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TrigonometryTrigonometry is derived from Greek words  trigonon (three angles) and metron ( measure).Trigonometry is the branch of 

h i   hi h d l   i h mathematics which deals with triangles, particularly triangles in a plane where one angle of the a plane where one angle of the triangle is 90 degrees

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TrigonometryTrigonometry means science of measuring triangles.specifically deals with the relationships between the sides 

d  h   l   f  i l  and the angles of triangles, Triangles on a sphere are also t di d  i   h i l studied, in spherical trigonometry. 

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History“T i  i     h   k  f   “Trigonometry is not the work of any one person or nation. Its history spans thousands of years and has spa s ousa ds o yea s a d astouched every major civilization.”The origins of trigonometry can be g g ytraced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley  more than 4000 years agoValley, more than 4000 years ago.

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HistoryTh  fi   d d    f The first recorded use of trigonometry came from the Hellenistic mathematician e e s c a e a c aHipparchus circa 150 BC, who compiled a trigonometric table using th   i  f   l i  t i lthe sine for solving triangles.The Sulba Sutras written in India, between 800 BC and 500 BC  between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for i li   h    ( h   i   f 

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circling the square (the opposite of squaring the circle).

ORIGIN OF ‘SINE’The first use of the idea of The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhatain A.D. 500.Aryabhata used the word ‘ardha-jya’ for the half chord which came to be known as which came to be known as ‘jiva’ in due course.Later  ‘jiva’ came to be Later,  jiva came to be known as ‘sinus’ and later as ‘sine’.

Aryabhata

A.D. 476-550

History

M   i t  th ti i  lik  Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunusand Al‐Kashi made significant gcontributions in this field(trigonometry).

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TRIGONOMETRY, as it is actually used in calculus TRIGONOMETRY, as it is actually used in calculus and science, is not just about solving triangles.It becomes the mathematical description of pthings that rotate or vibrate, such as light, sound, the paths of planets about the sun, or satellites about the earth. It is necessary therefore to have angles of any size, 

d    d    h   h   i   f  h  and to extend to them the meanings of the trigonometric functions.

AnglesAn angle is determined by An angle is determined by rotating a ray about its endpoint.pThe starting point of the ray is the initial side.

B

The ending position is the terminal side. 

Initial SideVERTEX(O) A

AnglesTh   d i t  f th    i  The endpoint of the ray is called the vertex of the angle.

A   l  i   t d d  iti  An angle in standard position has its initial side on the positive x‐axis and its vertex at positive x axis and its vertex at the origin.

Angle has three parts:

1. Initial side 2. Terminal side 3. vertex

Initial SideVERTEX

AnglesIf the rotation of the ray is counterclockwise the angle has positive measure.

f h f hIf the rotation of the ray is clockwise the angle has negative measuremeasure.

A l i f d b j i i h d iAn angle is formed by joining the endpoints of two half-lines called rays. The side you measure to is called the terminal sideThe side you measure to is called the terminal side.Angles measured counterclockwise are given a positive sign and angles measured clockwise arepositive sign and angles measured clockwise are given a negative sign.

P i i A lThis is a counterclockwise rotation

Positive Angle

Negative Angle

Initial Side

rotation.This is a clockwise rotation.

The side you measure from is called the initial side.

ANGLE:The measure or size of an angle is the amount of rotation from the initial ray to the terminal ray

the size of the angled    d d does not depend onthe lengths of its sides.

We commonly refer to the measure of the l (“ h ”)θangle as      (“theta”)θ

I G k T M !It’s Greek To Me!It is customary to use small letters in the GreekIt is customary to use small letters in the Greek alphabet to symbolize angle measurement.

α β γα β γalpha beta gamma

δθalpha beta gamma

δφθtheta φ

phi delta

We can use a coordinate system with angles by putting the initial side along the positive x-axis

ith th t t th i iwith the vertex at the origin.Quadrant I

angle Quadrant IIθ positive

I iti l Sidθ i

angle

Initial Sideθ negative

If the terminal side isQuadrant IV

angle

If the terminal side is along an axis it is called a quadrantal angle.

We say the angle lies in whatever quadrant the terminal side lies in.

a quadrantal angle.

If we start with the initial side and go all of the way around in a counterclockwise θ = 360° ydirection we have 360 degrees

θ = 90°9

If we went 1/4 of the way in a clockwise direction the

You are probably already familiar with a right angle h 1/4 f h

angle would measure -90°

θ   90° that measures 1/4 of the way around or 90°

θ = ‐ 90°

Let’s talk about degrees first. You are probably already somewhat familiar with degrees.

What is the measure of this angle?What is the measure of this angle?θ = ‐ 360° + 45° You could measure in the positive

direction and go around another rotation which would be another 360°

θ = ‐ 315°θ = 45°which would be another 360

You could measure in the positive direction θ = 360° + 45° = 405°

You could measure in the negativeYou could measure in the negative direction

There are many ways to express the given angle. y y p g gWhichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I.

