Editing Trigonometric tables (section)

== On-demand computation ==
[[File:Bernegger Manuale 137.jpg|thumb|right|200px|A page from a 1619 book of [[mathematical table]]s.]]

Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with [[Floating point|floating-point]] units, is to combine a [[polynomial]] or [[rational function|rational]] [[approximation theory|approximation]] (such as [[Chebyshev approximation]], best uniform approximation, [[Padé approximant|Padé approximation]], and typically for higher or variable precisions, [[Taylor series|Taylor]] and [[Laurent series]]) with range reduction and a table lookup &mdash; they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, but methods like [[Gal's accurate tables]], Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a [[hardware multiplier]], there is an algorithm called [[CORDIC]] (as well as related techniques) that is more efficient, since it uses only [[shift operator|shift]]s and additions. All of these methods are commonly implemented in [[computer hardware|hardware]] for performance reasons.

The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a [[minimax approximation algorithm]].

For [[arbitrary-precision arithmetic|very high precision]] calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the [[arithmetic-geometric mean]], which itself approximates the trigonometric function by the ([[complex number|complex]]) [[elliptic integral]] (Brent, 1976).

Trigonometric functions of angles that are [[rational number|rational]] multiples of 2π are [[algebraic number]]s. The values for ''a/b·2π'' can be found by applying [[de Moivre's identity]] for ''n = a'' to a ''b<sup>th</sup>'' [[root of unity]], which is also a root of the polynomial ''x<sup>b</sup> - 1'' in the [[complex plane]].  For example, the cosine and sine of 2π&nbsp;⋅&nbsp;5/37 are the [[real part|real]] and [[imaginary part]]s, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the [[degree of a polynomial|degree]]-37 polynomial ''x''<sup>37</sup>&nbsp;&minus;&nbsp;1. For this case, a [[root-finding algorithm]] such as [[Newton's method]] is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for [[transcendental number|transcendental]] trigonometric constants, however.