Trigonometric tables

<h2 id="using-a-trigonometry-table-involves-a-few-straightforward-steps">Using a trigonometry table involves a few straightforward steps</h2>
<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>
<h2 id="on-demand-computation">On-demand computation</h2>

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="/facts/Floating_point/eIckahxe">floating-point</a> units, is to combine a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> or <a href="/facts/Rational_function/2nJGu0Ko">rational</a> <a href="/facts/Approximation_theory/87AaPXLp">approximation</a> (such as <a href="/facts/Chebyshev_approximation/87AaPXLp">Chebyshev approximation</a>, best uniform approximation, <a href="/facts/Pad%25C3%25A9_approximant/cR5BjAb8">Padé approximation</a>, and typically for higher or variable precisions, <a href="/facts/Taylor_series/M0aTuftH">Taylor</a> and <a href="/facts/Laurent_series/sSMevFU8">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="/facts/Gal%2527s_accurate_tables/2Uzx2JKm">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="/facts/Hardware_multiplier/GfkwufDb">hardware multiplier</a>, there is an algorithm called <a href="/facts/CORDIC/sQqTjkd2">CORDIC</a> (as well as related techniques) that is more efficient, since it uses only <a href="/facts/Shift_operator/6c6lc0ks">shifts</a> and additions. All of these methods are commonly implemented in <a href="/facts/Computer_hardware/8jSjsCNv">hardware</a> for performance reasons.
The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a <a href="/facts/Minimax_approximation_algorithm/A6EEMLwY">minimax approximation algorithm</a>.
For <a href="/facts/Arbitrary-precision_arithmetic/cXxc8yB2">very high precision</a> calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the <a href="/facts/Arithmetic-geometric_mean/8PRN9UB4">arithmetic-geometric mean</a>, which itself approximates the trigonometric function by the (<a href="/facts/Complex_number/2w7mqI7e">complex</a>) <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a> (Brent, 1976).
Trigonometric functions of angles that are <a href="/facts/Rational_number/VJQmyY05">rational</a> multiples of 2π are <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a>. The values for a/b·2π can be found by applying <a href="/facts/De_Moivre%2527s_identity/uyG4k2uC">de Moivre's identity</a> for n = a to a bth <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>, which is also a root of the polynomial xb - 1 in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. For example, the cosine and sine of 2π ⋅ 5/37 are the <a href="/facts/Real_part/2w7mqI7e">real</a> and <a href="/facts/Imaginary_part/2w7mqI7e">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="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a>-37 polynomial x37 − 1. For this case, a root-finding algorithm such as <a href="/facts/Newton%2527s_method/VlRhI2FL">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="/facts/Transcendental_number/IrB7JppM">transcendental</a> trigonometric constants, however.

<h2 id="half-angle-and-angle-addition-formulas">Half-angle and angle-addition formulas</h2>
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="/facts/Trigonometric_identity/Xw9Jx8Vz">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="/facts/Ptolemy/htudGQTQ">Ptolemy</a>, who derived them in the <a href="/facts/Almagest/30qaBRAq">Almagest</a>, a treatise on <a href="/facts/History_of_astronomy/js3QcRnX">astronomy</a>. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which x lies):

cos
        ⁡
        
          (
          
            
              x
              2
            
          
          )
        
        =
        ±
        
          
            
              
                
                  1
                  2
                
              
            
            (
            1
            +
            cos
            ⁡
            x
            )
          
        
      
    
    {\displaystyle \cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}}

sin
        ⁡
        
          (
          
            
              x
              2
            
          
          )
        
        =
        ±
        
          
            
              
                
                  1
                  2
                
              
            
            (
            1
            −
            cos
            ⁡
            x
            )
          
        
      
    
    {\displaystyle \sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}}

sin
        ⁡
        (
        x
        ±
        y
        )
        =
        sin
        ⁡
        (
        x
        )
        cos
        ⁡
        (
        y
        )
        ±
        cos
        ⁡
        (
        x
        )
        sin
        ⁡
        (
        y
        )
        
      
    
    {\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,}

cos
        ⁡
        (
        x
        ±
        y
        )
        =
        cos
        ⁡
        (
        x
        )
        cos
        ⁡
        (
        y
        )
        ∓
        sin
        ⁡
        (
        x
        )
        sin
        ⁡
        (
        y
        )
        
      
    
    {\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,}

These were used to construct <a href="/facts/Ptolemy%2527s_table_of_chords/h6Ubj0sq">Ptolemy's table of chords</a>, 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 <a href="/facts/Versine/MllE4ISd">versine</a>.

