Showing papers on "Inverse trigonometric functions published in 2017"
TL;DR: In this paper, the authors developed a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations, which reduced the wave equations into a system of first-order ordinary differential equations (ODEs) in time variable.
Abstract: Purpose
The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc.
Design/methodology/approach
Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed.
Findings
A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations.
Originality/value
To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).
30 citations
TL;DR: A robust technique for the single-phase grid voltage fundamental amplitude, frequency, and phase angle estimation under distorted grid conditions based on a demodulation method tuned at a fixed frequency, which is less affected by dc offset, can provide faster frequency estimation, and also avoids interdependent loop, trigonometric, and inverse trig onometric functions operation.
Abstract: This paper proposes a robust technique for the single-phase grid voltage fundamental amplitude, frequency, and phase angle estimation under distorted grid conditions. It is based on a demodulation method tuned at a fixed frequency. It does not have stability issue due to an open-loop structure, does not require real-time evaluation of trigonometric and inverse trigonometric functions, and also avoids the use of look-up table. It can provide accurate estimation of the single-phase grid voltage fundamental parameters under dc offset and harmonics. When compared with a frequency adaptive demodulation technique, the proposed one is less affected by dc offset, can provide faster frequency estimation, and also avoids interdependent loop, trigonometric, and inverse trigonometric functions operation. Simulation and experimental results are presented to verify the performance of the proposed technique.
26 citations
TL;DR: The Sumudu transform integral equation is solved by continuous integration by parts to obtain its definition for trigonometric functions, and the obtained result is inverted to show the expansion of trig onometric functions as an infinite series.
24 citations
TL;DR: The mathematical description of the stowing and deployment geometry, as well as the forces inflicted by the mechanism provides an understanding of how exactly the membrane deploys and through which edges the deployment forces are transferred.
22 citations
TL;DR: In this paper, a method to sharpen bounds of both sinc(x) and $$\arcsin (x)$$ functions, and the inequalities in exponential form as well, is presented.
Abstract: This paper presents a new method to sharpen bounds of both sinc(x) and $$\arcsin (x)$$
functions, and the inequalities in exponential form as well. It also provides a method for finding two-sided bounds, which are also unsolved in previous state-of-art references.
18 citations
TL;DR: In this paper, the authors presented the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms, and applied the results to the derivation of generating functions and to the number of closed walks on a path and in a cycle.
Abstract: We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.
16 citations
TL;DR: An improved VLSI (Very Large Scale of Integration) architecture for real-time and high-accuracy computation of trigonometric functions with fixed-point arithmetic, particularly arctangent using CORDIC and fast magnitude estimation is presented.
Abstract: This paper presents an improved VLSI (Very Large Scale of Integration) architecture for real-time and high-accuracy computation of trigonometric functions with fixed-point arithmetic, particularly arctangent using CORDIC (Coordinate Rotation Digital Computer) and fast magnitude estimation. The standard CORDIC implementation suffers of a loss of accuracy when the magnitude of the input vector becomes small. Using a fast magnitude estimator before running the standard algorithm, a pre-processing magnification is implemented, shifting the input coordinates by a proper factor. The entire architecture does not use a multiplier, it uses only shift and add primitives as the original CORDIC, and it does not change the data path precision of the CORDIC core. A bit-true case study is presented showing a reduction of the maximum phase error from 414 LSB (angle error of 0.6355 rad) to 4 LSB (angle error of 0.0061 rad), with small overheads of complexity and speed. Implementation of the new architecture in 0.18 µm CMOS technology allows for real-time and low-power processing of CORDIC and arctangent, which are key functions in many embedded DSP systems. The proposed macrocell has been verified by integration in a system-on-chip, called SENSASIP (Sensor Application Specific Instruction-set Processor), for position sensor signal processing in automotive measurement applications.
14 citations
01 Jan 2017
TL;DR: Otero as discussed by the authors explored the genesis of the trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.
Abstract: F1. A Genetic Context for Understanding the Trigonometric Functions In this project, we explore the genesis of the trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The goal is to provide the typical student in a pre-calculus course some context for understanding these concepts that is generally missing from standard textbook developments. Trigonometry emerged in the ancient Greek world (and, it is suspected, independently in China and India as well) from the geometrical analyses needed to solve basic astronomical problems regarding the relative positions and motions of celestial objects. While the Greeks (Hipparchus, Ptolemy) recognized the usefulness of tabulating chords of central angles in a circle as aids to solving problems of spherical geometry, Hindu mathematicians, like Varahamahira (505–587), in his Pancasiddhantika [61], found it more expedient to tabulate half-chords, whence the use of the sine and cosine became popular. We will examine an excerpt from this work, wherein Varahamahira describes a few of the standard modern relationships between sine and cosine in the course of creating a sine table. In the 11th century, the Arabic scholar and expert on Hindu science Abu l-Rayhan Muhammad al-Biruni (973–1055) published The Exhaustive Treatise on Shadows (ca. 1021) [55]. In this work, we see how Biruni presents geometrical methods for the use of sundials; the relations within right triangles made by the gnomon of a sundial and the shadow cast on its face lead to the study and tabulation of values of the tangent and cotangent, secant and cosecant. Biruni also works out the relationships that these quantities have with the sines and cosines of the angles. However, the modern terminology for the standard trigonometric quantities is not established until the European Renaissance. Foremost in this development is the landmark On Triangles (1463) by Regiomontanus (Johannes Müller) [49]. Regiomontanus exposes trigonometry in a purely geometrical form and then applies the ideas to problems in circular and spherical geometry. We examine a few of the theorems that explore the trigonometric relations and which are used to solve triangle problems. This project is intended for courses in pre-calculus, trigonometry, the history of mathematics, or as a capstone course for teachers. Primary Author: Danny Otero.
12 citations
TL;DR: The Padé approximant is used to give the refinements of some remarkable inequalities involving inverse trigonometric functions, and it is shown that the new inequalities presented are more refined than that obtained in earlier papers.
Abstract: The Pade approximation is a useful method for creating new inequalities and improving certain inequalities. In this paper we use the Pade approximant to give the refinements of some remarkable inequalities involving inverse trigonometric functions, it is shown that the new inequalities presented in this paper are more refined than that obtained in earlier papers.
10 citations
TL;DR: In this paper, a new algorithm that does not require any of these parameters is presented with related software code and some application examples, which indicate a fair robustness of the estimates with respect to the noise, for the instantaneous phase and frequency.
Abstract: The instantaneous phase of a complex trace is normally computed as an inverse tangent function of the ratio between its imaginary and real parts. This algorithm often produces a discontinuous function in the range [−π, +π], as the principal value of the inverse tangent is assumed. The attempt of removing these discontinuities, called phase unwrapping, introduces uncertainties and some personal bias because it requires user-defined parameters. A new algorithm that does not require any of these parameters is presented with the related software code and some application examples. The results indicate a fair robustness of the estimates with respect to the noise, for the instantaneous phase and frequency. In addition, the median-averaged instantaneous frequency is a useful tool for the seismic stratigraphy analysis.
10 citations
TL;DR: In this paper, the authors proved two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873] using a method for proving inequalities of mixed trigonometric polynomial functions.
Abstract: In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873], using a method for proving inequalities of mixed trigonometric polynomial functions.
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TL;DR: In this article, the authors proposed an efficient numerical scheme for the approximate solution of the time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions.
Abstract: In this paper, we propose an efficient numerical scheme for the approximate solution of the time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation and the derivative in space is discretized using cubic trigonometric B-spline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with a finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straight forward interpolation and very less computational cost.
TL;DR: In this article, the free vibration of functionally graded (FG) beam subjected to all sets of boundary conditions has been investigated using Rayleighgh-Ritz method and different higher-order shear deformation beam theories have also been incorporated, out of which three SDBTs are proposed in the form of inverse trigonometric functions.
Abstract: In this paper, free vibration of functionally graded (FG) beam subjected to all sets of boundary conditions has been investigated using Rayliegh-Ritz method. Different higher-order shear deformation beam theories (SDBTs) have also been incorporated, out of which three SDBTs are proposed in the form of inverse trigonometric functions. The proposed deformation theories satisfy the transverse shear stress conditions at the bottom and top surfaces of the beam. The material properties of FG beam are assumed to vary along thickness direction in power-law form and trial functions denoting the displacement components are expressed as linear combination of algebraic polynomials. Rayleigh–Ritz method is used to estimate frequency parameters in order to handle to all sorts of boundary conditions at the edges with ease. Comparison of frequency parameters is carried out with the available literature in special cases and new results are also provided after checking the convergence of frequency parameters.
TL;DR: The authors gave new sharp refinements of Shafer-Fink's inequality, using suitable changes of variables, and showed that these refinements can be applied to any inequality inequality inequality.
Abstract: The aim of our work is to give new sharp refinements of Shafer-Fink’s inequality, using suitable changes of variables.
TL;DR: In this paper, two iterative algorithms are presented to solve the inverse problem of gas-turbine blade cooling, where the first one is about reducing the number of trigonometric functions, and the second one is regularized in the process of iteration.
TL;DR: In this article, the second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of R(t); using this definition, an ordinary differential equation is constructed.
Abstract: The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal axis as the variable t and the vertical axis as the variable r(t) are created by numerically solving the ordinary differential equation. These graphs show several regular waves with a specific periodicity and waveform depending on the natural number n. In this study, the functions that satisfy the ordinary differential equation are presented as the leaf functions. For n = 1, the leaf function become a trigonometric function. For n = 2, the leaf function becomes a lemniscatic elliptic function. These functions involve an additional theorem. In this paper, based on the additional theorem for n = 1 and n = 2, the double angle and additional theorem for n = 3 are presented.
ENEA1
TL;DR: In this paper, the generalized trigonometric functions are applied to problems in classical mechanics and to the theory of integral equations, and they make further progress in the generalization process by discussing the properties of Laguerre trigonometries along with the relevant link with the Bessel functions.
Abstract: We present some applications of the generalized trigonometric functions to problems in classical mechanics and to the theory of integral equations. We discuss how second and third order trigonometries are ideally suited tools to treat either damped harmonic oscillators and three dimensional rotational models. We make further progress in the generalization process by discussing the properties of Laguerre trigonometries along with the relevant link with the theory of Bessel functions.
TL;DR: In this article, the authors used rational functions, logarithms and inverse trigonometric functions to construct closed-form expressions of the Voigt function K (x, y ) in terms of rational functions and showed that these expressions are not competitive with other algorithms with respect to computational speed.
Abstract: Rational approximations for the Gauss function can be used to construct closed-form expressions of the Voigt function K ( x, y ) in terms of rational functions, logarithms and inverse trigonometric functions. The comparison with accurate reference values indicates a relative accuracy in the percent range for y ≳ 1, but serious problems for smaller y . Furthermore, these expressions are not competitive with other algorithms with respect to computational speed. Both accuracy and speed tests indicate that supposedly “good” approximations of the integrand do not necessarily provide good approximations of the integral, i.e. Voigt function.
01 Jul 2017
TL;DR: The authors identify two conceptions of angles, trigonometric functions, and inverse trigonometrical functions that rely on either a static or a dynamic definition of angle, and argue that transparency in making explicit how these conceptions can be bridged might be useful in understanding difficulties that emerge when solving problems with inverse trigonal functions.
Abstract: Using expository text and examples available in 10 college textbooks we identify two conceptions of angles, trigonometric functions, and inverse trigonometric functions that rely on either a static or a dynamic definition of angle. Although the textbooks favor a conception of trigonometric functions that is based on a dynamic conception of angle, they split in their definition of inverse trigonometric functions. We argue that transparency in making explicit how these conceptions can be bridged might be useful in understanding difficulties that emerge when solving problems with inverse trigonometric functions.
TL;DR: A novel harmonics extraction algorithm based on the principle of trigonometric orthogonal functions (TOF) is proposed, and its mathematical principle and physical meaning are introduced in detail and its implementation and superiority in terms of computation efficiency are analyzed.
Abstract: For a single-phase active power filter (APF), designing a more efficient algorithm to guarantee accurate and fast harmonics extraction with a lower computing cost is still a meaningful topic. The common idea still employs a IRPT-based Park transform, which was originally designed for 3-phase applications. Therefore, an additional virtual signal generation (VSG) link is necessary when it is used in the single-phase condition. This method, with virtual signal generation and transform, is obviously not the most efficient one. Regarding this problem, this paper proposes a novel harmonics extraction algorithm to further improve efficiency. The new algorithm is based on the principle of trigonometric orthogonal functions (TOF), and its mathematical principle and physical meaning are introduced in detail. Its implementation and superiority in terms of computation efficiency are analyzed by comparing it with conventional methods. Finally, its effectiveness is well validated through detailed simulations and laboratory experiments.
TL;DR: In this paper, series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions are presented, and some inequalities for trigonometrical functions are established.
Abstract: We present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. From these results, we establish some inequalities for trigonometric and hyperbolic functions. Mathematics subject classification (2010): 11B68, 26D05.
TL;DR: The generalized trigonometric functions (GTF) have been introduced using an appropriate redefinition of Euler type identities involving non-standard forms of imaginary numbers, realized by different types of matrices as discussed by the authors.
Abstract: The generalized trigonometric functions (GTF) have been introduced using an appropriate redefinition of Euler type identities involving non-standard forms of imaginary numbers, realized by different types of matrices. In this paper we use the GTF to get parameterization of practical interest for non-singular matrices. The possibility of using this procedure to deal with applications in electron transport is also touched on.
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TL;DR: In this paper, an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals is proposed.
Abstract: We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used as a cardinality measure for fine comparison between infinite countable sets in metric space which respects the intuitive notion of the set's size.
TL;DR: In this article, Lagrange interpolation was used for replacing the arctangent function used in the LC-CFO estimator, and the steady-state mean square errors were very close to the Cramer-Rao bounds at median to high signal-to-noise ratios.
