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{{Trigonometry}}
In [[mathematics]], the '''trigonometric functions''' (also called circular functions) are [[function (mathematics)|function]]s of an [[angle]]. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
 
The most familiar trigonometric functions are the sine, cosine, and tangent. The sine function takes an angle and tells the length of the ''y''-component (rise) of that triangle. The cosine function takes an angle and tells the length of ''x''-component (run) of a triangle. The tangent function takes an angle and tells the slope (''y''-component divided by the ''x''-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as [[ratio]]s of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a [[unit circle]]. More modern definitions express them as [[Series (mathematics)|infinite series]] or as solutions of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.
 
Trigonometric functions have a wide range of uses including computing unknown lengths and angles in [[triangle]]s (often [[right triangle]]s). In this use, trigonometric functions are used for instance in navigation, engineering, and physics. A common use in elementary physics is resolving a [[vector]] into [[Cartesian]] coordinates. The sine and cosine functions are also commonly used to model [[periodic|periodic function]] phenomena such as [[sound]] and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.
 
In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another. Especially in the case of 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 ==
[[Image:Trigonometry triangle.svg|right|thumb|A [[right triangle]] always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.]]
 
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 [[similarity (geometry)|similar triangles]] maintain the same ratios between their sides. That is, for any similar triangle the ratio of the [[hypotenuse]] (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.
 
In order 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&nbsp;'''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&nbsp;'''a'''.
* The ''adjacent side'' is the side that is in contact with (adjacent to) both the angle we are interested in (angle ''A'') and the right angle, in this case side&nbsp;'''b'''.
 
In ordinary [[Euclidean geometry]], the inside angles of every triangle total 180[[degree (angle)|°]] (π [[radian]]s). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of 0°–90°. The following definitions apply to angles in this 0°–90° range. (They can be extended to the full set of real arguments by using the [[unit circle]], or by requiring certain symmetries and that they be [[periodic function]]s.){{Clarify|date=April 2009}}
 
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.
 
{| class=wikitable style="margin-left:1em"
! style="text-align:left" | '''Function'''
! style="text-align:left" | '''Abbreviation'''
! style="text-align:left" | '''Description'''
! style="text-align:left" | '''[[List of trigonometric identities|Identities]] (using [[radian]]s)'''
|- style="background-color:#FFFFFF"
| '''Sine'''
| sin
| <math>\frac {\textrm{opposite}} {\textrm{hypotenuse}} </math>
| <math>\sin \theta \equiv \cos \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\csc \theta}</math>
|- style="background-color:#FFFFFF"
| '''Cosine'''
| cos
| <math>\frac {\textrm{adjacent}} {\textrm{hypotenuse}} </math>
| <math>\cos \theta \equiv \sin \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\sec \theta}\,</math>
|- style="background-color:#FFFFFF"
| '''Tangent'''
| tan (or&nbsp;tg)
|align=center| <math>\frac {\textrm{opposite}} {\textrm{adjacent}} </math>
| <math>\tan \theta \equiv \frac{\sin \theta}{\cos \theta} \equiv \cot \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\cot \theta} </math>
|- style="background-color:#FFFFFF"
| '''Cotangent'''
| cot (or&nbsp;ctg or&nbsp;ctn)
|align=center| <math>\frac {\textrm{adjacent}} {\textrm{opposite}} </math>
| <math>\cot \theta \equiv \frac{\cos \theta}{\sin \theta} \equiv \tan \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\tan \theta} </math>
|- style="background-color:#FFFFFF"
| '''Secant'''
| sec
| <math>\frac {\textrm{hypotenuse}} {\textrm{adjacent}} </math>
| <math>\sec \theta \equiv \csc \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\cos \theta} </math>
|- style="background-color:#FFFFFF"
| '''Cosecant'''
| csc (or&nbsp;cosec)
| <math>\frac {\textrm{hypotenuse}} {\textrm{opposite}} </math>
| <math>\csc \theta \equiv \sec \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\sin \theta} </math>
|}
{{FixBunching|beg}}
[[Image:Unitcircledefs.svg|250px|thumb|right|The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number ''θ'' is the length of the curve; thus angles are being measured in [[radian]]s. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. ("Fixed" in this context means not moving as ''θ'' changes; "moving" means depending on&nbsp;''θ''.)
Thus, as ''θ'' goes from&nbsp;0 up to a right angle, sin&nbsp;''θ'' goes from 0 to&nbsp;1, tan&nbsp;''θ'' goes from 0 to&nbsp;∞, and sec&nbsp;''θ'' goes from 1 to&nbsp;∞.]]
{{FixBunching|mid}}
[[Image:Unitcirclecodefs.svg|250px|thumb|right|The cosine, cotangent, and cosecant functions of an angle&nbsp;''&theta;'' constructed geometrically in terms of a unit circle. The functions whose names have the prefix&nbsp;''co-'' use horizontal lines where the others use vertical lines.]]
{{FixBunching|end}}
 
=== Sine ===
The '''sine''' of an angle is the ratio of the length of the opposite [[Cathetus|side]] to the length of the hypotenuse. In our case
:<math>\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.</math>
Note that this ratio does not depend on size of the particular right triangle chosen, as long as it contains the angle ''A'', since all such triangles are [[similarity (geometry)|similar]].
 
