Sine


In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle.
More generally, the definition of sine can be extended to any real value in terms of the length of a certain line segment in a unit circle. 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.
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.
The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word "sine" 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.

Right-angled triangle definition

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 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 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.

Unit circle definition

In trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system.
Let a line through the origin intersect the unit circle, making an angle of θ with the positive half of the x-axis. The x- and y-coordinates of this point of intersection are equal to and, respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when 0° < θ < 90°. Because the length of the hypotenuse of the unit circle is always 1,. The length of the opposite side of the triangle is simply the y-coordinate. A similar argument can be made for the cosine function to show that when 0° < θ < 90°, even under the new definition using the unit circle. is then defined as, or, equivalently, as the gradient of the line segment.
Using the unit circle definition has the advantage that the angle can take any real argument. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

Identities

Exact identities :
These apply for all values of.

Reciprocal

The reciprocal of sine is cosecant, i.e., the reciprocal of is, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

Inverse

The inverse function of sine is arcsine or inverse sine. As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example,, but also, etc. It follows that the arcsine function is multivalued:, but also,, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression will evaluate only to a single value, called its principal value.
k is some integer:
Or in one equation:
Arcsin satisfies:
and

Calculus

For the sine function:
The derivative is:
The antiderivative is:
C denotes the constant of integration.

Other trigonometric functions

It is possible to express any trigonometric function in terms of any other.
Sine in terms of the other common trigonometric functions:
For all equations which use plus/minus, the result is positive for angles in the first quadrant.
The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:
where sin2 means 2.

Sine squared function

The graph shows both the sine function and the function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.
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:

Properties relating to the quadrants

The table below displays many of the key properties of the sine function, arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity of the sine function.
QuadrantDegreesRadiansValueSignMonotonyConvexity
1st Quadrantincreasingconcave
2nd Quadrantdecreasingconcave
3rd Quadrantdecreasingconvex
4th Quadrantincreasingconvex


The following table gives basic information at the boundary of the quadrants.
DegreesRadiansPoint type
Root, Inflection
Maximum
Root, Inflection
Minimum

Series definition

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.
Using the reflection from the calculated geometric derivation of the sine is with the -th derivative at the point 0:
This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x :
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
The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that
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
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.
In general, mathematically important relationships between the sine and cosine functions and the exponential function are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.
A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.

Continued fraction

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

Fixed point

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.

Arc length

The arc length of the sine curve between and is.
This integral is an elliptic integral of the second kind.
The arc length for a full period is
where is the gamma function.
The arc length of the sine curve from 0 to x is the above number divided by times x, plus a correction
that varies periodically in x with period. The Fourier series for this correction
can be written in closed form using special functions, but it is perhaps more instructive to write the decimal approximations of the Fourier coefficients.
The sine curve arc length from 0 to x is
The leading term in the above equation, and the limit of arc length to distance ratio is given by:

Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
This is equivalent to the equality of the first three expressions below:
where R is the triangle's circumradius.
It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Special values

For certain integral numbers x of degrees, the value of sin is particularly simple. A table of some of these values is given below.
90 degree increments:
Other values not listed above:

Relationship to complex numbers

Sine is used to determine the imaginary part of a complex number given in polar coordinates :
the imaginary part is:
r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is a complex number.
Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take a complex number as an argument.

Sine with a complex argument

The definition of the sine function for complex arguments z:
where i 2 = −1, and sinh is hyperbolic sine. This is an entire function. Also, for purely real x,
For purely imaginary numbers:
It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:

Partial fraction and product expansions of complex sine

Using the partial fraction expansion technique in complex analysis, one can find that the infinite series
both converge and are equal to. Similarly, one can show that
Using product expansion technique, one can derive
Alternatively, the infinite product for the sine can be proved using complex Fourier series.
Using complex Fourier series, the function can be decomposed as
Setting yields
Therefore we get
The function is the derivative of. Furthermore, if, then the function such that the emerged series converges is, which can be proved using the Weierstrass M-test. It follows that
Exponentiating gives
Since and, we have. Hence
Let. Since converges uniformly on any closed disk, converges uniformly on any closed disk as well. Therefore the infinite product for the sine is valid for all, which completes the proof.

Usage of complex sine

sin is found in the functional equation for the Gamma function,
which in turn is found in the functional equation for the Riemann zeta-function,
As a holomorphic function, sin z is a 2D solution of Laplace's equation:
The complex sine function is also related to the level curves of pendulums.

Complex graphs

History

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. The chord function was discovered by Hipparchus of Nicaea and Ptolemy of Roman Egypt.
The function of sine and versine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin.
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. With the exception of the sine, the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant. Al-Khwārizmī produced tables of sines, cosines and tangents. Muhammad ibn Jābir al-Harrānī al-Battānī discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.
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. 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.
In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x. Roger Cotes computed the derivative of sine in his Harmonia Mensurarum. Leonhard Euler's Introductio in analysin infinitorum 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.

Etymology

, the word sine derives from the Sanskrit word for chord, jiva*. This was transliterated in Arabic as jiba جيب, which however is meaningless in that language and abbreviated jb جب. Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جيب, which means "bosom". When the Arabic texts were translated in the 12th century into Latin by Gerard of Cremona, he used the Latin equivalent for "bosom", . 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.

Software implementations

There is no standard algorithm for calculating sine. IEEE 754-2008, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine. Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin.
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. This allowed results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.
The CORDIC algorithm is commonly used in scientific calculators.
The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to sin.
Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
In programming languages, sin is typically either a built-in function or found within the language's standard math library.
For example, the C standard library defines sine functions within math.h: sin, sinf, and sinl. The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine.
Similarly, Python defines math.sin within the built-in math module. Complex sine functions are also available within the cmath module, e.g. cmath.sin. CPython's math functions call the C math library, and use a double-precision floating-point format.

Turns based implementations

Some software libraries provide implementations of sine using the input angle in half-turns, a half-turn being an angle of 180 degrees or radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.
EnvironmentFunction nameAngle units
MATLABsinpihalf-turns
OpenCLsinpihalf-turns
Rsinpihalf-turns
Juliasinpihalf-turns
CUDAsinpihalf-turns
ARMsinpihalf-turns

The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing,, and in binary floating-point or binary scaled fixed-point always involves a loss of accuracy.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo involves inaccuracies in representing.
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution. If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to would be incurred.

Citations