# Sine

(Redirected from Sine function)

For other uses, see Sine (disambiguation). Sine_sentence_0

Not to be confused with sign or sign (mathematics). Sine_sentence_1

Sine_table_infobox_0

More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. Sine_sentence_2

More modern definitions express the sine as an infinite series, or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Sine_sentence_3

The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. Sine_sentence_4

The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin. Sine_sentence_5

The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha. Sine_sentence_6

## Right-angled triangle definition Sine_section_0

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 α in triangle ABC is the angle of interest. Sine_sentence_7

The three sides of the triangle are named as follows: Sine_sentence_8

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• The opposite side is the side opposite to the angle of interest, in this case side a.Sine_item_0_0
• The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.Sine_item_0_1
• The adjacent side is the remaining side, in this case side b. It forms a side of (and is adjacent to) both the angle of interest (angle A) and the right angle.Sine_item_0_2

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: Sine_sentence_9

The other trigonometric functions of the angle can be 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. Sine_sentence_10

## Unit circle definition Sine_section_1

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system. Sine_sentence_11

Using the unit circle definition has the advantage that the angle can be extended to any real argument. Sine_sentence_12

This can also be achieved by requiring certain symmetries, and that sine be a periodic function. Sine_sentence_13

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• Sine_item_1_3

## Identities Sine_section_2

Main article: List of trigonometric identities Sine_sentence_14

### Reciprocal Sine_section_3

The reciprocal of sine is cosecant, i.e., the reciprocal of sin(A) is csc(A), or cosec(A). Sine_sentence_16

Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side: Sine_sentence_17

### Inverse Sine_section_4

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin). Sine_sentence_18

As sine is non-injective, it is not an exact inverse function, but a partial inverse function. Sine_sentence_19

For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. Sine_sentence_20

It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. Sine_sentence_21

When only one value is desired, the function may be restricted to its principal branch. Sine_sentence_22

With this restriction, for each x in the domain, the expression arcsin(x) will evaluate only to a single value, called its principal value. Sine_sentence_23

where (for some integer k): Sine_sentence_24

Or in one equation: Sine_sentence_25

By definition, arcsine satisfies the equation: Sine_sentence_26

and Sine_sentence_27

### Calculus Sine_section_5

See also: List of integrals of trigonometric functions and Differentiation of trigonometric functions Sine_sentence_28

For the sine function: Sine_sentence_29

The derivative is: Sine_sentence_30

The antiderivative is: Sine_sentence_31

where C denotes the constant of integration. Sine_sentence_32

### Other trigonometric functions Sine_section_6

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_sentence_33

The following table documents how sine can be expressed in terms of the other common trigonometric functions: Sine_sentence_34

For all equations which use plus/minus (±), the result is positive for angles in the first quadrant. Sine_sentence_35

The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity: Sine_sentence_36

where sin(x) means (sin(x)). Sine_sentence_37

### Sine squared function Sine_section_7

The graph shows both the sine function and the function, with the sine in blue and sine squared in red. Sine_sentence_38

Both graphs have the same shape, but with different ranges of values, and different periods. Sine_sentence_39

Sine squared has only positive values, but twice the number of periods. Sine_sentence_40

The sine squared function can be expressed as a modified sine wave from the Pythagorean identity and power reduction—by the cosine double-angle formula: Sine_sentence_41

## Properties relating to the quadrants Sine_section_8

The following table gives basic information at the boundary of the quadrants. Sine_sentence_42

## Series definition Sine_section_9

Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine. Sine_sentence_43

Using the reflection from the calculated geometric derivation of the sine is with the (4n+k)-th derivative at the point 0: Sine_sentence_44

This gives the following Taylor series expansion at x = 0. Sine_sentence_45

One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians): Sine_sentence_46

If x were expressed in degrees then the series would contain factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so Sine_sentence_47

The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that Sine_sentence_48

The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that Sine_sentence_49

The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients. Sine_sentence_50

In general, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Sine_sentence_51

Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians. Sine_sentence_52

A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator. Sine_sentence_53

### Continued fraction Sine_section_10

The sine function can also be represented as a generalized continued fraction: Sine_sentence_54

The continued fraction representation can be derived from Euler's continued fraction formula and expresses the real number values, both rational and irrational, of the sine function. Sine_sentence_55

## Fixed point Sine_section_11

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin(0) = 0. Sine_sentence_56

## Arc length Sine_section_12

The leading term in the above equation, and the limit of arc length to distance ratio is given by: Sine_sentence_57

