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General information
General definition	$\begin{align} &\sin(\alpha) = \frac {\textrm{opposite}} {\textrm{hypotenuse}} \\[8pt] &\cos(\alpha) = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} \\[8pt] \end{align}$
Fields of application	Trigonometry, Fourier series, etc.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $\theta$ , the sine and cosine functions are denoted as $\sin \theta$ and $\cos \theta$ .

The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine and cosine functions are 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. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.

Notation
Definitions
- Right-angled triangle definitions
History
- Etymology
Software implementations
- Turns based implementations
See also

Notation

Sine and cosine are written using functional notation with the abbreviations sin and cos.

Often, if the argument is simple enough, the function value will be written without parentheses, as $sin θ$ rather than as $sin(θ)$ .

Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.

Definitions

Right-angled triangle definitions

For the angle

α

, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

To define the sine and cosine 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. The three sides of the triangle are named as follows:

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

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, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse: ${\displaystyle \sin(\alpha) = \frac {\textrm{opposite}} {\textrm{hypotenuse}} \qquad \cos(\alpha) = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} }$ The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.

As stated, the values $\sin(\alpha)$ and $\cos(\alpha)$ appear 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 ratios are the same for each of them.

History

Quadrant from 1840s Ottoman Turkey with axes for looking up the sine and versine of angles

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 (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

The sine and cosine functions can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), 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 (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. Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. 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°.

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). 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 (1722). 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.

Etymology

The word sine is derived, indirectly, from the Sanskrit word jyā 'bow-string' or more specifically its synonym jīvá (both adopted from Ancient Greek χορδή 'string'), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā). This was transliterated in Arabic as jība, which is meaningless in that language and written as jb (جب). Since Arabic is written without short vowels, jb was interpreted as the homograph jayb (جيب), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent sinus (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a toga over the breast'). 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 word cosine derives from an abbreviation of the Latin complementi sinus 'sine of the complementary angle' as cosinus in Edmund Gunter's Canon triangulorum (1620), which also includes a similar definition of cotangens.

Software implementations

There is no standard algorithm for calculating sine and cosine. IEEE 754, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.

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(10²²).

A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or linearly interpolate between the 2 closest values to approximate it. This allows 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 and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to sin and cos.

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

In programming languages, sin and cos are 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(double), sinf(float), and sinl(long double). 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 (sinh).

Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). 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 and cosine using the input angle in half-turns, a half-turn being an angle of 180 degrees or $\pi$ radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases. In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these functions are called sinpi and cospi. For example, sinpi(x) would evaluate to $\sin(\pi x),$ where $x$ is expressed in half-turns, and consequently the final input to the function, $πx$ can be interpreted in radians by $sin$ .

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 $2\pi$ , $\pi$ , and ${\textstyle \frac{\pi}{2}}$ in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.

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 ${\textstyle \frac{\pi}{2}}$ involves inaccuracies in representing ${\textstyle \frac{\pi}{2}}$ .

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 ${\textstyle \frac{\pi}{2048}}$ would be incurred.