Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications

Abstract

Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10−16, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals.

Schröder-Based Inverse Function Approximation

“…Consider an initial approximation of f −1 0 for the inverse function f −1 . For a given value of y, the initial approximation of x 0 to f −1 (y) is given by f −1 0 (y), and the first-order approximation for f −1 , consistent with (15), is…”

“…Point-based approximations such as Taylor series expansions, for example, often do not lead to suitable initial approximations as such approx-imations of a fixed order, whilst having a low error at the point of approximation, generally have an increasing error, and potentially an increasing relative error, as the distance from the point of approximation increases. This situation is illustrated in Figure 2 of [15], where the relative errors in Taylor series approximations for arcsine are detailed.…”

“…Naturally, there are many approximations for arcsine, and an overview of published approximations and new results for arcsine, arccosine and arctangent is provided in [15]. Graphs of arcsine and arccosine are shown in Figure 4.…”

“…Consider the published approximations for arcsine [17], [15] (Equations ( 10) and ( 31)) and [13] (p. 81, Equation (4.4.46)):…”

“…The approximation f −1 0,3 given by ( 48) is a lower bound for arcsine [15] (Equation ( 112)). Simulation results indicate that the first-to fourth-order approximations, as given by ( 42) to (45), and based on f −1 0,3 , are lower bounds with improved accuracy and with the relative error bounds detailed in Table 1.…”