Approximations for Trigonometric Functions

References Char, B. W., et al., Maple V Language Reference Manual, 

Springer-Verlag, New York (1991) Wolfram, S., The Mathematics Book, 5th ed., Wolfram Media (2003). 

An Example of Functional Notation Suppose that a storage warehouse of 16,000 ft3 is required. The construction costs per square foot are 10, 3, and 2 for walls, roof, and floor respectively. What are the minimum cost dimensions Thus, with h = height, x = width, and y = length, the respective costs are 

In this form it can be shown that the minimum cost will occur for x = y therefore 

By evaluation, the smallest cost will occur when x = 40. 

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Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

On the other hand, the comments made for the case of nonutilization of an auxiliary elastic element in relation to the different types of response are also pertinent in the present context. In fact, if the trigonometric functions appearing in Eq. (17.293) are approximated by... 

Since Equation (3) is nonlinear and transcendental i.e. invokes trigonometric functions), the usual linear version of least-squares analysis can not be used. The method is to approximate the expression for as a Taylor series and only retain the first term ... 

The numerical value of the gamma function has been tabulated for all values of n between 1 and 2 to twelve decimal places. By the aid of such a table, the approximate value of all definite integrals reducible to gamma functions can be calculated as easily as ordinary trigonometrical functions, or logarithms. There are four cases ... 

Quite often one is interested in approximating a periodic function, /, over a period. Therefore, the trigonometric partial sum Sk which best approximates such a periodic function over one full period, p, is the one for which the total squared error... 

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