Trigonometry is derived from Greek words meaning "three angles" and "measure". It deals with relationships between sides and angles of triangles, especially right triangles. The document discusses the history of trigonometry dating back to ancient Egypt and Babylon, and how it advanced through the works of Greek astronomer Hipparchus and Ptolemy. It also discusses the six trigonometric ratios and their formulas, various trigonometric identities, and applications of trigonometry in fields like architecture, engineering, astronomy, music, optics, and more.

2. Trigonometry is derived from Greek words
trigonon (three angles) and matron ( measure).
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
base on these functions.

3. Trigonometry a beginning
The history of trigonometry dates back to the early ages of Egypt and
Babylon . Angles were then measured in degrees. History of trigonometry was
then advanced by the Greek astronomer Hipparchus who compiled a
trigonometry table that measured the length of the chord
subtending the various angles in a circle of a fixed radius.
This was done in
increasing degrees of 71.
In the 5th century, Ptolemy took this further by creating the
table of chords with increasing 1 degree. This was known
as Menelaus theorem which formed the foundation of trigonometry studies
for the next 3 centuries. Around the same period, Indian mathematicians
created the trigonometry system based on the sine function instead of the
chords. Note that this was not seen to be ratio but rather the opposite of the
angle in a right angle of fixed hypotenuse. The history of trigonometry also
included Muslim astronomers who compiled both the studies of the Greeks
and Indians .

4. 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

5. Values of trigonometric
function of Angle A
sin = a/c
cos = b/c
tan = a/b
cosec = c/a
sec = c/b
cot = b/a

6. Trigonometric ratios
The relationships between the angles and
the sides of a right triangle are expressed in
terms of TRIGONOMETRIC RATIOS.
The six trigonometric ratios for the angle 
Name of the ratio Abbreviation
Sine of  Sin 
Cosine of  Cos 
Tangent of  Tan 
Cotangent of  Cot 
Secant of  Sec 
Cosecant of  Csc 

7. Name of Ratio Abbreviation Explicit Formula Formula Memory Aid
Sin Of q. Sin q. SOH
Cosine Of q. Cos q. CAH
Tangent Of q. Tan q. TOA
Cotangent of q. Cot q.
Secant of q. Sec q.
Cosecant of q. Cos q.

8. RELATION WITH RATIOS
1)Relation with cosine
sin2 A + cos2A = 1
sin2 A = 1 - cos2A
Hence sin A = (1 - cos2A)1/2
Also
sin A = cos (90-A)
3) Relation with cosecant
sin A = 1/ cosec A
(cosecant is the reciprocal of sine)
5) Relation with cotangent
sin2A (1 + tan2A) = tan2A
sin2A (1 + 1/cot2A) = 1/cot2A
sin2A (cot2A + 1)/cot2A = 1/cot2A
sin A = (cot2A + 1)-1/ 2
2) Relation with secant
sin2A = 1 - cos2A
= 1 - 1/sec2A (
secant is the reciprocal of
cosine)
= (sec2A - 1)/sec2A
4) Relation with Tangent
sin2A = cos2A tan2A
= (1-sin2A)*tan2A
sin2A + sin2A*tan2A =
tan2A
sin2A (1 + tan2A) = tan2A
sin A = {tan2A/(1 + tan2A)
}1/2

9. • sin2A + cos2A = 1
• 1 + tan2A = sec2A
• 1 + cot2A = cosec2A
• 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)
• 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)}
Standard Identities

10. Trigonometric ratios of complementary angles
Trigonometrical
ratio of angle
Trigonometrical
ratio of complementary
angle
Formulas

11. 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

12. Calculator
This Calculates the values of trigonometric
functions of different angles.
First Enter whether you want to enter the
angle in radians or in degrees. Radian
gives a bit more accurate value than
Degree.
Then Enter the required trigonometric
function in the format given below:
Enter 1 for sin.
Enter 2 for cosine.
Enter 3 for tangent.
Enter 4 for cosecant.
Enter 5 for secant.
Enter 6 for cotangent.
Then enter the magnitude of angle.

