Trigonometry has an enormous variety of applications. The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as navigation, land surveying, building, and the like. It is also used extensively in a number of academic fields, primarily mathematics, science and engineering. Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... Surveyor at work with a leveling instrument. ... For other uses, see Building (disambiguation). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Part of a scientific laboratory at the University of Cologne. ... Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ...

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 in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics. For other uses, see Music (disambiguation). ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... The Fourier transform, named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... This article is about the field of statistics. ...

Some fields to which trigonometry is applied

Among the scientific fields that make use of trigonometry are these:

acoustics, architecture, astronomy (and hence navigation, 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 of financial markets), electrical engineering, electronics, land surveying 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, probability theory, psychology, seismology, statistics, and visual perception.

Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... This article is about building architecture. ... For other uses, see Astronomy (disambiguation). ... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the spheres interior). ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... For other uses, see Chemistry (disambiguation). ... The Falkirk Wheel in Scotland. ... For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ... ‹ The template below has been proposed for deletion. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... In finance, financial markets facilitate: The raising of capital (in the capital markets); The transfer of risk (in the derivatives markets); and International trade (in the currency markets). ... Electrical Engineers design power systems… … and complex electronic circuits. ... This article is about the engineering discipline. ... Surveyor at work with a leveling instrument. ... Geodetic pillar (1855); Ostend, Belgium Archive with lithography plates for maps of Bavaria in the Landesamt für Vermessung und Geoinformation in Munich Geodesy (IPA North American English ; British, Australian English etc. ... Physical science is a encompassing term for the branches of natural science, and science, that study non-living systems, in contrast to the biological sciences. ... Mechanical Engineering is an engineering discipline that involves the application of principles of physics for analysis, design, manufacturing, and maintenance of mechanical systems. ... A lathe is a common tool used in machining. ... Medical imaging designates the ensemble of techniques and processes used to create images of the human body (or parts thereof) for clinical purposes (medical procedures seeking to reveal, diagnose or examine disease) or medical science (including the study of normal anatomy and function). ... CAT apparatus in a hospital Computed axial tomography (CAT), computer-assisted tomography, computed tomography, CT, or body section roentgenography is the process of using digital processing to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around... Ultrasound is a form of cyclic sound pressure with a frequency greater than the upper limit of human hearing, this limit being approximately 20 kilohertz (20,000 hertz). ... // Meteorology (from Greek: μετέωρον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... Music theory is a field of study that investigates the nature or mechanics of music. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ... Thermohaline circulation Oceanography (from Ocean + Greek γράφειν = write), also called oceanology or marine science, is the branch of Earth Sciences that studies the Earths oceans and seas. ... For the book by Sir Isaac Newton, see Opticks. ... Pharmacology (in Greek: pharmakon (φάρμακον) meaning drug, and lego (λέγω) to tell (about)) is the study of how drugs interact with living organisms to produce a change in function. ... Phonetics (from the Greek word φωνή, phone meaning sound, voice) is the study of the sounds of human speech. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Psychology (from Greek: ψυχή, psukhē, spirit, soul; and λόγος, logos, knowledge) is both an academic and applied discipline involving the scientific study of mental processes and behavior. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the propagation of elastic waves through the Earth. ... This article is about the field of statistics. ... This does not cite any references or sources. ...

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 that Pythagoras was the earliest known contributor to the mathematical theory of music. For other uses, see Music (disambiguation). ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ...

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 some waves and the graph of the sine function is also not coincidental. In some other fields, among them climatology, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine functions. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time,[1] and is a branch of the atmospheric sciences. ...

Fourier series

Many fields make use of trigonometry in a more advanced way 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. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ...

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, when you 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). For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... This article does not cite any references or sources. ... Magnified view of refined sugar crystals. ...

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, is diffusion. Among others are: the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenberg's inequality. Image compression is the application of Data compression on digital images. ... Audio compression can mean two things: Audio data compression - in which the amount of data in a recorded waveform is reduced for transmission. ... Video compression refers to making a digital video signal use less data, without noticeably reducing the quality of the picture. ... For other uses, see Telephone (disambiguation). ... Broadcasting is the distribution of audio and/or video signals which transmit programs to an audience. ... For the scientific and engineering discipline studying computer networks, see Computer networking. ... This article does not cite any references or sources. ... In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. ... Isoperimetry literally means having an equal perimeter. In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. ... Example of eight random walks in one dimension starting at 0. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...

Fourier transforms

A 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 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 the carrying capacity. This kind of relationship is called a differential 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, or resonance are encountered, Fourier transforms or Fourier series are nearby. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... Carrying capacity usually refers to the biological carrying capacity of a population level that can be supported for an organism, given the quantity of food, habitat, water and other life infrastructure present. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... The word spectrum (plural, spectra) has many uses: // Common nouns The Spectrum article explains why so many things are called by this name The spectrum of activity of a drug The political spectrum of opinion The economic spectrum The bipolar spectrum, in psychology The autistic spectrum, in psychology In the... This article is about the components of sound. ... This article is about resonance in physics. ...

Statistics, including mathematical psychology

Some psychologists have claimed that intelligence quotients are distributed according to the celebrated 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 functions). That is one of a variety of applications of Fourier transforms to statistics. Psychology (from Greek: ψυχή, psukhē, spirit, soul; and λόγος, logos, knowledge) is both an academic and applied discipline involving the scientific study of mental processes and behavior. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... This article is about the field of statistics. ...

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 (unpolarized 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 of light that gets through decreases until the two lenses are at right angles to each other, when no light gets through. When the angle through which the one pair is rotated is θ, what fraction of the light that penetrates when the angle is 0, gets through? Answer: it is cos² θ. 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 degrees is 1/2. This article treats polarization in electrodynamics. ... Ray-Ban Wayfarer sunglasses (RB2132 901L) Sunglasses are a visual aid, variously termed spectacles or glasses, which feature lenses that are coloured or darkened to prevent strong light from reaching the eyes. ... This article does not cite any references or sources. ... This article is about angles in geometry. ...

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 Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

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 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 applying Fourier analysis to number theory. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...