Abstract: We modify a LC-CFO estimator in the literature to make it easier to implement. By using the Lagrange interpolation, three low-order functions are used as candidates for replacing the arctangent function used in the LC-CFO estimator. The simplest one among these low-order functions sets its output directly equal to its input. In consequence, up to 20 iterations are needed for convergence with the carrier frequency offset up to 0.5. Moreover, the steady-state mean square errors are very close to the Cramer–Rao bounds at median-to-high signal-to-noise ratios. Those performances are about the same as when using a conventional LC-CFO estimator.
01 Jul 2017
TL;DR: The error sources and the total computational error of CORDIC algorithm in calculating arctangent, sine, and cosine functions are discussed and some conclusions for error propagation mechanism are revealed, which are not quite consistent with existing literature.
Abstract: Coordinate rotation digital computer (CORDIC) algorithm can be used to implement a variety of transcendental functions. Using theory and techniques of uncertainty evaluation, we discuss the error sources and the total computational error of CORDIC algorithm in calculating arctangent, sine, and cosine functions. Besides, we reveal some conclusions for error propagation mechanism, which are not quite consistent with existing literature. Numerical simulation results are presented as well, which can verify the proposed expressions of uncertainty.
01 Jul 2017
TL;DR: The experimental results show that the arctangent approach based on CORDIC can measure the real-time phase difference accurately and it can be demonstrated that thephase difference between two sinusoidal signals can be evaluated with the relative measurement errors smaller than 1.4% within the phase difference range of 45°.
Abstract: A digital correlation method to measure the phase difference between two sinusoidal signals is presented. In order to meet the requirements of real-time in practice application, a FPGA is employed to implement the measurement algorithm. In the correlation analysis by FPGA, a CORDIC algorithm is adopted to solve inverse trigonometric function, and thereby the phase difference can be conveniently derived. The experimental results show that the arctangent approach based on CORDIC can measure the real-time phase difference accurately. And it can be demonstrated that the phase difference between two sinusoidal signals can be evaluated with the relative measurement errors smaller than 1.4% within the phase difference range of 45°.
TL;DR: In this paper, the accuracy and precision of a trigonometric experiment using entirely mechanical tools is compared to one using electronic tools, such as a smartphone clinometer application and a laser pointer.
Abstract: Smartphones used as tools provide opportunities for the teaching of the concepts of accuracy and precision and the mathematical concept of arctan. The accuracy and precision of a trigonometric experiment using entirely mechanical tools is compared to one using electronic tools, such as a smartphone clinometer application and a laser pointer. This research has demonstrated how two classroom activities based on tool-making can enhance student measurement and application of accuracy and precision considerations through a trigonometric activity investigating arctan.
TL;DR: In this paper, the second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of R(t); using this definition, an ordinary differential equation is constructed.
Abstract: The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal axis as the variable t and the vertical axis as the variable r(t) are created by numerically solving the ordinary differential equation. These graphs show several regular waves with a specific periodicity and waveform depending on the natural number n. In this study, the functions that satisfy the ordinary differential equation are presented as the leaf functions. For n = 1, the leaf function become a trigonometric function. For n = 2, the leaf function becomes a lemniscatic elliptic function. These functions involve an additional theorem. In this paper, based on the additional theorem for n = 1 and n = 2, the double angle and additional theorem for n = 3 are presented.
01 Nov 2017
TL;DR: This paper proposes an accurate and low complex cell histogram generation by bypass the gradient of pixel computation by employing the bin's boundary angle method to determine the two quantized angles.
Abstract: Histogram of Oriented Gradient (HOG) is a popular feature description for the purpose of object detection. However, HOG algorithm requires a high performance system because of its complex operation set. In HOG algorithm, the cell histogram generation is one of the most complex part, it uses inverse tangent, square, square root, floating point multiplication. In this paper, we propose an accurate and low complex cell histogram generation by bypass the gradient of pixel computation. It employs the bin's boundary angle method to determine the two quantized angles. However, instead of choosing an approximate value of tan, the nearest greater and the nearest smaller of each tan value from the ratios between pixel's derivative in y and x direction are used. The magnitudes of two bins are the solutions of a system of two equations, which represents the equality of the gradient of a pixel and its two bins in both vertical and horizontal direction. The proposed method spends only 30 addition and 40 shift operations to caculate two bins of a pixel. Simulation results show that the percentage of error when reconstructing the differences in x and y direction are always less than 2% with 8-bit length of the fractional part. Additionally, manipulating the precision of gradient magnitude is very simple by pre-defined sine and cosine values of quantized angles. The synthesis results of a hardware implementation of the proposed method occupy 3.57 KGEs in 45nm NanGate standard cell library. The hardware module runs at the maximum frequency of 400 MHz, and the throughput is 0.4 pixel/ns for a single module. It is able to support 48 fps with 4K UHD resolution.