"to nine decimal places, ""i.e."" formula_12 (of 3.1415926535897...). The development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) was carried out by mathematicians of the Kerala School in the 15th century CE. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). Shēr Shāh of northern India issued silver currency bearing Islamic motifs, later imitated by the Mughal empire. The Chinese merchant Ma Huan (1413–51) noted that gold coins, known as ""fanam"", were issued in Cochin and weighed"
"""Euler's formula"", as well as near-modern abbreviations (""sin."", ""cos."", ""tang."", ""cot."", ""sec."", and ""cosec.""). A few functions were common historically, but are now seldom used, such as the chord (), the versine () (which appeared in the earliest tables), the coversine (), the haversine (), the exsecant (), and the excosecant (). See List of trigonometric identities for more relations between these functions. The word ""sine"" derives from Latin ""sinus"", meaning ""bend; bay"", and more specifically ""the hanging fold of the upper part of a toga"", ""the bosom of a garment"", which was chosen as the translation of what was"
"terms of rational functions of a new variable . These identities are known collectively as the tangent half-angle formulae because of the definition of . These identities can be useful in calculus for converting rational functions in sine and cosine to functions of in order to find their antiderivatives. Technically, the existence of the tangent half-angle formulae stems from the fact that the circle is an algebraic curve of genus 0. One then expects that the ""circular functions"" should be reducible to rational functions. Geometrically, the construction goes like this: for any point (cos φ, sin φ) on the unit"
"1400) made early strides in the analysis of trigonometric functions in terms of infinite series. The terms ""tangent"" and ""secant"" were first introduced by the Danish mathematician Thomas Fincke in his book ""Geometria rotundi"" (1583). The first published use of the abbreviations ""sin"", ""cos"", and ""tan"" is probably by the 16th century French mathematician Albert Girard. In a paper published in 1682, Leibniz proved that is not an algebraic function of . 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"
"the excosecant function is related to the cotangent function by The exsecant function is related to the sine function by In analogy, the excosecant function is related to the cosine function by The exsecant and excosecant functions can be extended into the complex plane. Exsecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations. The"
"(for example, formula_114, formula_115 and formula_116 are the angles of a triangle), Proof: If formula_119 quarter circle, Proof: Replace each of formula_121, formula_122, and formula_123 with their complementary angles, so cotangents turn into tangents and vice versa. Given so the result follows from the triple tangent identity. First, start with the sum-angle identities: By adding these together, Similarly, by subtracting the two sum-angle identities, Let formula_133 and formula_134, Substitute formula_115 and formula_116 Therefore, Similarly for cosine, start with the sum-angle identities: Again, by adding and subtracting Substitute formula_115 and formula_116 as before, The figure at the right shows a sector"
"sine"" aka ""area hyperbolic sine"" (Latin: ""Area sinus hyperbolicus""): The domain is the whole real line. ""Inverse hyperbolic cosine"" aka ""area hyperbolic cosine"" (Latin: ""Area cosinus hyperbolicus""): The domain is the closed interval . ""Inverse hyperbolic tangent"" aka ""area hyperbolic tangent"" (Latin: ""Area tangens hyperbolicus""): The domain is the open interval . ""Inverse hyperbolic cotangent"" aka ""area hyperbolic cotangent"" (Latin: ""Area cotangens hyperbolicus""): The domain is the union of the open intervals and . ""Inverse hyperbolic secant"" aka ""area hyperbolic secant"" (Latin: ""Area secans hyperbolicus""): The domain is the semi-open interval . ""Inverse hyperbolic cosecant"" aka ""area hyperbolic cosecant"" (Latin:"
"""fold""). Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage. The English form ""sine"" was introduced in the 1590s. The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to codice_1. Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387. In programming languages, codice_1 is typically either a built-in function or found within the language's standard math library. For example, the C"
"as follows: The ""sine"" of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case 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 ""cosine"" of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case The ""tangent"" of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case"
"trigonometric functions from a unit circle, such as was used historically. sec(""θ"") is the secant line , and the exsecant is the portion of this secant that lies ""exterior"" to the circle (""ex"" is Latin for ""out of""). A related function is the excosecant or coexsecant, also known as exterior, external, outward or outer cosecant and abbreviated as excosec, coexsec, excsc or exc, the exsecant of the complementary angle: Important in fields such as surveying, railway engineering (for example to lay out railroad curves and superelevation), civil engineering, astronomy, and spherical trigonometry up into the 1980s, the exsecant function is"
"of the rail tracks and it is the basis of the Hallade method for rail surveying. The term ""sagitta"" (often abbreviated ""sag"") is used similarly in optics, for describing the surfaces of lenses and mirrors. Versine The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. The versine or versed sine is a trigonometric function already"
"of cosines may be used to prove Heron's formula, which is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths ""a"", ""b"", and ""c"", and if the semiperimeter is then the area of the triangle is: where R is the radius of the circumcircle of the triangle. The law of tangents: Euler's formula, which states that formula_24, produces the following analytical identities for sine, cosine, and tangent in terms of ""e"" and the imaginary unit ""i"": Trigonometry Trigonometry (from Greek ""trigōnon"", ""triangle"" and ""metron"", ""measure"")"
"the expression will evaluate only to a single value, called its principal value. ""k"" is some integer: Or in one equation: Arcsin satisfies: and For the sine function: The derivative is: The antiderivative is: ""C"" denotes the constant of integration. It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function). Sine in terms of the other common trigonometric functions: Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant. The basic relationship between the sine and"
"defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides. As stated, the value formula_2 appears to depend on the choice of right triangle containing an angle of measure ""α"". However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them. In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system. Let a line through the"
"geometry and trigonometry courses, the notation sin, cos, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse. The law of sines, or sine rule, states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle"
"function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle ""θ"" considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve but in general the solution for the cosine of the one-third angle involves the cube root of a complex number (giving ""casus irreducibilis""). In practice, all values of sines, cosines, and tangents not found in this article are approximated using"
"Versine The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. It is written as versin(""θ""), sinver(""θ""), vers(""θ""), ver(""θ"") or siv(""θ""). In Latin, it is known as the ""sinus versus"" (flipped sine), ""versinus"", ""versus"" or the ""sagitta"" (arrow)."
