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Trigonometry
“Trig” redirects here. For other uses, see Trig (disam- tive curvature, in elliptic geometry (a fundamental part
biguation).
of astronomy and navigation). Trigonometry on surfaces
Trigonometry (from Greek trigōnon, “triangle” and of negative curvature is part of hyperbolic geometry.
Trigonometry basics are often taught in schools, either as
a separate course or as a part of a precalculus course.
d
cr
Fc
excsc H ot
A
cvs
G
csc
1
sin
tan
1 History
arc
θ C
O cos versin D
exsec
Main article: History of trigonometry
Sumerian astronomers studied angle measure, using a di-
E
sec
B
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
metron, “measure”[1] ) is a branch of mathematics that
studies relationships involving lengths and angles of
triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to
astronomical studies.[2]
The 3rd-century astronomers first noted that the lengths
of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at
least the length of one side and the value of one angle
is known, then 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 and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across
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 surveying.
Hipparchus, credited with compiling the first trigonometric table,
is known as “the father of trigonometry”.[3]
vision of circles into 360 degrees.[4] 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.[5]
Trigonometry is most simply associated with planar rightangle triangles (each of which is 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 relate to right-angle triangles. One exception to this is spherical trigonometry, the
study of triangles on spheres, surfaces of constant posi-
In the 3rd century BC, Hellenistic Greek mathematicians
such as Euclid (from Alexandria, Egypt) and Archimedes
(from Syracuse, Sicily) 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
1
2
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.[9] 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. At about the same time,
Chinese mathematicians developed trigonometry independently, although it was not a major field of study
for them. 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 Arabic astronomers such as Al Battani and
Nasir al-Din al-Tusi.[10] 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.[11] At the same time another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.[12] 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.[13] Bartholomaeus Pitiscus was the first to
use the word, publishing his Trigonometria in 1595.[14]
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 de-
OVERVIEW
velopment of trigonometric series.[15] Also in the 18th
century, Brook Taylor defined the general Taylor series.[16]
2 Overview
Main article: Trigonometric function
If one angle of a triangle is 90 degrees and one of the
B
c
Hy
A
e
us
en
t
po
Adjacent
b
Opposite
than algebraically. In 140 BC Hipparchus (from Iznik,
Turkey) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.[6] In the
2nd century AD the Greco-Egyptian astronomer Ptolemy
(from Alexandria, Egypt) printed detailed trigonometric
tables (Ptolemy’s table of chords) in Book 1, chapter 11
of his Almagest.[7] Ptolemy used chord length to define
his trigonometric functions, a minor difference from the
sine convention we use today.[8] (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.
2
a
C
In this right triangle: sin A = a/ c; cos A = b/ c; tan A = a/ b.
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 =
a
opposite
= .
hypotenuse
c
• Cosine function (cos), defined as the ratio of the
adjacent leg to the hypotenuse.
cos A =
b
adjacent
= .
hypotenuse
c
• Tangent function (tan), defined as the ratio of the
opposite leg to the adjacent leg.
2.2
Mnemonics
tan A =
3
opposite
a
a c
a b
sin A
=
= ∗ = / =
.
adjacent
b
c b
c c
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. 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).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot),
Fig. 1a – Sine and cosine of an angle θ defined using the unit
respectively:
circle.
csc A =
1
hypotenuse
c
=
= ,
sin A
opposite
a
sec A =
1
hypotenuse
c
=
= ,
cos A
adjacent
b
cot A =
1
adjacent
cos A
b
=
=
= .
tan A
opposite
sin A
a
from calculus and infinite series. With these definitions
the trigonometric functions can be defined for complex
numbers. The complex exponential function is particularly useful.
ex+iy = ex (cos y + i sin y).
The inverse functions are called the arcsine, arccosine, See Euler’s and De Moivre’s formulas.
and arctangent, respectively. There are arithmetic re• Graphing process of y = sin(x) using a unit circle.
lations between these functions, which are known as
trigonometric identities. The cosine, cotangent, and cose• Graphing process of y = csc(x), the reciprocal of
cant are so named because they are respectively the sine,
sine, using a unit circle.
tangent, and secant of the complementary angle abbrevi• Graphing process of y = tan(x) using a unit circle.
ated 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.
2.1
Extending the definitions
2.2 Mnemonics
Main article: Mnemonics in trigonometry
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 SOHCAH-TOA:[17]
The above definitions only apply to angles between 0 and
Sine = Opposite ÷ Hypotenuse
90 degrees (0 and π/2 radians). Using the unit circle,
one can extend them to all positive and negative arguCosine = Adjacent ÷ Hypotenuse
ments (see trigonometric function). The trigonometric
Tangent = Opposite ÷ Adjacent
functions are periodic, with a period of 360 degrees or
2π radians. That means their values repeat at those in- One way to remember the letters is to sound them out
tervals. The tangent and cotangent functions also have a phonetically (i.e., SOH-CAH-TOA, which is pronounced
shorter period, of 180 degrees or π radians.
