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Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
The polynomials, exponential function e x, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan.
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [ 1 ] In the table below, the label "Undefined" represents a ratio 1 : 0. {\displaystyle 1:0.}
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...
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 ...
Over the complex numbers, P n splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals, P n splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the ...
CORDIC (coordinate rotation digital computer), Volder's algorithm, Digit-by-digit method, Circular CORDIC (Jack E. Volder), [1] [2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), [3] [4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), [5] [6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots ...
Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives ...