This document provides an introduction to trigonometry, discussing its definitions, history, and essential concepts such as trigonometric ratios and identities. It highlights the origins of trigonometry from ancient civilizations and its applications in mathematics, particularly in calculating distances and analyzing right triangles. The document concludes by emphasizing the ongoing research and significance of trigonometry in various fields.

1. INTRODUCTION TO
TRIGNOMETRY
MADE BY: KRISHNARAJ MISHRA
SUBJECT:MATHS
SUBMITTED TO: NM GIRI SIR

2. 1) Introduction to trignometry
2) History
3) Trignometric ratios
4) Values of trignometric function
5) Trignometric ratios of some specific angles
6) Trignometric ratios of some complementry angles
7) Trignometric identities
8) Conclusion

3. INTRODUCTION
• The distances or heights can be
found by using some
mathematical techniques , which
come under a branch of
mathematics called
‘trignometry’.
• The word ‘trignometry’ is
derived from Greek words ‘tri’
(meaning three),’gon’(meaning
sides) and metron(meaning
measure).

4. INTRODUCTION
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.

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. 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

7. 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

8. TRIGONOMETRIC RATIOS
 Sine(sin) opposite side/hypotenuse
Cosine(cos) adjacent side/hypotenuse
Tangent(tan) opposite side/adjacent side
Cosecant(cosec) hypotenuse/opposite side
 Secant(sec) hypotenuse/adjacent side
Cotangent(cot) adjacent side/opposite side

9. sin = a/c
cos = b/c
tan = a/b
cosec = c/a
sec = c/b
cot = b/a

10. • In Δ ABC, right-angled at B, if one angle is
45°, then the other angle is also 45°, i.e., ∠ A =
∠ C = 45° .
• Suppose BC = AB = a.
• Then by Pythagoras Theorem, AC2 = AB2
+ BC2
= a2 + a2 = 2a2,
Therefore, AC = 2 a

11. Trigonometric Ratios of 45°
 sin 45° = side opposite to angle 45° / hypotenuse
=BC/AC = a/a√2 = 1/ √2
 cos 45° = side adjacent to angle 45°/ hypotenuse
=AB/AC = a/a √2 = 1/ √2
 tan 45° =side opposite to angle 45°/ side adjacent to
angle 45°
=BC/AB = a/a = 1
 cosec 45°=1/sin 45°= √2
 sec 45°=1/cos 45°= √2
 cot 45°=1/tan 45°= 1

12.  cosec 30°=1/sin 30° = 2
 sec 30°=1/cos 30° = 2/√3
 cot 30°=1/tan 30° = √3
 sin 60°= a√3/2a = √3/2
 cos 60°= ½
 tan 60°= √3
 cos 60°= 2/√3
 sec 60°= 2
 cot 60°= 1/√3

13. Trigonometric Ratios of 0°And 90°
Sin 0⁰ = 0
cos 0⁰ = 1
Sin 90⁰ = 1
Cos 90⁰ =0

14. VALUES OF TRIGONOMETRIC
FUNCTION
0 30 45 60 90
Sine 0 0.5 1/2 3/2 1
Cosine 1 3/2 1/2 0.5 0
Tangent 0 1/ 3 1 3 Not defined
Cosecant Not defined 2 2 2/ 3 1
Secant 1 2/ 3 2 2 Not defined
Cotangent Not defined 3 1 1/ 3 0

15. RELATION BETWEEN DIFFERENT
TRIGONOMETRIC IDENTITIES
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent

16.  sin (90⁰-A) = cos A
 tan (90⁰-A) = cot A
 sec (90⁰-A) = cosec A
 cos (90⁰-A) = sin A
 cot (90⁰-A) = tan A
 cosec (90⁰-A) = sec A

17. Trigonometric identities
Osin2A + cos2A = 1
O1 + tan2A = sec2A
O1 + cot2A = cosec2A
Osin(A+B) = sinAcosB + cosAsin B
Ocos(A+B) = cosAcosB – sinAsinB
Otan(A+B) = (tanA+tanB)/(1 – tanAtan B)
Osin(A-B) = sinAcosB – cosAsinB
Ocos(A-B)=cosAcosB+sinAsinB
Otan(A-B)=(tanA-tanB)(1+tanAtanB)

18. Trigonometric identities

 sin2A =2sinAcosA
 cos2A=cos2A - sin2A
 tan2A=2tanA/(1-tan2A)
 sin(A/2) = ±{(1-cosA)/2}
 Cos(A/2)= ±{(1+cosA)/2}
 Tan(A/2)= ±{(1-cosA)/(1+cosA)}

19. Conclusion
Trigonometry is a branch of Mathematics with
several important and useful applications.
Hence it attracts more and more research with
several theories published year after year.

20. Thank You……..