Trigonometry is the branch of mathematics that deals with triangles and their angles. It originated over 4000 years ago in ancient Egypt, Mesopotamia, and India, where it was used to calculate sundials and circle squares. Key contributors include Hipparchus, who compiled trigonometric tables using sines, and ancient Indian mathematicians who computed sine values. Trigonometry defines functions like sine, cosine, and tangent that relate a triangle's angles and sides. It has many applications, including astronomy, navigation, engineering, acoustics, and more.

1. TRIGONOMETRY

2. What is trigonometry?
 Trigonometry is the branch
of
mathematics which deals
with
triangles, particularly
triangles in a plane where
one angle of the triangle is
90 degrees. Trigonometry
is derived from Greek
words
trigonon (three angles) and
metron ( measure).

3. History of
trigonometry• 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).

4. In a right triangle, the shorter sides are called legs and the longest side (which is the one
opposite the right angle) is called the hypotenuse
First let’s look at the three basic functions.
SINE
They are abbreviated using their first 3 letters
COSINE
TANGENT
hypotenuse
opposite
sin 
hypotenuse
adjacent
cos 
adjacent
opposite
tan 

5. 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

6. Easy way to learn trigonometric
ratios

7. Values of Trigonometric
function0 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

8. Trigonometric ratios of
complementary
angles
• Sin(90-A)= cos A,
Cos(90- A)= sin A
• Tan(90-a)= cot A,
Cot(90-A)= tan A
• Sec(90-A)= cosec A,
cosec(90-A)= sec A

9. Trigonometric
identities• Identity1- sin2A + cos2A = 1
• PROOF:
• IN TRIANGLE ABC,
• AB2 + BC2 = AC2 ( BY PYTHAGORAS THEOREM) (1)
• DIVIDING EACH TERM OF (1) BY AC
AB2 / AC2 + BC2 / AC2 = AC2 / AC2
(cos A) 2 + (sin A) 2 = 1
→ cos2 A + sin2 A = 1
A
B
C
2

10. • Identity2 : 1 + tan2A =
sec2A
• Dividing eq (1) by
AB2
(AB/AB) 2 + (BC/AB2
=(AC/AB) 2
=> 1+ Tan2 A = sec2
A
• Identity3: 1 + cot2A =
cosec2A
• Dividing eq (1) by BC2
• (AB/BC) 2 + (BC/BC) 2
= (AC/BC) 2
• => Cot2A + 1 = cosec2
A

11. Solving a Problem with
the Tangent Ratio
60º
53 ft
h = ?
1
2 3
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
ft92353
531
3
53
60tan



h
h
h
adj
opp

12. Applications of
Trigonometry • This field of mathematics can be applied in
astronomy, navigation, music theory,
acoustics, optics, analysis of financial
markets, electronics, probability theory,
statistics, biology, medical imaging (CAT
scans and ultrasound), pharmacy,
chemistry, number theory (and hence
cryptology), seismology, meteorology,
oceanography, many physical sciences,
land surveying and geodesy, architecture,
phonetics, economics, electrical
engineering, mechanical engineering, civil
engineering, computer graphics,
cartography, crystallography and game
development.

13. Applications of Trigonometry in
Astronomy
Since ancient times trigonometry was used
in astronomy.
The technique of triangulation is used to
measure the distance to nearby stars.
In 240 B.C., a mathematician named
Eratosthenes discovered the radius of the
Earth using trigonometry and geometry.
In 2001, a group of European astronomers
did an experiment that started in 1997
about the distance of Venus from the Sun.
Venus was about 105,000,000 kilometers
away from the Sun .

14. Application of
Trigonometry in
ArchitectureMany modern buildings have
beautifully curved surfaces.
Making these curves out of steel,
stone, concrete or glass is extremely
difficult, if not impossible.
One way around to address this
problem is to piece the surface
together out of many flat panels,
each sitting at an angle to the one
next to it, so that all together they
create what looks like a curved
surface.
The more regular these shapes, the
easier the building process.

15. Thank you
Made by-
Divya pandey
X-A