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id="toc-On-demand_computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#On-demand_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>On-demand computation</span> </div> </a> <ul id="toc-On-demand_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Half-angle_and_angle-addition_formulas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Half-angle_and_angle-addition_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Half-angle and angle-addition formulas</span> </div> </a> <ul id="toc-Half-angle_and_angle-addition_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_quick,_but_inaccurate,_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_quick,_but_inaccurate,_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>A quick, but inaccurate, approximation</span> </div> </a> <ul id="toc-A_quick,_but_inaccurate,_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_better,_but_still_imperfect,_recurrence_formula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_better,_but_still_imperfect,_recurrence_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>A better, but still imperfect, recurrence formula</span> </div> </a> <ul id="toc-A_better,_but_still_imperfect,_recurrence_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> 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.sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="background:#ccccff;"><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File"><a href="/wiki/File:Sinus_und_Kosinus_am_Einheitskreis_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/250px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/375px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/500px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 2x" data-file-width="410" data-file-height="410" /></a></span></td></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Outline_of_trigonometry" title="Outline of trigonometry">Outline</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">History</a></li> <li><a href="/wiki/Uses_of_trigonometry" title="Uses of trigonometry">Usage</a></li></ul> <ul><li><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">Functions</a> (<a href="/wiki/Sine_and_cosine" title="Sine and cosine">sin</a>, <a href="/wiki/Sine_and_cosine" title="Sine and cosine">cos</a>, <a href="/wiki/Trigonometric_functions#tangent" title="Trigonometric functions">tan</a>, <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse</a>)</li> <li><a href="/wiki/Generalized_trigonometry" title="Generalized trigonometry">Generalized trigonometry</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Reference</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">Identities</a></li> <li><a href="/wiki/Exact_trigonometric_values" title="Exact trigonometric values">Exact constants</a></li> <li><a class="mw-selflink selflink">Tables</a></li> <li><a href="/wiki/Unit_circle" title="Unit circle">Unit circle</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Laws and theorems</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Law_of_sines" title="Law of sines">Sines</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Cosines</a></li> <li><a href="/wiki/Law_of_tangents" title="Law of tangents">Tangents</a></li> <li><a href="/wiki/Law_of_cotangents" title="Law of cotangents">Cotangents</a></li></ul> <ul><li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">Trigonometric substitution</a></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">Integrals</a> (<a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse functions</a>)</li> <li><a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Derivatives</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Mathematicians</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Habash_al-Hasib_al-Marwazi" class="mw-redirect" title="Habash al-Hasib al-Marwazi">al-Hasib</a></li> <li><a href="/wiki/Al-Battani" title="Al-Battani">al-Battani</a></li> <li><a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viète</a></li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Trigonometry" title="Template:Trigonometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Trigonometry" title="Template talk:Trigonometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Trigonometry" title="Special:EditPage/Template:Trigonometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, tables of <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> are useful in a number of areas. Before the existence of <a href="/wiki/Calculator" title="Calculator">pocket calculators</a>, <b>trigonometric tables</b> were essential for <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Science" title="Science">science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>. The calculation of <a href="/wiki/Mathematical_table" title="Mathematical table">mathematical tables</a> was an important area of study, which led to the development of the <a href="/wiki/History_of_computing" title="History of computing">first mechanical computing devices</a>. </p><p>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 <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> method. Interpolation of simple look-up tables of trigonometric functions is still used in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, where only modest accuracy may be required and speed is often paramount. </p><p>Another important application of trigonometric tables and generation schemes is for <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) algorithms, where the same trigonometric function values (called <i>twiddle factors</i>) 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). </p><p>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.