A Case Study On Applications Of Trigonometry In Oceanography
IJSRD - International Journal for Scientific Research & Development Vol. 5, Issue 12, 2018 ISSN (online): 2321-0613A Case Study on Applications of Trigonometry in OceanographySuvitha M.1 Kalaivani M.21,2Student1,2Department of Mathematics1,2Sri Krishna Arts and Science College, Coimbatore, Tamil Nadu, IndiaAbstract— This article introduces a trigonometric field thatextends in the field of real numbers adding two elements: sinand cos satisfying an axiom sin² cos² 1.It is shown thatassigning meaningful names to particular elements to thefield, All know trigonometric identities may be introducedand proved. The main objective of this study is aboutoceanography how the oceanographers guide everyone withthe awareness of the Tsunami waves and protect the marineanimals hitting the ships and cliffs.Key words: Trigonometric Identities, Trigonometric Ratios,Trigonometric Functionsbisected to create two right-angle triangles, most problemscan be reduced to calculations on right-angle triangles. Thusthe majority of applications relate to right-angle triangles.One exception to this is spherical trigonometry, the study oftriangles on spheres, surfaces of constant positive curvature,in elliptic geometry (a fundamental part of astronomy andnavigation). Trigonometry on surfaces of negative curvatureis part of hyperbolic geometry.II. HISTORY OF TRIGONOMETRYI. INTRODUCTIONA. TrigonometryFig. 1:All of the trigonometric functions of an angle θ can beconstructed geometrically in terms of a unit circle centered atO.Trigonometry (from Greek trigonon, "triangle" andmetron, "measure") is a branch of mathematics that studiesrelationships involving lengths and angles of triangles. Thefield emerged in the Hellenistic world during the 3rd centuryBC from applications of geometry to astronomical studies.The 3rd-century astronomers first noted that the lengths of thesides of a right-angle triangle and the angles between thosesides have fixed relationships: that is, if at least the length ofone side and the value of one angle is known, then all otherangles and lengths can be determined algorithmically. Thesecalculations soon came to be defined as the trigonometricfunctions and today are pervasive in both pure and appliedmathematics: fundamental methods of analysis such as theFourier transform, for example, or the wave equation, usetrigonometric functions to understand cyclical phenomenaacross many applications in fields as diverse as physics,mechanical and electrical engineering, music and acoustics,astronomy, ecology, and biology. Trigonometry is also thefoundation of surveying. Trigonometry is most simplyassociated with planar right-angle triangles (each of which isa two-dimensional triangle with one angle equal to 90degrees). The applicability to non-right-angle triangles exists,but, since any non-right-angle triangle (on a flat plane) can beFig. 2:Hipparchus, credited with compiling the first trigonometric,is known as "the father of trigonometry".A thick ring-like shell object found at the IndusValley Civilization site of Lothal, with four slits each in twomargins served as a compass to measure angles on planesurfaces or in the horizon in multiples of 40degrees,upto 360degrees. Such shell instruments were probably invented tomeasure 8–12 whole sections of the horizon and sky,explaining the slits on the lower and upper margins.Archaeologists consider this as evidence that the Lothalexperts had achieved an 8–12 fold division of horizon andsky, as well as an instrument for measuring angles andperhaps the position of stars, and for navigation. Sumerianastronomers studied angle measure, using a division of circlesinto 360 degrees. They, and later the Babylonians, studied theratios of the sides of similar triangles and discovered someproperties of these ratios but did not turn that into a systematicmethod for finding sides and angles of triangles. The ancientNubians used the same method. In the 3rd century BC,Hellenistic mathematicians such as Euclid and Archimedesstudied the properties of chords and inscribed angles incircles, and they proved theorems that are equivalent tomodern trigonometric formulae, although they presentedthem geometrically rather than algebraically. In 140 BC,Hipparchus (from Nicaea, Asia Minor) gave the first tables ofchords, analogous to modern tables of sine values, and usedthem to solve problems in trigonometry and sphericaltrigonometry. In the 2nd century AD, the Greco-Egyptianastronomer Ptolemy (from Alexandria, Egypt) printeddetailed trigonometric tables (Ptolemy's table of chords, hisAll rights reserved by www.ijsrd.com955