Angles can be measured in 3 ways:Angles can be measured in 3 ways:1)  Degrees  (360 parts to a rotation)

used for triangle applications

)  R di2)  Radians (2∏ parts to a rotation)used for circular applications

3)  Grads (100 parts to a rotation)used by the militaryused by the military

DEGREE MEASURETo measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal 3 qparts, and we call each of those equal parts a "degree." Its symbol is a small 0: ° "  d "  Th  f ll  i l   h   ill b  1° ‐‐"1 degree."  The full circle, then, will be 360°.  But why the number 360? What is so special about it? Why not 100° or 1000°?about it? Why not 100 or 1000 ?

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• First, 360 has many divisors, it will have many whole number partsit will have many whole number parts. It has an exact half and an exact third ‐‐ which a power of 10 does not havea power of 10 does not have. 360 has a fourth part, a fifth, a sixth, and so on Those are natural divisions of the circleon. Those are natural divisions of the circle, and it is very convenient for their measures to be whole numbers (Even the ancients didn'tbe whole numbers.  (Even the ancients didn t like fractions)

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Secondly, 360 is close to the number of days in the astronomical year: 365days in the astronomical year: 365.

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To say that angle BAC is 30° means 3that its sides enclose 30

f  h   l of those equal divisions. Arc BC is 30/360is 30/360of the entire circumference.

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So, when 360° is the measure of a full circle, then 180°will be half a i l   °  circle.  90° ‐‐ one right angle ‐‐ will be a quarter of a circle; a quarter of a circle; and 270° will be three quarters of a qcircle:  three right angles.

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ANGLE: One degree, minute and secondsIf a rotation from initial side 

/to terminal side is 1/360 th of a revolution , the angle is said

fto have a measure of one Degree written as 10

E h d  i  Each degree is divided into 60 equal parts called minutes

0 ( ’)10 =60 minutes(60’)Each minute is further divided into 60 equal 

  ll d  dparts called seconds1’=60 seconds(60’’)

Radian MeasureR ll f  G t                             Recall from Geometry…                            Ratio of circumference of a circle to its diameter is constant.

The Greeks named this constant number as ∏.∏ ≈ 3.14159      ∏ ≈  22/7∏  is an irrational number.          It  t b   itt      t i ti     ti  It cannot be written as a terminating or repeating 

decimal.

Radian Measure

RADIAN MEASURE

OB

θr

Angle of a whole turn = Circumference of a circleAngle of a whole turn   Circumference of a circle

Radius

= 2 rπ= 

2 d

r

= 2π rad

We also know that the angle at the centre of a cicrle is 360 d360 degrees.

Therefore, 360° = 2π rad

Radian MeasureAn angle

measured in radiansis the ratio of the

arc length of the circlearc length of the circleto the radius f th   i l

radius

θ

of the circle.

Radian MeasureRadian measure does not use any artificial unit, it l th ti f l th t diit only uses the ratio of arc length to radius.

θ

arc lengthRadian measure of θ =

arc lengthr

Radian measure of θ = radius r

A  radian: One radian is the measure of the angle gsubtended at the centre of a circle by an arc whose length is equal to the radius ofy g qthe circle A