'so-kə-toe-uh' /soʊkəˈtoʊə/). Another method is to exThe trigonometric functions can be defined in other ways pand the letters into a sentence, such as "Some Old
besides the geometrical definitions above, using tools Hippie Caught Another Hippie Trippin' On Acid”.[18]
4
2.3
6
Calculating trigonometric functions
Main article: Trigonometric tables
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.
COMMON FORMULAE
etry is essential) and hence navigation (on the oceans,
in aircraft, and in space), music theory, audio synthesis, acoustics, optics, electronics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry,
number theory (and hence cryptology), seismology,
meteorology, oceanography, many physical sciences,
land surveying and geodesy, architecture, image compression, phonetics, economics, electrical engineering,
mechanical engineering, civil engineering, computer
graphics, cartography, crystallography and game development.
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 some- 4 Pythagorean identities
times gradians. Most computer programming languages
provide function libraries that include the trigonometric Identities are those equations that hold true for any value.
functions. The floating point unit hardware incorporated
into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometsin2 A + cos2 A = 1
ric functions.[19]
(The following two can be derived from the first.)
3
Applications of trigonometry
sec2 A − tan2 A = 1
csc2 A − cot2 A = 1
5 Angle transformation formulae
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
Sextants 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.
tan(A ± B) =
tan A ± tan B
1 ∓ tan A tan B
cot(A ± B) =
cot A cot B ∓ 1
cot B ± cot A
Main article: Uses of trigonometry
6 Common formulae
There 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.
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
Fields that use trigonometry or trigonometric functions a triangle and a, b and c are the lengths of sides of the
include astronomy (especially for locating apparent po- triangle opposite the respective angles (as shown in the
sitions of celestial objects, in which spherical trigonom- diagram).
6.3
Law of tangents
5
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=
1
(a + b + c),
2
then the area of the triangle is:
Area = ∆ =
√
abc
s(s − a)(s − b)(s − c) =
4R
where R is the radius of the circumcircle of the triangle.
Triangle with sides a,b,c and respectively opposite angles A,B,C
6.3 Law of tangents
6.1
Law of sines
The law of sines (also known as the “sine rule”) for an
arbitrary triangle states:
a
b
c
abc
=
=
= 2R =
,
sin A
sin B
sin C
2∆
The law of tangents:
[
]
tan 12 (A − B)
a−b
[1
]
=
a+b
tan 2 (A + B)
6.4 Euler’s formula
where ∆ is the area of the triangle and R is the radius of
Euler’s formula, which states that eix = cos x + i sin x
the circumscribed circle of the triangle:
, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
abc
.
R= √
(a + b + c)(a − b + c)(a + b − c)(b + c − a)
eix − e−ix
eix + e−ix
i(e−ix − eix )
sin x =
,
cos x =
,
tan x = ix
Another law involving sines can be used to calculate the
2i
2
e + e−ix
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 7 See also
of the angle between the two sides:
• Aryabhata’s sine table
Area = ∆ =
6.2
1
ab sin C.
2
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:
• Generalized trigonometry
• Lénárt sphere
• List of triangle topics
• List of trigonometric identities
• Rational trigonometry
• Skinny triangle
• Small-angle approximation
c2 = a2 + b2 − 2ab cos C,
or equivalently:
a2 + b2 − c2
cos C =
.
2ab
• Trigonometric functions
• Trigonometry in Galois fields
• Unit circle
• Uses of trigonometry
6
8
10
References
[1] “trigonometry”. Online Etymology Dictionary.
[2] R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The
Gale Group (2002)
EXTERNAL LINKS
9 Bibliography
• Boyer, Carl B. (1991). A History of Mathematics
(Second ed.). John Wiley & Sons, Inc. ISBN 0471-54397-7.
[3] Boyer (1991). “Greek Trigonometry and Mensuration”.
A History of Mathematics. p. 162.
• Hazewinkel, Michiel, ed. (2001), “Trigonometric
functions”, Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4
[4] Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-38795136-9
• Christopher M. Linton (2004). From Eudoxus to
Einstein: A History of Mathematical Astronomy .
Cambridge University Press.
[5] Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. pp. 744–. ISBN
978-3-540-06995-9.
[6] Thurston, pp. 235–236.
• Weisstein, Eric W., “Trigonometric Addition Formulas”, MathWorld.
10 External links
[7] Toomer, G. J. (1998), Ptolemy’s Almagest, Princeton University Press, ISBN 0-691-00260-6
• Khan Academy: Trigonometry, free online micro
lectures
[8] Thurston, pp. 239–243.
• Trigonometry by Alfred Monroe Kenyon and Louis
Ingold, The Macmillan Company, 1914. In images,
full text presented.
[9] Boyer p. 215
[10] Boyer pp. 237, 274
• Benjamin Banneker’s Trigonometry Puzzle at
Convergence
[11] http://www-history.mcs.st-and.ac.uk/Biographies/
Regiomontanus.html
• Dave’s Short Course in Trigonometry by David
Joyce of Clark University
[12] N.G. Wilson, From Byzantium to Italy. Greek Studies in the
Italian Renaissance, London, 1992. ISBN 0-7156-2418-0
• Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free
Documentation License
[13] Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W.
