Cosine

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They 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 positive and negative values and even to complex numbers. All of these approaches will be presented below. In modern usage, there are six basic trigonometric functions, which are tabulated below 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 equally define them geometrically or by other means and derive the relations.) A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as:
- versed sine (versin = 1 − cos)
- exsecant (exsec = sec − 1). Many more relations between these functions are listed in the article about trigonometric identities.

History

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin²(A/2) = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day. The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th–5th century), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees. The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values. Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhata) was transliterated by the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, probably because jiba (جب) and jaib (جب) are written the same in Arabic (many vowels are excluded from words written in the Arabic alphabet). All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (1464) of Regiomontanus (1436–1476), as well as his later Tabulae directionum (which included the tangent function, unnamed). The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596. The Introductio in analysin infinitorum (1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" e^ix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec..

Right triangle definitions

Euler's formula 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°). Then, 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

. Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar. The set of zeroes of sine 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 zeroes 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\

. 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

Mnemonics

There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a" or "sock-a toe-a" depending upon which side of the Atlantic you hail from. Can also be read as "soccer tour"). It means:
- SOH ... sin = opposite/hypotenuse
- CAH ... cos = adjacent/hypotenuse
- TOA ... tan = opposite/adjacent. Many other such words and phrases have been contrived. For more see: trigonometry mnemonics.

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, this gives rise to the following matchings: # 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, which is converting between the two ways of telling how slanted a line is: angles and slopes. 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

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 trig 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 used so far. 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 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. circle 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 trig functions can be defined by: :

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

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

integer Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown at 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 (see below). 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

exsec Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). 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 opposite of sine. 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: :

\tan x = x + \frac + \frac + \frac + \cdots = \sum_^\infty \frac, \quad \left | x \right | < \frac

\csc x = \frac   + \frac   + \frac   + \frac   + \cdots = \frac   + \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac

\sec x = 1 + \frac   + \frac   + \frac   + \cdots = 1+ \sum_^\infty \frac, \quad \left | x \right | < \frac

\cot x = \frac   - \frac  - \frac   - \frac   - \cdots = \frac   - \sum_^\infty \frac, \quad 0 < \left | x \right | < \frac

where :

E_n \,

is the nth Euler number, and :

U_n \,

is the nth up/down number.

Relationship to exponential function

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 relationship was first noted by Euler and the 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^ix, 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^ \, = \,  = -\imath \sinh \left( \imath z\right)

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

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

: $\cos x \, = \, \mbox (e^)$

: $\sin x \, = \, \mbox (e^)$

It is also shown 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

i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector 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. See the trigonometric identity article for this technique.

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 [http://www.usfca.edu/vca/PDF/vca-preface.pdf].

The significance of radians
Radians 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,
: $f(x) = \sin(kx); k \ne 0, k \ne 1 \,$
then the derivatives will scale by amplitude.
: $f'(x) = k\cos(kx) \,$ .

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 \,,
but
: $y$ = -k^2y \,;
similarly for cosine.

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.

Other definitions

Theorem: There exists exactly one pair of real functions
s, c with the following properties:

For any $x, y \in\mathbb$ :

: $s(x)^2 + c(x)^2 = 1,\,$

: $s(x+y) = s(x)c(y) + c(x)s(y),\,$

: $c(x+y) = c(x)c(y) - s(x)s(y),\,$

: $0 < xc(x) < s(x) < x\ \mathrm\ 0 < x < 1.$

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. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by 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). See also: Generating trigonometric tables.

Modern computers use a variety of techniques (Kantabutra, 1996). One common method, especially on higher-end processors with floating point units, is to combine a polynomial approximation (such as a Taylor series or a rational function) with 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 for performance reasons.

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 π/60 radians (three degrees) can be found exactly by hand.

Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45 degrees). 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 π/4 radians (45 degrees) can then be found using the Pythagorean theorem:

: $c = \sqrt = \sqrt2$

Therefore:

: $\sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos \left(\pi / 4 \right) = \cos \left(45^\circ\right) =$
: $\tan \left(\pi / 4 \right) = \tan \left(45^\circ\right) = = 1$

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:

: $\sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos \left(\pi / 3 \right) = \cos \left(60^\circ\right) =$
: $\cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin \left(\pi / 3 \right) = \sin \left(60^\circ\right) =$
: $\tan \left(\pi / 6 \right) = \tan \left(30^\circ\right) = \cot \left(\pi / 3 \right) = \cot \left(60^\circ\right) =$

See also: Exact trigonometric constants

Inverse functions

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. 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 & -\frac < y < \frac, y \ne 0,
& y = \arccot(x) & \mbox & x = \cot(y) \\

\end$

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 are sometimes confused with the multiplicative inverses of the functions. Our notation avoids such confusion.

The following series definition may be obtained:

: $\begin
\arcsin z & = & z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots\\
& = & \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arccos z & = & \frac - \arcsin z \\
& = & \frac - (z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots ) \\
& = & \frac - \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arctan z & = & z - \frac +\frac -\frac +\cdots \\
& = & \sum_^\infty \frac
\end
\ , \quad \left| z \right| < 1$

: $\begin
\arccsc z & = & \arcsin\left(z^\right) \\
& = & z^ + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac +\cdots \\
& = & \sum_^\infty \left( \frac \right) \frac \end
\ , \quad \left| z \right| > 1$

: $\begin
\arcsec z & = & \arccos\left(z^\right) \\
& = & \frac - (z^ + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots ) \\
& = & \frac - \sum_^\infty \left( \frac \right) \frac
\end
\ , \quad \left| z \right| > 1$

: $\begin
\arccot z & = & \frac - \arctan z \\
& = & \frac - ( z - \frac +\frac -\frac +\cdots ) \\
& = & \frac - \sum_^\infty \frac
\end
\ , \quad \left| z \right| < 1$

These functions may also be defined by proving that they are antiderivatives of other functions.

: $\arcsin\left(x\right) =
\int_0^x \frac 1 \,\mathrmz, \quad |x| < 1$
: $\arccos\left(x\right) =
\int_x^1 \frac \,\mathrmz,\quad |x| < 1$
: $\arctan\left(x\right) =
\int_0^x \frac 1 \,\mathrmz,
\quad \forall x \in \mathbb$
: $\arccot\left(x\right) =
\int_x^\infty \frac \,\mathrmz,
\quad z > 0$
: $\arcsec\left(x\right) =
\int_x^1 \frac 1 \,\mathrmz, \quad x > 1$
: $\arccsc\left(x\right) =
\int_x^\infty \frac \,\mathrmz, \quad x > 1$

Inverse trigonometric functions can be generalized to complex arguments using the complex logarithm.

: $\arcsin (z) = -i \log \left( i \left( z + \sqrt\right) \right)$
: $\arccos (z) = -i \log \left( z + \sqrt\right)$
: $\arctan (z) = \frac \log\left(\frac\right)$

Note: arcsec can also mean arcsecond.

Identities

: $\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y$

: $\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y$

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

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

: $\sin x+\sin y=2\sin \left( \frac \right) \cos \left( \frac \right)$

: $\sin x-\sin y=2\cos \left( \frac \right) \sin \left( \frac \right)$

: $\cos x+\cos y=2\cos \left( \frac \right) \cos \left( \frac \right)$

: $\cos x-\cos y=-2\sin \left( \frac \right)\sin \left( \frac \right)$

: $\tan x+\tan y=\frac$

: $\tan x-\tan y=\frac$

: $\cot x+\cot y=\frac$

: $\cot x-\cot y=\frac$

See also trigonometric identity.

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 for an arbitrary triangle states:

: $\frac = \frac = \frac$

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. 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:

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

Again, this theorem can be proven by dividing the triangle into two right ones. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known.

If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Cosine law.

Law of tangents

There is also a law of tangents:

: $\frac = \frac$

law of tangents The trigonometric functions are also important outside of the study of triangles. They are periodic functions with characteristic wave patterns as graphs, useful for modelling 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.

The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curves can be described by a Fourier series. Its equation is:
: $(x(\theta),\,y(\theta)) = \sum_^\infty \frac (\sin(\theta\cdot F(n)),\, \cos(\theta\cdot F(n)))$
where F(n) is the nth Fibonacci number.

