Fontem paginae Trigonometria inspicere

[[Fasciculus:Circle-trig6.svg|300px|thumb|Omnes [[functiones trigonometricae]] anguli ''θ'' fieri possunt cum [[circulus|circulo]] cuius centrum est ''O''.]] 

'''Trigonometria''' (de [[Graece|Graecis]] verbis ''τρίγωνον trigōnon'' '[[triangulum]]', et ''μέτρον metron'' '[[mensura]]') est ramus [[mathematica]]e qui spectat in [[triangulum|triangula]], praecipue in [[triangulum rectum|triangula recta]]. Trigonometria explicat comparationes inter latera et [[angulus|angulos]] trianguli.

==Historia==
Fertur [[Bartholomaeus Pitiscus|Bartholomaeum Pitiscum]] primum introduxisse verbum ''trigonometriam'' in [[lingua|linguis]] vulgaribus (eae sunt Anglica, Francogallica, etc.), in suo ''Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus'', anno [[1595]] edito.<ref>[https://web.archive.org/web/19991004035216/http://members.aol.com/jeff570/t.html Primi usus appellationum mathematicarum,] members.aol.com.</ref>
[[Fasciculus:Table_of_Trigonometry,_Cyclopaedia,_Volume_2.jpg|thumb|"Table of Trigonometry" in ''[[Cyclopaedia]]'' (1728).]]

Trigonometria verisimiliter creata est ad [[ars navigandi|navigandum]], ut [[astronomia]].<ref>C. M. Linton, From Exodus to Einstein: A History of Mathematical Astronomy (anglice:Ab Exodo ad Einstein: Historia astronomiae mathematicalis), Cambridge Universitas Impressionis Ufficium, 2004</ref> Primi qui eam usi sunt [[Aegyptus|Aegyptii]], [[Mesopotamia|Mesopotameni]] [[India|Indi]] fuerunt, ante quattuor milia annorum. Angulos gradibus, minuta, secondaque metiri commentus est a [[Babylon|Babyloniis]]. <!-- The common practice of measuring angles in degrees, minutes and seconds comes from the [[Babylonia]]n's [[sexagesimal|base sixty]] system of numeration. The [[Sulba Sutras]] written in India, between 800 BC and 500 BC, correctly computes the sine of [[π]]/4 (45[[Degree (angle)|°]]) as 1/√2 in a procedure for circling the square (the opposite of [[squaring the circle]]).

The first recorded use of trigonometry came from the [[Greek mathematics|Hellenistic mathematician]] [[Hipparchus]][http://www.math.rutgers.edu/~cherlin/History/Papers2000/hunt.html] circa 150 BC, who compiled a trigonometric [[Mathematical table|table]] using the [[sine]] for solving triangles. [[Ptolemy]] further developed trigonometric calculations circa 100 AD.

The ancient [[Sinhalese people|Sinhalese]] in [[Sri Lanka]], when constructing reservoirs in the [[Anuradhapura]] kingdom, used trigonometry to calculate the gradient of the water flow.  Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.<ref>http://www.lankalibrary.com/phpBB/viewtopic.php?p=1236&sid=88ab8557d1c7c641bb9da7821eded404</ref>

The Indian mathematician [[Aryabhata]] in 499, gave tables of half chords which are now known as [[sine]] tables, along with [[cosine]] tables. He used ''zya'' for sine, ''kotizya'' for cosine, and ''otkram zya'' for inverse sine, and also introduced the [[versine]]. Another Indian mathematician, [[Brahmagupta]] in 628, used an [[interpolation]] formula to compute values of sines, up to the second order of the [[Isaac Newton|Newton]]-[[James Stirling (mathematician)|Stirling]] interpolation formula.

In the 10th century, the Persian mathematician and astronomer [[Abul Wáfa]] introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (''a'' + ''b''), and discovered the sine formula for spherical geometry:

:<math>\frac{\sin(A)}{\sin(a)} = \frac{\sin(B)}{\sin(b)} = \frac{\sin(C)}{\sin(c)}.</math>

Also in the late 10th and early 11th centuries, the Egyptian astronomer [[Ibn Yunus]] performed many careful trigonometric calculations and demonstrated the formula

:<math>\cos(a) \cos(b) = \frac{\cos(a+b) + \cos(a-b)}{2}.</math>.

[[Indian mathematics|Indian mathematicians]] were the pioneers of variable computations [[algebra]] for use in astronomical calculations along with trigonometry. [[Lagadha]] (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his ''[[Vedanga Jyotisha]]''.

[[Persian people|Persian]] [[Islamic mathematics|mathematician]] [[Omar Khayyám]] (1048-1131) combined trigonometry and [[approximation theory]] to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation <math>x^3 + 200 x = 20 x^2 + 2000</math> and found a positive root of this cubic by considering the intersection of a rectangular [[hyperbola]] and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician [[Bhaskara]] in 1150, along with some sine and cosine formulae. Bhaskara also developed [[spherical trigonometry]].