=== Cosine ===
The '''cosine''' of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case
:<math>\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.</math>
 
=== Tangent ===
The '''tangent''' of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
:<math>\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.</math>
 
=== Reciprocal functions ===
The remaining three functions are best defined using the above three functions.
 
The '''cosecant''' csc(''A''), or cosec(''A''), is the [[multiplicative inverse|reciprocal]] of sin(''A''), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
:<math>\csc A = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}. </math>
 
The '''secant''' sec(''A'') is the [[multiplicative inverse|reciprocal]] of cos(''A''), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
 
:<math>\sec A = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}. </math>
 
The '''cotangent''' cot(''A'') is the [[multiplicative inverse|reciprocal]] of tan(''A''), i.e. the ratio of the length of the adjacent side to the length of the opposite side:
:<math>\cot A = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}. </math>
 
=== Slope definitions ===
Equivalent to the right-triangle definitions the trigonometric functions can be defined in terms of the ''rise'', ''run'', and ''[[slope]]'' of a line segment relative to some horizontal line. 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, tangent. With a [[unit circle]], the following correspondence of definitions exists:
 
# Sine is first, rise is first. Sine takes an angle and tells the rise when the length of the line is 1.<!-- TANGENT "tells the rise" when the RUN is 1. -->
# Cosine is second, run is second. Cosine takes an angle and tells the run when the length of the line is 1.
# Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope, and tells the rise when the 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. (Note that the arctangent or "inverse tangent" is not to be confused with the ''cotangent,'' which is cos divided by sin.)
 
While the radius of the circle 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, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1° is 5&nbsp;cos(1°)
 
== Unit-circle definitions ==
[[Image:Unit circle angles.svg|right|thumb|360px|The [[unit circle]]]]
The six trigonometric functions can also be defined in terms of the [[unit circle]], the [[circle]] of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.
 
The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 radians.
 
It also provides a single visual picture that encapsulates at once all the important triangles. From the [[Pythagorean theorem]] the equation for the unit circle is:
 
: <math>x^2 + y^2 = 1. \, </math>
 
In the picture, some common angles, measured in radians, are given. Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles.
 
Let a line through the origin, making an angle of ''θ'' with the positive half of the ''x''-axis, intersect the unit circle. The ''x''- and ''y''-coordinates of this point of intersection are equal to cos&nbsp;''θ'' and sin&nbsp;''θ'', respectively.
 
The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin&nbsp;''θ'' = ''y''/1 and cos&nbsp;''θ'' = ''x''/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to&nbsp;1.
 
Note that these values can easily be memorized in the form
: <math>\tfrac{1}{2}\sqrt{0},\quad \tfrac{1}{2}\sqrt{1},\quad \tfrac{1}{2}\sqrt{2},\quad \tfrac{1}{2}\sqrt{3},\quad \tfrac{1}{2}\sqrt{4}.</math>
{{clr}}
 
[[Image:Sine cosine plot.svg|300px|right|thumb|The sine and cosine functions graphed on the Cartesian plane.]]
For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine and cosine are [[periodic function]]s with period 2π:
: <math>\sin\theta = \sin\left(\theta + 2\pi k \right),\,</math>
: <math>\cos\theta = \cos\left(\theta + 2\pi k \right),\,</math>
for any angle ''θ'' and any [[integer]] ''k''.
 
The ''smallest'' positive period of a periodic function is called the ''primitive period'' of the function.
 
The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.
{{clr}}
 
[[Image:Trigonometric functions.svg|right|thumb|300px|Trigonometric functions:
<span style="color:#00A">Sine</span>,
<span style="color:#0A0">Cosine</span>,
<span style="color:#A00">Tangent</span>,
<span style="color:#AA0">Cosecant</span>,
<span style="color:#A0A">Secant</span>,
<span style="color:#0AA">Cotangent</span>]]
Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:
: <math>\tan\theta = \frac{\sin\theta}{\cos\theta},\ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}\,</math>
: <math>\sec\theta = \frac{1}{\cos\theta},\ \csc\theta = \frac{1}{\sin\theta}\,</math>
 
So :
* The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.
* The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.
 
To the right is an image that displays a noticeably different graph of the trigonometric function f(θ)= tan(θ) graphed on the Cartesian plane.
* Note that its ''x''-intercepts correspond to that of sin(''θ'') while its undefined values correspond to the ''x''-intercepts of the cos(''θ'').
* Observe that the function's results change slowly around angles of ''k''π, but change rapidly at angles close to (''k''&nbsp;+&nbsp;1/2)π.
* The graph of the tangent function also has a vertical [[asymptote]] at ''θ''&nbsp;=&nbsp;(''k''&nbsp;+&nbsp;1/2)π.
* This is the case because the function approaches infinity as ''θ'' approaches (''k''&nbsp;+&nbsp;1/2)π from the left and minus infinity as it approaches (''k''&nbsp;+&nbsp;1/2)π from the right.
{{clr}}
 
