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trigonometry
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{{short description|In geometry, study of the relationship between angles and lengths}}{{redirect|Trig}}{{pp-move-indef}}File:Circle-trig6.svg|thumb|350px|right|All of the trigonometric functiontrigonometric function{{Trigonometry}}Trigonometry (from Greek (wikt:τρίγωνον|trigōnon), "triangle" and (wikt:μέτρον|metron), "measure"WEB,weblink trigonometry, Online Etymology Dictionary, ) is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002) In particular, 3rd-century astronomers first noted{{cn|reason=probably already known in ancient Egypt|date=April 2019}} that the ratio of the lengths of two sides of a right-angled triangle depends only on one acute angle of the triangle. These dependencies are now called trigonometric functions.Trigonometry is the foundation of all applied geometry, including geodesy, surveying, celestial mechanics, solid mechanics, navigation.Trigonometric functions have been extended as functions of a real or complex variable, which are today pervasive in all mathematics.

History

File:Hipparchos 1.jpeg|thumb|upright|left|Hipparchus, credited with compiling the first trigonometric table, has been described as "the father of trigonometry".BOOK, Boyer, Carl Benjamin Boyer, A History of Mathematics, 1991, Greek Trigonometry and Mensuration, 162, ]]Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.Aaboe, Asger (2001). Episodes from the Early History of Astronomy. New York: Springer. {{isbn|0-387-95136-9}} They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.BOOK, Otto Neugebauer, A history of ancient mathematical astronomy. 1,weblink 1975, Springer-Verlag, 978-3-540-06995-9, 744–, In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.Thurston, pp. 235–236. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.{{Citation|title=Ptolemy's Almagest|last1=Toomer|first1=G.|authorlink=Gerald J. Toomer|publisher=Princeton University Press|year= 1998|isbn =978-0-691-00260-6}} Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.Thurston, pp. 239–243. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.Boyer p. 215 These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.{{Citation needed|date=November 2011}} The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.WEB,weblink Al-Tusi_Nasir biography, www-history.mcs.st-andrews.ac.uk, 2018-08-05, One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth., WEB,weblink the cambridge history of science, October 2013, WEB,weblink ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica, electricpulp.com, www.iranicaonline.org, en, 2018-08-05, His major contribution in mathematics (Nasr, 1996, pp. 208-214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles., Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.WEB, trigonometry,weblink Encyclopædia Britannica, 2008-07-21, He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.BOOK, J. Lennart, Berggren, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Mathematics in Medieval Islam, Princeton University Press, 2007, 978-0-691-11485-9, 518, Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.Boyer pp. 237, 274 One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years.WEB,weblink Regiomontanus biography, History.mcs.st-and.ac.uk, 2017-03-08, At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.N.G. Wilson (1992). From Byzantium to Italy. Greek Studies in the Italian Renaissance, London. {{isbn|0-7156-2418-0}} Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.BOOK, Grattan-Guinness, Ivor, 1997, The Rainbow of Mathematics: A History of the Mathematical Sciences, W.W. Norton, 978-0-393-32030-5, Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.BOOK, Robert E. Krebs, Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance,weblink 2004, Greenwood Publishing Group, 978-0-313-32433-8, 153–, Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.William Bragg Ewald (2007). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. p. 93. {{isbn|0-19-850535-3}} Also in the 18th century, Brook Taylor defined the general Taylor series.Kelly Dempski (2002). Focus on Curves and Surfaces. p. 29. {{isbn|1-59200-007-X}}

Overview

(File:TrigonometryTriangle.svg|thumb|245px|In this right triangle: {{math|1= sin A = a/c;}} {{math|1= cos A = b/c;}} {{math|1= tan A = a/b.}})If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
  • Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.


sin A=frac{textrm{opposite}}{textrm{hypotenuse}}=frac{a}{c}.
  • Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.


cos A=frac{textrm{adjacent}}{textrm{hypotenuse}}=frac{b}{c}.
  • Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.


tan A=frac{textrm{opposite}}{textrm{adjacent}}=frac{a}{b}=frac{a/c}{b/c}=frac{sin A}{cos A}.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively.(see below under Mnemonics).The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
csc A=frac{1}{sin A}=frac{textrm{hypotenuse}}{textrm{opposite}}=frac{c}{a} ,
sec A=frac{1}{cos A}=frac{textrm{hypotenuse}}{textrm{adjacent}}=frac{c}{b} ,
cot A=frac{1}{tan A}=frac{textrm{adjacent}}{textrm{opposite}}=frac{cos A}{sin A}=frac{b}{a} .
The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are 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 questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Extending the definitions

(File:Sin-cos-defn-I.png|right|thumb|240px|Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.)The above definitions only apply to angles between 0 and 90 degrees (0 and π/2 radians). Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
e^{x+iy} = e^x(cos y + i sin y).
See Euler's and De Moivre's formulas.Image:Sine curve drawing animation.gif|Graphing process of y = sin(x) using a unit circle.Image:csc drawing process.gif|Graphing process of y = csc(x), the reciprocal of sine, using a unit circle.Image:tan drawing process.gif|Graphing process of y = tan(x) using a unit circle.