"of the Arabic ""jiba"", which is a transliteration of the Sanskrit word for half the chord, ""jya-ardha"". To define the sine function of an acute angle ""α"", start with a right triangle that contains an angle of measure ""α""; in the accompanying figure, angle ""A"" in triangle ""ABC"" is the angle of interest. The three sides of the triangle are named as follows: Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, or: The other trigonometric functions of the angle can be"
"frequency and its higher harmonics. The resonant frequencies of a string are determined by the length between the fixed ends and the tension of the string. Sine wave A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (""t"") is: where: The sine wave is important in"
"quadrant is a straight return, second quadrant is read from pi/2-x, third and fourth are negatives of the first and second respectively). For cosine, we only have to return the angle shifted by pi/2 (i.e. x+pi/2). For tangent, we divide the sine by the cosine (divide-by-zero handling may be needed depending on implementation): When using interpolation, the size of the lookup table can be reduced by using ""nonuniform sampling"", which means that where the function is close to straight, we use few sample points, while where it changes value quickly we use more sample points to keep the approximation close"
"summarized in the following table and described in more detail below. The angle is the angle between the hypotenuse and the adjacent line – the angle at ""A"" in the accompanying diagram. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin ""sinus"" for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle ""opens"". In our case: The cosine (""sine complement"", Latin: ""cosinus"", ""sinus complementi"") of an angle is the ratio of the"
"cosine and sine functions: Substituting these expressions into the spherical law of cosines nets: or after simplifying: The big O terms for and are dominated by as and get small, so we can write this last expression as: Spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a ""spherical triangle"" on the surface of the sphere is defined by the great circles connecting three points ,"
"of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. In 1342, Levi ben Gershon, known as Gersonides, wrote ""On Sines, Chords and Arcs"", in particular proving the sine law for plane triangles and giving five-figure sine tables. A simplified trigonometric table, the """"toleta de marteloio"""", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco. In the 15th century,"
"length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle. Because the angle sum of a triangle is radians, the co-angle is equal to ; so . In our case: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin ""linea tangens"" or touching line (cf. ""tangere"","
"Fourier sine and cosine series In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. In this article, ""f"" denotes a real valued function on formula_1 which is periodic with period 2""L"". The value of term c is L which is always taken as half of the total interval function ""f"" is defined in or we are concerned in .. In here we are using L instead of c . If f(x) is an odd function, then the Fourier Half Range sine series of f is"
"particular instance) or as solutions of differential equations. For example the sine and the cosine functions are the solutions of the linear differential equation such that When working with complex numbers different types of functions are used: The study of complex functions is a vast subject in mathematics with many applications, and that can claim to be an ancestor to many other areas of mathematics, like homotopy theory, and manifolds. In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector"
"Differentiation of trigonometric functions The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin(""x""), cos(""x"") and tan(""x""). For example, the derivative of ""f""(""x"") = sin(""x"") is represented as ""f"" ′(""a"") = cos(""a""). ""f"" ′(""a"") is the rate of change of sin(""x"") at a particular point ""a"". All derivatives of circular trigonometric functions can be found using those of sin(""x"") and cos(""x""). The quotient rule is then implemented to differentiate the resulting expression. Finding the derivatives of the inverse"
"be considered as the unit complex numbers, i.e., the set of complex numbers of the form for all (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as phase factor. The trigonometric functions cosine and sine of angle may be defined on the unit circle as follows: If is a point on the unit circle, and if the ray from the origin (0, 0) to makes an angle from the positive -axis, (where counterclockwise turning is positive), then The equation gives the relation The unit circle also demonstrates that sine and cosine are periodic"
"antiderivatives of other functions. The arcsine, for example, can be written as the following integral: Analogous formulas for the other functions can be found at inverse trigonometric functions. Using the complex logarithm, one can generalize all these functions to complex arguments: In an inner product space, the angle between two non-zero vectors is defined to be The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results. The law of sines states that for an arbitrary triangle with sides , , and and angles opposite those sides , and : where"
"both lengths are small integer multiples of a common length. The resemblance between the shape of a vibrating string and the graph of the sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some waves and the graph of the sine function is also not coincidental. In some other fields, among them climatology, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine function. Many fields make use of trigonometry in more advanced ways than can be discussed in a single article. Often those"
"leave little doubt that they once existed. Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the ""Aryabhatiya"". The ""Siddhantas"" and the ""Aryabhatiya"""
"sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard Euler in 1748 in his Introduction to the Analysis of the Infinite. These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola ""xy"" = 1 by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced ""The Quadrature of the Parabola""."
"is the first of the sine rules. See curved variations of the Law of Sines to see details of this derivation. Applying the cosine rules to the polar triangle gives (Todhunter, Art.47), ""i.e."" replacing ""A"" by π–""a"", ""a"" by π–""A"" etc., The six parts of a triangle may be written in cyclic order as (""aCbAcB""). The cotangent, or four-part, formulae relate two sides and two angles forming four ""consecutive"" parts around the triangle, for example (""aCbA"") or (""BaCb""). In such a set there are inner and outer parts: for example in the set (""BaCb"") the inner angle is ""C"", the"