[[Image:Circle-trig6.svg|right|thumb|300px|All of the trigonometric functions of the angle ''θ'' can be constructed geometrically in terms of a unit circle centered at ''O''.]]
Alternatively, ''all'' of the basic trigonometric functions can be defined in terms of a unit circle centered at ''O'' (as shown in the picture to the right), and similar such geometric definitions were used historically.
* In particular, for a chord ''AB'' of the circle, where ''θ'' is half of the subtended angle, sin(''θ'') is ''AC'' (half of the chord), a definition introduced in [[India]]<ref name=boyer/> (see [[history of trigonometry|history]]).
* cos(''θ'') is the horizontal distance ''OC'', and [[versine|versin]](''θ'')&nbsp;=&nbsp;1&nbsp;−&nbsp;cos(''θ'') is ''CD''.
* tan(''θ'') is the length of the segment ''AE'' of the tangent line through ''A'', hence the word ''[[tangent]]'' for this function. cot(''θ'') is another tangent segment, ''AF''.
* sec(''θ'')&nbsp;=&nbsp;''OE'' and csc(''θ'')&nbsp;=&nbsp;''OF'' are segments of [[secant line]]s (intersecting the circle at two points), and can also be viewed as projections of ''OA'' along the tangent at ''A'' to the horizontal and vertical axes, respectively.
* ''DE'' is [[exsecant|exsec]](''θ'') = sec(''θ'')&nbsp;−&nbsp;1 (the portion of the secant outside, or ''ex'', the circle).
* From these constructions, it is easy to see that the secant and tangent functions diverge as ''θ'' approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as ''θ'' approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.<ref>See Maor (1998)</ref>)
{{clr}}
 
== Series definitions ==
[[Image:Taylorsine.svg|300px|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]
 
Using only geometry and properties of [[limit of a function|limits]], it can be shown that the [[derivative]] of sine is cosine and the derivative of cosine is the negative of sine. (Here, and generally in [[calculus]], all angles are measured in [[radian]]s; see also [[#The significance of radians|the significance of radians]] below.) One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'':<ref>See Ahlfors, pages 43–44.</ref>
 
:<math>
\begin{align}
\sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt]
& = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}, \\[8pt]
\cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\[8pt]
& = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}.
\end{align}
</math>
 
These identities are sometimes taken as the ''definitions'' of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (''e.g.,'' in [[Fourier series]]), since the theory of [[Series (mathematics)|infinite series]] can be developed from the foundations of the [[real number|real number system]], independent of any geometric considerations. The [[derivative|differentiability]] and [[continuous function|continuity]] of these functions are then established from the series definitions alone.
 
Combining these two series gives [[Euler's formula]]: cos ''x'' + ''i'' sin ''x'' = ''e''<sup>''ix''</sup>.
 
Other series can be found.<ref>Abramowitz; Weisstein.</ref> For the following trigonometric functions:
: ''U''<sub>''n''</sub> is the ''n''th [[up/down number]],
: ''B''<sub>''n''</sub> is the ''n''th [[Bernoulli number]], and
: ''E''<sub>''n''</sub> (below) is the ''n''th [[Euler number]].
<!-- ''It would be nice if someone adds how the infinite series expressions are deduced.'' -->
 
'''Tangent'''
: <math>
\begin{align}
\tan x & {} = \sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!} \\[8pt]
& {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} x^{2n-1}}{(2n)!} \\[8pt]
& {} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315} + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>
 
When this series for the tangent function is expressed in a form in which the denominators are the corresponding factorials, and the numerators, called the "tangent numbers", have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets of odd cardinality.{{Citation needed|date=June 2008}}
 
'''Cosecant'''
: <math>
\begin{align}
\csc x & {} = \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n} x^{2n-1}}{(2n)!} \\[8pt]
& {} = \frac {1} {x} + \frac {x} {6} + \frac {7 x^3} {360} + \frac {31 x^5} {15120} + \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>
 
<!-- extra blank line between [[TeX]] displays for legibility -->
'''Secant'''
: <math>
\begin{align}
\sec x & {} = \sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!}
= \sum_{n=0}^\infty \frac{(-1)^n E_{2n} x^{2n}}{(2n)!} \\[8pt]
& {} = 1 + \frac {x^2} {2} + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>
 
When this series for the secant function is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "secant numbers", have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets of even cardinality.{{Citation needed|date=June 2008}}
 
'''Cotangent'''
: <math>
\begin{align}
\cot x & {} = \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!} \\[8pt]
& {} = \frac {1} {x} - \frac {x}{3} - \frac {x^3} {45} - \frac {2 x^5} {945} - \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>
 
From a theorem in [[complex analysis]], there is a unique [[analytic continuation]] of this real function to the domain of complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.
 
===Relationship to exponential function and complex numbers===
[[Image:Sine and Cosine fundamental relationship to Circle (and Helix).gif|thumb|300px|[[Euler's formula]] illustrated with the three dimensional [[helix]], starting with the 2-D [[Projection (linear algebra)|orthogonal]] components of the [[unit circle]], [[sine wave|sine]] and cosine (using ''&theta;''&nbsp;=&nbsp;''t''&nbsp;).]]
It can be shown from the series definitions<ref> For a demonstration, see [[Euler's_formula#Using Taylor series]]</ref> that the sine and cosine functions are the [[complex number|imaginary]] and real parts, respectively, of the [[exponential function#On the complex plane|complex exponential function]] when its argument is purely imaginary:
 
:<math> e^{i \theta} = \cos\theta + i\sin\theta. \, </math>
 
This identity is called [[Euler's formula]]. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the [[complex plane]], defined by ''e''<sup>&nbsp;''ix''</sup>, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.
 
