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| [[File:STS-114 Steve Robinson on Canadarm2.jpg|thumb|The [[Canadarm2]] robotic manipulator on the [[International Space Station]] is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.]]
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| {{Trigonometry}}
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| ''Trigonometry'' (from [[Ancient Greek|Greek]] ''[[wikt:τρίγωνον|trigōnon]]'', "triangle" + ''[[wikt:μέτρον|metron]]'', "measure"<ref>{{cite web|url=http://www.etymonline.com/index.php?term=trigonometry|title=trigonometry|publisher=Online Etymology Dictionary}}</ref>) is a branch of [[mathematics]] that studies relationships involving lengths and [[angles]] of [[triangle]]s. The field emerged during the 3rd century BC from applications of [[geometry]] to astronomical studies.<ref>R. Nagel (ed.), ''Encyclopedia of Science'', 2nd Ed., The Gale Group (2002)</ref>
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| The 3rd century astronomers first noted that the lengths of the sides of a right angle triangle and the [[angle]]s between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the [[trigonometric functions]] and today are pervasive in both [[pure mathematics|pure]] and [[applied mathematics|applied]] mathematics: fundamental methods of analysis such as the [[Fourier Transform]], for example, or the [[wave equation]], use trigonometric functions to understand [[periodic function|cyclical]] phenomena across a great many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology and biology. Trigonometry is also the foundation of the practical art of [[surveying]].
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| Trigonometry is most simply associated with [[planar]] [[right angle]] triangles (a two dimensional triangle with one angle equal to 90 degrees). The applicability to non-right angle triangles exists but, since any non-right angle triangle (on a flat plane) can be bisected to create two right angle triangles, most problems can be reduced to calculations on right angle triangles. Thus the majority of applications relates to right angle triangles. One exception to this is [[Spherical trigonometry]], the study of triangles on [[sphere]]s, surfaces of constant positive [[curvature]], in [[elliptic geometry]] (a fundamental part of [[astronomy]] and [[navigation]]). Trigonometry on surfaces of negative curvature is part of [[hyperbolic geometry]].
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| Trigonometry basics are often taught in [[school]] either as a separate course or as part of a [[precalculus]] course.
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| ==History==
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| {{main|History of trigonometry}}
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| [[File:Hipparchos 1.jpeg|thumb|180px|left|The first [[Generating trigonometric tables|trigonometric table]] was apparently compiled by [[Hipparchus]], who is now consequently known as "the father of trigonometry."<ref>{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Greek Trigonometry and Mensuration|page=162}}</ref>]]
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| [[Sumer]]ian astronomers studied angle measure, using a division of circles into 360 degrees.<ref>Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9</ref> They, and later the [[Babylonians]], studied the ratios of the sides of [[Similarity (geometry)|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 [[Nubia|ancient Nubians]] used a similar method.<ref>{{cite book|author=Otto Neugebauer |title=A history of ancient mathematical astronomy. 1 |url=http://books.google.com/books?id=vO5FCVIxz2YC&pg=PA744 |year=1975 |publisher=Springer-Verlag |isbn=978-3-540-06995-9 |pages=744–}}</ref> The [[ancient Greeks]] transformed trigonometry into an ordered science.<ref>"[http://www.math.rutgers.edu/%7Echerlin/History/Papers2000/hunt.html The Beginnings of Trigonometry]". Rutgers, The State University of New Jersey.</ref>
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| In the 3rd century BCE, classical [[Greek mathematics|Greek mathematicians]] (such as [[Euclid]] and [[Archimedes]]) studied the properties of [[chord (geometry)|chords]] and [[inscribed angle]]s in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. [[Ptolemy|Claudius Ptolemy]] expanded upon [[Hipparchus]]' ''Chords in a Circle'' in his ''[[Almagest]]''.<ref>Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004). ''[http://books.google.com/books?id=BKRE5AjRM3AC&pg=PA36 Sherlock Holmes in Babylon: and other tales of mathematical history]''. [[Mathematical Association of America|MAA]]. p. 36. ISBN 0-88385-546-1</ref>
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| The modern [[trigonometric functions|sine function]] was first defined in the ''[[Surya Siddhanta]]'', and its properties were further documented by the 5th century (CE) [[Indian mathematics|Indian mathematician]] and astronomer [[Aryabhata]].<ref>Boyer p. 215</ref> These Greek and Indian works were translated and expanded by [[Mathematics in medieval Islam|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}} At about the same time, [[Chinese mathematics|Chinese]] mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached [[Europe]] via [[Latin translations of the 12th century|Latin translations]] of the works of [[Astronomy in medieval Islam|Persian and Arabic astronomers]] such as [[Muhammad ibn Jābir al-Harrānī al-Battānī|Al Battani]] and [[Nasir al-Din al-Tusi]].<ref>Boyer pp. 237, 274</ref> One of the earliest works on trigonometry by a European mathematician is ''De Triangulis'' by the 15th century [[Germany|German]] mathematician [[Regiomontanus]]. Trigonometry was still so little known in 16th-century Europe that [[Nicolaus Copernicus]] devoted two chapters of ''[[De revolutionibus orbium coelestium]]'' to explain its basic concepts.
