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Trigonometry is derived from Greek words Trigonometry is derived from Greek words trigonontrigonon (three angles) and (three angles) and metronmetron ( measure). ( measure).

Trigonometry is the branch of mathematics Trigonometry is the branch of mathematics which deals with triangles, particularly triangles which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 in a plane where one angle of the triangle is 90 degreesdegrees

Triangles on a sphere are also studied, in Triangles on a sphere are also studied, in spherical trigonometry. spherical trigonometry.

Trigonometry specifically deals with the Trigonometry specifically deals with the relationships between the sides and the angles of relationships between the sides and the angles of triangles, that is, on the trigonometric functions, triangles, that is, on the trigonometric functions, and with calculations based on these functions.and with calculations based on these functions.

TrigonometryTrigonometry

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HistoryHistory

The origins of trigonometry can be traced to the civilizations The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.than 4000 years ago.

Some experts believe that trigonometry was originally Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the invented to calculate sundials, a traditional exercise in the oldest booksoldest books

The first recorded use of trigonometry came from the The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving compiled a trigonometric table using the sine for solving triangles.triangles.

The Sulba Sutras written in India, between 800 BC and 500 The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45BC, correctly compute the sine of π/4 (45°°) as 1/√2 in a ) as 1/√2 in a procedure for circling the square (the opposite of squaring the procedure for circling the square (the opposite of squaring the circle).circle).

Many ancient mathematicians like Aryabhata, Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry).contributions in this field(trigonometry).

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Right TriangleRight Triangle

A triangle in which one angle A triangle in which one angle is equal to 90is equal to 90 is called right is called right triangle.triangle.

The side opposite to the right The side opposite to the right angle is known as hypotenuse.angle is known as hypotenuse.

AB is the hypotenuseAB is the hypotenuse

The other two sides are The other two sides are known as legs.known as legs.

AC and BC are the legsAC and BC are the legs

Trigonometry deals with Right Triangles

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Pythagoras TheoremPythagoras Theorem

In any right triangle, the area of the square whose In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legsthe squares whose sides are the two legs..

In the figureIn the figureABAB22 = BC = BC22 + AC + AC22

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Trigonometric ratiosTrigonometric ratios

Sine(sin) opposite side/hypotenuseSine(sin) opposite side/hypotenuse Cosine(cos) adjacent side/hypotenuse Cosine(cos) adjacent side/hypotenuse Tangent(tan) opposite side/adjacent side Tangent(tan) opposite side/adjacent side Cosecant(cosec) hypotenuse/opposite sideCosecant(cosec) hypotenuse/opposite side Secant(sec) hypotenuse/adjacent sideSecant(sec) hypotenuse/adjacent side Cotangent(cot) adjacent side/opposite side Cotangent(cot) adjacent side/opposite side

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Values of trigonometric function Values of trigonometric function of Angle Aof Angle A

sin = a/c

cos = b/c

tan = a/b

cosec = c/a

sec = c/b

cot = b/a

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Values of Trigonometric functionValues of Trigonometric function

00 3030 4545 6060 9090

SineSine 00 0.50.5 1/1/22 3/23/2 11

CosineCosine 11 3/23/2 1/1/22 0.50.5 00

TangentTangent 00 1/ 1/ 33 11 33 Not definedNot defined

CosecantCosecant Not definedNot defined 22 22 2/ 2/ 33 11

SecantSecant 11 2/ 2/ 33 22 22 Not definedNot defined

CotangentCotangent Not definedNot defined 33 11 1/ 1/ 33 00

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CalculatorCalculator

This Calculates the values of trigonometric functions of This Calculates the values of trigonometric functions of different angles.different angles.

First Enter whether you want to enter the angle in First Enter whether you want to enter the angle in radians or in degrees. Radian gives a bit more accurate radians or in degrees. Radian gives a bit more accurate value than Degree.value than Degree.

Then Enter the required trigonometric function in the Then Enter the required trigonometric function in the format given below:format given below:

Enter 1 for sin.Enter 1 for sin. Enter 2 for cosine.Enter 2 for cosine. Enter 3 for tangent.Enter 3 for tangent. Enter 4 for cosecant.Enter 4 for cosecant. Enter 5 for secant.Enter 5 for secant. Enter 6 for cotangent.Enter 6 for cotangent. Then enter the magnitude of angle.Then enter the magnitude of angle.

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Trigonometric identitiesTrigonometric identities sinsin22A + cosA + cos22A = 1A = 1 1 + tan1 + tan22A = secA = sec22AA 1 + cot1 + cot22A = cosecA = cosec22AA sin(A+B) = sinAcosB + cosAsin Bsin(A+B) = sinAcosB + cosAsin B cos(A+B) = cosAcosB – sinAsinBcos(A+B) = cosAcosB – sinAsinB tan(A+B) = (tanA+tanB)/(1 – tanAtan B)tan(A+B) = (tanA+tanB)/(1 – tanAtan B) sin(A-B) = sinAcosB – cosAsinBsin(A-B) = sinAcosB – cosAsinB cos(A-B)=cosAcosB+sinAsinBcos(A-B)=cosAcosB+sinAsinB tan(A-B)=(tanA-tanB)(1+tanAtanB)tan(A-B)=(tanA-tanB)(1+tanAtanB) sin2A =2sinAcosAsin2A =2sinAcosA cos2A=coscos2A=cos22A - sinA - sin22AA tan2A=2tanA/(1-tantan2A=2tanA/(1-tan22A)A) sin(A/2) = sin(A/2) = ±±{(1-cosA)/2}{(1-cosA)/2} Cos(A/2)= Cos(A/2)= ±±{(1+cosA)/2}{(1+cosA)/2} Tan(A/2)= Tan(A/2)= ±±{(1-cosA)/(1+cosA)}{(1-cosA)/(1+cosA)}

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Relation between different Relation between different Trigonometric IdentitiesTrigonometric Identities

SineSine CosineCosine TangentTangent CosecantCosecant SecantSecant CotangentCotangent

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Angles of Elevation and Angles of Elevation and DepressionDepression

Line of sight: The line from our eyes to the Line of sight: The line from our eyes to the object, we are viewing.object, we are viewing.

