# Dictionary Definition

sine n : ratio of the opposite side to the
hypotenuse of a right-angled triangle [syn: sin]

# User Contributed Dictionary

see Sine

## English

### Etymology

From sinus.### Noun

- In the context of "trigonometry|mathematics": In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.

#### Usage notes

In various branches of mathematics, the sine of an angle is determined in various ways, including the following:- The y-coordinate of the point on the unit circle at the given anticlockwise angle from the x axis.
- The sum of the real or complex power series
- \sum_1^\inftyx^

#### Synonyms

- Symbol: sin

#### Derived terms

#### Translations

trigonometric function

### See also

## Irish

### Pronunciation

- IPA: /ˈʃinʲə/

### Adjective

sine- comparative of sean

### Noun

sine#### Declension

## Latin

### Preposition

sine (+ )#### Derived terms

## Scottish Gaelic

### Pronunciation

- IPA: /ˈʃinʲə/

### Adjective

sine- comparative of sean

### Noun

sine# Extensive Definition

In mathematics, the
trigonometric functions (also called circular functions) are
functions
of an angle. They are
important in the study of triangles
and modeling periodic
phenomena, among many other applications. Trigonometric
functions are commonly defined as ratios 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 infinite
series or as solutions of certain differential
equations, allowing their extension to arbitrary positive and
negative values and even to complex
numbers.

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.

## History

The notion that there should be some standard correspondence between the length of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that 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.Trigonometric functions were studied by Hipparchus of
Nicaea
(180–125 BC), Ptolemy of Egypt (90–165 AD),
Aryabhata
(476–550), Varahamihira,
Brahmagupta, ,
Abū al-Wafā' al-Būzjānī, Omar
Khayyam, Bhāskara
II, Nasir
al-Din al-Tusi, Ghiyath
al-Kashi (14th century), Ulugh Beg (14th
century), Regiomontanus
(1464), Rheticus,
and Rheticus' student Valentin Otho.

Madhava
of Sangamagramma (c. 1400) made early strides in the analysis
of trigonometric functions in terms of infinite
series. 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.

A few functions were common historically (and
appeared in the earliest tables), but are now seldom used, such as
the chord
(crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1
− cos(θ) = 2 sin2(θ/2)), the haversine (haversin(θ) =
versin(θ) / 2 = sin2(θ/2)), the exsecant (exsec(θ) = sec(θ)
− 1) and the excosecant (excsc(θ) =
exsec(π/2 − θ) = csc(θ) − 1). Many more
relations between these functions are listed in the article about
trigonometric identities.

## Right triangle definitions

In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A:We use the following names for the sides of the
triangle:

- The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
- The opposite side is the side opposite to the angle we are interested in, in this case a.
- The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

All triangles are taken to exist in the Euclidean
plane so that the inside angles of each triangle sum to π
radians (or 180°);
therefore, for a right triangle the two non-right angles are
between zero and π/2 radians (or 90°). The
reader should note that the following definitions, strictly
speaking, only define the trigonometric functions for angles in
this range. We extend them to the full set of real arguments by
using the unit circle,
or by requiring certain symmetries and that they be periodic
functions.

1) The sine of an angle is the ratio of the
length of the opposite side to the length of the hypotenuse. In our
case

- \sin A = \frac = \frac .

The set of zeroes of sine (i.e., the values of x
for which \sin x =0) is

- \left\.

2) The cosine of an angle is the ratio of the
length of the adjacent side to the length of the hypotenuse. In our
case

- \cos A = \frac = \frac .

The set of zeros of cosine is

- \left\.

3) 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

- \tan A = \frac = \frac .

The set of zeroes of tangent is

- \left\.

- \tan A = \frac .

The remaining three functions are best defined
using the above three functions.

4) The cosecant csc(A) is the multiplicative
inverse of sin(A), i.e. the ratio of the length of the
hypotenuse to the length of the opposite side:

- \csc A = \frac = \frac .