The 13th century [[Persian people|Persian]] [[Islamic mathematics|mathematician]] [[Nasir al-Din Tusi]], along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his ''Treatise on the Quadrilateral'' was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

In the 14th century, Persian mathematician [[al-Kashi]] and [[Timurid]] mathematician [[Ulugh Beg]] (grandson of [[Timur]]) produced tables of trigonometric functions as part of their studies of astronomy.

The mathematician [[Bartholemaeus Pitiscus]] published an influential work on trigonometry in 1595 which may have coined the word "trigonometry".

==Overview==
[[Image:TrigonometryTriangle.svg|thumb|240px|In this right triangle: {{mbox|= sin ''A'' = ''a''/''c'';}} {{mbox|= cos ''A'' = ''b''/''c'';}} {{mbox|= tan ''A'' = ''a''/''b''.}}]]
By definition, one [[angle]] of a right triangle is 90 degrees and If one of the other angles is known, the third can be calculated since all three angles of any triangle must add up to 180 degrees. The [[shape]] of a right triangle is completely determined, up to [[similarity (geometry)|similarity]], by the angles. This means that once one of the other angles is known, the [[ratio]]s of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following [[trigonometric function]]s of the known angle ''A'':

*The '''sine''' function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
:<math>\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}</math>
*The '''cosine''' function (cos), defined as the ratio of the adjacent leg to the hypotenuse. 
:<math>\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}</math>
*The '''tangent''' function (tan), defined as the ratio of the opposite leg to the adjacent leg.
:<math>\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{\sin A}{\cos A}</math>

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]]).

The [[Multiplicative inverse|reciprocals]] of these functions are named the '''cosecant''' (csc or cosec), '''secant''' (sec) and '''cotangent''' (cot), respectively. 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]].

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 an 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===
[[Image:Sine cosine plot.svg|300px|right|thumb|Graphs of the functions sin(''x'') and  cos(''x''), where the angle ''x'' is measured in radians.]]
[[Image:sin_drawing_process.gif|250px|thumb|right|Graphing process of ''y'' = sin(''x'') using a unit circle.]]
[[Image:tan_drawing_process.gif|250px|thumb|right|Graphing process of ''y'' = tan(''x'') using a unit circle.]]
[[Image:csc_drawing_process.gif|250px|thumb|right|Graphing process of ''y'' = csc(''x'') using a unit circle.]]
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 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 function '''cis''' is particularly useful

: <math>\operatorname{cis} (x) = \cos x + i\sin x \! = e^{ix} </math>

See [[Euler's formula|Euler's]] and [[De Moivre's formula|De Moivre's]] formulas.

===Mnemonics===
Students often use [[mnemonic]]s 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, as in SOH-CAH-TOA.<br />

:'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse<br />
:'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse<br />
:'''T'''angent = '''O'''pposite ÷ '''A'''djacent

Alternatively, one can devise sentences which consist of words beginning with the letters to be remembered. For example, to recall that Tan = Opposite/Adjacent, the letters T-O-A must be remembered. Any memorable phrase constructed of words beginning with the letters T-O-A will serve.

Another type of mnemonic describes facts in a simple, memorable way, such as "Plus to the right, minus to the left; positive height, negative depth," which refers to trigonometric functions generated by a revolving line.

===Calculating trigonometric functions===
{{main|Generating trigonometric tables}}
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. 

Today [[scientific calculator]]s 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, [[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.

==Applications of trigonometry ==
{{main|Uses of trigonometry}}
There are an enormous number of [[uses of trigonometry|applications 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.

Fields which make use of trigonometry or trigonometric functions include [[astronomy]] (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence [[navigation]] (on the oceans, in aircraft, and in space), [[music theory]], [[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]].

[[Image:Frieberger drum marine sextant.jpg|thumb|250px|Marine [[sextant]]s like this 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 then be determined from several such measurements.]]

==Common formulae==
:{{main|Trigonometric identity}}
:{{main|Trigonometric function}}
Certain equations involving trigonometric functions are true for all angles and are known as ''trigonometric identities.'' Many express important geometric relationships. For example, the Pythagorean identities are an expression of the [[Pythagorean Theorem]]. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see [[trigonometric identity]].

===Trigonometric identities===
{{Trigonometry}}
====Pythagorean identities====
:<math>\begin{align}
\sin^2 A + \cos^2 A &= 1 \\
\tan^2 A + 1 &= \sec^2 A \\
1+\cot^2 A &= \csc^2 A \end{align}</math>

====Sum and product identities====

=====Sum to product:=====
:<math>\begin{align}
\sin A \pm \sin B &= 2\sin \left( \frac{A \pm B}{2}\right)\cos \left(\frac{A \mp B}{2} \right)\\
\cos A + \cos B &= 2\cos \left(\frac{A + B}{2} \right)\cos  \left(\frac{A - B}{2}\right)\\
\cos A - \cos B &= -2\sin \left(\frac{A + B}{2} \right) \sin \left(\frac{A - B}{2}\right)
\end{align}</math>

=====Product to sum:=====
:<math>\begin{align}
\cos A \,\cos B &= \frac{1}{2}[\cos(A + B) + \cos (A - B)]\\
\sin A \,\sin B &= -\frac{1}{2}[\cos(A + B) - \cos (A - B)]\\
\cos A \,\sin B &= \frac{1}{2}[\sin(A + B) - \sin (A - B)]\\
\sin A \,\cos B &= \frac{1}{2}[\sin(A + B) + \sin (A - B)]
\end{align}</math>

=====Sine, cosine, and tangent of a sum=====
Detailed, diagramed proofs of the first two of these formulas
are available<br /> for download as a four-page PDF document
at [[Image:Sine_Cos_Proofs.pdf]].

