426:) 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).
36:
112:
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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
406:
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
501:) with range reduction and a table lookup — 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
811:
724:
1514:
469:
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
1027:
919:
448:
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.
617:
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
1362:= −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.
430:
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.
256:
1714:
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:
1966:
65:
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is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for
262:
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Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and
505:, Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a
354:
1684:). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing
1945:
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starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer
2018:
1419:
87:
1386:
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The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a
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Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/
525:
154:
1952:
532:
2035:
Gal, Shmuel and
Bachelis, Boris (1991) "An accurate elementary mathematical library for the IEEE floating point standard",
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48:
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calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the
58:
52:
44:
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247:
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1831:) in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes.
1397:
635:. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which
1979:
1840:
536:
1937:
Manfred Tasche and
Hansmartin Zeuner (2002) "Improved roundoff error analysis for precomputed twiddle factors",
69:
160:
502:
1855:
593:, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the
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James C. Schatzman (1996) "Accuracy of the discrete
Fourier transform and the fast Fourier transform",
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Vitit
Kantabutra (1996) "On hardware for computing exponential and trigonometric functions,"
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Choose the trigonometric function you're interested in from the vertical axis (first column).
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method. Interpolation of simple look-up tables of trigonometric functions is still used in
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217:
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806:{\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}}
719:{\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}}
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Another important application of trigonometric tables and generation schemes is for
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1711:) (in both the worst and average cases), where ε is the floating-point precision.
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Determine the specific angle for which you need to find the trigonometric values.
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1509:{\displaystyle e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }}
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1962:
513:(as well as related techniques) that is more efficient, since it uses only
415:, where only modest accuracy may be required and speed is often paramount.
1993:
1974:
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A simple recurrence formula to generate trigonometric tables is based on
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This leads to the following recurrence to compute trigonometric values
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422:(FFT) algorithms, where the same trigonometric function values (called
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601: − 1. For this case, a root-finding algorithm such as
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1404:. Statements consisting only of original research should be removed.
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was an important area of study, which led to the development of the
585:. For example, the cosine and sine of 2π ⋅ 5/37 are the
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Locate this angle along the horizontal axis (top row) of the table.
1891:"Trigonometry Table: Learning of trigonometry table is simplified"
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1065:
517:
and additions. All of these methods are commonly implemented in
1022:{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}
914:{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}
539:, which itself approximates the trigonometric function by the (
434:
Using a trigonometry table involves a few straightforward steps
1703:
table in exact arithmetic, but has errors in finite-precision
1369:
1051:
A quick, but inaccurate, algorithm for calculating a table of
29:
1069:
1963:
Fast
Multiple-Precision Evaluation of Elementary Functions
1823:). The errors of this method are much smaller, O(ε √
375:
are useful in a number of areas. Before the existence of
1355:= 1024, the maximum error in the sine values is ~0.015 (
1393:
1344:= 256 the maximum error in the sine values is ~0.061 (
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1445:
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1939:
493:, and typically for higher or variable precisions,
1967:Journal of the Association for Computing Machinery
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1895:Yogiraj notes | General study and Law study Notes
1707:arithmetic. In fact, the errors grow as O(ε
1366:A better, but still imperfect, recurrence formula
1688:in the complex plane to solve for the primitive
57:but its sources remain unclear because it lacks
1351:= −1.0368 instead of −0.9757). For
1036:, which was applied to astronomical problems.
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8:
2026:Radian reduction for trigonometric functions
2011:Software Manual for the Elementary Functions
550:Trigonometric functions of angles that are
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2037:ACM Transactions on Mathematical Software
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1975:"On Computing The Fast Fourier Transform"
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27:Lists of values of mathematical functions
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1314:(0) = 1, whose analytical solution is
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613:Half-angle and angle-addition formulas
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2009:William J. Cody Jr., William Waite,
1946:SIAM Journal on Scientific Computing
2032:SIGNUM Newsletter 18: 19-24, 1983.