13. 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.

14. A few examples of this include
Soccer ball begin kicked
A baseball begin thrown
An athlete long jumping
Fireworks and
Water fountains
Projectile motion refers to
the motion of an object
projected into the air at an
angle.

15. cos = hypotenuse
adjacent side

16. Applications of Trigonometry in
Astronomy
The technique used is
triangulation ).
By looking at a star one day
and then looking at it again
6 months later, an
astronomer can see a
difference in the viewing
angle.
With a little trigonometry,
the different angles yield a
distance.

17. Application of Trigonometry in
Architecture
• Many 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.
• Regular flat shapes like squares, pentagons and hexagons, can
be made out of triangles, and so trigonometry plays an
important role in architecture.

18. Engineers of various types use the
fundamentals of trigonometry
to build structures/systems,
design bridges and
solve scientific problems.
Here the height of the
building is determined
by using the function
Tan
tan = =adjacent side H
opposite side D
Some structures that were built using
trigonometry are:
Leaning tower of Pisa
Eiffel Tower
Qutub Minar

19. 19
• The graphs of the functions sin(x) and cos(x) look like waves. Sound
travels in waves, although these are not necessarily as regular as
those of the sine and cosine functions.
• However, a few hundred years ago, mathematicians realized that
any wave at all is made up of sine and cosine waves. This fact lies at
the heart of computer music.
• Since a computer cannot listen to music as we do, the only way to
get music into a computer is to represent it mathematically by its
constituent sound waves.
• This is why sound engineers, those who research and develop the
newest advances in computer music technology, and sometimes
even composers have to understand the basic laws of trigonometry.
• Waves move across the oceans, earthquakes produce shock waves
and light can be thought of as traveling in waves. This is why
trigonometry is also used in oceanography, seismology, optics and
many other fields like meteorology and the physical sciences.
Waves

20. When you think trigonometry you should think triangles -- not just geometric triangles but
musical triangles -- because trigonometry is the mathematics of sound and music.
Sound is the variation of air pressure.
The simplest sounds, called pure tones are represented by
f(t) = A sin(2 pi w t)

21. Computer generation of complex imagery is
made possible by the use of geometrical patterns
that define the precise location and color of each
of the infinite points on the image to be created.
The edges of the triangles that form the image
make a wire frame of the object to be created and
contribute to a realistic picture.
In medicine it s use in CAT and MRI scans.
Digital imaging-

22. Whenlight moves from a dense toa
less dense medium, suchas from
water to air,
At this point, light is reflected in the
incident medium, knownas internal
reflection.
Beforethe ray totally internally
reflects, thelight refractsat
the critical angle
When θ1 > θ crit, no refracted ray appears,
and the incident ray undergoes total
internal reflection from the interface
medium.
SNELL’S LAW
Sine i
Sine r
= Refractive index

23. Once the ray reaches the viewer’s eye,
it interprets it as if it traces back along a
perfectly straight "line of sight".
This line is however at a tangent to the
path the ray takes at the point it reaches
the eye.
The result is that an "inferior image" of
the sky above appears on the ground.
Cold air is denser than warm air
As light passes from colder air, the
light rays bend away from the
direction
When light rays pass from hotter to
colder, they bend toward the direction
of the gradient.

24. Here, if the angle of elevation between the line of sight
and ground is known, the Distance between can be
measured using the Tangent function

25. Among the scientific fields that make use of trigonometry are these:
Acoustics
Architecture
Astronomy (and hence navigation, on the oceans, in aircraft, and in space)
Biology
Cartography
Chemistry
Civil Engineering
Computer graphics
Geophysics
Crystallography
Economics
Electrical engineering
Electronics
Land surveying
Geodesy
Physical sciences
Mechanical engineering, machining
Medical imaging (CAT
scans and ultrasound)
Meteorology
Music theory
Number theory (and hence
cryptography),
Oceanography
Optics
Pharmacology
Phonetics
Probability theory
Psychology
Seismology
Statistics
Visual perception
So Where Ever You Go Trig Will Follow You

26. 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