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{{redirect|Trig}}{{pp-move-indef}}{{short description|In geometry, study of the relationship between angles and lengths}}{{Trigonometry}}Trigonometry (from Greek (wikt:Ï„ÏÎ¯Î³Ï‰Î½Î¿Î½|trigÅnon), "triangle" and (wikt:Î¼Î­Ï„ÏÎ¿Î½|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) The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.BOOK, Carl Benjamin, Boyer, Carl Benjamin Boyer, A History of Mathematics, 2nd, John Wiley & Sons, Inc., 1991, 978-0-471-54397-8,weblink Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.BOOK, Charles William Hackley, A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables,weblink 1853, G. P. Putnam, Trigonometry is known for its many identities,BOOK, Mary Jane Sterling, Trigonometry For Dummies,weblink 24 February 2014, John Wiley & Sons, 978-1-118-82741-3, 185, BOOK, P.R. Halmos, I Want to be a Mathematician: An Automathography,weblink 1 December 2013, Springer Science & Business Media, 978-1-4612-1084-9, which are equations used for rewriting trigonometrical expressions to solve equations, to find a more useful expression, or to discover new relationships.BOOK, Ron Larson, Robert P. Hostetler, Trigonometry,weblink 10 March 2006, Cengage Learning, 0-618-64332-X, 230,

Trigonometric ratios

(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.}})Trigonometric ratios are the ratios between edges of a right triangle. 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.

• 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.Since any two right triangles with the same acute angle A are similarBOOK, James Stewart, Lothar Redlin, Saleem Watson, Algebra and Trigonometry,weblink 16 January 2015, Cengage Learning, 978-1-305-53703-3, 448, , the value of a trigonometric ratio depends only on the angle A.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 cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".BOOK, Dick Jardine, Amy Shell-Gellasch, Mathematical Time Capsules: Historical Modules for the Mathematics Classroom,weblink 2011, MAA, 978-0-88385-984-1, 182, With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.BOOK, Krystle Rose Forseth, Christopher Burger, Michelle Rose Gilman, Deborah J. Rumsey, Pre-Calculus For Dummies,weblink 7 April 2008, John Wiley & Sons, 978-0-470-16984-1, 218, 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.

{{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 Ã· Hypotenuse Cosine = Adjacent Ã· Hypotenuse Tangent = Opposite Ã· 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|oÊŠ|k|Ã¦|Ëˆ|t|oÊŠ|É™}}). 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,

The unit circle and common trigonometric values

(File:Sin-cos-defn-I.png|right|thumb|Fig. 1a â€“ Sine and cosine of an angle Î¸ defined using the unit circle.)Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.BOOK, David Cohen, Lee B. Theodore, David Sklar, Precalculus: A Problems-Oriented Approach, Enhanced Edition,weblink 17 July 2009, Cengage Learning, 1-4390-4460-0, In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos A and y = sin A . This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:BOOK, W. Michael Kelley, The Complete Idiot's Guide to Calculus,weblink 2002, Alpha Books, 978-0-02-864365-6, 45, {| class="wikitable"! Function! 0! pi/6! pi/4! pi/3! pi/2! 2pi/3! 3pi/4! 5pi/6! pi
! sine| 0| 1/2| sqrt{2}/2| sqrt{3}/2| 1| sqrt{3}/2| sqrt{2}/2| 1/2| 0
! cosine| 1| sqrt{3}/2| sqrt{2}/2| 1/2| 0| -1/2| -sqrt{2}/2| -sqrt{3}/2| -1
! tangent| 0| sqrt{3}/3| 1| sqrt{3}| undefined| -sqrt{3}| -1| -sqrt{3}/3| 0
! secant| 1| 2sqrt{3}/3| sqrt{2}| 2| undefined| -2| -sqrt{2}| -2sqrt{3}/3| -1
! cosecant| undefined| 2| sqrt{2}| 2sqrt{3}/3| 1| 2sqrt{3}/3| sqrt{2}| 2| undefined
! cotangent| undefined| sqrt{3}| 1| sqrt{3}/3| 0| -sqrt{3}/3| -1| -sqrt{3}| undefined

Trigonometric functions of real or complex variables

Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative argumentsBOOK, Jenny Olive, Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students,weblink 18 September 2003, Cambridge University Press, 978-0-521-01707-7, 175, (see trigonometric function).