Furthermore, this allows for the definition of the trigonometric functions for complex arguments ''z'':
 
: <math>\sin z = \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \frac{e^{i z} - e^{-i z}}{2i}\, = \frac{\sinh \left( i z\right) }{i} </math>
 
: <math>\cos z = \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n} = \frac{e^{i z} + e^{-i z}}{2}\, = \cosh \left(i z\right) </math>
 
where ''i''<sup>&nbsp;2</sup>&nbsp;=&nbsp;−1. Also, for purely real ''x'',
 
: <math>\cos x = \mbox{Re } (e^{i x}) \,</math>
 
: <math>\sin x = \mbox{Im } (e^{i x}) \,</math>
 
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.
 
: <math>\sin (x + iy) = \sin x \cosh y + i \cos x \sinh y,\,</math>
 
: <math>\cos (x + iy) = \cos x \cosh y - i \sin x \sinh y.\,</math>
 
This exhibits a deep relationship between the complex sine and cosine functions and their real and real hyperbolic counterparts.
 
====Complex graphs====
In the following graphs, the domain is the complex plane pictured, and the range values are indicated at each point by color. Brightness indicates the size (absolute value) of the range value, with black being zero. Hue varies with argument, or angle, measured from the positive real axis. ([[:Image:Complex coloring.jpg|more]])
 
{| style="text-align:center"
|+ '''Trigonometric functions in the complex plane'''
|[[Image:Complex sin.jpg|1000x140px|none]]
|[[Image:Complex cos.jpg|1000x140px|none]]
|[[Image:Complex tan.jpg|1000x140px|none]]
|[[Image:Complex Cot.jpg|1000x140px|none]]
|[[Image:Complex Sec.jpg|1000x140px|none]]
|[[Image:Complex Csc.jpg|1000x140px|none]]
|-
|<math>
\sin z\,
</math>
|<math>
\cos z\,
</math>
|<math>
\tan z\,
</math>
|<math>
\cot z\,
</math>
|<math>
\sec z\,
</math>
|<math>
\csc z\,
</math>
|}
 
==Definitions via differential equations==
Both the sine and cosine functions satisfy the [[differential equation]]
: <math>y'' = -y.\,</math>
 
That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional [[function space]] ''V'' consisting of all solutions of this equation,
* the sine function is the unique solution satisfying the initial condition <math>\scriptstyle \left( y'(0), y(0) \right) = (1, 0)\,</math> and
* the cosine function is the unique solution satisfying the initial condition <math>\scriptstyle \left( y'(0), y(0) \right) = (0, 1)\,</math>.
 
Since the sine and cosine functions are linearly independent, together they form a [[basis (linear algebra)|basis]] of ''V''. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See [[linear differential equation]].) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the [[List of trigonometric identities|trigonometric identities]] for the sine and cosine functions.
 
Further, the observation that sine and cosine satisfies <math>\scriptstyle y'' = -y\,</math> means that they are [[eigenfunction]]s of the second-derivative operator.
 
The tangent function is the unique solution of the nonlinear differential equation
: <math>y' = 1 + y^2\,</math>
satisfying the initial condition <math>\scriptstyle y(0) = 0\,</math>. There is a very interesting visual proof that the tangent function satisfies this differential equation; see Needham's ''Visual Complex Analysis.''<ref>{{cite web|url=ix|title=INSERT TITLE|last=Needham|first=p}}</ref>
 
=== The significance of radians ===
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only those sines and cosines which map radians to ratios satisfy the differential equations which classically describe them. If an argument to sine or cosine in radians is scaled by frequency,
 
: <math>f(x) = \sin kx, \,</math>
 
then the derivatives will scale by ''amplitude''.
 
: <math>f'(x) = k\cos kx. \,</math>
 
Here, ''k'' is a constant that represents a mapping between units. If ''x'' is in degrees, then
 
: <math>k = \frac{\pi}{180^\circ}.</math>
 
This means that the second derivative of a sine in degrees satisfies not the differential equation
 
: <math>y'' = -y\,</math>
 
but rather
 
: <math>y'' = -k^2 y.\,</math>
 
The cosine's second derivative behaves similarly.
 
This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.
 
== Identities ==
{{Main|List of trigonometric identities}}
Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the '''Pythagorean identity,''' which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the [[Pythagorean theorem]]. In symbolic form, the Pythagorean identity is written
 
:<math>\sin^2 x + \cos^2 x = 1, \, </math>
 
where sin<sup>2</sup>&nbsp;''x''&nbsp;+&nbsp;cos<sup>2</sup>&nbsp;''x'' is standard notation for (sin&nbsp;''x'')<sup>2</sup>&nbsp;+&nbsp;(cos&nbsp;''x'')<sup>2</sup>.
 
Other key relationships are the '''sum and difference formulas,''' which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to [[Ptolemy]]; one can also produce them algebraically using Euler's formula.
{{col-start}}
{{col-2}}
:<math>\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y, \,</math>
:<math>\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y, \,</math>
{{col-2}}
:<math>\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y, \,</math>
:<math>\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y. \,</math>
{{col-end}}
When the two angles are equal, the sum formulas reduce to simpler equations known as the '''double-angle formulae.'''
 
These identities can also be used to derive the [[List of trigonometric identities#Product-to-sum_and_sum-to-product identities|product-to-sum identities]] that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations, much like the [[logarithm|logarithm function]].
 
