Uses of trigonometry
Encyclopedia
Amongst 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
.
may perhaps know nothing of mathematics, but would probably know that Pythagoras
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 some wave
s 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 function.
, 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:
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).
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 walk
s, quadratic reciprocity
, the central limit theorem
, Heisenberg's inequality.
. Fourier transforms involve integral
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 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.
s. That is one of a variety of applications of Fourier transforms to statistics
.
Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.
that gets through decreases until the two lenses are at right 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 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.
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 is 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.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
; still other uses are more technical, such as in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. The mathematical topics of Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
.
Some fields in which trigonometry is applied
Scientific fields that make use of trigonometry include:- acousticsAcousticsAcoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, architectureArchitectureArchitecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
, astronomyAstronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, cartographyCartographyCartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...
, civil engineeringCivil engineeringCivil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
, geophysicsGeophysicsGeophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
, crystallographyCrystallographyCrystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
, electrical engineeringElectrical engineeringElectrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, electronicsElectronicsElectronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, land surveyingSurveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...
and geodesyGeodesyGeodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...
, many physical sciencePhysical sciencePhysical science is an encompassing term for the branches of natural science and science that study non-living systems, in contrast to the life sciences...
s, mechanical engineeringMechanical engineeringMechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, machiningMachiningConventional machining is a form of subtractive manufacturing, in which a collection of material-working processes utilizing power-driven machine tools, such as saws, lathes, milling machines, and drill presses, are used with a sharp cutting tool to physical remove material to achieve a desired...
, medical imagingMedical imagingMedical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
, number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, oceanographyOceanographyOceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...
, opticsOpticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, pharmacologyPharmacologyPharmacology is the branch of medicine and biology concerned with the study of drug action. More specifically, it is the study of the interactions that occur between a living organism and chemicals that affect normal or abnormal biochemical function...
, probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, seismologySeismologySeismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...
, statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, and visual perceptionVisual perceptionVisual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision...
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 musicMusic
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
may perhaps know nothing of mathematics, but would probably know that Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
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
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
function is no mere coincidence. In oceanography, the resemblance between the shapes of some wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
s and the graph of the sine function is also not coincidental. In some other fields, among them climatology
Climatology
Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...
, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine function.
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 seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
, 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:
where each of the squares (
) is a different number, and one is adding infinitely many terms. Fourier used these for studying heatHeat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
flow and diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
(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).
Fourier series are also applicable to subjects whose connection with wave motion is far from obvious. One ubiquitous example is digital compression
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....
whereby images
Image compression
The objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form.- Lossy and lossless compression :...
, audio and video data are compressed into a much smaller size which makes their transmission feasible over telephone
Telephone
The telephone , colloquially referred to as a phone, is a telecommunications device that transmits and receives sounds, usually the human voice. Telephones are a point-to-point communication system whose most basic function is to allow two people separated by large distances to talk to each other...
, internet
Internet
The Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
and broadcast
Broadcasting
Broadcasting is the distribution of audio and video content to a dispersed audience via any audio visual medium. Receiving parties may include the general public or a relatively large subset of thereof...
networks
Computer network
A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow sharing of resources and information....
. Another example, mentioned above, is diffusion. Among others are: the geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
, isoperimetric problems
Isoperimetry
The isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Isoperimetric literally means "having the same perimeter"...
, recurrence of random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s, quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
, the central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
, Heisenberg's inequality.
Fourier transforms
A more abstract concept than Fourier series is the idea of Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. Fourier transforms involve integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
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 the carrying capacity
Carrying capacity
The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...
. This kind of relationship is called a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
. 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
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
, or resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
are encountered, Fourier transforms or Fourier series are nearby.
Statistics, including mathematical psychology
Intelligence 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, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 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. Why the ubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence trigonometric functionTrigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s. That is one of a variety of applications of Fourier transforms to statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
.
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 lightLight
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
that gets through decreases until the two lenses are at right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
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 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.
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 properties rather than quantitative properties of 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 sequence of fractionsDiscard 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 is 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
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
evaluated at 42.) This hints at the possibility of applying Fourier analysis to number theory.