A Case Study on Applications of Trigonometry in Oceanography(IJSRD/Vol. 5/Issue 12/2018/253)Almagest. Ptolemy used chordlength to define histrigonometric functions, a minor difference from the sineconvention we use today. (The value we call sin(θ) can befound by looking up the chord length for twice the angle ofinterest (2θ) in Ptolemy's table, and then dividing that valueby two.) Centuries passed before more detailed tables wereproduced, and Ptolemy's treatise remained in use forperforming trigonometric calculations in astronomythroughout the next 1200 years in the medieval Byzantine,Islamic, and, later, Western European worlds.The modern sine convention is first attested in theSurya Siddhanta, and its properties were further documentedby the 5th century (AD) Indian mathematician andastronomer Aryabhata. These Greek and Indian works icians. By the 10th century, Islamic mathematicianswere using all six trigonometric functions, had tabulated theirvalues, and were applying them to problems in geometry. Atabout the same time, Chinese mathematicians developedtrigonometry independently, although it was not a major fieldof study for them. Knowledge of trigonometric functions andmethods reached Western Europe via Latin translations ofPtolemy's Greek Almagest as well as the works of Persian andArabic astronomers such as Al Battani and Tusi. One of theearliest works on trigonometry by a northern Europeanmathematician is De Triangulis by the 15th century Germanmathematician Regiomontanus, who was encouraged towrite, and provided with a copy of the Almagest, by theByzantine Greek scholar cardinal Basilios Bessarion withwhom he lived for several years At the same time, anothertranslation of the Almagest from Greek into Latin wascompleted by the Cretan Trebizond. Trigonometry was stillso little known in 16th-century northern Europe that NicolausCopernicus devoted two chapters of De revolution busorbiumcoelestium to explain its basic concepts Driven by thedemands of navigation and the growing need for accuratemaps of large geographic areas, trigonometry grew into amajor branch of mathematics. Bartholomaeus was the first touse the word, publishing his Trigonometria in 1595.GemmaFrisius described for the first time the method of triangulationstill used today in surveying. It was Leonhard Euler who fullyincorporated complex numbers into trigonometry. The worksof the Scottish mathematicians James Gregoryin the 17thcentury and Colin Maclaurin in the 18th century wereinfluential in the development of series. Also in the 18thcentury, Brook Taylor defined the general Taylor series.A. Trigonometric FunctionsIf one angle of a triangle is 90 degrees and one of the otherangles is known, the third is thereby fixed, because the threeangles of any triangle add up to 180 degrees. The two acuteangles therefore add up to 90 degrees: they arecomplementary. The shape of a triangle is completelydetermined, except for similarity, by the angles. Once theangles are known, the ratios of the sides are determined,regardless of the overall size of the triangle. If the length ofone of the sides is known, the other two are determined. Theseratios are given by the following trigonometric functions ofthe known angle A, where a, b and c refer to the lengths ofthe sides.1) In this Right TriangleabaSin A ; cos A ; tan A ccb Sine function (sin), defined as the ratio of the sideopposite the angle to the hypotenuse.OppositeaSinA Cosine function (cos), defined as the ratio of the adjacentleg to the hypotenuse.AdjacentbCosA Tangent function (tan), defined as the ratio of theopposite leg to the adjacent leg.Oppositea a c a bSinATanA * / HypotenuseHypotenuseccHypotenuse b c b c ccOSAThe hypotenuse is the side opposite to the 90 degreeangle in a right triangle; it is the longest side of the triangleand one of the two sides adjacent to angle A. The adjacent legis the other side that is adjacent to angle A. The opposite sideis the side that is opposite to angle A. The terms perpendicularand base are sometimes used for the opposite and adjacentsides respectively. The reciprocals of these functions arenamed the cosecant (csc or cosec), secant (sec), and cotangent(cot), respectively1HypotenusecCosecA SinASecA CotA 1oppositeaFig. 3:Hypotenusec CosAAdjacentb1AdjacentCosAbTanA Opposite SinA aThe inverse functions are called the arcsine,arccosine, and arctangent, respectively. There are arithmeticrelations between these functions, which are known astrigonometric identities. The cosine, cotangent, and cosecantare so named because they are respectively the sine, tangent,and secant of the complementary angle abbreviated to "co".With these functions, one can answer virtually all questionsabout arbitrary triangles by using the law of sine’s and the lawof cosines. These laws can be used to compute the remainingangles and sides of any triangle as soon as two sides and theirincluded angle or two angles and a side or three sides areknown. These laws are useful in all branches of geometry,since every polygon may be described as a finite combinationof triangles.III. PYTHAGOREAN IDENTITIESThe following identities are related to the Pythagoreantheorem and hold for any value.All rights reserved by www.ijsrd.com956