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| 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.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 0-393-32030-8}}</ref> [[Bartholomaeus Pitiscus]] was the first to use the word, publishing his ''Trigonometria'' in 1595.<ref>{{cite book|author=Robert E. Krebs |title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissnce |url=http://books.google.com/books?id=MTXdplfiz-cC&pg=PA153 |year=2004 |publisher=Greenwood Publishing Group |isbn=978-0-313-32433-8 |pages=153–}}</ref> [[Gemma Frisius]] described for the first time the method of [[triangulation]] still used today in surveying. It was [[Leonhard Euler]] who fully incorporated [[complex number]]s into trigonometry. The works of [[James Gregory (astronomer and mathematician)|James Gregory]] in the 17th century and [[Colin Maclaurin]] in the 18th century were influential in the development of [[trigonometric series]].<ref>William Bragg Ewald (2008). ''[http://books.google.com/books?id=AcuF0w-Qg08C&pg=PA93 From Kant to Hilbert: a source book in the foundations of mathematics]''. [[Oxford University Press US]]. p. 93. ISBN 0-19-850535-3</ref> Also in the 18th century, [[Brook Taylor]] defined the general [[Taylor series]].<ref>Kelly Dempski (2002). ''[http://books.google.com/books?id=zxdigX-KSZYC&pg=PA29 Focus on Curves and Surfaces]''. p. 29. ISBN 1-59200-007-X</ref>
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| ==Overview==
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| [[File:TrigonometryTriangle.svg|thumb|245px|In this right triangle: {{nowrap|1= sin ''A'' = ''a''/''c'';}} {{nowrap|1= cos ''A'' = ''b''/''c'';}} {{nowrap|1= tan ''A'' = ''a''/''b''.}}]]
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| 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 (geometry)|similarity]], by the angles. Once the angles are known, the [[ratio]]s 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 function]]s of the known angle ''A'', where ''a'', '' b'' and ''c'' refer to the lengths of the sides in the accompanying figure:
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| *'''[[Sine]]''' function (sin), defined as the ratio of the side opposite the angle to the [[hypotenuse]].
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| :: <math>\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.</math>
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| *'''[[Cosine]]''' function (cos), defined as the ratio of the [[adjacent side (right triangle)|adjacent]] leg to the hypotenuse.
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| :: <math>\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.</math>
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| *'''[[Tangent]]''' function (tan), defined as the ratio of the opposite leg to the adjacent leg.
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| :: <math>\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.</math>
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| 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. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under [[#Mnemonics|Mnemonics]]).
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| The [[Multiplicative inverse|reciprocals]] of these functions are named the '''cosecant''' (csc or cosec), '''secant''' (sec), and '''cotangent''' (cot), respectively:
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| :<math>\csc A=\frac{1}{\sin A}=\frac{c}{a} ,</math>
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| :<math>\sec A=\frac{1}{\cos A}=\frac{c}{b} ,</math>
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| :<math>\cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}=\frac{b}{a} .</math>
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| The [[Inverse trigonometric function|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-".
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| 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.