5) The secant sec(A) is the multiplicative
inverse of cos(A), i.e. the ratio of the length of the
hypotenuse to the length of the adjacent side:

- \sec A = \frac = \frac .

6) The cotangent cot(A) is the multiplicative
inverse of tan(A), i.e. the ratio of the length of the adjacent
side to the length of the opposite side:

- \cot A = \frac = \frac .

### 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.
- Cosine is second, run is second. Cosine takes an angle and tells the run.
- Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope.

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 doesn't 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 cos(1°)

## Unit-circle definitions

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:

- x^2 + y^2 = 1 \,

In the picture, some common angles, measured in
radians, are given. Measurements in the counter clockwise 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 θ and sin 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 θ = y/1 and cos 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 1.

For angles greater than 2π or less than
−2π, simply continue to rotate around the circle. In this
way, sine and cosine become periodic
functions with period 2π:

- \sin\theta = \sin\left(\theta + 2\pi k \right)
- \cos\theta = \cos\left(\theta + 2\pi k \right)

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, cosine, secant, or cosecant is 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. Above, only sine and cosine were defined directly by the
unit circle, but the other four trigonometric functions can be
defined by:

- \tan\theta = \frac \quad \sec\theta = \frac

- \csc\theta = \frac \quad \cot\theta = \frac

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 kπ, but change rapidly at
angles close to (k + 1/2)π. The graph of the tangent function also
has a vertical asymptote at θ = (k + 1/2)π.
This is the case because the function approaches infinity as θ
approaches (k + 1/2)π from the left and minus infinity as it
approaches (k + 1/2)π from the right.

Alternatively, all of the basic trigonometric
functions can be defined in terms of a unit circle centered at O
(shown at right, near the top of the page), 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 (see
above). cos(θ) is the horizontal distance OC, and versin(θ) = 1 − 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) = OE and csc(θ) = OF
are segments of secant lines
(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 exsec(θ) = sec(θ) − 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.)

## Series definitions

Using only geometry and properties of 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 radians; see also
the significance of radians below.) One can then use the theory
of Taylor
series to show that the following identities hold for all
real
numbers x:

- \sin x = x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac

- \cos x = 1 - \frac + \frac - \frac + \cdots = \sum_^\infty \frac

These identities are often 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 infinite
series can be developed from the foundations of the real number
system, independent of any geometric considerations. The
differentiability
and continuity
of these functions are then established from the series definitions
alone.

Other series can be found:

\begin \tan x & = \sum_^\infty \frac \\ &
= \sum_^\infty \frac \\ & = x + \frac + \frac + \frac + \cdots,
\qquad \text |x|

where

- Un is the nth up/down number,
- Bn is the nth Bernoulli number, and
- En (below) is the nth Euler number.

When this is expressed in a form in which the
denominators are the corresponding factorials, and the numerators,
called the "tangent numbers", have a combinatorial
interpretation: they enumerate alternating
permutations of finite sets of odd cardinality.

\begin \csc x & = \sum_^\infty \frac \\ &
= \frac + \frac + \frac + \frac + \cdots, \qquad \text 0

\begin \sec x & = \sum_^\infty \frac =
\sum_^\infty \frac \\ & = 1 + \frac + \frac + \frac + \cdots,
\qquad \text |x|

When this is expressed in a form in which the
denominators are the corresponding factorials, the numerators,
called the "secant numbers", have a combinatorial
interpretation: they enumerate alternating
permutations of finite sets of even cardinality.

\begin \cot x & = \sum_^\infty \frac \\ &
= \frac - \frac - \frac - \frac - \cdots, \qquad \text 0

From a theorem in complex
analysis, there is a unique analytic extension of this real
function to the 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

It can be shown from the series definitions that
the sine and cosine functions are the imaginary
and real parts, respectively, of the complex
exponential function when its argument is purely
imaginary:

- e^ = \cos\theta + i\sin\theta. \,

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 eix, 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:

- \sin z = \sum_^\fracz^ \, = \, = -i \sinh \left( i z\right)

- \cos z = \sum_^\fracz^ \, = \, = \cosh \left(i z\right)

where
i 2 = −1. Also, for purely
real x,

- \cos x = \mbox (e^)

- \sin x = \mbox (e^)

It is also known that exponential processes are
intimately linked to periodic behavior.