{{anchor|SOHCAHTOA}}Mnemonics

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:{{MathWorld|title=SOHCAHTOA|urlname=SOHCAHTOA}}
Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-ka-toe-uh' {{IPAc-en|s|oʊ|k|æ|ˈ|t|oʊ|ə}}). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".A sentence more appropriate for high schools is "Some Old Horse Came A'Hopping Through Our Alley". BOOK, Memory: A Very Short Introduction, Jonathan K., Foster, Oxford, 2008, 978-0-19-280675-8, 128,

Calculating trigonometric functions

Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.Today, scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.BOOK, Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C, 2013, Intel,weblink

Applications

{{More citations needed section|date=April 2019}}File:Frieberger drum marine sextant.jpg|thumb|200px|Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometermarine chronometerThere is an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves.Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, electronics, biology, medical imaging (CT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, image compressionWEB,weblink JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81), 1993, International Telecommunications Union, 6 April 2019, , phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

Pythagorean identities

The following identities are related to the Pythagorean theorem and hold for any value:BOOK, Technical Mathematics with Calculus, illustrated, John C., Peterson, Cengage Learning, 2004, 978-0-7668-6189-3, 856,weblink Extract of page 856
sin^2 A + cos^2 A = 1
tan^2 A + 1 = sec^2 A
cot^2 A + 1 = csc^2 A

Angle transformation formulae

sin (A pm B) = sin A cos B pm cos A sin B
cos (A pm B) = cos A cos B mp sin A sin B
tan (A pm B) = frac{ tan A pm tan B }{ 1 mp tan A tan B}
cot (A pm B) = frac{ cot A cot B mp 1}{ cot B pm cot A }

Common formulae

{{Anchor|Triangle identities|Common formulas}}(File:Triangle ABC with Sides a b c 2.png|thumb|240px|right|Triangle with sides a,b,c and respectively opposite angles A,B,C)Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).Lecture 3 | Quantum Entanglements, Part 1 (Stanford), Leonard Susskind, trigonometry in five minutes, law of sin, cos, euler formula 2006-10-09.

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states:
frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R = frac{abc}{2Delta},
where Delta is the area of the triangle and R is the radius of the circumscribed circle of the triangle:
R = frac{abc}{sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.
Another law involving sines can be used to calculate the area of a triangle. Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:
mbox{Area} = Delta = frac{1}{2}a bsin C.

Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
c^2=a^2+b^2-2abcos C ,
or equivalently:
cos C=frac{a^2+b^2-c^2}{2ab}.
The law of cosines may be used to prove Heron's formula, which is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
s=frac{1}{2}(a+b+c),
then the area of the triangle is:
mbox{Area} = Delta = sqrt{s(s-a)(s-b)(s-c)} = frac{abc}{4R},
where R is the radius of the circumcircle of the triangle.

Law of tangents

The law of tangents:
frac{a-b}{a+b}=frac{tanleft[tfrac{1}{2}(A-B)right]}{tanleft[tfrac{1}{2}(A+B)right]}

Euler's formula

Euler's formula, which states that e^{ix} = cos x + i sin x, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
sin x = frac{e^{ix} - e^{-ix}}{2i}, qquad cos x = frac{e^{ix} + e^{-ix}}{2}, qquad tan x = frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.

See also

{{div col|colwidth=22em}} {{div col end}}

References

{{reflist}}

Bibliography

  • BOOK, Carl B., Boyer, Carl Benjamin Boyer, A History of Mathematics, Second, John Wiley & Sons, Inc., 1991, 978-0-471-54397-8,
  • {{Springer |title=Trigonometric functions |id=p/t094210}}
  • Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
  • {{MathWorld|title=Trigonometric Addition Formulas|urlname=TrigonometricAdditionFormulas|author=Weisstein, Eric W.}}

External links

{{Sister project links|Trigonometry}}{{Library resources box |by=no |onlinebooks=no |others=no |about=yes |label=Trigonometry}} {{Areas of mathematics | state=collapsed}}{{Authority control}}

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