=== Calculus ===
For [[integral]]s and [[derivative]]s of trigonometric functions, see the relevant sections of [[List of differentiation identities]], [[Lists of integrals]] and [[list of integrals of trigonometric functions]]. Below is the list of the derivatives and integrals of the six basic trigonometric functions.
 
:{| class="wikitable"
|-
| <math>\ \ \ \ f(x)</math>
| <math>\ \ \ \ f'(x)</math>
| <math>\int f(x)\,dx</math>
|-
| <math>\,\ \sin x</math>
| <math>\,\ \cos x</math>
| <math>\,\ -\cos x + C</math>
|-
| <math>\,\ \cos x</math>
| <math>\,\ -\sin x</math>
| <math>\,\ \sin x + C</math>
|-
| <math>\,\ \tan x</math>
| <math>\,\ \sec^{2} x = 1+\tan^{2} x</math>
| <math>-\ln \left |\cos x\right | + C</math>
|-
| <math>\,\ \cot x</math>
| <math>\,\ -\csc^{2} x = -(1+\cot^{2} x)</math>
| <math>\ln \left |\sin x\right | + C</math>
|-
| <math>\,\ \sec x</math>
| <math>\,\ \sec{x}\tan{x}</math>
| <math>\ln \left |\sec x + \tan x\right | + C</math>
|-
| <math>\,\ \csc x</math>
| <math>\,\ -\csc{x}\cot{x}</math>
| <math>\ -\ln \left |\csc x + \cot x\right | + C</math>
|}
 
=== Definitions using functional equations ===
In [[mathematical analysis]], one can define the trigonometric functions using [[functional equation]]s based on properties like the sum and difference formulas. Taking as given these formulas and the Pythagorean identity, for example, one can prove that only two [[real analysis|real function]]s satisfy those conditions. Symbolically, we say that there exists exactly one pair of real functions&nbsp;— <math>\scriptstyle \sin\,</math> and <math>\scriptstyle \cos\,</math>&nbsp;— such that for all real numbers <math>\scriptstyle x\,</math> and <math>\scriptstyle y\,</math>, the following equations hold:{{Citation needed|date=March 2008}}
: <math>\sin^2 x + \cos^2 x = 1\,</math>
: <math>\sin(x\pm y) = \sin x\cos y \pm \cos x\sin y\,</math>
: <math>\cos(x\pm y) = \cos x\cos y \mp \sin x\sin y\,</math>
with the added condition that <math>\scriptstyle 0 < x\cos x < \sin x < x\,</math> for <math>\scriptstyle 0 < x < 1\,</math>.
 
Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers.
As an example, this derivation can be used to define [[trigonometry in Galois fields]].
 
== Computation ==
The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of [[computer]]s and [[scientific calculator]]s that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found.
 
The first step in computing any trigonometric function is range reduction — reducing the given angle to a "reduced angle" inside a small range of angles, say 0 to ''π''/2, using the periodicity and symmetries of the trigonometric functions.
 
{{Main|Generating trigonometric tables}}
 
Prior to computers, people typically evaluated trigonometric functions by [[interpolation|interpolating]] from a detailed table of their values, calculated to many [[significant figures]]. Such tables have been available for as long as trigonometric functions have been described (see [[#History|History]] below), and were typically generated by repeated application of the half-angle and angle-addition [[List of trigonometric identities|identities]] starting from a known value (such as sin(''π''/2)&nbsp;=&nbsp;1).
 
Modern computers use a variety of techniques.<ref>Kantabutra.</ref> One common method, especially on higher-end processors with [[floating point]] units, is to combine a [[polynomial]] or [[rational function|rational]] [[approximation theory|approximation]] (such as [[Approximation theory|Chebyshev approximation]], best uniform approximation, and [[Padé approximant|Padé approximation]], and typically for higher or variable precisions, [[Taylor series|Taylor]] and [[Laurent series]]) with range reduction and a [[lookup table|table lookup]] — they first look up the closest angle in a small table, and then use the polynomial to compute the correction.<ref>However, doing that while maintaining precision is nontrivial, and methods like [[Gal's accurate tables]], Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used.</ref> On devices that lack [[Arithmetic logic unit|hardware multipliers]], an algorithm called [[CORDIC]] (as well as related techniques) which uses only addition, subtraction, [[bitwise operation|bitshift]] and [[lookup table|table lookup]], is often used. All of these methods are commonly implemented in [[Personal computer hardware|hardware]] [[floating-point unit]]s for performance reasons.
 
For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the [[arithmetic-geometric mean]], which itself approximates the trigonometric function by the ([[complex number|complex]]) [[elliptic integral]].<ref>{{cite web|url=http://doi.acm.org/10.1145/321941.321944|title=R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM '''23''', 242 (1976).}}</ref>
 
{{Main|Exact trigonometric constants}}
 
Finally, for some simple angles, the values can be easily computed by hand using the [[Pythagorean theorem]], as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of <math>\pi / 60</math> [[radian]]s (3°) can be found [[Exact trigonometric constants|exactly by hand]].
 