## Definitions via differential equations

Both the sine and cosine functions satisfy the
differential
equation

- y=-y.

That is to say, each is the negative 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 conditions
y(0) = 0 and y′(0) = 1, and the cosine function is the unique
solution satisfying the initial conditions y(0) = 1 and y′(0) = 0.
Since the sine and cosine functions are linearly independent,
together they form a 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 trigonometric
identities for the sine and cosine functions. Further, the
observation that sine and cosine satisfies y=-y means that they are
eigenfunctions of
the second-derivative operator.

The tangent function is the unique solution of
the nonlinear differential equation

- y'=1+y^2

satisfying the initial condition y(0) = 0. There
is a very interesting visual proof that the tangent function
satisfies this differential equation; see Needham's Visual Complex
Analysis.''

### 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,- f(x) = \sin(kx) \,

- f'(x) = k\cos(kx). \,

Here, k is a constant that represents a mapping
between units. If x is in degrees, then

- k = \frac.

This means that the second derivative of a sine
in degrees satisfies not the differential equation

- y = -y, \,

- y = -k^2y. \;

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

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 always 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 reads,

- \left(\sin x\right)^2 + \left(\cos x\right)^2 = 1,

- \sin^2\left( x\right) + \cos^2\left(x\right) = 1.

In some cases the inner parentheses may be
omitted.

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.

- \sin \left(x+y\right)=\sin x \cos y + \cos x \sin y
- \cos \left(x+y\right)=\cos x \cos y - \sin x \sin y
- \sin \left(x-y\right)=\sin x \cos y - \cos x \sin y
- \cos \left(x-y\right)=\cos x \cos y + \sin x \sin y

These identities can also be used to derive the
product-to-sum identities that were used in antiquity to
transform the product of two numbers in a sum of numbers and
greatly speed operations, much like the logarithm
function.

### Calculus

For integrals and derivatives of trigonometric functions, see the relevant sections of table of derivatives, table of integrals, and list of integrals of trigonometric functions. Below is the list of the derivatives and integrals of the six basic trigonometric functions.### Definitions using functional equations

In mathematical
analysis, one can define the trigonometric functions using
functional
equations 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
functions satisfy those conditions. Symbolically, we say that
there exists exactly one pair of real functions \,\ \sin and \,\
\cos such that for all real numbers x and y, the following
equations hold: \sin^2(x) + \cos^2(x) = 1,\,

- \sin(x\pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y),\,
- \cos(x\pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y),\,

- 0

## Computation

The computation of trigonometric functions is a
complicated subject, which can today be avoided by most people
because of the widespread availability of computers and scientific
calculators 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.

Prior to computers, people typically evaluated
trigonometric functions by 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 above),
and were typically generated by repeated application of the
half-angle and angle-addition identities
starting from a known value (such as sin(π/2) = 1).

Modern computers use a variety of techniques. One
common method, especially on higher-end processors with floating
point units, is to combine a polynomial or rational
approximation
(such as Chebyshev
approximation, best uniform approximation, and Padé
approximation, and typically for higher or variable precisions,
Taylor and
Laurent
series) with range reduction and a table lookup
— they first look up the closest angle in a small table,
and then use the polynomial to compute the correction. On simpler
devices that lack
hardware multipliers, there is an algorithm called CORDIC (as well as
related techniques) that is more efficient, since it uses only
shifts and
additions. All of these methods are commonly implemented in
hardware
floating
point units 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)
elliptic
integral.

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 \pi / 60 radians (3°) can be found
exactly by hand.

Consider a right triangle where the two other
angles are equal, and therefore are both \pi / 4 radians (45°).
Then the length of side b and the length of side a are equal; we
can choose a = b = 1. The values of sine, cosine and tangent of an
angle of \pi / 4 radians (45°) can then be found using the
Pythagorean theorem:

- c = \sqrt = \sqrt2.