Norton. ISBN 0-393-32030-8.
[14] Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages
and the Renaissance. Greenwood Publishing Group. pp.
153–. ISBN 978-0-313-32433-8.
[15] William Bragg Ewald (2008). From Kant to Hilbert: a
source book in the foundations of mathematics. Oxford
University Press US. p. 93. ISBN 0-19-850535-3
[16] Kelly Dempski (2002). Focus on Curves and Surfaces. p.
29. ISBN 1-59200-007-X
[17] Weisstein, Eric W., “SOHCAHTOA”, MathWorld.
[18] A sentence more appropriate for high schools is "'Some
Old Horse Came A''Hopping Through Our Alley”. Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0-19-280675-0.
[19] Intel® 64 and IA-32 Architectures Software Developer’s
Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and
3C (PDF). Intel. 2013.
7
11
11.1
Text and image sources, contributors, and licenses
Text
• Trigonometry Source: https://en.wikipedia.org/wiki/Trigonometry?oldid=698136325 Contributors: AxelBoldt, Zundark, Andre Engels,
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ParisianBlade, Prancibaldfpants, Echion2, NuclearWarfare, CylonSix, Cenarium, Aurora2698, Jotterbot, Eustress, Razorflame, Dekisugi,
DatDoo, BOTarate, Thingg, Kruusamägi, Ubardak, DumZiBoT, Kiensvay, Crazy Boris with a red beard, Bgreise24, BendersGame,
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Catherinemandot, CanadianLinuxUser, Mohammed farag, Cst17, Mohamed Magdy, MrOllie, Glane23, Favonian, Kyle1278, 5 albert
square, Sirturtle1990, Squandermania, Manorupa108, Ehrenkater, Timyxp, Tide rolls, Zorrobot, TeH nOmInAtOr, Quantumobserver,
HerculeBot, Crt, Cooookie3001, Ben Ben, Math Champion, Luckas-bot, Yobot, Tohd8BohaithuGh1, Senator Palpatine, Gobbleswoggler,
Mmxx, Ajh16, Goodberry, Wikihelper100, AnomieBOT, IRP, Proevaholic, AdjustShift, Kingpin13, Halfs, Materialscientist, Citation
bot, Akilaa, E2eamon, Frankenpuppy, ArthurBot, Xqbot, Sathimantha, TinucherianBot II, Techsmoke15, JimVC3, Nasnema, Ksagittariusr, Tyrol5, Gap9551, Sabio101, Omnipaedista, RibotBOT, Charvest, Doulos Christos, WhizzWizard, Raulshc, Sophus Bie, GhalyBot,
Shadowjams, Skepticalmouse, Nfgsurf7, Griffinofwales, Prari, FrescoBot, Tobby72, Pepper, Dan1232112321, Wikipe-tan, Oldlaptop321,
Routerone, Biscuit555, Rigaudon, Swaying tree, TheStudentRoom, Jamesooders, Wireless Keyboard, Cannolis, HamburgerRadio, 4pi,
Cobness, Dirtyasianmen, Watsup101, Pinethicket, I dream of horses, Boulaur, Hendvi, Kiefer.Wolfowitz, 01000100 W 01000010, Calmer
Waters, A8UDI, POOEATZA, Jschnur, Impala2009, Nosoup4uNOOB, SpaceFlight89, Σ, Rohitphy, Prin09, Maheshnathwani, Dude1818,
Dysfunktionall, Ykhatana1, FoxBot, TobeBot, PorkHeart, Vrenator, Leebian, Diannaa, Ammodramus, Pedro999999999, Stupid geometric shapes, Uber smart man, Sideways713, Minerva27, TjBot, Beyond My Ken, Hajatvrc, Saulth, Balph Eubank, Ctdpmet2, Jack Schlederer, Onef9day, EmausBot, Stryn, Immunize, Gfoley4, Ibbn, Faolin42, NotAnonymous0, Wikipelli, Pjboonstra, MonoALT, Sardar0987,
JSquish, Fæ, Nicajelejunsutuc, Rosswante, Monterey Bay, Openstrings, Mlpearc Public, Colin.campbell.27, L Kensington, Donner60,
Chewings72, Herebo, ChuispastonBot, Nicepepper, PaulAg54, ClueBot NG, Smtchahal, Kingofpoptarts, Mblum116, Arnavmishra2050,
8
11
TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
Gareth Griffith-Jones, Gvsip, Wcherowi, Zg001, Movses-bot, Churrrp, Krishnan.adarsh, Muon, Marechal Ney, Widr, Jorgenev, Helpful
Pixie Bot, Thisthat2011, පසිඳු කාවින්ද, Ajoygphilip, Nightenbelle, KLBot2, DBigXray, Adamdude16, TarekHammad, SRWikis, John
Cummings, Leonxlin, Hemil962, Cwm9, Stelpa, Wiki douglasm, SHUBHANKAN DAS, Mark Arsten, Shivanker1412, Altaïr, Theboorb,
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Xtarification, 1764cameron, Phasedeeznuts, Bossabdalla01, Salsbury saturday and Anonymous: 1403
11.2
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11.3
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