Graphs of trigonometric functions

The following table summarizes the properties of the graphs of the six main trigonometric functions:BOOK, Mary P Attenborough, Mathematics for Electrical Engineering and Computing,weblink 30 June 2003, Elsevier, 978-0-08-047340-6, 418, BOOK, Ron Larson, Bruce H. Edwards, Calculus of a Single Variable,weblink 10 November 2008, Cengage Learning, 0-547-20998-3, 21, {| class="wikitable"! Function! Period! Domain! Range! Graph
! sine| 2pi| (-infty,infty)| [-1,1]
Image:Sine one period.svg>200 px
! cosine| 2pi| (-infty,infty)| [-1,1]
Image:Cosine one period.svg>200 px
! tangent| pi| x neq pi/2+npi| (-infty,infty)
Image:Tangent-plot.svg>200 px
! secant| 2pi| x neq pi/2 + npi| (-infty,-1] cup [1,infty)
Image:Secant.svg>200 px
! cosecant| 2pi| x neq npi| (-infty,-1] cup [1,infty)
Image:Cosecant.svg>200 px
! cotangent| pi| x neq npi| (-infty,infty)
Image:Cotangent.svg>200 px

Inverse trigonometric functions

Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.BOOK, Elizabeth G. Bremigan, Ralph J. Bremigan, John D. Lorch, Mathematics for Secondary School Teachers,weblink 2011, MAA, 978-0-88385-773-1, {{rp|48ff}}The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:{{rp|48ff}}BOOK, Martin Brokate, Pammy Manchanda, Abul Hasan Siddiqi, Calculus for Scientists and Engineers,weblink 3 August 2019, Springer, 9789811384646, {{rp|521ff}}{| class="wikitable" style="text-align:center" !Name!Usual notation!Definition!Domain of x for real result!Range of usual principal value (radians)!Range of usual principal value (degrees)
arcsine >y = {{math>arcsin(x)}} x = {{mathsine>sin(y)}} âˆ’1 â‰¤ x â‰¤ 1 âˆ’{{sfrac2}} â‰¤ y â‰¤ {{sfrac2}} âˆ’90Â° â‰¤ y â‰¤ 90Â°
arccosine >y = {{math>arccos(x)}} x = {{mathcosine>cos(y)}} âˆ’1 â‰¤ x â‰¤ 1 0 â‰¤ y â‰¤ {{pi}} 0Â° â‰¤ y â‰¤ 180Â°
< y < {{sfrac< y < 90Â°
arctangent >y = {{math>arctan(x)}} x = {{mathTangent (trigonometry)>tan(y)}} all real numbers âˆ’{{sfrac2}} 2}} âˆ’90Â°
< y < 180Â°
arccotangent >y = {{math>arccot(x)}} x = {{mathcotangent>cot(y)}} all real numbersy < {{pi}} >| 0Â°
< {{sfrac< y â‰¤ {{pi}} < 90Â° or 90Â° < y â‰¤ 180Â°
arcsecant >y = {{math>arcsec(x)}} x = {{mathSecant (trigonometry)>sec(y)}} x â‰¤ âˆ’1 or 1 â‰¤ x 0 â‰¤ y 2}} or {{sfrac2}} 0Â° â‰¤ y
< 0 or 0 < y â‰¤ {{sfrac< 0Â° or 0Â° < y â‰¤ 90Â°
arccosecant >y = {{math>arccsc(x)}} x = {{mathcosecant>csc(y)}} x â‰¤ âˆ’1 or 1 â‰¤ x âˆ’{{sfrac2}} â‰¤ y 2}} âˆ’90Â° â‰¤ y

Power series representations

When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations:BOOK, Serge Lang, Complex Analysis,weblink 14 March 2013, Springer, 978-3-642-59273-7, 63,
begin{align}sin x & = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots & = sum_{n=0}^infty frac{(-1)^n x^{2n+1}}{(2n+1)!} end{align}
begin{align}cos x & = 1 - frac{x^2}{2!} + frac{x^4}{4!} - frac{x^6}{6!} + cdots & = sum_{n=0}^infty frac{(-1)^n x^{2n}}{(2n)!}.end{align}With these definitions the trigonometric functions can be defined for complex numbers.BOOK, Silvia Maria Alessio, Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications,weblink 9 December 2015, Springer, 978-3-319-25468-5, 339, When extended as functions of real or complex variables, the following formula holds for the complex exponential:
e^{x+iy} = e^x(cos y + i sin y).
This complex exponential function, written in terms of trigonometric functions, is particularly useful.BOOK, K. RAJA RAJESWARI, B. VISVESVARA RAO, SIGNALS AND SYSTEMS,weblink 24 March 2014, PHI Learning, 978-81-203-4941-4, 263, BOOK, John Stillwell, Mathematics and Its History,weblink 23 July 2010, Springer Science & Business Media, 978-1-4419-6053-5, 313,

Other Trigonometric Functions

In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord ({{math|1=crd(Î¸) = 2 sin({{sfrac|Î¸|2}})}}), the versine ({{math|1=versin(Î¸) = 1 âˆ’ cos(Î¸) = 2 sin2({{sfrac|Î¸|2}})}}) (which appeared in the earliest tables{{harvtxt|Boyer|1991|pp=xxiiiâ€“xxiv}}), the coversine ({{math|1=coversin(Î¸) = 1 âˆ’ sin(Î¸) = versin({{sfrac|{{pi}}|2}} âˆ’ Î¸)}}), the haversine ({{math|1=haversin(Î¸) = {{sfrac|1|2}}versin(Î¸) = sin2({{sfrac|Î¸|2}})}}),{{harvtxt|Nielsen|1966|pp=xxiiiâ€“xxiv}} the exsecant ({{math|1=exsec(Î¸) = sec(Î¸) âˆ’ 1}}), and the excosecant ({{math|1=excsc(Î¸) = exsec({{sfrac|{{pi}}|2}} âˆ’ Î¸) = csc(Î¸) âˆ’ 1}}). See List of trigonometric identities for more relations between these functions.