A Case Study on Applications of Trigonometry in Oceanography(IJSRD/Vol. 5/Issue 12/2018/253)Another law involving sines can be used to calculatethe area of a triangle. Given two sides a and b and the anglebetween the sides C, the area of the triangle is given by halfthe product of the lengths of two sides and the sine of theangle between the two sides.𝟏Area ab SinC𝟐D. Law of CosinesThe law of cosines (known as the cosine formula, or the "cosrule") is an extension of the Pythagorean Theorem to arbitrarytriangles.c² a² b²-2abCosCOr EquivalentlyFig. 4:sin²A cos²B 1tan²A 1 sec²Acot²A 1 cosec²A𝐚² 𝐛² 𝐜²A. Trigonometric RatioThe trigonometric functions are functions of an angle. Theyrelate the angles of a triangle to the lengths of its sidesTable 1:CosC 𝟐𝐚𝐛The law of cosines may be used to prove Heron'sformula, which is another method that may be used tocalculate the area of a triangle. This formula states that if atriangle has sides of lengths a, b, and c, and if the semiperimeter isS ½(a b c)Then the area of the triangle is𝐚𝐛𝐜Area s(s-a)(s-b)(s-c) 𝟒𝐑Where R is the radius of the circumcircle of the triangle.IV. APPLICATIONS IN TRIGONOMETRYB. Angle Transformation FormulaSin(A B) sinAcosB cosAsinBCos(A B) cosAcosB sinAsinBtan (A B) tanA tan/ 1 tanA tanBcot(A B) cotAcotB 1/ cotB cotAC. Law of SinesLaw of sines is a sine rule, it is an equation relatingthe lengths of the sides of a triangle to the sines of itsangles.abCabc 2R SinBSinC2 Where the area of the triangle and R is the radiusof the circumscribed circle of the triangle.abcR SinA (a b c)(a b c)(a b c)(b c a)Fig. 5:Sextants are used to measure the angle of the sun or stars withrespect to the horizon. Using trigonometry and a marinechronometer, the position of the ship can be determined fromsuch measurements.There is an enormous number of uses oftrigonometry and trigonometric functions. For instance, thetechnique of triangulation is used in astronomy to measurethe distance to nearby stars, in geography to measuredistances between landmarks, and in satellite navigationsystems. The sine and cosine functions are fundamental to theAll rights reserved by www.ijsrd.com957
A Case Study on Applications of Trigonometry in Oceanography(IJSRD/Vol. 5/Issue 12/2018/253)theory of periodic, such as those that describe soundand light waves. Fields that use trigonometry ortrigonometric functions include astronomy (especially forlocating apparent positions of celestial objects, in whichspherical trigonometry is essential) and hence navigation (onthe oceans, in aircraft, and in space), music theory, audiosynthesis, acoustics, optics, electronics, biology, medicalimaging (CAT scans and ultrasound), pharmacy, chemistry,number theory (and hence cryptology),seismology,meteorology, oceanography, many physical sciences, landsurveying and geodesy, architecture, image compression,phonetics, economics, electrical engineering, mechanicalengineering, civil engineering, graphics, cartography,crystallography and game development.A. Uses of Trigonometry in AstronomyTrigonometry is often used to find distances to nearbystars and other celestial objects using a method ofparallax. Parallax can be defined as the apparent shift ofa nearby star against the fixed background that can benoticed as the earth orbits the sun.B. Uses of Trigonometry in Game DevelopmentTrigonometry is used extensively in game developmentin order for the game to function. Trigonometry is usedin writing the program for games so that objects canmove . Also used for designing object, characters and sets.C. Uses of Trigonometry in ArchitectureTranscript of how trigonometry is used in architecturefor example architects have to calculate exact angles ofintersection for components of their structure to ensurestability and safety. Example of trigonometric use inarchitecture include arches, domes, support beams andsuspension bridges.D. Uses of Trigonometry in Music TheoryTrigonometry plays a major role in musical theory andproduction. Sound waves travel in repeating wave pattern,which can be represented graphically by sine and cosinefunction. A single note can be modeled on sine curve,and chord can be modeled with multiple sine curves usedin conjunction with one another.E. Uses of Trigonometry in NavigationTrigonometry was developed for use in sailing as anavigation method used with astronomy. It is the branchof trigonometry concerned with the measurement of theangles and sides of spherical triangles. It is used forplanning long distances routes around the world.F. Uses of Trigonometry in ChemistryIn chemistry, chemist use trigonometry when accuratelydescribing the angles that are created when atoms bondtogether to form molecules geometry. Trigonometricfunctions such as sine cosines are essential to describematerials in their three dimensions.G. Uses of Trigonometry in CrystallographyTrigonometric functions, trigonometry comes up in manyaspects of biology One example is x-ray crystallography,a technique used to determine the three dimensionalstructure of molecule etc.V. OCEANOGRAPHYA. Introduction of OceanographyOceanography also known as Oceanology is the study of thephysical and biological aspects of the ocean. Oceanographycovers a wide range of topics including marine life andecosystems, ocean circulation, waves, plate tectonics and thegeology of the sea