its all about trignometry

1. MADE BY :-
ultron
INTRODUCTION
TO
TRIGONOMETRY

2. •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).

3. The word trigonometry is derived from the
ancient Greek language and means
measurement of triangles.
trigonon “triangle”
+
metron “measure”
=
Trigonometry
Trigonometry...?????

4. a b
c
B A
C
Trigonometry ...
is all about
Triangles…

5. 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.
5

6. A right-angled triangle (the right angle is
shown by the little box in the corner) has
names for each side:
Adjacent is adjacent to the angle "θ“
Opposite is opposite the angle
 The longest side is the Hypotenuse.
Right Angled Triangle
Opposite
Adjacent
θ

7. ANGLES
Angles (such as the angle "θ" ) can be
in Degrees or Radians.
Here are some examples:
Angle Degree Radians
Right Angle 90° π/2
Straight Angle 180° π
Full Rotation 360° 2π

8. Trigonometric ratios ..

9. •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

10. B A
C
Sin /
Cosec 
P
(pandit)
H
(har)
Cos /
Sec 
B
(badri)
H
(har)
Tan /
Cot 
P
(prasad)
B
(bole)
This is
pretty
easy!
BASE (B)
PERPENDICULAR (P)

10

11.  A 0  3 0  4 5  6 0  9 0 
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
11

12. • 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
• Trigonometric Ratios of Complementary
Angles

13. TRIGONOMETRIC IDENTITIES

14.  Reciprocal Identities
sin u = 1/csc u
cos u = 1/sec u
tan u = 1/cot u
csc u = 1/sin u
sec u = 1/cos u
cot u = 1/tan u
 Quotient Identities
tan u = sin u /cos u
cot u =cos u /sin u

15. 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 
15

16. •Trigonometric identities
• sin(A+B) = sinAcosB + cosAsin B
• cos(A+B) = cosAcosB – sinAsinB
• tan(A+B) = (tanA+tanB)/(1 – tanAtan B)
• sin(A-B) = sinAcosB – cosAsinB
• cos(A-B)=cosAcosB+sinAsinB
• tan(A-B)=(tanA-tanB)(1+tanAtanB)

17. • 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)}
•Trigonometric identities

18. •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.

19. Thank You……..