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609:trigonometric constants, however.
401:first mechanical computing devices
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2024:Mary H. Payne, Robert N. Hanek,
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526:minimax approximation algorithm
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1973:Singleton, Richard C (1967).
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1400:the claims made and adding
461:A page from a 1619 book of
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1851:Exact trigonometric values
1827:) on average and O(ε
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625:, who derived them in the
248:Trigonometric substitution
1980:Communications of the ACM
1034:Ptolemy's table of chords
562:can be found by applying
537:arithmetic-geometric mean
521:for performance reasons.
1928:A History of Mathematics
1306:with initial conditions
1296:{\displaystyle dc/dt=-s}
619:trigonometric identities
161:Generalized trigonometry
43:This article includes a
2013:, Prentice-Hall, 1980,
1815:where α = 2 sin(π/
1696: − 1).
1252:{\displaystyle ds/dt=c}
487:Chebyshev approximation
474:units, is to combine a
373:trigonometric functions
72:more precise citations.
1841:Aryabhata's sine table
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1994:10.1145/363717.363771
1932:John Wiley & Sons
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564:de Moivre's identity
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268:Trigonometric series
1208:This is simply the
533:very high precision
507:hardware multiplier
463:mathematical tables
397:mathematical tables
383:were essential for
230:Pythagorean theorem
2058:Numerical analysis
1861:Numerical analysis
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1385:possibly contains
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491:Padé approximation
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45:list of references
1948:17(5): 1150–1166.
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545:elliptic integral
413:computer graphics
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200:Laws and theorems
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78:December 2018
71:
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54:
50:
46:
41:
32:
31:
19:
2053:Trigonometry
2025:
2010:
1984:
1978:
1969:23: 242–251.
1938:
1927:
1909:
1898:. Retrieved
1894:
1885:
1866:Plimpton 322
1828:
1824:
1820:
1816:
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1798:
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1384:
1356:
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1334:
1332:
1327:
1323:
1319:
1315:
1311:
1310:(0) = 0 and
1307:
1305:
1210:Euler method
1207:
1202:
1198:
1194:
1190:
1188:
1182:
1178:
1174:
1169:
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1159:
1155:
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1132:
1126:
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1104:
1097:
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1038:
1031:
636:
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616:
598:
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559:
549:
530:
523:
468:
428:
423:
417:
405:
380:
371:, tables of
366:
186:
104:Trigonometry
84:
75:
64:Please help
56:
1959:R. P. Brent
1941:4(1): 1–18.
393:engineering
369:mathematics
294:Brahmagupta
263:Derivatives
192:Unit circle
70:introducing
2047:Categories
1900:2023-11-02
1877:References
1654:= 0, ...,
1537:as above:
1394:improve it
476:polynomial
385:navigation
304:al-Battani
284:Hipparchus
223:Cotangents
177:Identities
1680:= sin(2π/
1667:= cos(2π/
1502:θ
1499:Δ
1488:×
1483:θ
1464:Δ
1458:θ
1398:verifying
1288:−
1007:
992:
986:∓
974:
959:
944:±
935:
899:
884:
878:±
866:
851:
836:±
827:
793:
787:−
764:±
740:
706:
677:±
653:
633:astronomy
485:(such as
319:de Moivre
253:Integrals
169:Reference
139:Functions
1961:(1976) "
1835:See also
1585:−
1193:= 0,...,
1173:−
628:Almagest
552:rational
519:hardware
480:rational
299:al-Hasib
239:Calculus
218:Tangents
2003:6287781
1926:(1991)
1392:Please
1177:×
1144:×
1041:versine
639:lies):
623:Ptolemy
581:in the
541:complex
389:science
329:Fourier
289:Ptolemy
255: (
213:Cosines
155:inverse
141: (
127:History
122:Outline
66:improve
2017:
2001:
1846:CORDIC
1671:) and
1326:= cos(
1322:) and
1318:= sin(
1100:) is:
1079:) and
1070:π
595:degree
560:a/b·2π
515:shifts
511:CORDIC
495:Taylor
187:Tables
1999:S2CID
1701:exact
1201:= 2π/
579:x - 1
570:to a
568:n = a
324:Euler
314:Viète
208:Sines
132:Usage
51:, or
2015:ISBN
1690:root
1650:for
1528:and
1189:for
1088:for
1064:for
589:and
587:real
566:for
531:For
497:and
391:and
2030:ACM
1989:doi
1965:",
1733:= 0
1724:= 1
1692:of
1556:= 0
1547:= 1
1396:by
1360:803
1349:202
1330:).
