In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of 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. Trigonometric calculations played an important role in the early study of astronomy. Early tables were constructed by repeatedly applying trigonometric identities (like the half-angle and angle-sum identities) to compute new values from old ones. 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.
Using a trigonometry table 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 floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) 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 Gal's accurate tables, Cody and Waite range reduction, and Payne and Hanek degree and radian reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons. The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a minimax approximation algorithm. For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral (Brent, 1976). Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers. The values for a/b·2π can be found by applying de Moivre's identity for n = a to a bth root of unity, which is also a root of the polynomial xb - 1 in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the degree-37 polynomial x37 − 1. For this case, a root-finding algorithm such as Newton's method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however.
Example To compute the sine function of 75 degrees, 9 minutes, 50 seconds using a table of trigonometric functions such as the Bernegger table from 1619 illustrated above, one might simply round up to 75 degrees, 10 minutes and then find the 10 minute entry on the 75 degree page, shown above-right, which is 0.9666746. However, this answer is only accurate to four decimal places. If one wanted greater accuracy, one could interpolate linearly as follows: From the Bernegger table:
sin (75° 10′) = 0.9666746 sin (75° 9′) = 0.9666001 The difference between these values is 0.0000745. Since there are 60 seconds in a minute of arc, we multiply the difference by 50/60 to get a correction of (50/60)*0.0000745 ≈ 0.0000621; and then add that correction to sin (75° 9′) to get :
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