<h2 id="a-quick-but-inaccurate-approximation">A quick, but inaccurate, approximation</h2>
A quick, but inaccurate, algorithm for calculating a table of N approximations sn for <a href="/facts/Sine/gQ6LXK8z">sin</a>(2<a href="/facts/Pi/vC0UBj1q">π</a>n/N) and cn for <a href="/facts/Cosine/czej7ur0">cos</a>(2πn/N) is:

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 <a href="/facts/Numerical_ordinary_differential_equations/rbrYBco3">Euler method</a> for integrating the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a>:

d
        s
        
          /
        
        d
        t
        =
        c
      
    
    {\displaystyle ds/dt=c}

d
        c
        
          /
        
        d
        t
        =
        −
        s
      
    
    {\displaystyle 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.

<h2 id="a-better-but-still-imperfect-recurrence-formula">A better, but still imperfect, recurrence formula</h2>

A simple recurrence formula to generate trigonometric tables is based on <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> and the relation:

e
          
            i
            (
            θ
            +
            Δ
            )
          
        
        =
        
          e
          
            i
            θ
          
        
        ×
        
          e
          
            i
            Δ
            θ
          
        
      
    
    {\displaystyle e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }}

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 <a href="/facts/Newton%2527s_method/VlRhI2FL">Newton's method</a> in the complex plane to solve for the primitive <a href="/facts/Root_of_unity/EqH0ho1Y">root</a> of zN − 1).
This method would produce an exact table in exact arithmetic, but has errors in finite-precision <a href="/facts/Floating-point/eIckahxe">floating-point</a> 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 significant improvement is to use the following modification to the above, a trick (due to Singleton<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>) 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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Aryabhata%2527s_sine_table/HcDsEuDj">Aryabhata's sine table</a></li>
<li><a href="/facts/CORDIC/sQqTjkd2">CORDIC</a></li>
<li><a href="/facts/Exact_trigonometric_values/zYE1BVLF">Exact trigonometric values</a></li>
<li><a href="/facts/Madhava%2527s_sine_table/3KDWEd0o">Madhava's sine table</a></li>
<li><a href="/facts/Numerical_analysis/QT9cYa9z">Numerical analysis</a></li>
<li><a href="/facts/Plimpton_322/bfIIEP8k">Plimpton 322</a></li>
<li><a href="/facts/Prosthaphaeresis/u1FKPxpj">Prosthaphaeresis</a></li></ul>

<ul><li><a href="/facts/Carl_B._Boyer/H58yXLK2">Carl B. Boyer</a> (1991) A History of Mathematics, 2nd edition, <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>.</li>
<li>Manfred Tasche and Hansmartin Zeuner (2002) "Improved roundoff error analysis for precomputed twiddle factors", Journal for Computational Analysis and Applications 4(1): 1–18.</li>
<li>James C. Schatzman (1996) "Accuracy of the discrete Fourier transform and the fast Fourier transform", <a href="/facts/SIAM_Journal_on_Scientific_Computing/waxjpMla">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="/facts/IEEE_Transactions_on_Computers/jjc4KLle">IEEE Transactions on Computers</a> 45(3): 328–339 .</li>
<li><a href="/facts/R._P._Brent/JYLmuA0M">R. P. Brent</a> (1976) "<a href="http://doi.acm.org/10.1145/321941.321944">Fast Multiple-Precision Evaluation of Elementary Functions</a>", <a href="/facts/Journal_of_the_Association_for_Computing_Machinery/qkIF9LLS">Journal of the Association for Computing Machinery</a> 23: 242–251.</li>
<li>Singleton, Richard C (1967). <a href="https://doi.org/10.1145%2F363717.363771">"On Computing The Fast Fourier Transform"</a>. <a href="/facts/Communications_of_the_ACM/Ix1DvEAk">Communications of the ACM</a>. 10 (10): 647–654. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F363717.363771">10.1145/363717.363771</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:6287781">6287781</a>.</li>
<li>William J. Cody Jr., William Waite, Software Manual for the Elementary Functions, Prentice-Hall, 1980, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-822064-6.</li>
<li>Mary H. Payne, Robert N. Hanek, Radian reduction for trigonometric functions, <a href="/facts/Association_for_Computing_Machinery/abI2atlP">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="/facts/ACM_Transactions_on_Mathematical_Software/3Ev3LTR8">ACM Transactions on Mathematical Software</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Trigonometry Table: Learning of trigonometry table is simplified". Yogiraj notes | General study and Law study Notes. Retrieved 2023-11-02. <a href="https://www.yogiraj.co.in/trigonometry-table" target="_blank">https://www.yogiraj.co.in/trigonometry-table</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Singleton 1967 - Singleton, Richard C (1967). "On Computing The Fast Fourier Transform". Communications of the ACM. 10 (10): 647–654. doi:10.1145/363717.363771. S2CID 6287781. <a href="https://doi.org/10.1145%2F363717.363771" target="_blank">https://doi.org/10.1145%2F363717.363771</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Trigonometric tables open-in-new

Trigonometric tables