## Law of sines Sine_section_13

Main article: Law of sines Sine_sentence_58

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C: Sine_sentence_59

This is equivalent to the equality of the first three expressions below: Sine_sentence_60

where R is the triangle's circumradius. Sine_sentence_61

It can be proven by dividing the triangle into two right ones and using the above definition of sine. Sine_sentence_62

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. Sine_sentence_63

This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. Sine_sentence_64

## Special values Sine_section_14

For certain integral numbers x of degrees, the value of sin(x) is particularly simple. Sine_sentence_66

A table of some of these values is given below. Sine_sentence_67

90 degree increments: Sine_sentence_68

Other values not listed above: Sine_sentence_69

## Relationship to complex numbers Sine_section_15

Main article: Trigonometric functions § Relationship to exponential function (Euler's formula) Sine_sentence_70

Sine is used to determine the imaginary part of a complex number given in polar coordinates (r, φ): Sine_sentence_71

the imaginary part is: Sine_sentence_72

r and φ represent the magnitude and angle of the complex number respectively. Sine_sentence_73

i is the imaginary unit. Sine_sentence_74

z is a complex number. Sine_sentence_75

Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine_sentence_76

Sine can also take a complex number as an argument. Sine_sentence_77

### Sine with a complex argument Sine_section_16

The definition of the sine function for complex arguments z: Sine_sentence_78

where i = −1, and sinh is hyperbolic sine. Sine_sentence_79

This is an entire function. Sine_sentence_80

Also, for purely real x, Sine_sentence_81

For purely imaginary numbers: Sine_sentence_82

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument: Sine_sentence_83

#### Partial fraction and product expansions of complex sine Sine_section_17

Using the partial fraction expansion technique in complex analysis, one can find that the infinite series Sine_sentence_84

Using product expansion technique, one can derive Sine_sentence_85

Alternatively, the infinite product for the sine can be proved using complex Fourier series. Sine_sentence_86

#### Usage of complex sine Sine_section_18

sin(z) is found in the functional equation for the Gamma function, Sine_sentence_87

which in turn is found in the functional equation for the Riemann zeta-function, Sine_sentence_88

As a holomorphic function, sin z is a 2D solution of Laplace's equation: Sine_sentence_89

The complex sine function is also related to the level curves of pendulums. Sine_sentence_90

## History Sine_section_20

Main articles: Trigonometric functions § History, and History of trigonometry Sine_sentence_91

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. Sine_sentence_92

The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). Sine_sentence_93

The function of sine and versine (1 - cosine) can be traced to the jyā and koṭi-jyā functions used in Gupta period (320 to 550 CE) Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. Sine_sentence_94

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. Sine_sentence_95

With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant. Sine_sentence_96

Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Sine_sentence_97

Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. Sine_sentence_98

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician Albert Girard; these were further promulgated by Euler (see below). Sine_sentence_99

The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe 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. Sine_sentence_100

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x. Sine_sentence_101

Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722). Sine_sentence_102

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Sine_sentence_103

### Etymology Sine_section_21

Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). Sine_sentence_104

This was transliterated in Arabic as jiba جيب, which however is meaningless in that language and abbreviated jb جب . Sine_sentence_105

Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جيب, which means "bosom". Sine_sentence_106

When the Arabic texts were translated in the 12th century into Latin by Gerard of Cremona, he used the Latin equivalent for "bosom", (which means "bosom" or "bay" or "fold"). Sine_sentence_107

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. Sine_sentence_108

The English form sine was introduced in the 1590s. Sine_sentence_109

## Software implementations Sine_section_22

There is no standard algorithm for calculating sine. Sine_sentence_111

IEEE 754-2008, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine. Sine_sentence_112

Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. Sine_sentence_113

This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin(10). Sine_sentence_114

A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. Sine_sentence_115

This allowed results to be looked up from a table rather than being calculated in real time. Sine_sentence_116

With modern CPU architectures this method may offer no advantage. Sine_sentence_117

The CORDIC algorithm is commonly used in scientific calculators. Sine_sentence_118

The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. Sine_sentence_119

In computing, it is typically abbreviated to sin. Sine_sentence_120

Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387. Sine_sentence_121

In programming languages, sin is typically either a built-in function or found within the language's standard math library. Sine_sentence_122

For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double). Sine_sentence_123

The parameter of each is a floating point value, specifying the angle in radians. Sine_sentence_124

Each function returns the same data type as it accepts. Sine_sentence_125

Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh). Sine_sentence_126

Similarly, Python defines math.sin(x) within the built-in math module. Sine_sentence_127

Complex sine functions are also available within the cmath module, e.g. cmath.sin(z). Sine_sentence_128

CPython's math functions call the C math library, and use a double-precision floating-point format. Sine_sentence_129

### Turns based implementations Sine_section_23

Sine_table_general_1