Trigonometric tables
This article includes a improve this article by introducing more precise citations. (December 2018) ) |
Trigonometry |
---|
Reference |
Laws and theorems |
Calculus |
Mathematicians |
In
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 is still used in computer graphics, where only modest accuracy may be required and speed is often paramount.
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 trigonometry table is essentially a reference chart that presents the values of sine, cosine, tangent, and other trigonometric functions for various angles. These angles are usually arranged across the top row of the table, while the different trigonometric functions are labeled in the first column on the left. To locate the value of a specific trigonometric function at a certain angle, you would find the row for the function and follow it across to the column under the desired angle.[1]
Utilising a trigonometry table involves a few straightforward steps
- Determine the specific angle for which you need to find the trigonometric values.
- Locate this angle along the horizontal axis (top row) of the table.
- Choose the trigonometric function you're interested in from the vertical axis (first column).
- Trace across from the function and down from the angle to the point where they intersect on the table; the number at this intersection provides the value of the trigonometric function for that angle.
On-demand computation
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
The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a minimax approximation algorithm.
For
Trigonometric functions of angles that are
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
These were used to construct Ptolemy's table of chords, which was applied to astronomical problems.
Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine.
A quick, but inaccurate, approximation
A quick, but inaccurate, algorithm for calculating a table of N approximations sn for
- s0 = 0
- c0 = 1
- sn+1 = sn + d × cn
- cn+1 = cn − d × sn
for n = 0,...,N − 1, where d = 2π/N.
This is simply the
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
This article possibly contains original research. (December 2018) |
A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation:
This leads to the following recurrence to compute trigonometric values sn and cn as above:
- c0 = 1
- s0 = 0
- cn+1 = wr cn − wi sn
- sn+1 = wi cn + wr sn
for n = 0, ..., N − 1, where wr = cos(2π/N) and wi = 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 zN − 1).
This method would produce an exact table in exact arithmetic, but has errors in finite-precision
A significant improvement is to use the following modification to the above, a trick (due to Singleton[2]) often used to generate trigonometric values for FFT implementations:
- c0 = 1
- s0 = 0
- cn+1 = cn − (α cn + β sn)
- sn+1 = sn + (β cn − α sn)
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
- Aryabhata's sine table
- CORDIC
- Exact trigonometric values
- Madhava's sine table
- Numerical analysis
- Plimpton 322
- Prosthaphaeresis
References
- ^ "Trigonometry Table: Learning of trigonometry table is simplified". Yogiraj notes | General study and Law study Notes. Retrieved 2023-11-02.
- ^ Singleton 1967
- John Wiley & Sons.
- Manfred Tasche and Hansmartin Zeuner (2002) "Improved roundoff error analysis for precomputed twiddle factors", Journal for Computational Analysis and Applications 4(1): 1–18.
- James C. Schatzman (1996) "Accuracy of the discrete Fourier transform and the fast Fourier transform", SIAM Journal on Scientific Computing 17(5): 1150–1166.
- Vitit Kantabutra (1996) "On hardware for computing exponential and trigonometric functions," IEEE Transactions on Computers 45(3): 328–339 .
- Journal of the Association for Computing Machinery23: 242–251.
- Singleton, Richard C (1967). "On Computing The Fast Fourier Transform". S2CID 6287781.
- William J. Cody Jr., William Waite, Software Manual for the Elementary Functions, Prentice-Hall, 1980, ISBN 0-13-822064-6.
- Mary H. Payne, Robert N. Hanek, Radian reduction for trigonometric functions, ACM SIGNUM Newsletter 18: 19-24, 1983.
- Gal, Shmuel and Bachelis, Boris (1991) "An accurate elementary mathematical library for the IEEE floating point standard", ACM Transactions on Mathematical Software.