMETRY.
 
7
 
Hipparchus is considered by some as the
greatest astronomer. He was the first
Greek to develop quantitative  and
accurate models for the  motion of the
Sun and the 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.
 
8
 
Right Triangle
 
The hypotenuse of a right triangle
 is always the side opposite the
 right angle. It is the longest side
 in a right triangle.
 
The adjacent leg is the other side
that is adjacent to angle θ (theta).
It is also sometimes called as base.
 
The opposite side is the side that is
opposite to angle θ (theta). It is also
sometimes called  perpendicular
.
 
9
 
The sides are always defined with respect to acute
angle ‘A’ or angle ‘C’.
 
Right Triangle
 
10
 
To find the height of the clock tower and the tree
 
11
 
Sine Function/Ratio(Sin)
 
12
 
13
 
Tangent Function/Ratio (Tan)
 
14
 
The  other trigonometric ratios are cosec
Ɵ
, sec
Ɵ
 and cot
Ɵ
.
 
 The ratios 
 cosec
Ɵ
, sec
Ɵ
 and cot
Ɵ 
are  the reciprocals of the
ratios sin
Ɵ,
 cos
Ɵ
  and tan
Ɵ
   respectively.
 
Reciprocal Functions/Ratios
 
15
 
Secant 
(Sec) 
Function/Ratio
 
16
 
The cosecant or cosec(A), is the reciprocal of sin(A); i.e. the
ratio of the length of the hypotenuse to the length of the
opposite side.
 
 
  
Cosecant 
(Cosec)
Function/Ratio
 
17
 
 
 
 
 
The cotangent cot(A) is the reciprocal of tan(A); i.e. the ratio
of the length of the adjacent side to the length of the
opposite side.
 
 
Cotangent
(Cot) 
Function/ Ratio
 
18
 
Example
 
19
 
20
 
RECAPITULATION
 
Using the given figure, find all the trigonometric ratios for
angle A and B.
 
21
 
22
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Trigonometry, derived from Greek words, is the study of triangle sides and angles. This ancient discipline plays a crucial role in various real-life applications, such as astronomy, navigation, and defense. Delve into the historical background of trigonometry, from its early origins to the advancements by Greek astronomer Hipparchus. Discover the significance of trigonometric ratios, identities, and the practical importance of triangles in precise measurements. Unravel the mathematical beauty and practicality of trigonometry in our daily lives and scientific endeavors.


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  1. Module 1 of 3 Ratna Rathor AECS-2 Mumbai 1

  2. Trigonometric Ratios 2

  3. The word 'trigonometry' is derived from the Greek words 'tri' (meaning three), 'gon' (meaning sides) and 'metron' (meaning measure). In fact, trigonometry is the study of the relationship between the sides and angles of a triangle. tri gon Metron Three Sides Measure 3

  4. OBJECTIVES At the end of this lesson, you will be able to- State the trigonometric ratios of an acute angle in a right-angled triangle. Calculate the trigonometric ratios of an acute angle using Pythagoras Theorem. Establish some identities involving these ratios, called Trigonometric Identities. 4

  5. Trigonometry is said to be the most important mathematical relationship ever discovered. Triangles are one of the most simple forms found in Nature, but their mathematics has vital importance, especially where precise distance measurements are needed. 5

  6. Applications in Real Life In ancient timed, it was used for astronomy in finding the distance of stars. Finding the radius of the earth Finding the height of hills, buildings, trees etc. Navigation Airplane, Ships etc. Defence 6

  7. Historical Background The history of trigonometry dates back to the early age of Egypt and Babylon. Angles were then measured in degrees. It was then advanced by the Greek astronomer Hipparchus in the second century B.C. He compiled a trigonometric table that measured the length of a chord subtending various angles in a circle of a fixed radius r. He is known as the father of TRIGONOMETRY. 7

  8. Hipparchus is considered by some as the greatest astronomer. He was the first Greek to develop quantitative and accurate models for the motion of the Sun and the 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. 8

  9. Right Triangle The hypotenuseof a right triangle is always the side opposite the right angle. It is the longest side in a right triangle. The adjacent leg is the other side that is adjacent to angle (theta). It is also sometimes called as base. The opposite side is the side that is opposite to angle (theta). It is also sometimes called perpendicular. 9

  10. Right Triangle The sides are always defined with respect to acute angle A or angle C . 10

  11. To find the height of the clock tower and the tree 11

  12. Sine Function/Ratio(Sin) 12

  13. Cosine Function/Ratio(Cos) 13

  14. Tangent Function/Ratio (Tan) 14

  15. Reciprocal Functions/Ratios The other trigonometric ratios are cosec , sec and cot . The ratios cosec , sec and cot are the reciprocals of the ratios sin , cos and tan respectively. 15

  16. Secant (Sec) Function/Ratio The secant sec(A) is the reciprocal of cos(A); i.e. the ratio of the length of the hypotenuse to the length of the adjacent side. ??? ? = ?????????? ? ???????? = ? ???? = ? 16

  17. Cosecant (Cosec)Function/Ratio The cosecant or cosec(A), is the reciprocal of sin(A); i.e. the ratio of the length of the hypotenuse to the length of the opposite side. ??? ? = ?????????? ? ???????? = ? ?????? = ? 17

  18. Cotangent(Cot) Function/ Ratio The cotangent cot(A) is the reciprocal of tan(A); i.e. the ratio of the length of the adjacent side to the length of the opposite side. ??? ? = ???????? ? ???????? = ? ???? = ? 18

  19. Example 19

  20. 20

  21. RECAPITULATION Using the given figure, find all the trigonometric ratios for angle A and B. 21

  22. THANK YOU 22

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