Therefore:

- \sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos \left(\pi / 4 \right) = \cos \left(45^\circ\right) = ,
- \tan \left(\pi / 4 \right) = \tan \left(45^\circ\right) = = \cdot = = 1.

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:

- \sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos \left(\pi / 3 \right) = \cos \left(60^\circ\right) = ,
- \cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin \left(\pi / 3 \right) = \sin \left(60^\circ\right) = ,
- \tan \left(\pi / 6 \right) = \tan \left(30^\circ\right) = \cot \left(\pi / 3 \right) = \cot \left(60^\circ\right) = .

## Inverse functions

The trigonometric functions are periodic, and hence not 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 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:- \begin

\mbox & -\frac \le y \le \frac, & y =
\arcsin(x) & \mbox & x = \sin(y) \\ \\ \mbox & 0 \le y
\le \pi, & y = \arccos(x) & \mbox & x = \cos(y) \\ \\
\mbox & -\frac < y < \frac, & y = \arctan(x) &
\mbox & x = \tan(y) \\ \\ \mbox & -\frac \le y \le \frac, y
\ne 0, & y = \arccsc(x) & \mbox & x = \csc(y) \\ \\
\mbox & 0 \le y \le \pi, y \ne \frac, & y = \arcsec(x)
& \mbox & x = \sec(y) \\ \\ \mbox & 0

For inverse trigonometric functions, the
notations sin−1 and cos−1 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 "arcsecond".

Just like the sine and cosine, the inverse
trigonometric functions can also be defined in terms of infinite
series. For example, \arcsin z = z + \left( \frac \right) \frac +
\left( \frac \right) \frac + \left( \frac \right) \frac + \cdots
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: \arcsin\left(x\right) =
\int_0^x \frac 1 \,\mathrmz, \quad |x| Analogous formulas for the
other functions can be found at
Inverse trigonometric function. Using the complex
logarithm, one can generalize all these functions to complex
arguments: \arcsin (z) = -i \log \left( i z + \sqrt \right)

\arccos (z) = -i \log \left( z +
\sqrt\right)

\arctan (z) = \frac \log\left(\frac\right)

## Properties and applications

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:

- \frac = \frac = \frac

- \frac = \frac = \frac = 2R

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:

- c^2=a^2+b^2-2ab\cos C \,

- \cos C=\frac

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 is mostly used to determine a
side of a triangle if two sides and an angle are known, although in
some cases there can be two positive solutions as in the SSA
ambiguous case. And can also be used to find the cosine of an
angle (and consequently the angle itself) if all the sides are
known.

### Other useful properties

There is also a law of
tangents:

- \frac = \frac

### Periodic functions

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe the 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 the uniform circular motion.Trigonometric functions also prove to be useful
in the study of general periodic
functions. These functions have characteristic wave patterns as
graphs, useful for modeling recurring phenomena such as sound or
light waves. Every signal
can be written as a (typically infinite) sum of sine and cosine
functions of different frequencies; this is the basic idea of
Fourier
analysis, where trigonometric series are used to solve a
variety of boundary-value problems in partial differential
equations. For example the square wave,
can be written as the Fourier
series

- x_(t) = \frac \sum_^\infty .

In the animation on the right it can be seen that
just a few terms already produce a fairly good approximation.

## See also

- 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
- Continued fraction of Gauss A continued fraction definition for the tangent function

## Notes

## References

- Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York. (1964). ISBN 0-486-61272-4.
- Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
- 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, Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
- Needham, Tristan, "Preface"" to Visual Complex Analysis. Oxford University Press, (1999). ISBN 0-19-853446-9.
- O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics Archive. (1996).
- O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2000).
- Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).
- Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.

## External links

- Visionlearning Module on Wave Mathematics
- GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions

sine in Arabic: تابع مثلثي

sine in Asturian: Función trigonométrica

sine in Bosnian: Trigonometrijske funkcije

sine in Bulgarian: Тригонометрична функция

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