Applications

Astronomy

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,BOOK, Olinthus Gregory, Elements of Plane and Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations,weblink 1816, Baldwin, Cradock, and Joy, predicting eclipses, and describing the orbits of the planets.Neugebauer, Otto. "Mathematical methods in ancient astronomy." Bulletin of the American Mathematical Society 54.11 (1948): 1013-1041.In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,BOOK, Michael Seeds, Dana Backman, Astronomy: The Solar System and Beyond,weblink 5 January 2009, Cengage Learning, 0-495-56203-3, 254, as well as in satellite navigation systems.

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 chronometerHistorically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.BOOK, John Sabine, The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc,weblink 1800, 1, Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.BOOK, Mordechai Ben-Ari, Francesco Mondada, Elements of Robotics,weblink 25 October 2017, Springer, 978-3-319-62533-1, 16,

Surveying

In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.BOOK, George Roberts Perkins, Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables,weblink 1853, D. Appleton & Company, On a larger scale, trigonometry is used in geography to measure distances between landmarks,BOOK, Charles W. J. Withers, Hayden Lorimer, Geographers: Biobibliographical Studies,weblink 14 December 2015, A&C Black, 978-1-4411-0785-5, 6,

Periodic functions

(File:Fourier series and transform.gif|frame|right|Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, S(f) (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).)The sine and cosine functions are fundamental to the theory of periodic functions,BOOK, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie, Fourier and Laplace Transforms,weblink 7 August 2003, Cambridge University Press, 978-0-521-53441-3, 61, such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanicsBOOK, Bernd Thaller, Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena,weblink 8 May 2007, Springer Science & Business Media, 978-0-387-22770-2, 15, and communicationsBOOK, M. Rahman, Applications of Fourier Transforms to Generalized Functions,weblink 2011, WIT Press, 978-1-84564-564-9, , among other fields.

Optics and Acoustics

Trigonometry is useful in many physical sciences,BOOK, Lawrence Bornstein, Basic Systems, Inc, Trigonometry for the Physical Sciences,weblink 1966, Appleton-Century-Crofts, including acoustics,BOOK, John J. Schiller, Marie A. Wurster, College Algebra and Trigonometry: Basics Through Precalculus,weblink 1988, Scott, Foresman, 978-0-673-18393-4, and optics. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.BOOK, Dudley H. Towne, Wave Phenomena,weblink 5 May 2014, Dover Publications, 978-0-486-14515-0,

Identities

(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)Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.BOOK, Dugopolski, Trigonometry I/E Sup,weblink July 2002, Addison Wesley, 978-0-201-78666-8, Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,BOOK, V&S EDITORIAL BOARD, CONCISE DICTIONARY OF MATHEMATICS,weblink 6 January 2015, V&S Publishers, 978-93-5057-414-0, 288, relate both the sides and angles of a given triangle.

Triangle identities

{{Anchor|Triangle identities|Common formulas}}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)}}.

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}.

Law of tangents

The law of tangents, developed by FranÃ§ois ViÃ¨te, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables.BOOK, Ron Larson, Trigonometry,weblink 29 January 2010, Cengage Learning, 1-4390-4907-6, 331, It is given by:
frac{a-b}{a+b}=frac{tanleft[tfrac{1}{2}(A-B)right]}{tanleft[tfrac{1}{2}(A+B)right]}

Area

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:BOOK, Cynthia Y. Young, Precalculus,weblink 19 January 2010, John Wiley & Sons, 978-0-471-75684-2, 435, Heron's formula 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:BOOK, Richard N. Aufmann, Vernon C. Barker, Richard D. Nation, College Trigonometry,weblink 5 February 2007, Cengage Learning, 0-618-82507-X, 306,
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.
mbox{Area} = Delta = frac{1}{2}a bsin C.

Trigonometric identities

Pythagorean identities

The following trigonometric 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

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}}.

Other trigonometric identities

Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.

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{{reflist}}

Bibliography

• BOOK, Carl B., Boyer, Carl Benjamin Boyer, A History of Mathematics, Second, John Wiley & Sons, Inc., 1991, 978-0-471-54397-8, registration,weblink
• {{Springer |title=Trigonometric functions |id=p/t094210}}
• Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
• {{citation |last1=Nielsen |first1=Kaj L. |title=Logarithmic and Trigonometric Tables to Five Places |edition=2nd |location=New York, USA |publisher=Barnes & Noble |date=1966 |lccn=61-9103}}

{{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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