floor, and the chemical and physicalproperties of the ocean. Oceanography is the application ofall science to the phenomena of the ocean. To trulyunderstand the ocean and how it works, one must knowsomething about almost all fields of science and theirrelationship to the marine environment. Thus, oceanographyis no single science but rather a combination of varioussciences. The objective of oceanography, at least to thescientist, is to increase human understanding of all aspects ofthe world’s ocean and of the processes. This is encompassesthe subsidiary aim of describing as many marine features aspossible of the many scientific disciplines that make upoceanography, one would be justified in calling the study ofthe oceans a very board science, and however in another senseoceanography is restrictive. It is not a universal science likephysics or chemistry; where in the physical laws governingmatter appear to have application throughout our universe.B. Scope1) Chemical OceanographyChemical reactions that occur both in the ocean and on thesea floor.2) Biologcial OceanographyBiological oceanography deals with the distribution andenvironmental aspect of life in the ocean3) Physical OceanographyPhysical reactions such as changes and motion of the ocean.4) Geological OceanographyGeological oceanography is used to study the sediments andtopography of the ocean floor.5) Ocean EngineeringOcean engineering concerns with the development oftechnology for oceanographic research and exploitation.6) Marine PolicyMarine policy considers the application of social and politicalsciences such as economics, laws and policy towards the useand management of the ocean.7) Satellite OceanographyIt deals with the measurements of the ocean color, which canbe used assessments of phytoplankton biomass and are ofgreat interest to marine biologists.C. History of OceanographyHumans first acquired knowledge of the waves and currentsof the seas and oceans in pre-historic times. Observations ontides were recorded by Aristotle and Strabo. Early explorationof the oceans was primarily for cartography and mainlylimited to its surfaces and of the animals that fishermenbrought up in nets, though depth soundings by lead line weretaken. Although Juan Ponce de Leon in 1513 first identifiedthe Gulf Stream, and the current was well known to mariners,All rights reserved by www.ijsrd.com958
A Case Study on Applications of Trigonometry in Oceanography(IJSRD/Vol. 5/Issue 12/2018/253)Benjamin Franklin made the first scientific study of it andgave it its name. Franklin measured water temperaturesduring several Atlantic crossings and correctly explained theGulf Stream’s cause. Franklin and Timothy Folger printed thefirst map of the Gulf Stream in 1769-1770.Fig. 6:Map of the Gulf Stream by Benjamin Franklin,1769-1770. Courtesy of the NOVAA photo libraryInformation on the currents of the Pacific Ocean was gatheredby explorers of the late 18th century including James Cookand Louis Antoine De Bougainville. James Rennell wrote thefirst scientific textbooks on oceanography, detailing thecurrent flows of the Atlantic and Indian oceans. During avoyage around the Cape of Good Hope in 1777, he mapped“the banks and currents at the Lagullas”. He was also the firstto understand the nature of the intermittent current near theIsles of Scilly, (now known as Rennell”s Current).Fig. 7:1799 Map of the currents in the Atlantic and Indianoceans, by James Rennell Sir James Clark Ross took the firstmodern sounding in deep sea in 1840, and Charles Darwinpublished a paper on reefs and the formation of atolls as aresult of the second voyage of HMS Beagle in 1831-6. RobertFitzRoy Published a four-volume report of the Beagle’s threevoyages. In 1841-1842 Edward Forbes undertook dredging inthe Aegean Sea that founded marine ecology. The firstsuperintendent of the United States naval observatory (18421861), Matthew Fontaine Maury devoted his time to the studyof marine meteorology, navigation, and charting prevailingwinds and currents.His 1855 textbook physical geography of the seawas one of the first comprehensive oceanography studies.Many nations sent oceanographic observations to Maury atthe naval observatory, where he and his colleagues evaluatedthe information and distributed the results worldwide.D. Modern OceanographyHuman knowledge of the oceans remained confined to thetopmost few fathoms of the water and a small amount of thebottom, mainly in shallow areas. Almost nothing was knownof the ocean depths. The Royal Navy's efforts to chart all ofthe world's coastlines in the mid-19th century reinforced thevague idea that most of the ocean was very deep, althoughlittle more was known. As explorati
A Case Study on Applications of Trigonometry in Oceanography Suvitha M.1 Kalaivani M.2 1,2Student 1,2Department of Mathematics 1,2Sri Krishna Arts and Science College, Coimbatore, Tamil Nadu, India Abstract— This article introduces a trigonometric field that extends in the field of real numbers adding two elements: sin
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