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FORMATTIVEASSESSMENT

MATHS PROJECT

WORK

4



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TRIGONOMETRY •Introducti

on

•Trigonometric

Identities

• TrigonometricRatios

•

History



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•

INTRODUCTION •To estimate the height, H, of a building,

measure the distance, D, from the pointof observation to the base of thebuilding and the angle, (theta), shownθ

in the diagram. The ratio of the height H

to the distance D is equal to thetrigonometric function tangent (H/D =θ

tan ). To calculate H, multiply tangentθ

by the distance D (H = D tan ). Theθ θ

angle can be roughly estimated bypointing one arm at the base of thebuilding and the other arm at the roof and judging whether the angle formed is

close to 15°, 30°, 45°, 60°, or 75°. Theangle can be estimated more accuratelywith a protractor and a plumb bob madeof a pencil hanging from a string. Hangthe plumb bob from the zero point in themiddle of the straight edge of theprotractor. Sight along the edge of theprotractor at the roof of the building.Measure the angle formed by thestraight edge of the protractor and the



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•Trigonometry, branch of mathematics that deals withthe relationships between the sides and angles of

triangles and with the properties and applications of thetrigonometric functions of angles. The two branches of trigonometry are plane trigonometry, which deals withfigures lying wholly in a single plane, and sphericaltrigonometry, which deals with triangles that aresections of the surface of a sphere.

•The earliest applications of trigonometry were inthe fields of navigation, surveying, and astronomy,in which the main problem generally was todetermine an inaccessible distance, such as thedistance between the earth and the moon, or of a

distance that could not be measured directly, suchas the distance across a large lake. Otherapplications of trigonometry are found in physics,chemistry, and almost all branches of engineering,particularly in the study of periodic phenomena,

such as vibration studies of sound, a bridge, or abuilding, or the flow of alternating current.



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•

TRIGONOMETRICRATIOS• If is one of the acutθ

e angles of a righttriangle, the definitions

of the trigonometricfunctions given abovecan be applied to asθ

follows (Fig. 4). Imaginethe vertex A is placed atthe intersection of the x -axis and y -axis in Fig. 3,that AC extends alongthe positive x -axis, and

that B is the point P, sothat AB = AP = r. Then



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•The numerical values of thetrigonometric functions of a

few angles can be readilyobtained; for example, eitheracute angle of an isoscelesright triangle is 45°, as shownin Fig. 4. Therefore, it follows

that;



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• SINE

•Sine, one of the fundamental ratios of trigonometry. A ratio is a proportionalrelationship between two numberscalculated by dividing one number by theother. Sine embodies the relationshipbetween the lengths of the sides of a

right triangle (a triangle with a 90° angle)and the magnitudes of its angles. Thisrelationship means that varying onevalue, such as the length of a side,requires another value, such as themagnitude of an angle, to change in apredictable way.• The sine, usually abbreviated sin, of oneof the acute (less than 90°) angles of aright triangle is equal to the length of theside opposite the acute angle divided bythe length of the longest side, called thehypotenuse:

.



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• COSECA

NT•Cosecant, one of the sixfundamental ratios of trigonometry. The others are

sine, cosine, secant, tangent,and cotangent. A ratio is aproportional relationshipbetween two numberscalculated by dividing one

number by the other. Cosecantembodies such a relationshipbetween the magnitudes of theangles of a right triangle (a

triangle with one 90° angle)and the lengths of the sides.



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•Tange

nt• Tangent (trigonometry), one of the sixfundamental ratios of trigonometry. The othersare sine, cosine, secant, cosecant, and cotangent.A ratio is a proportional relationship between twonumbers calculated by dividing one number bythe other. Tangent embodies such a relationshipbetween the magnitudes of the angles of a righttriangle (a triangle having one 90° angle) and thelengths of the sides. Varying one value, such asthe magnitude of an angle, requires the relatedvalue, such as the length of a side, to change in apredictable way.• The tangent, usually abbreviated tan, of one of

the acute (less than 90°) angles of a right triangleis found by dividing the length of the sideopposite the angle by the length of the shorter of the two sides adjacent to the angle. (The otheradjacent side is called the hypotenuse, and is thetriangle’s longest side.)



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•Cotangent, one of the six fundamentalratios of trigonometry, along with tangent,sine, cosine, secant, and cosecant. (A ratiois a proportional relationship between twonumbers calculated by dividing one

number by the other.) Cotangent embodiesthe relationship between the magnitudes of the angles of a right triangle—a trianglewith one 90° angle—and the lengths of itssides. Varying one of the numbers, such asthe length of a side, requires the relatednumber, such as the magnitude of an

angle, to change in a predictable way.• The cotangent, usually abbreviated cot, of one of the acute (less than 90°) angles of aright triangle is equal to the length of theshorter of the two sides adjacent to theangle divided by the length of the sideopposite the angle:

•Cotangent



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•Cosine, one of the sixfundamental ratios of trigonometry: cosine, sine, secant,cosecant, tangent, and cotangent.(A ratio is a proportionalrelationship between two numberscalculated by dividing one numberby the other.) Cosine embodies the

relationship between themagnitudes of the angles of a righttriangle—a triangle with one 90°angle—and the lengths of its sides.Varying one value, such as themagnitude of an angle, requires

