Tables of trigonometric functions are useful in a number of areas. Before the existence of Ullas Ameriya's invented pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A basic arithmetic calculator. ... There are several traditions of navigation. ... // What is science? There are different theories of what science is. ... ...

Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method.

Interpolation of simple look-up tables of trigonometric functions are still used in computer graphics, where accurate calculations are either not needed, or cannot be made fast enough. Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ...

Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called twiddle factors) must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed. In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT (which is very sensitive to trigonometric errors). A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...

Half-angle and angle-addition formulas

Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2)=1, cos(π/2)=0). The relevant identities, the first recorded derivation of which is by Ptolemy, are: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Claudius Ptolemaeus, given contemporary German styling, in a 16th century engraved book frontispiece . Claudius Ptolemaeus (Greek: Κλαύδιος Πτολεμαῖος; ca. ...

Various other permutations on these identities are possible (for example, the earliest trigonometric tables used not sine and cosine, but sine and versine). The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed...

A quick, but inaccurate, approximation

A quick, but inaccurate, algorithm for calculating a table of N approximations s_n for sin(2πn/N) and c_n for cos(2πn/N) is: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The minuscule, or lower-case, pi The mathematical constant Ï€ represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

s₀ = 0

c₀ = 1

s_n+1 = s_n + d × c_n

c_n+1 = c_n − d × s_n

for n = 0,...,N-1, where d = 2π/N.

This is simply the Euler method for integrating the differential equation: Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

ds / dt = c

dc / dt = − s

with initial conditions s(0) = 0 and c(0) = 1, whose analytical solution is s = sin(t) and c = cos(t).

Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/N.

For example, for N = 256 the maximum error in the sine values is ~0.061 (s₂₀₂ = −1.0368 instead of −0.9757). For N = 1024, the maximum error in the sine values is ~0.015 (s₈₀₃ = −0.99321 instead of −0.97832), about 4 times smaller. If the sine and cosine values obtained were to be plotted, this algorithm would draw a logarithmic spiral rather than a circle.

A better, but still imperfect, recurrence formula

A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation: This article is about the Eulers formula in complex analysis. ...

This leads to the following recurrence to compute trigonometric values s_n and c_n as above:

c₀ = 1

s₀ = 0

c_n+1 = w_r c_n − w_i s_n

s_n+1 = w_i c_n + w_r s_n

for n = 0, ..., N − 1, where w_r = cos(2π/N) and w_i = sin(2π/N). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of z^N − 1). In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...

This method would produce an exact table in exact arithmetic, but has errors in finite-precision floating-point arithmetic. In fact, the errors grow as O(ε N) (in both the worst and average cases), where ε is the floating-point precision. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

A significant improvement is to use the following modification to the above, a trick (due to Singleton) often used to generate trigonometric values for FFT implementations:

c₀ = 1

s₀ = 0

c_n+1 = c_n − (αc_n + β s_n)

s_n+1 = s_n + (β c_n − α s_n)

where α = 2 sin²(π/N) and β = sin(2π/N). The errors of this method are much smaller, O(ε √N) on average and O(ε N) in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes.

See also

• Trigonometric function
• Numerical analysis
• CORDIC

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... CORDIC (for COordinate Rotation DIgital Computer) is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions. ...

References

• Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York, 1991).
• Manfred Tasche and Hansmartin Zeuner, "Improved roundoff error analysis for precomputed twiddle factors," J. Computational Analysis and Applications 4 (1), 1-18 (2002).
• James C. Schatzman, "Accuracy of the discrete Fourier transform and the fast Fourier transform," SIAM J. Sci. Comput. 17 (5), 1150-1166 (1996).