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| ===Extending the definitions===
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| [[File:Sin-cos-defn-I.png|right|thumb|240px|Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.]]
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| The above definitions apply to angles between 0 and 90 degrees (0 and π/2 [[radian]]s) only. Using the [[unit circle]], one can extend them to all positive and negative arguments (see [[trigonometric function]]). The trigonometric functions are [[periodic function|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.
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| 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 number]]s. The complex exponential function is particularly useful.
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| : <math>e^{x+iy} = e^x(\cos y + i \sin y).</math>
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| See [[Euler's formula|Euler's]] and [[De Moivre's formula|De Moivre's]] formulas.
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| <gallery>
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| Image:Sine_curve_drawing_animation.gif|Graphing process of ''y'' = sin(''x'') using a unit circle.
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| Image:tan drawing process.gif|Graphing process of ''y'' = tan(''x'') using a unit circle.
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| Image:csc drawing process.gif|Graphing process of ''y'' = csc(''x'') using a unit circle.
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| </gallery> | |
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| ==={{anchor|SOHCAHTOA}}Mnemonics===
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| <!-- Note: | |
| One mnemonic is enough, and the others were absolutely ridiculous. DO NOT ADD IN YOUR OWN MNEMONICS, or other ridiculous ones such as "Sex on holidays Comes after having Tons of alcohol" etcetera. Such edits DO NOT improve the article, but add a method of obfuscation and in some cases, subversive vandalism through a "learning mnemonic", and WILL BE REVERTED.-->
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| {{main|Mnemonics in trigonometry}}
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| A common use of [[mnemonic]]s 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 as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:<ref>{{MathWorld|title=SOHCAHTOA|urlname=SOHCAHTOA}}</ref>
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| :'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse
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| :'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse
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| :'''T'''angent = '''O'''pposite ÷ '''A'''djacent
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| One way to remember the letters is to sound them out phonetically (i.e., ''SOH-CAH-TOA'', which is pronounced 'so-kə-'''toe'''-uh' {{IPAc-en|s|oʊ|k|ə|ˈ|t|oʊ|ə}}). Another method is to expand the letters into a sentence, such as "'''S'''ome '''O'''ld '''H'''ippy '''C'''aught '''A'''nother '''H'''ippy '''T'''rippin' '''O'''n '''A'''cid".<ref>A sentence more appropriate for high schools is "Some old horse came a'hopping through our alley". {{cite book |title=Memory: A Very Short Introduction|first=Jonathan K.|last=Foster|publisher=Oxford|year=2008|isbn=0-19-280675-0|page=128}}</ref>
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| ===Calculating trigonometric functions===
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| {{main|Generating trigonometric tables}}
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| Trigonometric functions were among the earliest uses for [[mathematical table]]s. 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 rule]]s had special scales for trigonometric functions.
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| Today [[scientific calculator]]s have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes [[Euler's formula|cis]] and their inverses). Most allow a choice of angle measurement methods: degrees, radians, and sometimes [[grad (angle)|grad]]. Most computer [[programming language]]s provide function libraries that include the trigonometric functions. The [[floating point unit]] hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.<ref name=Intel2013>{{cite book |title=Intel® 64 and IA-32 Architectures Software Developer’s Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C |year=2013 |publisher=Intel |url=http://download.intel.com/products/processor/manual/325462.pdf}}</ref>
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| ==Applications of trigonometry==
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| [[File:Frieberger drum marine sextant.jpg|thumb|200px|[[Sextant]]s are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a [[marine chronometer]], the position of the ship can be determined from such measurements.]]
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| {{main|Uses of trigonometry}}
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| There are 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 system]]s. The sine and cosine functions are fundamental to the theory of [[periodic function]]s such as those that describe sound and [[light]] waves.
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| 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]], analysis of financial markets, [[electronics]], [[probability theory]], [[statistics]], [[biology]], [[medical imaging]] ([[CAT scan]]s and [[ultrasound]]), [[pharmacy]], [[chemistry]], [[number theory]] (and hence [[cryptology]]), [[seismology]], [[meteorology]], [[oceanography]], many [[physical science]]s, land [[surveying]] and [[geodesy]], [[architecture]], [[phonetics]], [[economics]], [[electrical engineering]], [[mechanical engineering]], [[civil engineering]], [[computer graphics]], [[cartography]], [[crystallography]] and [[game development]].