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Using_a_trigonometry_table_involves_a_few_straightforward_steps">Using a trigonometry table involves a few straightforward steps</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=1" title="Edit section: Using a trigonometry table involves a few straightforward steps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Determine the specific angle for which you need to find the trigonometric values.</li> <li>Locate this angle along the horizontal axis (top row) of the table.</li> <li>Choose the trigonometric function you're interested in from the vertical axis (first column).</li> <li>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.</li></ol> <div class="mw-heading mw-heading2"><h2 id="On-demand_computation">On-demand computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=2" title="Edit section: On-demand computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bernegger_Manuale_137.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Bernegger_Manuale_137.jpg/200px-Bernegger_Manuale_137.jpg" decoding="async" width="200" height="282" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Bernegger_Manuale_137.jpg/300px-Bernegger_Manuale_137.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Bernegger_Manuale_137.jpg/400px-Bernegger_Manuale_137.jpg 2x" data-file-width="1481" data-file-height="2087" /></a><figcaption>A page from a 1619 book of <a href="/wiki/Mathematical_table" title="Mathematical table">mathematical tables</a>.</figcaption></figure> <p>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 <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating-point</a> units, is to combine a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> or <a href="/wiki/Rational_function" title="Rational function">rational</a> <a href="/wiki/Approximation_theory" title="Approximation theory">approximation</a> (such as <a href="/wiki/Chebyshev_approximation" class="mw-redirect" title="Chebyshev approximation">Chebyshev approximation</a>, best uniform approximation, <a href="/wiki/Pad%C3%A9_approximant" title="Padé approximant">Padé approximation</a>, and typically for higher or variable precisions, <a href="/wiki/Taylor_series" title="Taylor series">Taylor</a> and <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a>) 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 <a href="/wiki/Gal%27s_accurate_tables" title="Gal's accurate tables">Gal's accurate tables</a>, Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a <a href="/wiki/Hardware_multiplier" class="mw-redirect" title="Hardware multiplier">hardware multiplier</a>, there is an algorithm called <a href="/wiki/CORDIC" title="CORDIC">CORDIC</a> (as well as related techniques) that is more efficient, since it uses only <a href="/wiki/Shift_operator" title="Shift operator">shifts</a> and additions. All of these methods are commonly implemented in <a href="/wiki/Computer_hardware" title="Computer hardware">hardware</a> for performance reasons. </p><p>The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a <a href="/wiki/Minimax_approximation_algorithm" title="Minimax approximation algorithm">minimax approximation algorithm</a>. </p><p>For <a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">very high precision</a> calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the <a href="/wiki/Arithmetic-geometric_mean" class="mw-redirect" title="Arithmetic-geometric mean">arithmetic-geometric mean</a>, which itself approximates the trigonometric function by the (<a href="/wiki/Complex_number" title="Complex number">complex</a>) <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integral</a> (Brent, 1976). </p><p>Trigonometric functions of angles that are <a href="/wiki/Rational_number" title="Rational number">rational</a> multiples of 2π are <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>. The values for <i>a/b·2π</i> can be found by applying <a href="/wiki/De_Moivre%27s_identity" class="mw-redirect" title="De Moivre's identity">de Moivre's identity</a> for <i>n = a</i> to a <i>b<sup>th</sup></i> <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a>, which is also a root of the polynomial <i>x<sup>b</sup> - 1</i> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. For example, the cosine and sine of 2π ⋅ 5/37 are the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real</a> and <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary parts</a>, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a>-37 polynomial <i>x</i><sup>37</sup> − 1. For this case, a root-finding algorithm such as <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a> trigonometric constants, however. </p> <div class="mw-heading mw-heading2"><h2 id="Half-angle_and_angle-addition_formulas">Half-angle and angle-addition formulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=3" title="Edit section: Half-angle and angle-addition formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 <a href="/wiki/Trigonometric_identity" class="mw-redirect" title="Trigonometric identity">trigonometric identities</a> starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>, who derived them in the <i><a href="/wiki/Almagest" title="Almagest">Almagest</a></i>, a treatise on <a href="/wiki/History_of_astronomy" title="History of astronomy">astronomy</a>. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which <i>x</i> lies): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c4dc49d708fb64dcee0005f7e3c520fd7da155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.581ex; height:4.843ex;" alt="{\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4316afaf4d485354de1870d09d89939eff341d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.325ex; height:4.843ex;" alt="{\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>±</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>±</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a622d6c8d04c640fbd1f0790e151290b2942c4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.232ex; height:2.843ex;" alt="{\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>±</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a087e7829135a573deb56dce39371aee9dcb86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.487ex; height:2.843ex;" alt="{\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}"></span></dd></dl> <p>These were used to construct <a href="/wiki/Ptolemy%27s_table_of_chords" title="Ptolemy's table of chords">Ptolemy's table of chords</a>, which was applied to astronomical problems. </p><p>Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and <a href="/wiki/Versine" title="Versine">versine</a>. </p> <div class="mw-heading mw-heading2"><h2 id="A_quick,_but_inaccurate,_approximation"><span id="A_quick.2C_but_inaccurate.2C_approximation"></span>A quick, but inaccurate, approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=4" title="Edit section: A quick, but inaccurate, approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A quick, but inaccurate, algorithm for calculating a table of <i>N</i> approximations <i>s</i><sub><i>n</i></sub> for <a href="/wiki/Sine" class="mw-redirect" title="Sine">sin</a>(2<a href="/wiki/Pi" title="Pi">π</a><i>n</i>/<i>N</i>) and <i>c</i><sub><i>n</i></sub> for <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cos</a>(2π<i>n</i>/<i>N</i>) is: </p> <dl><dd><i>s</i><sub>0</sub> = 0</dd> <dd><i>c</i><sub>0</sub> = 1</dd> <dd><i>s</i><sub><i>n</i>+1</sub> = <i>s</i><sub><i>n</i></sub> + <i>d</i> × <i>c</i><sub><i>n</i></sub></dd> <dd><i>c</i><sub><i>n</i>+1</sub> = <i>c</i><sub><i>n</i></sub> − <i>d</i> × <i>s</i><sub><i>n</i></sub></dd></dl> <p>for <i>n</i> = 0,...,<i>N</i> − 1, where <i>d</i> = 2π/<i>N</i>. </p><p>This is simply the <a href="/wiki/Numerical_ordinary_differential_equations#Euler_method" class="mw-redirect" title="Numerical ordinary differential equations">Euler method</a> for integrating the <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds/dt=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds/dt=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93631283595520dd7691739a36dbf618d1a7a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.63ex; height:2.843ex;" alt="{\displaystyle ds/dt=c}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dc/dt=-s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dc/dt=-s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ac27718738683773e591d2f86a1543d15e03f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.438ex; height:2.843ex;" alt="{\displaystyle dc/dt=-s}"></span></dd></dl> <p>with initial conditions <i>s</i>(0) = 0 and <i>c</i>(0) = 1, whose analytical solution is <i>s</i> = sin(<i>t</i>) and <i>c</i> = cos(<i>t</i>). </p><p>Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/<i>N</i>. </p><p>For example, for <i>N</i> = 256 the maximum error in the sine values is ~0.061 (<i>s</i><sub>202</sub> = −1.0368 instead of −0.9757). For <i>N</i> = 1024, the maximum error in the sine values is ~0.015 (<i>s</i><sub>803</sub> = −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. </p> <div class="mw-heading mw-heading2"><h2 id="A_better,_but_still_imperfect,_recurrence_formula"><span id="A_better.2C_but_still_imperfect.2C_recurrence_formula"></span>A better, but still imperfect, recurrence formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=5" title="Edit section: A better, but still imperfect, recurrence formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Trigonometric_tables&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">December 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A simple recurrence formula to generate trigonometric tables is based on <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> and the relation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>×</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi mathvariant="normal">Δ</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec4a50d26c14208220beea5be65dcbef56d77bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.197ex; height:2.843ex;" alt="{\displaystyle e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }}"></span></dd></dl> <p>This leads to the following recurrence to compute trigonometric values <i>s</i><sub><i>n</i></sub> and <i>c</i><sub><i>n</i></sub> as above: </p> <dl><dd><i>c</i><sub>0</sub> = 1</dd> <dd><i>s</i><sub>0</sub> = 0</dd> <dd><i>c</i><sub><i>n</i>+1</sub> = <i>w</i><sub><i>r</i></sub> <i>c</i><sub><i>n</i></sub> − <i>w</i><sub><i>i</i></sub> <i>s</i><sub><i>n</i></sub></dd> <dd><i>s</i><sub><i>n</i>+1</sub> = <i>w</i><sub><i>i</i></sub> <i>c</i><sub><i>n</i></sub> + <i>w</i><sub><i>r</i></sub> <i>s</i><sub><i>n</i></sub></dd></dl> <p>for <i>n</i> = 0, ..., <i>N</i> − 1, where <i>w</i><sub><i>r</i></sub> = cos(2π/<i>N</i>) and <i>w</i><sub><i>i</i></sub> = sin(2π/<i>N</i>). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> in the complex plane to solve for the primitive <a href="/wiki/Root_of_unity" title="Root of unity">root</a> of <i>z</i><sup><i>N</i></sup> − 1). </p><p>This method would produce an <i>exact</i> table in exact arithmetic, but has errors in finite-precision <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">floating-point</a> arithmetic. In fact, the errors grow as O(ε <i>N</i>) (in both the worst and average cases), where ε is the floating-point precision. </p><p>A significant improvement is to use the following modification to the above, a trick (due to Singleton<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>) often used to generate trigonometric values for FFT implementations: </p> <dl><dd><i>c</i><sub>0</sub> = 1</dd> <dd><i>s</i><sub>0</sub> = 0</dd> <dd><i>c</i><sub><i>n</i>+1</sub> = <i>c</i><sub><i>n</i></sub> − (α <i>c</i><sub><i>n</i></sub> + β <i>s</i><sub><i>n</i></sub>)</dd> <dd><i>s</i><sub><i>n</i>+1</sub> = <i>s</i><sub><i>n</i></sub> + (β <i>c</i><sub><i>n</i></sub> − α <i>s</i><sub><i>n</i></sub>)</dd></dl> <p>where α = 2 sin<sup>2</sup>(π/<i>N</i>) and β = sin(2π/<i>N</i>). The errors of this method are much smaller, O(ε √<i>N</i>) on average and O(ε <i>N</i>) in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Aryabhata%27s_sine_table" class="mw-redirect" title="Aryabhata's sine table">Aryabhata's sine table</a></li> <li><a href="/wiki/CORDIC" title="CORDIC">CORDIC</a></li> <li><a href="/wiki/Exact_trigonometric_values" title="Exact trigonometric values">Exact trigonometric values</a></li> <li><a href="/wiki/Madhava%27s_sine_table" title="Madhava's sine table">Madhava's sine table</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a></li> <li><a href="/wiki/Prosthaphaeresis" title="Prosthaphaeresis">Prosthaphaeresis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trigonometric_tables&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.yogiraj.co.in/trigonometry-table">"Trigonometry Table: Learning of trigonometry table is simplified"</a>. <i>Yogiraj notes | General study and Law study Notes</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-11-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Yogiraj+notes+%7C+General+study+and+Law+study+Notes&rft.atitle=Trigonometry+Table%3A+Learning+of+trigonometry+table+is+simplified&rft_id=https%3A%2F%2Fwww.yogiraj.co.in%2Ftrigonometry-table&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+tables" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFSingleton1967">Singleton 1967</a></span> </li> </ol></div></div> <ul><li><a href="/wiki/Carl_B._Boyer" class="mw-redirect" title="Carl B. Boyer">Carl B. Boyer</a> (1991) <i>A History of Mathematics</i>, 2nd edition, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>.</li> <li>Manfred Tasche and Hansmartin Zeuner (2002) "Improved roundoff error analysis for precomputed twiddle factors", <i>Journal for Computational Analysis and Applications</i> 4(1): 1–18.</li> <li>James C. Schatzman (1996) "Accuracy of the discrete Fourier transform and the fast Fourier transform", <a href="/wiki/SIAM_Journal_on_Scientific_Computing" title="SIAM Journal on Scientific Computing">SIAM Journal on Scientific Computing</a> 17(5): 1150–1166.</li> <li>Vitit Kantabutra (1996) "On hardware for computing exponential and trigonometric functions," <a href="/wiki/IEEE_Transactions_on_Computers" title="IEEE Transactions on Computers">IEEE Transactions on Computers</a> 45(3): 328–339 .</li> <li><a href="/wiki/R._P._Brent" class="mw-redirect" title="R. P. Brent">R. P. Brent</a> (1976) "<a rel="nofollow" class="external text" href="http://doi.acm.org/10.1145/321941.321944">Fast Multiple-Precision Evaluation of Elementary Functions</a>", <a href="/wiki/Journal_of_the_Association_for_Computing_Machinery" class="mw-redirect" title="Journal of the Association for Computing Machinery">Journal of the Association for Computing Machinery</a> 23: 242–251.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingleton1967" class="citation journal cs1">Singleton, Richard C (1967). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363717.363771">"On Computing The Fast Fourier Transform"</a>. <i><a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>. <b>10</b> (10): 647–654. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363717.363771">10.1145/363717.363771</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6287781">6287781</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=On+Computing+The+Fast+Fourier+Transform&rft.volume=10&rft.issue=10&rft.pages=647-654&rft.date=1967&rft_id=info%3Adoi%2F10.1145%2F363717.363771&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6287781%23id-name%3DS2CID&rft.aulast=Singleton&rft.aufirst=Richard+C&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F363717.363771&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrigonometric+tables" class="Z3988"></span></li> <li>William J. Cody Jr., William Waite, <i>Software Manual for the Elementary Functions</i>, Prentice-Hall, 1980, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-822064-6" title="Special:BookSources/0-13-822064-6">0-13-822064-6</a>.</li> <li>Mary H. Payne, Robert N. Hanek, <i>Radian reduction for trigonometric functions</i>, <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">ACM</a> SIGNUM Newsletter 18: 19-24, 1983.</li> <li>Gal, Shmuel and Bachelis, Boris (1991) "An accurate elementary mathematical library for the IEEE floating point standard", <a href="/wiki/ACM_Transactions_on_Mathematical_Software" title="ACM Transactions on Mathematical Software">ACM Transactions on Mathematical Software</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Trigonometric_tables&oldid=1239775740">https://en.wikipedia.org/w/index.php?title=Trigonometric_tables&oldid=1239775740</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Trigonometry" title="Category:Trigonometry">Trigonometry</a></li><li><a href="/wiki/Category:Numerical_analysis" title="Category:Numerical analysis">Numerical analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_December_2018" title="Category:Articles lacking in-text citations from December 2018">Articles lacking in-text citations from December 2018</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Articles_that_may_contain_original_research_from_December_2018" title="Category:Articles that may contain original research from December 2018">Articles that may contain original research from December 2018</a></li><li><a href="/wiki/Category:All_articles_that_may_contain_original_research" title="Category:All articles that may contain original research">All articles that may contain original research</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 11 August 2024, at 13:24<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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