Showing papers on "Inverse trigonometric functions published in 2008"
Patent•
15 Apr 2008
TL;DR: In this paper, a light flux elliptical polarization modulator with polarization modulation parameters determined by inverse trigonometric functions such as arctg, arcctg and arcsin (or their combinations) from algebraic relations between squared amplitudes of image temporal scanning signals, that permits with help of spatially-periodic polarization selector to obtain the images of the left and the right views in left and right observation areas.
Abstract: The invention relates to stereoscopic video equipment and can be used to design stereoscopic TV sets and monitors with a possibility to observe stereo images both glasses-free and with use of passive glasses while keeping a possibility to observe monoscopic images. These possibilities are provided in the method and in the device by means of a light flux elliptical polarization modulator with polarization modulation parameters determined by inverse trigonometric functions such as arctg, arcctg, arccos and arcsin (or their combinations) from algebraic relations between squared amplitudes of image temporal scanning signals, that permits with help of spatially-periodic polarization selector to obtain the images of the left and the right views in the left and right observation areas.
28 citations
TL;DR: In this paper, the power series quotient monotone rule is used to prove Shafer-Fink inequalities for the inverse sine to arc hyperbolic sine.
Abstract: In this paper, we extend some Shafer-Fink-type inequalities for the inverse sine to arc hyperbolic sine, and give two simple proofs of these inequalities by using the power series quotient monotone rule.
27 citations
TL;DR: Inverse Trigonometric Functions Arctan and Arccot This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arcta and arCCot.
Abstract: Summary. This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several dierentiation formulas of arctan and arccot.
12 citations
TL;DR: Several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized Borel transforms have been presented in this article, which are obtained through certain limit procedures which involve the multiparameter binomial inequalities, integral inequalities for continuous functions, and weighted norm inequalities for analytic functions.
11 citations
TL;DR: In this paper, an innovative technique is developed for obtaining infinite product representations for some elementary functions, based on the comparison of alternative expressions of Green's functions constructed by two different methods.
10 citations
TL;DR: 3D FCT and IFCT are symmetric and relatively fast, they can be used in any application requiring a real-time symmetric codec, such as video conferencing, online multiparty video games, and three-dimensional graphics rendering.
Abstract: A fast three-dimensional discrete cosine transform algorithm (3D FCT) and a fast 3D inverse cosine transform (3D IFCT) algorithm are presented, suitable for analysis of 3D data points. Many existing algorithms for three-dimensional data points make use of either the 1D cosine transform or both the 2D and 1D cosine transforms. Existing algorithms based on the 1D discrete cosine transform (DCT) apply the separable 1D transform to the data points in the $x$, $y$, and $z$ directions, respectively, while those based on 2D and 1D transforms apply the 2D cosine transform for the $x$-$y$ planes and then the 1D cosine transform in the $z$ direction. The proposed 3D DCT algorithms handle the 3D data points directly and have been shown to be computationally efficient. They involve a 3D decomposition process where a data volume is recursively decomposed in each dimension until unit data cubes are obtained. The algorithms are presented in the form of a signal flow graph which captures the various computations involved. A complexity analysis along with empirical results is included, demonstrating the performance of the proposed direct 3D DCT algorithms. As 3D FCT and IFCT are symmetric and relatively fast, they can be used in any application requiring a real-time symmetric codec, such as video conferencing, online multiparty video games, and three-dimensional graphics rendering.
7 citations
TL;DR: Combining numerical integration method and rapidly convergent iterative methods, a hybrid method of great efficiency is constructed for finding an initial approximation to a real root of nonlinear equation f ( x ) = 0.
7 citations
TL;DR: In this paper, the identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function were established.
Abstract: By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established.
5 citations
TL;DR: In this article, the authors consider the inverse problem of designing an array of superconducting Josephson junctions that has a given maximum static current pattern as a function of the applied magnetic field.
Abstract: We consider the inverse problem of designing an array of superconducting Josephson junctions that has a given maximum static current pattern as a function of the applied magnetic field. Such devices are used for magnetometry and as terahertz oscillators. The model is a 2D semilinear elliptic operator with Neuman boundary conditions so the direct problem is difficult to solve because of the nonlinearity and the multiplicity of solutions. For an array of small junctions in a passive region, the model can be reduced to a 1D linear partial differential equation with Dirac distribution sine nonlinearities. For small junctions and a symmetric device, the maximum current is the absolute value of a cosine Fourier series whose coefficients (respectively wave numbers) are proportional to the areas (respectively the positions) of the junctions. The inverse problem is solved by inverse cosine Fourier transform after choosing the area of the central junction. We present several examples and show that the reconstruction is robust and that its accuracy can be controlled. These new devices could then be tailored to meet specific applications.