Consider a right triangle where the two other angles are equal, and therefore are both <math>\pi / 4</math> radians (45°). Then the length of side ''b'' and the length of side ''a'' are equal; we can choose <math>a = b = 1</math>. The values of sine, cosine and tangent of an angle of <math>\pi / 4</math> radians (45°) can then be found using the Pythagorean theorem:
 
:<math>c = \sqrt { a^2+b^2 } = \sqrt2\,.</math>
 
Therefore:
 
:<math>\sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos \left(\pi / 4 \right) = \cos \left(45^\circ\right) = {1 \over \sqrt2},\,</math>
 
:<math>\tan \left(\pi / 4 \right) = \tan \left(45^\circ\right) = {{\sin \left(\pi / 4 \right)}\over{\cos \left(\pi / 4 \right)}} = {1 \over \sqrt2} \cdot {\sqrt2 \over 1} = {\sqrt2 \over \sqrt2} = 1. \,</math>
 
To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:
 
:<math>\sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos \left(\pi / 3 \right) = \cos \left(60^\circ\right) = {1 \over 2}\,,</math>
:<math>\cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin \left(\pi / 3 \right) = \sin \left(60^\circ\right) = {\sqrt3 \over 2}\,,</math>
:<math>\tan \left(\pi / 6 \right) = \tan \left(30^\circ\right) = \cot \left(\pi / 3 \right) = \cot \left(60^\circ\right) = {1 \over \sqrt3}\,.</math>
<!-- Please do not expand this section by adding more and more formulas for different angles, or more formulas for the same angles. We have a whole article on [[Exact trigonometric constants]], linked above. Go expand that. -->
 
== Inverse functions ==
{{Main|Inverse trigonometric functions}}
The trigonometric functions are periodic, and hence not [[injective function|injective]], so strictly they do not have an [[inverse function]]. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is [[bijection|bijective]]. In the following, the functions on the left are ''defined'' by the equation on the right; these are not proved identities. The principal inverses are usually defined as:
 
: <math> \begin{matrix}
 
\mbox{for} & -\frac{\pi}{2} \le y \le \frac{\pi}{2},
& y = \arcsin x & \mbox{if} & x = \sin y \,;\\ \\
\mbox{for} & 0 \le y \le \pi,
& y = \arccos x & \mbox{if} & x = \cos y \,;\\ \\
\mbox{for} & -\frac{\pi}{2} < y < \frac{\pi}{2},
& y = \arctan x & \mbox{if} & x = \tan y \,;\\ \\
\mbox{for} & -\frac{\pi}{2} \le y \le \frac{\pi}{2}, y \ne 0,
& y = \arccsc x & \mbox{if} & x = \csc y \,;\\ \\
\mbox{for} & 0 \le y \le \pi, y \ne \frac{\pi}{2},
& y = \arcsec x & \mbox{if} & x = \sec y \,;\\ \\
\mbox{for} & 0 < y < \pi,
& y = \arccot x & \mbox{if} & x = \cot y \,.
 
\end{matrix} </math>
 
For inverse trigonometric functions, the notations sin<sup>−1</sup> and cos<sup>−1</sup> are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "[[minute of arc|arcsecond]]".
 
Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,
:<math>
\arcsin z = z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^7} {7} + \cdots\,.</math>
These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:
:<math>
\arcsin z =
\int_0^z \frac 1 {\sqrt{1 - x^2}}\,dx, \quad |z| < 1\,.
</math>
Analogous formulas for the other functions can be found at [[Inverse trigonometric functions]]. Using the [[complex logarithm]], one can generalize all these functions to complex arguments:
:<math>
\arcsin z = -i \log \left( i z + \sqrt{1 - z^2} \right)\,,
</math>
:<math>
\arccos z = -i \log \left( z + \sqrt{z^2 - 1}\right)\,,
</math>
:<math>
\arctan z = \frac{i}{2} \log\left(\frac{1-iz}{1+iz}\right)\,.
</math>
<!-- (note: these should probably be presented as definite integrals, removing the ambiguity of the constant) -->
 
== Properties and applications ==
{{Main|Uses of trigonometry}}
 
The trigonometric functions, as the name suggests, are of crucial importance in [[trigonometry]], mainly because of the following two results.
 
===Law of sines===
The '''[[law of sines]]''' states that for an arbitrary [[triangle]] with sides ''a'', ''b'', and ''c'' and angles opposite those sides ''A'', ''B'' and ''C'':
 
:<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\,,</math>
 
or, equivalently,
 
:<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\,,</math>
 
where ''R'' is the triangle's [[circumscribed circle|circumradius]].
 
[[Image:Lissajous curve 5by4.svg|right|thumb|right|A [[Lissajous curve]], a figure formed with a trigonometry-based function.]]
 
It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
 
===Law of cosines===
The '''[[law of cosines]]''' (also known as the cosine formula) is an extension of the [[Pythagorean theorem]]:
 
:<math>c^2=a^2+b^2-2ab\cos C \,,</math>
also known as:
:<math>\cos C=\frac{a^2+b^2-c^2}{2ab}\,.</math>
 
In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the [[Pythagorean theorem]].
 
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
 
===Other useful properties===
There is also a '''[[law of tangents]]''':
 
:<math>\frac{a+b}{a-b} = \frac{\tan[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}\,.</math>
 
====Sine and cosine of sums of angles====
{{See also|Angle addition formula}}
Detailed, diagrammed construction proofs, by geometric construction, of formulas for the sine and cosine of the sum of two angles are available for download as a four-page PDF document at [[:File:Sine Cos Proofs.pdf]].
 