O

r

1 rad

r

O

B

Converting Between Degrees and RadiansWe have 360° = 2π radians

1 radian is180 57 2956°

°180° = π radians

57.2956π

≈ °90° = π/2 radians

1° = π /180 radians

π180

×Degrees Radians180

DR di

180π°

×

DegreesRadians

Radian Measure

S   θ     t ti     6 °   diSo  θ = 1 rotation = 360° = 2∏ radians

So  θ = ½ rotation = 180° = ∏ radians

Conversion  Factor: 180180

or ππ°

°180π

So, how to convert the measurements? • Radian Degree• Radian Degree

di ( × ) d180°x radians = (x × ) degrees180π

• Degree Radians

πx degrees = ( x × ) radians

180π°

Degree  Radian Conversion

Degrees  Radians:180

multiply by π°

Radians  Degrees: 180multiply by °Radians  Degrees: p y yπ

“What  you  want”   goes  on  top!y g p

Special AnglesSpecial Angles( )30

180π⎛ ⎞° =⎜ ⎟°⎝ ⎠30 _____ radians°=

( )180⎜ ⎟°⎝ ⎠Think…

( )45 π⎛ ⎞° =⎜ ⎟Think

45 _____ radians°=

( )45180

=⎜ ⎟°⎝ ⎠Think…

⎛ ⎞

60 di°

Think… ( )60180π⎛ ⎞° =⎜ ⎟°⎝ ⎠

60 _____ radians°=

Radian Measure30 _____ radians° =

( )30180π⎛ ⎞° =⎜ ⎟°⎝ ⎠

Think…

45 _____ radians° = ( )45180π⎛ ⎞° =⎜ ⎟°⎝ ⎠

Think…

Thi k ( ) π⎛ ⎞

60 _____ radians° =Think… ( )60

180π⎛ ⎞° =⎜ ⎟°⎝ ⎠

Think… ( )315 π⎛ ⎞° =⎜ ⎟315 _____ radians° =

Think… ( )315180

=⎜ ⎟°⎝ ⎠

Think… ( )120180π⎛ ⎞° =⎜ ⎟°⎝ ⎠120 _____ radians° =

180⎝ ⎠

Radian Measure3 _____4

radiansπ= °

3 1804π

π°⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠Think…

_____2

radiansπ= °

1802π

π°⎛ ⎞ ⎛ ⎞ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠Think…

_____2

3 diπ°

3 1802π

π°⎛ ⎞ ⎛ ⎞ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠Think…

_____2

5

radians = °5 1803π

π°⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠Think…

5 _____3

radiansπ= °

Changing Degree Measure to Radian Measure

Calculate the radian measure:a) 2100

8 0   d

b) 3150

1800 =  π rad

1180π

° =°

315 315180π

° = × °°

210 210180π

° = × °°

7

74π

= radians76π

=

= 3.67

Exact radians

Approximate radians

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3.67 pp

Changing Radian Measure to Degree Measure

Calculate the degree measure:π

a)

π rad   1800

b)23

radπ 12radπ

180°π rad = 1800

0

1801 radπ°

=

18012 12

radπ ππ°

= ×

= 150

c) 1.68 rad2 180 23 3π π

π°

= ×

= 1200

1801.68 1.68radπ°

= ×

= 96.260

Convert the following angles from degrees to radians. b) 112° 36‘a) 310°

° d

b) 112 36

180° = π rad180° = π rad

1°= π  rad8 °

180  π rad1°= π  rad

180°180°

310° = 310° × π8 °

112° 36‘ = 112.6°= 112.6° × π

180°

 5 411 rad

180°= 1.965 rad

= 5.411 rad

c) 85.7° d) 53° 120'

180° = π rad 180° = π rad°     d1°= π  rad

180°

1°= π  rad180°180

85.7°= 85.7° × π180°

53° 120‘ =  53.2°= 53.2° × π180°

= 1.496 rad

53180°

 0 9286 rad= 0.9286 rad

Convert following into degrees

b) 2.15 rad 

π rad   = 180°2.15 rad = 2.15  × 180°

π= 123.16°

b) 5π rad6

c) 1.36 rad

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d 180° π rad      = 180°π rad    = 180°5π rad = 5 π × 180° 1.36 rad = 1.36 × 180°

π6              6         π

π

= 77 92°= 150° = 77.92

Exercises :The angles of triangle are in the ratioThe angles of triangle are in the ratio

1:3:5. Find the angles in degrees and radians

The angles of triangle are such that one is the average of the other two and the greatest g gis twice the least. Express the angles in radian measure.

The angles of triangle are in A.P.. The greatest angle is three times the least. Find them in i)d   ii) dii)degrees  ii)radians.

What can you do with trig?

Historically, it was developed for astronomy d h b i i h b i i fand geography, but scientists have been using it for

centuries for other purposes, too. Besides other fi ld f h i i i d i h ifields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics,

i i d i i il i l l ( hi h itrig is used in primarily in calculus (which is perhaps its greatest application), linear algebra, and

i istatistics.

Applications of Trigonometry in AstronomyThe technique of triangulation is used to measure q gthe 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  as about  0 000 000 Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .

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A tAstronomy

Trigonometric tables were created over two thousand years ago for computations in astronomy. 

Engineering and physicsAlthough trigonometry was first applied to spheres  it has had greater application to spheres, it has had greater application to planes. Surveyors have used trigonometry f   i  E i  b h  ili  for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long. 

h i  l  h  d d    Physics lays heavy demands on trigonometry. Optics and statics are g y ptwo early fields of physics that use trigonometry, but all branches of trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding trigonometry aids in understanding space. Related fields such as physical h i t   t ll    t i  chemistry naturally use trig. 

Wavesa es

The graphs of the functions sin(x) and cos(x)g p ( ) ( )look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functionsas those of the sine and cosine functions.However, a few hundred years ago, mathematicians realized that any wave at all yis made up of sine and cosine waves. This fact lies at the heart of computer music. Si t t li t t i 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

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computer is to represent it mathematically by its constituent sound waves.

This is why sound engineers, those who y g ,research and develop the newest advances in computer music technology, and sometimes even composers have to 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 waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology y gyand the physical sciences.

ConclusionTrigonometry is a branch of Mathematics 

with several important and useful papplications.  Hence it attracts more and 

more research with several theories published year after year

Th k Y59

Thank You