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| ==Pythagorean identities==
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| [[Identity (mathematics)|Identities]] are those equations that hold true for any value.
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| :<math>\sin^2 A + \cos^2 A = 1 \ </math> | |
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| :<math>\sec^2 A - \tan^2 A = 1 \ </math>
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| :<math>\csc^2 A - \cot^2 A = 1 \ </math>
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| ==Angle transformation formulas==
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| :<math>\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B</math>
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| :<math>\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B</math>
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| :<math>\tan (A \pm B) = \frac{ \tan A \pm \tan B }{ 1 \mp \tan A \ \tan B}</math>
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| :<math>\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A } </math>
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| ==Common formulas==
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| [[File:Triangle ABC with Sides a b c.png|thumb|240px|left|Triangle with sides ''a'',''b'',''c'' and respectively opposite angles ''A'',''B'',''C'']]
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| 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.
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| 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.
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| ===Law of sines===
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| The '''[[law of sines]]''' (also known as the "sine rule") for an arbitrary triangle states:
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| :<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>
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| where ''R'' is the radius of the [[circumscribed circle]] of the triangle:
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| :<math>R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.</math>
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| Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
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| :<math>\mbox{Area} = \frac{1}{2}a b\sin C.</math>
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| [[File:Circle-trig6.svg|thumb|right|All of the [[trigonometric function]]s of an angle ''θ'' can be constructed geometrically in terms of a unit circle centered at ''O''.]]
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| ===Law of cosines===
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| The '''[[law of cosines]]''' (known as the cosine formula, or the "cos rule") is an extension of the [[Pythagorean theorem]] to arbitrary triangles:
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| :<math>c^2=a^2+b^2-2ab\cos C ,\,</math>
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| or equivalently: | |
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| :<math>\cos C=\frac{a^2+b^2-c^2}{2ab}.\,</math>
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| ===Law of tangents===
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| The '''[[law of tangents]]''':
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| :<math>\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}</math>
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| ===Euler's formula===
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| [[Euler's formula]], which states that <math>e^{ix} = \cos x + i \sin x</math>, produces the following [[mathematical analysis|analytical]] identities for sine, cosine, and tangent in terms of ''[[e (mathematics)|e]]'' and the [[imaginary unit]] ''i'':
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| :<math>\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}}.</math>
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| ==See also==
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| {{colbegin|3}}
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| * [[Aryabhata's sine table]]
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| * [[Generalized trigonometry]]
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| * [[Lénárt sphere]]
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| * [[List of triangle topics]]
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| * [[List of trigonometric identities]]
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| * [[Rational trigonometry]]
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| * [[Skinny triangle]]
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| * [[Small-angle approximation]]
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| * [[Trigonometric functions]]
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| * [[Trigonometry in Galois fields]]
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| * [[Unit circle]]
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| * [[Uses of trigonometry]]
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| {{colend}}
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| ==References==
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| {{reflist|35em}}
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| ===Bibliography===
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| *{{cite book|first=Carl B. |last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7}}
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| *{{springer|title=Trigonometric functions|id=p/t094210}}
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| *Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.
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| *Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld. Weiner.
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| ==External links==
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| {{Sister project links|Trigonometry}}
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| {{Library resources box
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| * [http://www.khanacademy.org/math/trigonometry Khan Academy: Trigonometry, free online micro lectures]
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| * [http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights], by [[Eli Maor]], Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
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| * [http://baqaqi.chi.il.us/buecher/mathematics/trigonometry/index.html Trigonometry] by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
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| * [http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=212&bodyId=81 Benjamin Banneker's Trigonometry Puzzle] at [http://mathdl.maa.org/convergence/1/ Convergence]
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| * [http://www.clarku.edu/~djoyce/trig/ Dave's Short Course in Trigonometry] by David Joyce of [[Clark University]]
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| *[http://www.mecmath.net/trig/trigbook.pdf Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License]
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| {{Mathematics-footer}}
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| [[Category:Trigonometry| ]]
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