- Uses of trigonometry
has an enormous variety of applications. The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such asTrigonometry navigation , landsurveying ,building , and the like. It is also used extensively in a number of academic fields, primarilymathematics ,science andengineering .Among the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of

music ; still other uses are more technical, such as innumber theory . The mathematical topics ofFourier series andFourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, includingstatistics .**ome fields to which trigonometry is applied**Among the scientific fields that make use of trigonometry are these:

:

acoustics ,architecture ,astronomy (and hencenavigation , on the oceans, in aircraft, and in space; in this connection, see great circle distance),biology ,cartography ,chemistry ,civil engineering ,computer graphics ,geophysics ,crystallography ,economics (in particular in analysis offinancial markets ),electrical engineering ,electronics , landsurveying andgeodesy , manyphysical science s,mechanical engineering ,machining ,medical imaging (CAT scan s andultrasound ),meteorology ,music theory ,number theory (and hencecryptography ),oceanography ,optics ,pharmacology ,phonetics ,probability theory ,psychology ,seismology ,statistics , andvisual perception .**How these fields interact with trigonometry**The fact that these fields make use of trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It "does" mean that "some" things in these fields cannot be understood without trigonometry. For example, a professor of

music may perhaps know nothing of mathematics, but would probably know thatPythagoras was the earliest known contributor to the mathematical theory of music.In "some" of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. 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 are consonant 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 somewave s and the graph of the sine function is also not coincidental. In some other fields, among themclimatology , biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine functions.**Fourier series**Many fields make use of trigonometry in more advanced ways than can be discussed in a single article. Often those involve what are called

Fourier series , after the 18th- and 19th-century French mathematician and physicist Joseph Fourier. Fourier series have a surprisingly diverse array of applications in many scientific fields, in particular in all of the phenomena involving seasonal 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:

:$color\{white\}\; underbrace\{color\{black\}square\}\_\{0\}$ $\{\}+\; underbrace\{squarecos\; heta+squaresin\; heta\}\_\{1\}$ $\{\}+\; underbrace\{squarecos(2\; heta)+squaresin(2\; heta)\}\_\{2\}$ $\{\}+\; underbrace\{squarecos(3\; heta)+squaresin(3\; heta)\}\_\{3\}$ $\{\}+\; color\{white\}underbrace\{color\{black\}cdots\}\_\{infin\}$

where each of the squares ($square$) is a different number, and one is adding infinitely many terms. Fourier used these for studying

heat flow anddiffusion (diffusion is the process whereby, when you drop asugar 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 from obvious. One ubiquitous example is digital compression whereby images, audio and video data are compressed into a much smaller size which makes their transmission feasible over

telephone ,internet and broadcast networks. Another example, mentioned above, isdiffusion . Among others are: thegeometry of numbers , isoperimetric problems, recurrence ofrandom walk s,quadratic reciprocity , thecentral limit theorem ,Heisenberg's inequality .**Fourier transforms**A more abstract concept than Fourier series is the idea of

Fourier transform . Fourier transforms involveintegral s 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 quantities themselves. 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 adifferential equation . If, given this information, we try to express population as a function of time, we are trying to "solve" the differential equation. Fourier transforms may be used to convert some differential equations to algebraic equations for which methods of solving them are known. Fourier transforms have many uses. In almost any scientific context in which the words spectrum,harmonic , orresonance are encountered, Fourier transforms or Fourier series are nearby.**tatistics, including mathematical psychology**Intelligence quotients are per definition distributed according to the bell-shaped curve. About 40% of the area under the curve is in the interval from 100 to 120; correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. About 9% of the area under the curve 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 to the "bell-shaped curve", including measurement errors in many physical measurements and the number of times you get heads when you toss a coin 10,000 times. Why the ubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence

trigonometric function s). That is one of a variety of applications of Fourier transforms tostatistics .Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.

**A simple experiment with polarized sunglasses**Get two pairs of identical polarized sunglasses (

**un**polarized sunglasses won't work here). Put the left lens of one pair atop the right lens of the other, both aligned identically. Slowly rotate one pair, and you observe that the amount oflight that gets through decreases until the two lenses are atright angle s to each other, when no light gets through. When the angle through which the one pair is rotated is θ, what fractions of the light that penetrates when the angle is 0, gets through? Answer: it is cos^{2}θ. For example, when the angle is 60 degrees, only 1/4 as much light penetrates the series of twolenses as when the angle is 0 degrees, since the cosine of 60 degrees is 1/2.**Number theory**There is a hint of a connection between trigonometry and

number theory . Loosely speaking, one could say that number theory deals with qualitative rather than quantitative properties of numbers. A central concept in number theory is "divisibility" (as in: 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 sequence of fractions:$frac\{1\}\{42\},\; qquad\; frac\{2\}\{42\},\; qquad\; frac\{3\}\{42\},\; qquaddotsdots,\; qquad\; frac\{39\}\{42\},\; qquad\; frac\{40\}\{42\},\; qquadfrac\{41\}\{42\}.$

Discard the ones that are not in lowest terms; keep only those that are in lowest terms:

:$frac\{1\}\{42\},\; qquad\; frac\{5\}\{42\},\; qquad\; frac\{11\}\{42\},\; qquaddots,\; qquad\; frac\{31\}\{42\},\; qquad\; frac\{37\}\{42\},\; qquadfrac\{41\}\{42\}.$

Then bring in trigonometry:

:$cosleft(2picdotfrac\{1\}\{42\}\; ight)+cosleft(2picdotfrac\{5\}\{42\}\; ight)+cdots+cosleft(2picdotfrac\{37\}\{42\}\; ight)+cosleft(2picdotfrac\{41\}\{42\}\; ight)$

The value of the sum is −1. How do we know that? Because 42 has an "odd" number of prime factors 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

Möbius function evaluated at 42.) This hints at the possibility of applyingFourier analysis to number theory.

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