Angle of Elevation:The angle through which Angle of Elevation:The angle through which our eyes move upwards to see an object our eyes move upwards to see an object above us.above us.

Angle of depression:The angle through Angle of depression:The angle through which our eyes move downwards to see an which our eyes move downwards to see an object below us. object below us.

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Problem solved using Problem solved using trigonometric ratiostrigonometric ratios

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Applications of TrigonometryApplications of Trigonometry This field of mathematics can be applied in astronomy,navigation, This field of mathematics can be applied in astronomy,navigation,

music theory, acoustics, optics, analysis of financial markets, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and civil engineering, computer graphics, cartography, crystallography and game development.game development.

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DerivationsDerivations Most Derivations heavily rely on Most Derivations heavily rely on

Trigonometry.Trigonometry.

Click the hyperlinks to view the derivationClick the hyperlinks to view the derivation A few such derivations are given below:-A few such derivations are given below:- Parallelogram law of addition of vectors.Parallelogram law of addition of vectors. Centripetal Acceleration.Centripetal Acceleration. Lens FormulaLens Formula Variation of Acceleration due to gravity due Variation of Acceleration due to gravity due

to rotation of earth.to rotation of earth. Finding angle between resultant and the Finding angle between resultant and the

vector.vector.

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Applications of Trigonometry Applications of Trigonometry in Astronomyin Astronomy

Since ancient times trigonometry was used in astronomy.Since ancient times trigonometry was used in astronomy. The technique of triangulation is used to measure the distance to nearby stars.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 In 240 B.C., a mathematician named Eratosthenes discovered the radius of the

Earth using trigonometry and geometry.Earth using trigonometry and geometry. In 2001, a group of European astronomers did an experiment that started in 1997 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 about the distance of Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .kilometers away from the Sun .

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Application of Trigonometry in Application of Trigonometry in ArchitectureArchitecture

Many modern buildings have beautifully curved surfaces. Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is Making these curves out of steel, stone, concrete or glass is

extremely difficult, if not impossible. extremely difficult, if not impossible. One way around to address this problem is to piece the One way around to address this problem is to piece the

surface together out of many flat panels, each sitting at an surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create angle to the one next to it, so that all together they create what looks like a curved surface.what looks like a curved surface.

The more regular these shapes, the easier the building The more regular these shapes, the easier the building process.process.

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

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WavesWaves

The graphs of the functions The graphs of the functions sin(x)sin(x) and and cos(x)cos(x) look like waves. Sound look like waves. Sound travels in waves, although these are not necessarily as regular as those travels in waves, although these are not necessarily as regular as those of the of the sinesine and and cosinecosine functions. functions.

However, a few hundred years ago, mathematicians realized that However, a few hundred years ago, mathematicians realized that anyany wave at all is made up of wave at all is made up of sinesine and and cosinecosine waves. This fact lies at the waves. This fact lies at the heart of computer music. heart of computer music.

Since a computer cannot listen to music as we do, the only way to get 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 music into a computer is to represent it mathematically by its constituent sound waves. constituent sound waves.

This is why sound engineers, those who research and develop the This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry.composers have to understand the basic laws of trigonometry.

Waves move across the oceans, earthquakes produce shock waves and Waves move across the oceans, earthquakes produce shock waves and light can be thought of as traveling in waves. This is why trigonometry light can be thought of as traveling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.like meteorology and the physical sciences.

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Digital ImagingDigital Imaging In theory, the computer needs an infinite amount of information to do In theory, the computer needs an infinite amount of information to do

this: it needs to know the precise location and colour of each of the this: it needs to know the precise location and colour of each of the infinitely many points on the image to be produced. In practice, this is infinitely many points on the image to be produced. In practice, this is of course impossible, a computer can only store a finite amount of of course impossible, a computer can only store a finite amount of information.information.

To make the image as detailed and accurate as possible, computer To make the image as detailed and accurate as possible, computer graphic designers resort to a technique called graphic designers resort to a technique called triangulationtriangulation. .

As in the architecture example given, they approximate the image by a As in the architecture example given, they approximate the image by a large number of triangles, so the computer only needs to store a finite large number of triangles, so the computer only needs to store a finite amount of data. amount of data.

The edges of these triangles form what looks like a wire frame of the The edges of these triangles form what looks like a wire frame of the object in the image. Using this wire frame, it is also possible to make object in the image. Using this wire frame, it is also possible to make the object move realistically.the object move realistically.

Digital imaging is also used extensively in medicine, for example in CAT Digital imaging is also used extensively in medicine, for example in CAT and MRI scans. Again, triangulation is used to build accurate images and MRI scans. Again, triangulation is used to build accurate images from a finite amount of information.from a finite amount of information.

It is also used to build "maps" of things like tumors, which help decide It is also used to build "maps" of things like tumors, which help decide how how xx-rays should be fired at it in order to destroy it. -rays should be fired at it in order to destroy it.

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ConclusionConclusion

Trigonometry is a branch of Mathematics with Trigonometry is a branch of Mathematics with several important and useful applications. several important and useful applications.

Hence it attracts more and more research with Hence it attracts more and more research with several theories published year after yearseveral theories published year after year