===Periodic functions===
[[Image:Synthesis square.gif|thumb|350px|right|Animation of the [[additive synthesis]] of a [[square wave]] with an increasing number of harmonics]]
[[File:Sawtooth Fourier Analysis.JPG|thumb|280px|Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top); the basis functions have [[wavelength]]s λ/''k'' (''k''=integer) shorter than the wavelength λ of the sawtooth itself (except for ''k''=1). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the [[Gibbs phenomenon]]]]
 
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe [[simple harmonic motion]], which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of [[uniform circular motion]].
 
Trigonometric functions also prove to be useful in the study of general [[periodic function]]s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light [[wave]]s.<ref name=Farlow> {{cite book |title=Partial differential equations for scientists and engineers |url=http://books.google.com/books?id=DLUYeSb49eAC&pg=PA82 |author=Stanley J Farlow |page=82 |isbn=048667620X |publisher=Courier Dover Publications |edition=Reprint of Wiley 1982 |year=1993}} </ref>
 
Under rather general conditions, a periodic function ''f(x)'' can be expressed as a sum of sine waves or cosine waves in a [[Fourier series]].<ref name=Folland>
 
See for example, {{cite book |author=Gerald B Folland |title=Fourier Analysis and its Applications |publisher=American Mathematical Society |edition=Reprint of Wadsworth & Brooks/Cole 1992 |url=http://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |pages = 77 ''ff'' |chapter=Convergence and completeness |year=2009 |isbn=0821847902}}
 
</ref> Denoting the sine or cosine [[basis functions]] by ''φ<sub>k</sub>'', the expansion of the periodic function ''f(t)'' takes the form:
 
:<math>f(t) = \sum _{k=1}^{\infty} c_k \varphi_k(t) \ . </math>
 
For example, the [[square wave]] can be written as the [[Fourier series]]
 
:<math> f_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ( (2k-1)t \right )}\over(2k-1)}.</math>
 
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a [[sawtooth wave]] are shown underneath.
 
== History ==
{{Main|History of trigonometric functions}}
 
The [[Chord (geometry)|chord]] function was discovered by [[Hipparchus]] of [[İznik|Nicaea]] (180–125 BC) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165 AD). The [[#Sine|sine]] and [[#Cosine|cosine]] functions were discovered by [[Aryabhata]] (476–550) and studied by [[Varahamihira]] and [[Brahmagupta]]. The [[#Tangent|tangent]] function was discovered by {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī]]}} (780–850), and the [[#Reciprocal functions|reciprocal functions]] of secant, cotangent and cosecant were discovered by [[Abū al-Wafā' Būzjānī]] (940–998). All six trigonometric functions were then studied by [[Omar Khayyám]], [[Bhāskara II]], [[Nasir al-Din al-Tusi]], [[Jamshīd al-Kāshī]] (14th century), [[Ulugh Beg]] (14th century), [[Regiomontanus]] (1464), [[Georg Joachim Rheticus|Rheticus]], and Rheticus' student [[Valentinus Otho]].{{Citation needed|date=June 2008}}
 
[[Madhava of Sangamagrama]] (c. 1400) made early strides in the [[mathematical analysis|analysis]] of trigonometric functions in terms of [[series (mathematics)|infinite series]].<ref name=mact-biog>{{cite web
| publisher=School of Mathematics and Statistics University of St Andrews, Scotland | title=Biography of Madhava
|author = J J O'Connor and E F Robertson
|url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html
| title=Madhava of Sangamagrama
| accessdate=2007-09-08
}}
</ref>
 
The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician [[Albert Girard]].
 
In a paper published in 1682, [[Gottfried Leibniz|Leibniz]] proved that sin ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994}}</ref><br> [[Leonhard Euler]]'s ''Introductio in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[[Euler's formula]]", as well as the near-modern abbreviations ''sin., cos., tang., cot., sec.,'' and ''cosec.''<ref name=boyer>See Boyer (1991).</ref>
 
A few functions were common historically, but are now seldom used, such as the [[chord (geometry)|chord]] (crd(''θ'') = 2&nbsp;sin(''θ''/2)), the [[versine]] (versin(''θ'') = 1&nbsp;−&nbsp;cos(''θ'') = 2&nbsp;sin<sup>2</sup>(''θ''/2)) (which appeared in the earliest tables <ref name=boyer/>), the [[Versine|haversine]] (haversin(''θ'') = versin(''θ'')&nbsp;/&nbsp;2 = sin<sup>2</sup>(''θ''/2)), the [[exsecant]] (exsec(''θ'') = sec(''θ'')&nbsp;−&nbsp;1) and the [[exsecant|excosecant]] (excsc(''θ'') = exsec(π/2&nbsp;−&nbsp;''θ'') = csc(''θ'')&nbsp;−&nbsp;1). Many more relations between these functions are listed in the article about [[List of trigonometric identities|trigonometric identities]].
 
[[Etymology|Etymologically]], the word ''sine'' derives from the [[Sanskrit]] word for half the chord, ''jya-ardha'', abbreviated to ''jiva''. This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as ''jiba'', written ''jb'', vowels not being written in Arabic. Next, this transliteration was mis-translated in the 12th century into [[Latin]] as ''[[wikt:sinus|sinus]]'', under the mistaken impression that ''jb'' stood for the word ''jaib'', which means "bosom" or "bay" or "fold" in Arabic, as does ''sinus'' in Latin.<ref>See Maor (1998), chapter 3, regarding the etymology.</ref> Finally, English usage converted the Latin word ''sinus'' to ''sine''.<ref>{{cite web|url=http://www.clarku.edu/~djoyce/trig/|title=Clark University}}</ref> The word ''tangent'' comes from Latin ''tangens'' meaning "touching", since the line ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin ''secans'' — "cutting" — since the line ''cuts'' the circle.
 
== See also ==
<div class="references" style="-moz-column-count:2; column-count:2;">
* [[Generating trigonometric tables]]
* [[Hyperbolic function]]
* [[Pythagorean theorem]]
* [[Unit vector]] (explains direction cosines)
* [[Table of Newtonian series]]
* [[List of trigonometric identities]]
* [[Proofs of trigonometric identities]]
* [[Euler's formula]]
* [[Polar sine]] — a generalization to vertex angles
* [[All Students Take Calculus]] — a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
* [[Gauss's continued fraction]] — a [[continued fraction]] definition for the tangent function
* [[Generalized trigonometry]]
</div>
 
== Notes ==
{{reflist}}
 
==References==
{{No footnotes|date=December 2008}}
{{refbegin}}
*Abramowitz, Milton and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]'', Dover, New York. (1964). ISBN 0-486-61272-4.
*[[Lars Ahlfors]], ''Complex Analysis: an introduction to the theory of analytic functions of one complex variable'', second edition, [[McGraw-Hill Book Company]], New York, 1966.
*[[Carl Benjamin Boyer|Boyer, Carl B.]], ''A History of Mathematics'', John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
* Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transaction on Mathematical Software (1991).
*Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. [[Penguin Books]], London. (2000). ISBN 0-691-00659-8.
* Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," ''IEEE Trans. Computers'' '''45''' (3), 328–339 (1996).
*Maor, Eli, ''[http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights]'', Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
*Needham, Tristan, [http://www.usfca.edu/vca/PDF/vca-preface.pdf "Preface"]" to ''[http://www.usfca.edu/vca/ Visual Complex Analysis]''. Oxford University Press, (1999). ISBN 0-19-853446-9.
*O'Connor, J.J., and E.F. Robertson, [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html "Trigonometric functions"], ''[[MacTutor History of Mathematics archive]]''. (1996).
*O'Connor, J.J., and E.F. Robertson, [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html "Madhava of Sangamagramma"], ''[[MacTutor History of Mathematics archive]]''. (2000).
*Pearce, Ian G., [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html "Madhava of Sangamagramma"], ''[[MacTutor History of Mathematics archive]]''. (2002).
*Weisstein, Eric W., [http://mathworld.wolfram.com/Tangent.html "Tangent"] from ''[[MathWorld]]'', accessed 21 January 2006.
{{refend}}
{{Wikibooks|Trigonometry}}
 
== External links ==
*[http://www.visionlearning.com/library/module_viewer.php?mid=131&l=&c3= Visionlearning Module on Wave Mathematics]
*[http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions
*[http://www.clarku.edu/~djoyce/trig/ Dave's draggable diagram.] (Requires java browser plugin)
 
[[Category:Trigonometry]]
[[Category:Elementary special functions]]
[[Category:Transcendental numbers|*]]
 
{{Link FA|ca}}
{{Link FA|km}}
 
[[ar:دوال مثلثية]]
[[ast:Función trigonométrica]]
[[bs:Trigonometrijske funkcije]]
[[bg:Тригонометрична функция]]
[[ca:Funció trigonomètrica]]
[[cs:Goniometrická funkce]]
[[da:Trigonometrisk funktion]]
[[de:Trigonometrische Funktion]]
[[el:Τριγωνομετρική συνάρτηση]]
[[es:Función trigonométrica]]
[[eo:Trigonometria funkcio]]
[[fa:سینوس (ریاضیات)]]
[[fr:Fonction trigonométrique]]
[[gl:Función trigonométrica]]
[[ko:삼각함수]]
[[io:Trigonometriala funciono]]
[[id:Fungsi trigonometrik]]
[[is:Hornafall]]
[[it:Funzione trigonometrica]]
[[lo:ຕຳລາໄຕມຸມ]]
[[lv:Trigonometriskās funkcijas]]
[[hu:Trigonometrikus függvények]]
[[nl:Goniometrische functie]]
[[ja:三角関数]]
[[no:Trigonometriske funksjoner]]
[[km:អនុគមន៍ ត្រីកោណមាត្រ]]
[[pl:Funkcje trygonometryczne]]
[[pt:Função trigonométrica]]
[[ro:Funcţie trigonometrică]]
[[ru:Тригонометрические функции]]
[[sq:Funksionet trigonometrike]]
[[simple:Trigonometric function]]
[[sk:Goniometrická funkcia]]
[[sl:Trigonometrična funkcija]]
[[sr:Тригонометријске функције]]
[[sh:Trigonometrijske funkcije]]
[[fi:Trigonometrinen funktio]]
[[sv:Trigonometrisk funktion]]
[[th:ฟังก์ชันตรีโกณมิติ]]
[[tg:Функсияҳои тригонометрӣ]]
[[tr:Trigonometrik fonksiyonlar]]
[[uk:Тригонометричні функції]]
[[vi:Hàm lượng giác]]
[[zh-classical:三角函數]]
[[zh:三角函数]]
Wedi dod o "https://cy.wikipedia.org/wiki/Wicipedia:Pwll_tywod"