:<math>\begin{align}
\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}
\end{align} </math>

====Double-angle identities====
:<math>\begin{align}
\sin 2A &= 2 \sin A \cos A \\
        &= \frac{2 \tan A}{1 + \tan^2 A}\\
\cos 2A &= \cos^2 A - \sin^2 A \\
        &= 2 \cos^2 A -1 \\
        &= 1-2 \sin^2 A \\
        &= {1 - \tan^2 A \over 1 + \tan^2 A}\\
\tan 2A &= \frac{2 \tan A}{1 - \tan^2 A}\\
        &= \frac{2 \cot A}{\cot^2 A - 1}\\
        &= \frac{2}{\cot A - \tan A}
\end{align}</math>

====Half-angle identities====
Note that <math>\pm</math> is correct, it means it may be either one, depending on the value of ''A/2''.

:<math>\begin{align}
\sin \frac{A}{2} &= \pm \sqrt{\frac{1-\cos A}{2}} \\
\cos \frac{A}{2} &= \pm \sqrt{\frac{1+\cos A}{2}} \\\tan \frac{A}{2} &= \pm \sqrt{\frac{1-\cos A}{1+\cos A}} = \frac {\sin A}{1+\cos A} = \frac {1-\cos A}{\sin A}
\end{align}</math>

====Stereographic ( or parametric ) identities====
:<math>\begin{align}
\sin A &= \frac{2T}{1+T^2} \\
\cos A &= \frac{1-T^2}{1+T^2}\\
\end{align}</math>

where  <math>T=\tan \frac{A}{2}</math>.

===Triangle identities===
[[Image:Triangle ABC with Sides a b c.png|thumb|240px|'''Laws of Sines and Cosines'''<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}</math><br /><br />
<math>\cos C=\frac{a^2+b^2-c^2}{2ab}</math>]]
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.

====Law of sines====

The '''[[law of sines]]''' (also know as the "sine rule") for an arbitrary triangle states:

:<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R</math>
where ''R'' is the radius of the [[circumcircle]] of the triangle.

====Law of cosines====
The '''[[law of cosines]]''' (also known as the cosine formula, or the "cos rule") is an extension of the [[Pythagorean theorem]] to arbitrary triangles:

:<math>c^2=a^2+b^2-2ab\cos C ,\,</math>

or equivalently:

:<math>\cos C=\frac{a^2+b^2-c^2}{2ab}\,</math>

====Law of tangents====
The '''[[law of tangents]]''':

:<math>\frac{a+b}{a-b}=\frac{\tan\left[\tfrac{1}{2}(A+B)\right]}{\tan\left[\tfrac{1}{2}(A-B)\right]}</math>

-->

==Notae==
<div class="references-small"><references/></div>

{{NexInt}}
*[[Aequatio trigonometrica]]
*[[Calculus infinitesimalis]]
*[[Functiones trigonometricae]]
*[[Metrum]]
*[[Sinus (mathematica)]]
*[[Theorema Pythagorae]]

==Nexus externi==
* [https://books.google.fr/books?id=O3haAAAAcAAJ&pg=PA32 Carl DOERFER Leçons de goniomètrie] (Francogallicae Latinaque, 1855)
* [http://www.pupress.princeton.edu/books/maor/ ''Trigonometric Delights,''] www.pupress.princeton.edu (liber ab Eli Maor scriptus, Princeton University P
ress, 1998; textus integer, PDF)
* [https://web.archive.org/web/20071104225720/http://baqaqi.chi.il.us/buecher/mathematics/trigonometry/index.html ''Trigonometry,''] baqaqi.chi.il.us (Alfred Monroe Kenyon et Louis Ingold, The Macmillan Company, 1914; textus integer, imagines)
* [http://www.mathwords.com/index_trig.htm ''Trigonometria,''] www.mathwords.com
* [https://web.archive.org/web/20070625162103/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=212&bodyId=81 Ludus trigonometricus a Benjamin Banneker creatus,] mathdl.maa.org
* [http://www.math10.com/en/algebra/sin-cos-tan-cot.html ''Trigonometria,''] www.math10.com
* [http://www.clarku.edu/~djoyce/trig/ ''Dave's Short Course in Trigonometry,''] www.clarku.edu (cursus scholasticus e David Joyce in [[Universitas Clarkensis|Universitate Clarkensis]])

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