For a compilation of many relations between the trigonometric functions, see trigonometric identities.

References

- Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York, 1991).

- Eli Maor, [http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights] (Princeton Univ. Press, 1998).

- "[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html Trigonometric functions]", MacTutor History of Mathematics Archive.

- Tristan Needham, Visual Complex Analysis, (Oxford University Press, 2000), ISBN 0198534469 [http://www.usfca.edu/vca Book website]

- Vitit Kantabutra, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328-339 (1996).

See also

- Generating trigonometric tables

- Hyperbolic function

- Pythagoras

- Pythagorean theorem

- Trigonometric identity

- Uses of trigonometry

- Direction cosines

- [http://wikisource.org/wiki/Trigonometric_functions_of_angles_0%C2%B0_to_90%C2%B0_by_degree Trigonometric functions of angles 0° to 90° by degree]

External links

- [http://www.walterzorn.com/grapher/grapher_e.htm Javascript function grapher] uses a javascript library to display functions. Works in nearly every modern browser.

- [http://www.geocities.com/SiliconValley/Garage/3323/aat/a_sin.html Sine and cosine function ] with an implementation in Rexx.
Category:Trigonometry
Category:Special functions

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Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.

Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.

The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History
:Main article: History of mathematics

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought.
In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.

Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.

Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts.

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales.

The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary.

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).

Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."

If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm]

In any case, mathematics shares much in common with many fields in the physical sciences, notably
the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).

The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.

The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.

The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.

:

:Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure.
:

:Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics.
:

:Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change
:Ways to express and handle change in mathematical functions, and changes between numbers.
:

:Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics.

:philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.

:

:Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems.

:Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike.

:See list of theorems for more

:Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures
See list of conjectures for more

:These are some of the major unsolved problems in mathematics.

:Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians
See also list of mathematics history topics
:History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields
:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:

- misunderstanding of the implications of mathematical rigour;

- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;

- lack of familiarity with, and therefore underestimation of, the existing literature.

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.

Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also

- Mathematical game

- Mathematical problem

- Mathematical puzzle

- Puzzle

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).

- Courant, R. and H. Robbins, What Is Mathematics? (1941);

- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.

- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.

- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.

- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.

- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).

- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot

- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.

- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.

- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.

- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.

- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]

- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.

- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.

- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...

- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.

- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics

- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.

- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.

- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.

- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.

- [http://www.physicsmathforums.com/ Physics Math Forums]

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Angle
This article is about angles in geometry. For other articles, see Angle (disambiguation)

----

An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry.

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

Units of measure for angles
In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.

- The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The SI system of units uses radians as the (derived) unit for angles. Because of the relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians have particular analytic properties, just as do exponential functions in the base e. (As we've discovered, this is no coincidence).

- The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. 2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians. Degrees are further broken down into minutes of arc and seconds of arc, which are 1/60th and 1/3600th of a degree, respectively. Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc were also used in navigation, and a nautical mile is roughly defined as one minute of arc of the earth's surface.

- The grad, also called grade, gradian or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.

- The point is used in navigation, and is defined as 1/32 of a circle, or exactly 11.25°.

- The full circle or full turns represents the number or fraction of complete full turns. For example, π/2 radians = 90° = 1/4 full circle

Conventions on measurement

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315.

In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

Types of angles

An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle.

Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:

- Angles smaller than a right angle are called acute angles

- Angles larger than a right angle are called obtuse angles.

- Angles equal to two right angles are called straight angles.

- Angles larger than two right angles are called reflex angles.

- The difference between an acute angle and a right angle is termed the complement of the angle

- The difference between an angle and two right angles is termed the supplement of the angle.

Some facts

In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) × π radians or (n − 2) × 180°.

If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles.

If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.

A formal definition
A Euclidean angle is completely determined by the corresponding right triangle. In particular, if $\theta$ is a Euclidean angle, it is true that

: $\cos \theta = \frac$

and

: $\sin \theta = \frac$

for two numbers $x$ and $y$ . So an angle can be legitimately given by two numbers $x$ and $y$ , or by a ratio $\frac$ . What's more, to any such ratio there corresponds exactly one angle, since

: $\frac = \frac = \frac$

(i.e., changing the ratio will necessarily change the sin and cos, which in the geometric range $0 < \theta < 2\pi$ are one-to-one - one sin or cos corresponds to one $\theta$ ).