5 citations
Book•
14 Feb 2008
TL;DR: In this paper, the authors present a review of the use of trigonometric functions in the context of calculus with questions about the limits of a function and its relation to its derivatives.
Abstract: 1. Functions and Limits 1.1 Functions and Their Graphs 1.2 Combining Functions Shifting and Scaling Graphs 1.3 Rates of Change and Tangents to Curves 1.4 Limit of a Function and Limit Laws 1.5 Precise Definition of a Limit 1.6 One-Sided Limits 1.7 Continuity 1.8 Limits Involving Infinity Questions to Guide Your Review Practice and Additional Exercises 2. Differentiation 2.1 Tangents and Derivatives at a Point 2.2 The Derivative as a Function 2.3 Differentiation Rules 2.4 The Derivative as a Rate of Change 2.5 Derivatives of Trigonometric Functions 2.6 Exponential Functions 2.7 The Chain Rule 2.8 Implicit Differentiation 2.9 Inverse Functions and Their Derivatives 2.10 Logarithmic Functions 2.11 Inverse Trigonometric Functions 2.12 Related Rates 2.13 Linearization and Differentials Questions to Guide Your Review Practice and Additional Exercises 3. Applications of Derivatives 3.1 Extreme Values of Functions 3.2 The Mean Value Theorem 3.3 Monotonic Functions and the First Derivative Test 3.4 Concavity and Curve Sketching 3.5 Parametrizations of Plane Curves 3.6 Applied Optimization 3.7 Indeterminate Forms and L'Hopital's Rule 3.8 Newton's Method 3.9 Hyperbolic Functions Questions to Guide Your Review Practice and Additional Exercises 4. Integration 4.1 Antiderivatives 4.2 Estimating with Finite Sums 4.3 Sigma Notation and Limits of Finite Sums 4.4 The Definite Integral 4.5 The Fundamental Theorem of Calculus 4.6 Indefinite Integrals and the Substitution Rule 4.7 Substitution and Area Between Curves Questions to Guide Your Review Practice and Additional Exercises 5. Techniques of Integration 5.1 Integration by Parts 5.2 Trigonometric Integrals 5.3 Trigonometric Substitutions 5.4 Integration of Rational Functions by Partial Fractions 5.5 Integral Tables and Computer Algebra Systems 5.6 Numerical Integration 5.7 Improper Integrals Questions to Guide Your Review Practice and Additional Exercises 6. Applications of Definite Integrals 6.1 Volumes by Slicing and Rotation About an Axis 6.2 Volumes by Cylindrical Shells 6.3 Lengths of Plane Curves 6.4 Exponential Change and Separable Differential Equations 6.5 Work and Fluid Forces 6.6 Moments and Centers of Mass Questions to Guide Your Review Practice and Additional Exercises 7. Infinite Sequences and Series 7.1 Sequences 7.2 Infinite Series 7.3 The Integral Test 7.4 Comparison Tests 7.5 The Ratio and Root Tests 7.6 Alternating Series, Absolute and Conditional Convergence 7.7 Power Series 7.8 Taylor and Maclaurin Series 7.9 Convergence of Taylor Series 7.10 The Binomial Series Questions to Guide Your Review Practice and Additional Exercises 8. Polar Coordinates and Conics 8.1 Polar Coordinates 8.2 Graphing in Polar Coordinates 8.3 Areas and Lengths in Polar Coordinates 8.4 Conics in Polar Coordinates 8.5 Conics and Parametric Equations The Cycloid Questions to Guide Your Review Practice and Additional Exercises 9. Vectors and the Geometry of Space 9.1 Three-Dimensional Coordinate Systems 9.2 Vectors 9.3 The Dot Product 9.4 The Cross Product 9.5 Lines and Planes in Space 9.6 Cylinders and Quadric Surfaces Questions to Guide Your Review Practice and Additional Exercises 10. Vector-Valued Functions and Motion in Space 10.1 Vector Functions and Their Derivatives 10.2 Integrals of Vector Functions 10.3 Arc Length and the Unit Tangent Vector T 10.4 Curvature and the Unit Normal Vector N 10.5 Torsion and the Unit Binormal Vector B 10.6 Planetary Motion Questions to Guide Your Review Practice and Additional Exercises 11. Partial Derivatives 11.1 Functions of Several Variables 11.2 Limits and Continuity in Higher Dimensions 11.3 Partial Derivatives 11.4 The Chain Rule 11.5 Directional Derivatives and Gradient Vectors 11.6 Tangent Planes and Differentials 11.7 Extreme Values and Saddle Points 11.8 Lagrange Multipliers Questions to Guide Your Review Practice and Additional Exercises 12. Multiple Integrals 12.1 Double and Iterated Integrals over Rectangles 12.2 Double Integrals over General Regions 12.3 Area by Double Integration 12.4 Double Integrals in Polar Form 12.5 Triple Integrals in Rectangular Coordinates 12.6 Moments and Centers of Mass 12.7 