1119:= 1
1110:= 0
1092:(2π
1090:cos
1066:sin
1004:sin
989:sin
971:cos
956:cos
932:cos
896:sin
881:cos
863:cos
848:sin
824:sin
790:cos
737:sin
703:cos
650:cos
478:or
367:In
151:tan
147:cos
143:sin
2049::
2028:,
1997:.
1985:10
1983:.
1977:.
1893:.
1784:=
1782:+1
1745:=
1743:+1
1630:+
1613:=
1611:+1
1568:=
1566:+1
1337:.
1216::
1205:.
1164:=
1162:+1
1140:+
1131:=
1129:+1
1068:(2
1043:.
543:)
528:.
403:.
387:,
379:,
153:,
149:,
145:,
55:,
47:,
2039:.
2021:.
2005:.
1991::
1934:.
1903:.
1829:N
1825:N
1821:N
1817:N
1811:)
1808:n
1804:s
1799:n
1795:c
1790:n
1786:s
1780:n
1776:s
1772:)
1769:n
1765:s
1760:n
1756:c
1751:n
1747:c
1741:n
1737:c
1731:0
1728:s
1722:0
1719:c
1709:N
1694:z
1682:N
1677:i
1673:w
1669:N
1664:r
1660:w
1656:N
1652:n
1644:n
1640:s
1636:r
1632:w
1627:n
1623:c
1619:i
1615:w
1609:n
1605:s
1599:n
1595:s
1591:i
1587:w
1582:n
1578:c
1574:r
1570:w
1564:n
1560:c
1554:0
1551:s
1545:0
1542:c
1534:n
1530:c
1525:n
1521:s
1496:i
1492:e
1480:i
1476:e
1472:=
1467:)
1461:+
1455:(
1452:i
1448:e
1423:)
1417:(
1412:)
1408:(
1390:.
1357:s
1353:N
1346:s
1342:N
1335:N
1328:t
1324:c
1320:t
1316:s
1312:c
1308:s
1291:s
1285:=
1282:t
1279:d
1275:/
1271:c
1268:d
1247:c
1244:=
1241:t
1238:d
1234:/
1230:s
1227:d
1203:N
1199:d
1195:N
1191:n
1183:n
1179:s
1175:d
1170:n
1166:c
1160:n
1156:c
1150:n
1146:c
1142:d
1137:n
1133:s
1127:n
1123:s
1117:0
1114:c
1108:0
1105:s
1098:N
1096:/
1094:n
1085:n
1081:c
1077:N
1075:/
1073:n
1061:n
1057:s
1053:N
1016:)
1013:y
1010:(
1001:)
998:x
995:(
983:)
980:y
977:(
968:)
965:x
962:(
953:=
950:)
947:y
941:x
938:(
908:)
905:y
902:(
893:)
890:x
887:(
875:)
872:y
869:(
860:)
857:x
854:(
845:=
842:)
839:y
833:x
830:(
799:)
796:x
784:1
781:(
775:2
772:1
761:=
757:)
752:2
749:x
744:(
712:)
709:x
700:+
697:1
694:(
688:2
685:1
674:=
670:)
665:2
662:x
657:(
637:x
599:x
572:b
465:.
356:e
349:t
342:v
259:)
157:)
91:)
85:(
80:)
76:(
62:.
20:)
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