Angles in different contexts

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

: $\mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.$

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>.

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

Two intersecting planes form an angle, called their dihedral angle. It is defined as the angle between two lines normal to the planes.

Also a plane and an intersecting line form an angle. This angle is equal to π/2 radians minus the angle between the intersecting line and the line that goes through the point of intersection and is perpendicular to the plane.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G,

: $\cos \theta = \frac
.$

Angles in astronomy

In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars.
Then the angle between those lines can be measured; this is the angular separation between the two stars.

Astronomers also measure the apparent size of objects.
For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth.
One could say, "The Moon subtends an angle of half a degree."
The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928.

The obsolete (but still commonly used) format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″

See also

- Central angle

- Complementary angles

- Inscribed angle

- Supplementary angles

- solid angle for a concept of angle in three dimensions.

- Astrological aspect

External links

- [http://www.unitconversion.org/unit_converter/angle.html Online Angle Converter - convert between various units of angle, such as degree, radian, grad, gon, minute, second, sign, mil, and so on]

- [http://www.unitconversion.org/unit_converter/angle-v.html Interactive Angle Conversion Table - convert selected unit to all other units of angle]

- [http://www.cut-the-knot.org/triangle/ABisector.shtml Angle Bisectors] at cut-the-knot

- [http://www.cut-the-knot.org/Curriculum/Geometry/PerpBiInQuadri.shtml Angle Bisectors and Perpendiculars in a Quadrilateral] at cut-the-knot

- [http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at cut-the-knot

- [http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at cut-the-knot

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]

Category:Elementary geometry
Category:Trigonometry

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Triangle (geometry)
:For alternative meanings, such as the musical instrument, see triangle (disambiguation).

A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.

Types of triangles
Triangles can be classified according to the relative lengths of their sides:

- In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon

- In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles.

- In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.

Equilateral triangle
Isosceles triangle
Scalene triangle

Equilateral Isosceles Scalene

Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.

- A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs of the triangle.

- An obtuse triangle has one internal angle larger than 90° (an obtuse angle).

- An acute triangle has internal angles that are all smaller than 90° (three acute angles).

Right triangle
Obtuse triangle
Acute triangle

Right Obtuse Acute

Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE.

A triangle is a polygon and a 2-simplex (see polytope).

Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.

In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides.

trigonometry

In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known.

trigonometry

A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as

: $c^2 = a^2 + b^2 \,$

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles.
The Pythagorean theorem can be generalized to the law of cosines:

: $c^2 = a^2 + b^2 - 2ab \cos\gamma \,$

which is valid for all triangles, even if γ is not a right angle.
The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known.

The law of sines states

: $\fraca=\fracb=\fracc=\frac1d$

where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions.

There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is : $;1:1:\sqrt$ . The "30-60-90 triangle" has sides in the ratio of $1:\sqrt:2$ .

Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

Menelaus' theorem

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above.

Thales' theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.

Thales' theorem

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse.
The three vertices together with the orthocenter are said to form an orthocentric system.

orthocentric system

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.

orthocentric system

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. This is also the triangle's center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side.

center of gravity

The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.

excircle

The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.

The center of the incircle is not in general located on Euler's line.

If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.

Using geometry

The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction.

area

To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is bh, the area of the given triangle must be ½bh.

parallelogram

Using vectors

The area of a parallelogram can also be calculated by the use of vectors. If AB and AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |AB × AC|, the magnitude of the cross product of vectors AB and AC. |AB × AC| is also equal to |h × AC|, where h represents the altitude h as a vector.

The area of triangle ABC is half of this, or S = ½|AB × AC|.

cross product

Using trigonometry

The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ.

It is of course no coincidence that the area of a parallelogram is ab sin γ.

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x₁, y₁) and C = (x₂, y₂), then the area S can be computed as 1/2 times the absolute value of the determinant

: $\beginx_1 & x_2 \\ y_1 & y_2 \end$

or S = ½ |x₁y₂ − x₂y₁|.