Triple Integrals in Cylindrical and Spherical Coordinates 12.8 Substitutions in Multiple Integrals Questions to Guide Your Review Practice and Additional Exercises 13. Integration in Vector Fields 13.1 Line Integrals 13.2 Vector Fields, Work, Circulation, and Flux 13.3 Path Independence, Potential Functions, and Conservative Fields 13.4 Green's Theorem in the Plane 13.5 Surface Area and Surface Integrals 13.6 Parametrized Surfaces 13.7 Stokes' Theorem 13.8 The Divergence Theorem and a Unified Theory Questions to Guide Your Review Practice and Additional Exercises Appendices 1. Real Numbers and the Real Line 2. Mathematical Induction 3. Lines, Circles, and Parabolas 4. Trigonometric Functions 5. Basic Algebra and Geometry Formulas 6. Proofs of Limit Theorems and L'Hopital's Rule 7. Commonly Occurring Limits 8. Theory of the Real Numbers 9. Convergence of Power Series and Taylor's Theorem 10. The Distributive Law for Vector Cross Products 11. The Mixed Derivative Theorem and the Increment Theorem 12. Taylor's Formula for Two Variables
4 citations
01 Jan 2008
Patent•
09 Oct 2008
TL;DR: In this article, the phase detector 5 has a segment specification section 27 for specifying which segment, of eight segments formed by dividing the range from 0 to 2π by π/4 from 0.
Abstract: PROBLEM TO BE SOLVED: To provide a phase detector capable of detecting the phase with a simple configuration. SOLUTION: The phase detector 5 has a segment specification section 27 for specifying which segment, of eight segments formed by dividing the range from 0 to 2π by π/4 from 0, includes the phase θ, based on the positive/negative of a sine wave signal Ssa, the positive/negative of a cosine wave signal Sca, and size relation between the absolute value of the sine wave signal Ssa and the absolute value of the cosine wave signal Sca. The phase detector 5 also has an arc tangent calculation section 37 for adjusting the positive/negative of the sine wave signal Ssa and cosine wave signal Sca so that the combination of the positive/negative of the sine wave signal, the positive/negative of the cosine wave signal, and size relation between the absolute value of the sine wave and the absolute value of the cosine wave become constant, setting one of the sine wave signal Ssa and cosine wave signal Sca as sine wave, setting the other as cosine wave, and calculating the inverse of the tangent based on the sine wave and the cosine wave by a procedure common in eight segments. COPYRIGHT: (C)2009,JPO&INPIT
01 Jan 2008
TL;DR: Using interval multivalued inverse functions as both a starting and rallying point, an overview of these constraint programming algorithms is presented, and parallels are drawn with classical numerical algorithms (most notably Gauss-Seidel).
Abstract: Given the relation y = cos x, where x lies in the interval [10,14], interval arithmetic will readily allow us to compute the possible values for y by considering the monotonic subdomains of the cosine function over [10, 14]: y \in [cos 10, 1] \approx [-0.84, 1]. On the other hand, what is the possible domain for an unknown x if the domain for y is [-0.3, 0.2]? Most interval arithmetic libraries will fix it at [acos 0.2, acos -0.3] \approx [1.36, 1.87] because they consider branch cuts of the multivalued inverse cosine to return principal values in the domain [0, \pi] only. Now, what if we know that x lies in the domain [20, 26]? The aforementioned inverse cosine interval function would not be of much help here, while considering a multivalued inverse cosine would permit restricting the domain of x to [6\pi + acos 0.2, 8\pi – acos 0.2] \approx [20.22, 23.77]. Such a use of relations between variables together with domains of possible values to infer tighter consistent domains is the core principle of Constraint Programming. Since Cleary's seminal work on relational arithmetic for Logic Programming languages, interval multivalued inverse functions have been repeatedly used in algorithms of increasing sophistication to solve systems of (in-)equations over real-valued variables. Using these functions as both a starting and rallying point, we present an overview of these constraint programming algorithms, and we draw parallels with classical numerical algorithms (most notably Gauss-Seidel). The implementation of interval multivalued inverse functions in the gaol C++ library is also discussed.
