Story of Trignometry

8/9/2019 Story of Trignometry

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BY:-Name: Rakshit GuptaClass:- Xth B

ROLL NO.:- 13


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INDEX

y

CLASSIC trigonometryy Basic Trigonometry Formulas

y Pythagorean Identities

y History of trigonometry

y Greek trigonometry

y Applications of trigonometry

y Calculating trigonometric functions


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y The branch of mathematics concerned with specificfunctions of angles and their application to

calculations. There are six functions of an anglecommonly used in trigonometry. Their names andabbreviations are sine (sin), cosine (cos), tangent(tan), cotangent (cot), secant (sec), and cosecant(cosec).


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EXAMPLESy For example, the triangle contains an angleA, and the

ratio of the side opposite toA and the side opposite to

the right angle (the hypotenuse) is called the sine ofA,or sin A; the other trigonometry functions are definedsimilarly.


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CLASSIC trigonometryy The word trigonometry comes from the Greek words trigonon

(triangle) and metron (to measure). Until about the 16thcentury, trigonometry was chief ly concerned with computing thenumerical values of the missing parts of a triangle (or any shapethat can be dissected into triangles) when the values of otherparts were given. For example, if the lengths of two sides of atriangle and the measure of the enclosed angle are known, thethird side and the two remaining angles can be calculated. Suchcalculations distinguish trigonometry from geometry,whichmainly investigates qualitative relations. Of course, this

distinction is not always absolute: the Pythagorean theorem,for example, is a statement about the lengths of the three sides ina right triangle and is thus quantitative in nature. Still, in itsoriginal form, trigonometry was by and large an offspring ofgeometry; it was not until the 16th century that the two becameseparate branches of mathematics.


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Basic Trigonometry Formulasi. Sin= OppositeSide/Hypotenuse

ii. Cos= AdjacentSide/Hypotenuse

iii. Tan= Sin/Cos = OppositeSide/Adjacent Side

iv. Cosec = 1/Sin=Hypotenuse/Opposite Side

v. Sec = 1/Cos =Hypotenuse/Adjacent Side

vi. Cot = 1/tan = Cos/Sin =Adjacent Side/Opposite Side


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Pythagorean Identitiesy sin2 + cos2 = 1

i. Cos2 =1- sin2

ii. sin2 =1-Cos2

y tan2 + 1 = sec2 ,

i. tan2 = sec2 -1

ii. sec2 - tan2 =1

y 1 + cot2 = cosec2

i. Cot2 = cosec2 -1

ii. Cosec2 -Cot2 =1


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For certain special right triangles the values of thefunctions may be calculated easily; e.g., in a right trianglewhose acute angles are 30 and 60 the sides are in theratio 1 :3 : 2, so that sin 30=cos 60=1/2, cos 30=sin60=3/2, tan 30=cot 60=1/3, cot 30=tan 60=3, sec30=cosec 60=2/3, and cosec 30=sec 60=2. For otherangles, the values of the trigonometric functions are

usually found from a set of tables or a scientificcalculator. For the limiting values of 0 and 90, thelength of one side of the triangle approaches zero whilethe other approaches that of the hypotenuse, resulting inthe values sin 0=cos 90=0, cos 0=sin 90=1, tan 0=cot

90=0, and sec 0=cosec 90=1; since division by zero isundefined, cot 0, tan 90, cosec 0, and sec 90 are allundefined, having infinitely large values.


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Greek trigonometryy Ancient Greek mathematicians such as Euclid and Archimedes studied the

properties of the chord of an angle and proved theorems that are equivalentto modern trigonometric formulae, although they presented themgeometrically rather than algebraically. Claudius Ptolemy expandedupon Hipparchus ' Chords in a Circle in his Almagest. The modern sine

function was first defined in the Surya Siddhanta, and its properties werefurther documented by the 5th century Indian mathematician andastronomer Aryabhatta. These Greek and Indian works were translated andexpanded by medieval Islamic mathematicians. By the 10th century, Islamicmathematicians were using all six trigonometric functions, had tabulatedtheir values, and were applying them to problems in spherical geometry . Atabout the same time, Chinese mathematicians developed trigonometryindependently, although it was not a major field of study for them.

Knowledge of trigonometric functions and methods reached Europe via Latintranslations of the works of Persian and Arabic astronomers such as AlBattani and Nasir al-Din al-Tusi

y Nasir al-Din al-Tusi. One of the earliest works on trigonometry by aEuropean mathematician is De Triangulis by the 15th century Germanmathematician Reiomontanus. Trigonometry was still so little known in 16thcentury Europe that Nicolaus Copernicus devoted two chapters of Derevolution bus erbium celestial to explaining its basic concepts.


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Calculating trigonometric

functionsy Trigonometric functions were among the earliest usesfor mathematical tables. Such tables were incorporated intomathematics textbooks and students were taught to look upvalues and how to interpolate between the values listed to get

higher accuracy. Slide rules had special scales for trigonometricfunctions.

y Today scientific calculators have buttons for calculating themain trigonometric functions (sin, cos, tan and sometimes cis)and their inverses. Most allow a choice of angle measurement

methods: degrees, radians and, sometimes, grad. Mostcomputer programming languages provide function librariesthat include the trigonometric functions. The floating pointunit hardware incorporated into the microprocessor chipsused in most personal computers have built-in instructions

for calculating trigonometric functions.


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Story of Trignometry