uses of trigonometry

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Uses of trigonometry

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Trigonometry has an enormous variety of applications. Theones mentioned explicitly in textbooks and courses ontrigonometry are its uses in practical endeavors such as

navigation , land surveying , building , and the like. It is also usedextensively in a number of academic fields, primarilymathematics , science and engineering .

Among the lay public of non-mathematicians and non-scientists,trigonometry is known chiefly for its application tomeasurement problems, yet is also often used in ways that arefar more subtle, such as its place in the theory of music ; stillother uses are more technical, such as in number theory . Themathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and findapplication in a number of areas, including statistics .

Contents 1 Some fields to which trigonometry is applied 2 How these fields interact with trigonometry 3 Fourier series 4 Fourier transforms 5 Statistics, including mathematical psychology 6 A simple experiment with polarized sunglasses 7 Number theory 8 Sources

Some fields to which trigonometry is appliedAmong the scientific fields that make use of trigonometry are these:

acoustics , architecture , astronomy (and hence navigation , on the oceans, in aircraft, andin space; in this connection, see great circle distance ), biology , cartography , chemistry , civil engineering , computer graphics , geophysics , crystallography , economics (in

Trigonometry

HistoryUsage

FunctionsInverse functionsFurther reading

Reference

List of identitiesExact constants

Generating trigonometrictables

Euclidean theory

Law of sinesLaw of cosinesLaw of tangents

Pythagorean theorem

Calculus

Trigonometric substitutionIntegrals of functions

Derivatives of functionsIntegrals of inverses

This article needs additional citationsfor verification .Please help improve this article by adding reliablereferences . Unsourced material may bechallenged and removed . (December 2008)
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http://en.wikipedia.org/wiki/Chemistryhttp://en.wikipedia.org/wiki/Civil_engineeringhttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Geophysicshttp://en.wikipedia.org/wiki/Crystallographyhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Trigonometry


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particular in analysis of financial markets ), electrical engineering , electronics , landsurveying and geodesy , many physical sciences , mechanical engineering , machining ,medical imaging (CAT scans and ultrasound ), meteorology , music theory , number theory (and hence cryptography ), oceanography , optics , pharmacology , phonetics , probabilitytheory , psychology , seismology , statistics , and visual perception .

How these fields interact with trigonometryThe fact that these fields make use of trigonometry does not mean knowledge of trigonometry isneeded in order to learn anything about them. It does mean that some things in these fieldscannot be understood without trigonometry. For example, a professor of music may perhapsknow nothing of mathematics, but would probably know that Pythagoras was the earliest knowncontributor to the mathematical theory of music.

In some of the fields of endeavor listed above it is easy to imagine how trigonometry could beused. For example, in navigation and land surveying, the occasions for the use of trigonometryare in at least some cases simple enough that they can be described in a beginning trigonometrytextbook. In the case of music theory, the application of trigonometry is related to work begun by

Pythagoras, who observed that the sounds made by plucking two strings of different lengths areconsonant if both lengths are small integer multiples of a common length. The resemblance between the shape of a vibrating string and the graph of the sine function is no mere coincidence.In oceanography, the resemblance between the shapes of some waves and the graph of the sinefunction is also not coincidental. In some other fields, among them climatology , biology, andeconomics, there are seasonal periodicities. The study of these often involves the periodic natureof the sine and cosine functions.

Fourier seriesMany fields make use of trigonometry in more advanced ways than can be discussed in a singlearticle. Often those involve what are called Fourier series , after the 18th- and 19th-centuryFrench mathematician and physicist Joseph Fourier . Fourier series have a surprisingly diversearray of applications in many scientific fields, in particular in all of the phenomena involvingseasonal periodicities mentioned above, and in wave motion, and hence in the study of radiation,of acoustics, of seismology, of modulation of radio waves in electronics, and of electric power engineering.

A Fourier series is a sum of this form:

where each of the squares ( ) is a different number, and one is adding infinitely many terms.Fourier used these for studying heat flow and diffusion (diffusion is the process whereby, whenyou drop a sugar cube into a gallon of water, the sugar gradually spreads through the water, or a

pollutant spreads through the air, or any dissolved substance spreads through any fluid).

Fourier series are also applicable to subjects whose connection with wave motion is far fromobvious. One ubiquitous example is digital compression whereby images , audio and video dataare compressed into a much smaller size which makes their transmission feasible over telephone ,internet and broadcast networks . Another example, mentioned above, is diffusion . Among others
http://en.wikipedia.org/wiki/Financial_marketshttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Surveyinghttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Physical_sciencehttp://en.wikipedia.org/wiki/Mechanical_engineeringhttp://en.wikipedia.org/wiki/Mechanical_engineeringhttp://en.wikipedia.org/wiki/Machininghttp://en.wikipedia.org/wiki/Medical_imaginghttp://en.wikipedia.org/wiki/Medical_imaginghttp://en.wikipedia.org/wiki/CAT_scanhttp://en.wikipedia.org/wiki/CAT_scanhttp://en.wikipedia.org/wiki/Ultrasoundhttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Music_theoryhttp://en.wikipedia.org/wiki/Music_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Cryptographyhttp://en.wikipedia.org/wiki/Oceanographyhttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Pharmacologyhttp://en.wikipedia.org/wiki/Phoneticshttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Seismologyhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Visual_perceptionhttp://en.wikipedia.org/wiki/Musichttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Climatologyhttp://en.wikipedia.org/wiki/Climatologyhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Jean_Baptiste_Joseph_Fourierhttp://en.wikipedia.org/wiki/Heathttp://en.wikipedia.org/wiki/Heathttp://en.wikipedia.org/wiki/Diffusionhttp://en.wikipedia.org/wiki/Sugar_cubehttp://en.wikipedia.org/wiki/Data_compressionhttp://en.wikipedia.org/wiki/Image_compressionhttp://en.wikipedia.org/wiki/Audio_compressionhttp://en.wikipedia.org/wiki/Video_compressionhttp://en.wikipedia.org/wiki/Telephonehttp://en.wikipedia.org/wiki/Internethttp://en.wikipedia.org/wiki/Broadcastinghttp://en.wikipedia.org/wiki/Computer_networkhttp://en.wikipedia.org/wiki/Diffusionhttp://en.wikipedia.org/wiki/Financial_marketshttp://en.wikipedia.org/wiki/Electrical_engineeringhttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Surveyinghttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Physical_sciencehttp://en.wikipedia.org/wiki/Mechanical_engineeringhttp://en.wikipedia.org/wiki/Machininghttp://en.wikipedia.org/wiki/Medical_imaginghttp://en.wikipedia.org/wiki/CAT_scanhttp://en.wikipedia.org/wiki/Ultrasoundhttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Music_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Cryptographyhttp://en.wikipedia.org/wiki/Oceanographyhttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Pharmacologyhttp://en.wikipedia.org/wiki/Phoneticshttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Psychologyhttp://en.wikipedia.org/wiki/Seismologyhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Visual_perceptionhttp://en.wikipedia.org/wiki/Musichttp://en.wikipedia.org/wiki/Pythagorashttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Climatologyhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Jean_Baptiste_Joseph_Fourierhttp://en.wikipedia.org/wiki/Heathttp://en.wikipedia.org/wiki/Diffusionhttp://en.wikipedia.org/wiki/Sugar_cubehttp://en.wikipedia.org/wiki/Data_compressionhttp://en.wikipedia.org/wiki/Image_compressionhttp://en.wikipedia.org/wiki/Audio_compressionhttp://en.wikipedia.org/wiki/Video_compressionhttp://en.wikipedia.org/wiki/Telephonehttp://en.wikipedia.org/wiki/Internethttp://en.wikipedia.org/wiki/Broadcastinghttp://en.wikipedia.org/wiki/Computer_networkhttp://en.wikipedia.org/wiki/Diffusion


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are: the geometry of numbers , isoperimetric problems , recurrence of random walks , quadraticreciprocity , the central limit theorem , Heisenberg's inequality .

Fourier transformsA more abstract concept than Fourier series is the idea of Fourier transform . Fourier transforms

involve integrals rather than sums, and are used in a similarly diverse array of scientific fields.Many natural laws are expressed by relating rates of change of quantities to the quantitiesthemselves. For example: The rate of change of population is sometimes jointly proportional to(1) the present population and (2) the amount by which the present population falls short of thecarrying capacity . This kind of relationship is called a differential equation . If, given thisinformation, we try to express population as a function of time, we are trying to "solve" thedifferential equation. Fourier transforms may be used to convert some differential equations toalgebraic equations for which methods of solving them are known. Fourier transforms havemany uses. In almost any scientific context in which the words spectrum , harmonic , or resonance are encountered, Fourier transforms or Fourier series are nearby.

Statistics, including mathematical psychologyIntelligence quotients are sometimes held to be distributed according to the bell-shaped curve .About 40% of the area under the curve is in the interval from 100 to 120; correspondingly, about40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under thecurve is in the interval from 120 to 140; correspondingly, about 9% of the population scores

between 120 and 140 on IQ tests, etc. Similarly many other things are distributed according tothe "bell-shaped curve", including measurement errors in many physical measurements. Why theubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence trigonometric functions ). That is one of a variety of applications of Fourier transforms to statistics .

Trigonometric functions are also applied when statisticians study seasonal periodicities, whichare often represented by Fourier series.

A simple experiment with polarized sunglassesGet two pairs of identical polarized sunglasses ( un polarized sunglasses won't work here). Put theleft lens of one pair atop the right lens of the other, both aligned identically. Slowly rotate one

pair, and you observe that the amount of light that gets through decreases until the two lenses areat right angles to each other, when no light gets through. When the angle through which the one

pair is rotated is , what fractions of the light that penetrates when the angle is 0, gets through?Answer: it is cos 2 . For example, when the angle is 60 degrees, only 1/4 as much light

penetrates the series of two lenses as when the angle is 0 degrees, since the cosine of 60 degreesis 1/2.

Number theoryThere is a hint of a connection between trigonometry and number theory. Loosely speaking, onecould say that number theory deals with qualitative properties rather than quantitative propertiesof numbers. A central concept in number theory is "divisibility" (example: 42 is divisible by 14

but not by 15). The idea of putting a fraction in lowest terms also uses the concept of divisibility:e.g., 15/42 is not in lowest terms because 15 and 42 are both divisible by 3. Look at the sequenceof fractions
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Discard the ones that are not in lowest terms; keep only those that are in lowest terms:

Then bring in trigonometry:

The value of the sum is 1. How do we know that? Because 42 has an odd number of primefactors and none of them are repeated: 42 = 2 3 7. (If there had been an even number of non-repeated factors then the sum would have been 1; if there had been any repeated prime factors(e.g., 60 = 2 2 3 5) then the sum would have been 0; the sum is the Mbius functionevaluated at 42.) This hints at the possibility of applying Fourier analysis to number theory.

SourcesUses of trigonometry. In State Master [Web]. Retrieved June 23, 2009, fromhttp://www.statemaster.com/encyclopedia/Uses-of-trigonometry
http://en.wikipedia.org/wiki/M%C3%B6bius_functionhttp://en.wikipedia.org/wiki/Fourier_analysishttp://en.wikipedia.org/wiki/Fourier_analysishttp://www.statemaster.com/encyclopedia/Uses-of-trigonometryhttp://en.wikipedia.org/wiki/M%C3%B6bius_functionhttp://en.wikipedia.org/wiki/Fourier_analysishttp://www.statemaster.com/encyclopedia/Uses-of-trigonometry

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