TL;DR: In this paper, a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series are discussed, and some specific cases are presented.
Abstract: We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.
Patent•
26 Mar 2008TL;DR: In this article, the authors proposed a phase detection device which can detect a phase by using a simple configuration, where the phase is detected by adjusting the signs of the sine wave and the cosine wave signal SSA and Sca, respectively.
Abstract: Provided is a phase detection device which can detect a phase by using a simple configuration. the phase detection device 5 has the section specifying part 27 specifying which of the eight sections obtained by dividing a range from 0 to 2π for each π/4 from 0 contains the phase θ based on the sign of the sine wave signal Ssa, the sign of the cosine wave signal Sca, and the magnitude relationship between the absolute value of the sine wave signal Ssa and the absolute value of the cosine wave signal Sca, and the arctangent calculation part 37 adjusting the signs of the sine wave signal Ssa and the cosine wave signal Sca and making one of the sine wave signal Ssa and the cosine wave signal Sca a sine wave and making the other a cosine wave so that the sign of the sine wave, the sign of the cosine wave, and the magnitude relationship between the absolute value of the sine wave and the absolute value of the cosine wave coincide regardless of which of the eight sections contains the phase θ, thereby calculating the arctangent of the tangent based on the sine wave and the cosine wave according to a procedure common to the eight parts.
25 Mar 2008
TL;DR: In this article, the authors consider a system of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand generalized functions and find locally integrable solutions.
Abstract: 【We consider a system of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand generalized functions. As a consequence we find locally integrable solutions of the n-dimensional trigonometric functional equation.】
Journal Article•
TL;DR: In this article, the authors introduced another way to fit-subsectional inverse tangent function fitting, and simulated the two ways of fitting by Matlab, the result reveals that polynomial fitting could not be employed and sectional inverse tangents function fitting is effective when available data is finite.
Abstract: Usually permalloy is adopted to be used for core in fluxgate sensors.To model a fluxgate sensor,it is necessary to fit the magnetic hysteresis loops of permalloy in the sensor.Traditionally,polynomial fitting is employed.But this way needs a lot of available data.This paper introduces another way to fit-subsectional inverse tangent function fitting.Simulated the two ways of fitting by Matlab,the result reveals that polynomial fitting could not be employed and sectional inverse tangent function fitting is effective when available data is finite.
01 Jan 2008
Posted Content•
TL;DR: In this paper, generalized definitions of exponential, trigonometric sine and cosine and hyperbolic Sine and Cosine functions are given, in the lowest order, these functions correspond to ordinary exponential sine etc. Some properties of the generalised functions are discussed.
Abstract: Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered.
TL;DR: In this article, a series of trigonometric functions is presented.Click on the link to view the abstract. But the authors do not discuss the relation between these functions and the series of functions.
Abstract: Click on the link to view the abstract. Keywords: Series of trigonometric functions Quaestiones Mathematicae 31(2008), 375–378
TL;DR: In this article, the graph method is used to obtain estimates for the derivatives of any order of inverse functions in terms of those of the original functions, and explicit asymptotics of the estimates obtained as the order of the derivative tends to infinity.
Abstract: We use the `graph' method to obtain estimates for the derivatives of any order of inverse functions in terms of those of the original functions. We construct explicit asymptotics of the estimates obtained as the order of the derivative tends to infinity. For analytic functions and functions in Gevrey's class, we obtain explicit estimates for all derivatives of the inverse functions.
Journal Article•
TL;DR: In this paper, the authors studied the stability problem for the pexider type trigonometric functional equation f(x+y)-f(x-y) = 2g(x}h(y), which is related to the d'Alembert, the Wilson, the sine, and the mixed trigonometrical functional equations.
Abstract: The aim of this paper is to study the stability problem for the pexider type trigonometric functional equation f(x+y)-f(x-y)=2g(x}h(y), which is related to the d'Alembert, the Wilson, the sine, and the mixed trigonometric functional equations.
TL;DR: Several Differentiation Formulas of Special Functions are proved involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions.
Abstract: Summary. In this article, we prove a series of dierentiation identities [2] involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions.