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.

Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side.

because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the

[8][9] Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions.

One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions.

[20] These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval.

The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves.

Both sine and cosine functions with multiple angles may appear as their linear combination, resulting in a polynomial.

The trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier series.

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates

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).

Since Arabic is written without short vowels, jb was interpreted as the homograph jayb (جيب), which means 'bosom', 'pocket', or 'fold'.

[39][40][41] 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.

[44] 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.

[45] 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).

[39][47] 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.

[50] In early 17th-century, the French mathematician Albert Girard published the first use of the abbreviations sin, cos, and tan; 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.

[52] 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.

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.

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

[54][55] These functions are called sinpi and cospi in MATLAB,[54] OpenCL,[56] R,[55] Julia,[57] CUDA,[58] and ARM.

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.

[60] 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