•Cosin

e



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•Secant

•Secant (trigonometry), one of thefundamental ratios of trigonometry. A ratio is a

proportional relationship betweentwo numbers calculated bydividing one number by the other.Secant embodies the relationship

between the lengths of the sidesof a right triangle (a triangle witha 90° angle) and the magnitudesof the angles. This relationshipmeans that varying one value,

such as the length of a side,



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•TRIGONOMETRICIDENTITIES

• The following formulas, calledidentities, which show therelationships between the

trigonometric functions, hold forall values of the angle , or of θtwo angles, and , for whichθ φthe functions involved are:



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•By repeated use of one or more of the formulas

in group V, which are known as reductionformulas, sin and cos can be expressed for anyθ θ

value of , in terms of the sine and cosine of θ

angles between 0° and 90°. By use of the formulasin groups I and II, the values of tan , cot , sec ,θ θ θ

and csc may be found from the values of sinθ θ

and cos . It is therefore sufficient to tabulate theθ

values of sin and cos for values of betweenθ θ θ

0° and 90°; in practice, to avoid tedious

calculations, the values of the other four functionsalso have been made available in tabulations forthe same range of .θ

•The variation of the values of the trigonometricfunctions for different angles may be representedby graphs, as in Fig. 5. It is readily ascertained



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•

HISTORY • The history of trigonometry goes back to the earliestrecorded mathematics in Egypt and Babylon. The Babylonians

established the measurement of angles in degrees, minutes,and seconds. Not until the time of the Greeks, however, did anyconsiderable amount of trigonometry exist. In the 2nd centurybc the astronomer Hipparchus compiled a trigonometric tablefor solving triangles. Starting with 7y° and going up to 180° bysteps of 7y°, the table gave for each angle the length of the

chord subtending that angle in a circle of a fixed radius r. Such atable is equivalent to a sine table. The value that Hipparchusused for r is not certain, but 300 years later the astronomerPtolemy used r = 60 because the Hellenistic Greeks hadadopted the Babylonian base-60 (sexagesimal) numeration

system .



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•In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for

steps of y°, from 0° to 180°, that is accurate to 1/3600 of a unit. He also explained hismethod for constructing his table of chords, and in the course of the book he gave manyexamples of how to use the table to find unknown parts of triangles from known parts.Ptolemy provided what is now known as Menelaus's theorem for solving spherical triangles,as well, and for several centuries his trigonometry was the primary introduction to thesubject for any astronomer. At perhaps the same time as Ptolemy, however, Indianastronomers had developed a trigonometric system based on the sine function rather thanthe chord function of the Greeks. This sine function, unlike the modern one, was not a ratio

but simply the length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.•Late in the 8th century, Muslim astronomers inherited both the Greek and the Indiantraditions, but they seem to have preferred the sine function. By the end of the 10th centurythey had completed the sine and the five other functions and had discovered and provedseveral basic theorems of trigonometry for both plane and spherical triangles. Severalmathematicians suggested using r = 1 instead of r = 60; this exactly produces the modernvalues of the trigonometric functions. The Muslims also introduced the polar triangle forspherical triangles. All of these discoveries were applied both for astronomical purposes andas an aid in astronomical time-keeping and in finding the direction of Mecca for the five dailyprayers required by Muslim law. Muslim scientists also produced tables of great precision.For example, their tables of the sine and tangent, constructed for steps of 1/60 of a degree,were accurate for better than one part in 700 million. Finally, the great astronomer Nasir al-Din al-Tusi wrote the Book of the Transversal Figure, which was the first treatment of planeand spherical trigonometry as independent mathematical sciences.



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• The Latin West became acquainted with Muslim trigonometry through translations of Arabicastronomy handbooks, beginning in the 12th century. The first major Western work on thesubject was written by the German astronomer and mathematician Johann Müller, known asRegiomontanus. In the next century the German astronomer Georges Joachim, known asRheticus introduced the modern conception of trigonometric functions as ratios instead of asthe lengths of certain lines. The French mathematician François Viète introduced the polartriangle into spherical trigonometry, and stated the multiple-angle formulas for sin(nq) andcos(nq) in terms of the powers of sin(q) and cos(q).• Trigonometric calculations were greatly aided by the Scottish mathematician John Napier,who invented logarithms early in the 17th century. He also invented some memory aids forten laws for solving spherical triangles, and some proportions (called Napier's analogies) for

solving oblique spherical triangles.•Almost exactly one half century after Napier's publication of his logarithms, Isaac Newtoninvented the differential and integral calculus. One of the foundations of this work wasNewton's representation of many functions as infinite series in the powers of x (see Sequenceand Series). Thus Newton found the series sin(x) and similar series for cos(x) and tan(x). Withthe invention of calculus, the trigonometric functions were taken over into analysis, wherethey still play important roles in both pure and applied mathematics.•Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the

trigonometric functions in terms of complex numbers (see Number). This made the wholesubject of trigonometry just one of the many applications of complex numbers, and showedthat the basic laws of trigonometry were simply consequences of the arithmetic of thesenumbers.



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THE END

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vishal singh x'a

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