Using Heron's formula

Yet another way to compute S is Heron's formula:
: $S = \sqrt$

where s = ½ (a + b + c) is the semiperimeter, or one half of the triangle's perimeter.

Using the side lengths and a numerically stable formula

Heron's formula is numerically unstable for triangles with a very small angle.
A stable alternative involves arranging the lengths of the sides so that:
:a ≥ b ≥ c
and computing
: $S = \frac\sqrt$

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Non-planar triangles

If any four of a triangle's elements (vertices, and/or elements of its sides) are plane to each other, the triangle is called plane. Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, he would find that its angles would be greater than 180°.

External links

- [http://ostermiller.org/calc/triangle.html Triangle Calculator] - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.

- [http://agutie.homestead.com/files/Napoleon0.htm Napoleon's theorem] A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"

- William Kahan: [http://http.cs.berkeley.edu/~wkahan/Triangle.pdf Miscalculating Area and Angles of a Needle-like Triangle].

- Clark Kimberling: [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]. Lists some 1600 interesting points associated with any triangle.

- Christian Obrecht: [http://perso.wanadoo.fr/obrecht/ Eukleides]. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.

- [http://www.cut-the-knot.org/triangle Triangle constructions, remarkable points and lines, and metric relations in a triangle] at cut-the-knot

- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1469&CurriculumID=4 Printable Worksheet on Types of Triangles]

- [http://www.vias.org/comp_geometry/geom_triangle.html Compendium Geometry] Analytical Geometry of Triangles

Category:Polygons

ko:삼각형

ja:三角形

th:รูปสามเหลี่ยม

Ratio
: For the use of ratio as a human capacity, see reason.

In algebra, a ratio is the relationship between two quantities. It is expressed as the quotient of two numbers, or as two numbers separated by a colon (pronounced "to"). A number that can be written as a ratio of two integers is a rational number. In physics, a ratio between two magnitudes of the same type of quantity gives a positive real number when the magnitudes are expressed relative to an absolute or natural zero. The ratio between a difference of two magnitudes to a third magnitude, such as a unit, gives a real number (i.e. positive or negative).

Examples

- If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.

- The ratio of heights of the Eiffel Tower (300 m) and the Great Pyramid (137 m) is 300:137, so one structure is more than twice the height of the other (or more precisely, 2.2 times).

- The ratio of the mass of Jupiter to the mass of the Earth is approximately 317.8:1.

- The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.

- If two axles are connected by gear wheels, the number of times one axle turns for each turn of the other is known as the gear ratio. The best example being the number of turns of the pedals of a bicycle compared with number of turns of the bicycle's rear wheel.

- The ratio of hydrogen atoms to oxygen in water is 2:1, or two parts to one.

Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are unitless; the units cancel out of the ratio. For example, the ingredients in a recipe that required 500 grams and 300 grams of each, would be in the ratio of 5:3, with no units.

Note also the difference between ratios and vulgar fractions. For example, if there are three raspberry candies and five blackcurrant candies, then the ratio of raspberry candies to blackcurrant candies is 3:5. This indicates that there are three fifths as many raspberry candies as blackcurrant candies. However the fraction of all the candies that are raspberry is three out of a total of all eight candies or 3/(3+5) = 3/8. Thus the chances of a randomly selected candy being raspberry are three in eight.

See also

- Analogy

- Conversion factor

- Financial ratio

- Golden ratio

- Odds

- Proportionality

- Ratio decidendi — the reasoning for a court of law's decision

- Rational number

Category:Algebra

ja:比

Infinite Series
In mathematics, a series is the sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them, e.g,

:1 + 2 + 3 + 4 + 5 + ...

which may or may not be meaningful.

In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.

Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.

Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

: $\sum_^k (an+b);$

and finite geometric series, a sum of a geometric progression, which can be written as:

: $\sum_^k a^.$

Infinite series

The sum of an infinite series is a limit of partial sums of infinitely many terms. Such a limit can have a finite value; if it has, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps
: $1+\frac+\frac+\frac+\frac+\cdots$
It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2 — in other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

: $\sum_^$}