## Trigonometric functions |

Reference |
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Laws and theorems |

In **trigonometric functions** (also called **circular functions**, **angle functions** or **goniometric functions**^{[1]}^{[2]}) are

The most familiar trigonometric functions are the

Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a

In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the *definitions* of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.

- right-angled triangle definitions
- unit-circle definitions
- algebraic values
- series definitions
- definitions via differential equations
- identities
- computation
- inverse functions
- properties and applications
- history
- etymology
- see also
- notes
- references
- external links

The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that

To define the trigonometric functions for the angle *A*, start with any right triangle that contains the angle *A*. The three sides of the triangle are named as follows:

- The
*hypotenuse*is the side opposite the right angle, in this case side*h*. The hypotenuse is always the longest side of a right-angled triangle. - The
*opposite side*is the side opposite to the angle we are interested in (angle*A*), in this case side*a*. - The
*adjacent side*is the side having both the angles of interest (angle*A*and right-angle*C*), in this case side*b*.

In ordinary *θ*) for angles θ, π − *θ*, π + *θ*, and 2π − *θ* depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at *A* in the accompanying diagram.

Function | Abbreviation | Description | |
---|---|---|---|

sine | sin | opposite/hypotenuse | |

cosine | cos | adjacent/hypotenuse | |

tangent | tan (or tg) | opposite/adjacent | |

cotangent | cot (or cotan or cotg or ctg or ctn) | adjacent/opposite | |

secant | sec | hypotenuse/adjacent | |

cosecant | csc (or cosec) | hypotenuse/opposite |

The **sine** of an angle is the ratio of the length of the opposite *sinus* for gulf or bay,^{[3]} since, given a unit circle, it is the side of the triangle on which the angle *opens*. In our case:

The **cosine** (*sine complement*, Latin: *cosinus*, *sinus complementi*) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle.^{[4]} Because the *B* is equal to π/2 − *A*; so cos *A* = sin *B* = sin(π/2 − *A*). In our case:

The **tangent** of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment *linea tangens* or touching line (cf. *tangere*, to touch).^{[5]} In our case:

Tangent may also be represented in terms of sine and cosine. That is:

These ratios do not depend on the size of the particular right triangle chosen, as long as the focus angle is equal, since all such triangles are

The acronyms "SOH-CAH-TOA" ("soak-a-toe", "sock-a-toa", "so-kah-toa") and "OHSAHCOAT" are commonly used

The remaining three functions are best defined using the three functions above and can be considered their

The **secant** of an angle is the reciprocal of its cosine, that is, the ratio of the length of the hypotenuse to the length of the adjacent side, so called because it represents the *cuts* the circle (from Latin: *secare*, to cut):^{[6]}

The **cosecant** (*secant complement*, Latin: *cosecans*, *secans complementi*) of an angle is the reciprocal of its sine, that is, the ratio of the length of the hypotenuse to the length of the opposite side, so called because it is the secant of the complementary or co-angle:

The **cotangent** (*tangent complement*, Latin: *cotangens*, *tangens complementi*) of an angle is the reciprocal of its tangent, that is, the ratio of the length of the adjacent side to the length of the opposite side, so called because it is the tangent of the complementary or co-angle:

Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the *rise*, *run*, and * slope* of a line segment relative to horizontal. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine and tangent. With a line segment length of 1 (as in a

- "Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise when the length of the line is 1.
- "Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run when the length of the line is 1.
- "Tangent combines the rise and run" meaning that Tangent takes the angle of the line segment and tells its slope, or alternatively, tells the vertical rise when the line segment's horizontal run is 1.

This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e. angles and slopes. (The arctangent or "inverse tangent" is not to be confused with the *cotangent*, which is cosine divided by sine.)

While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have a length of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the run at an angle of 7° is 5cos(7°).

Other Languages

العربية: دوال مثلثية

asturianu: Función trigonométrica

azərbaycanca: Triqonometrik funksiyalar

বাংলা: ত্রিকোণমিতিক অপেক্ষক

Bân-lâm-gú: Saⁿ-kak hâm-sò͘

башҡортса: Тригонометрик функциялар

беларуская: Трыганаметрычныя функцыі

български: Тригонометрична функция

bosanski: Trigonometrijska funkcija

català: Funció trigonomètrica

čeština: Goniometrická funkce

Cymraeg: Ffwythiannau trigonometreg

dansk: Trigonometrisk funktion

Deutsch: Trigonometrische Funktion

Ελληνικά: Τριγωνομετρική συνάρτηση

español: Función trigonométrica

Esperanto: Trigonometria funkcio

euskara: Funtzio trigonometriko

فارسی: توابع مثلثاتی

français: Fonction trigonométrique

galego: Función trigonométrica

한국어: 삼각함수

հայերեն: Եռանկյունաչափական ֆունկցիաներ

हिन्दी: त्रिकोणमितीय फलन

Bahasa Indonesia: Fungsi trigonometrik

íslenska: Hornafall

italiano: Funzione trigonometrica

עברית: פונקציות טריגונומטריות

ქართული: ტრიგონომეტრიული ფუნქციები

ລາວ: ຕຳລາໄຕມຸມ

Latina: Functiones trigonometricae

latviešu: Trigonometriskās funkcijas

magyar: Szögfüggvények

македонски: Тангенс

Bahasa Melayu: Fungsi trigonometri

Nederlands: Goniometrische functie

日本語: 三角関数

norsk: Trigonometrisk funksjon

norsk nynorsk: Trigonometrisk funksjon

occitan: Foncion trigonometrica

oʻzbekcha/ўзбекча: Trigonometrik funksiyalar

ភាសាខ្មែរ: អនុគមន៍ត្រីកោណមាត្រ

Piemontèis: Cosen

polski: Funkcje trygonometryczne

português: Função trigonométrica

română: Funcție trigonometrică

русский: Тригонометрические функции

Scots: Trigonometric functions

shqip: Funksionet trigonometrike

සිංහල: ත්රිකෝණමිතික ශ්රිත

Simple English: Trigonometric function

slovenčina: Goniometrická funkcia

slovenščina: Trigonometrična funkcija

کوردی: فانکشنە سێگۆشەیییەکان

српски / srpski: Тригонометријске функције

srpskohrvatski / српскохрватски: Trigonometrijske funkcije

suomi: Trigonometrinen funktio

svenska: Trigonometrisk funktion

தமிழ்: முக்கோணவியல் சார்புகள்

ไทย: ฟังก์ชันตรีโกณมิติ

тоҷикӣ: Функсияҳои тригонометрӣ

Türkçe: Trigonometrik fonksiyonlar

українська: Тригонометричні функції

اردو: مثلثیاتی دالہ

Tiếng Việt: Hàm lượng giác

文言: 三角函數

吴语: 三角函数

粵語: 三角函數

中文: 三角函数

Lingua Franca Nova: Funsionas trigonometrial