Showing papers on "Inverse trigonometric functions published in 2012"

Bicomplex Numbers and their Elementary Functions

Abstract: In this paper we introduce the algebra of bicomplex numbers as a generalization of the field of complex numbers. We describe how to define elementary functions in such an algebra (polynomials, exponential functions, and trigonometric functions) as well as their inverse functions (roots, logarithms, inverse trigonometric functions). Our goal is to show that a function theory on bicomplex numbers is, in some sense, a better generalization of the theory of holomorphic functions of one variable, than the classical theory of holomorphic functions in two complex variables.

Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions

Abstract: We establish sharp Wilker and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions.

Curve construction based on five trigonometric blending functions

Abstract: Five new trigonometric blending functions with two exponential shape parameters are given in this paper. Based on these blending functions, trigonometric Bezier curves analogous to the quartic Bezier curves, with two exponential shape parameters, are presented. The ellipses and parabolas can be represented exactly by using the trigonometric Bezier curves. Based on the blending functions, trigonometric B-spline curves with three local shape parameters and a global shape parameter are also constructed. The obtained spline curves can be C2∩FC2k+3 (k∈ℤ+) continuous by fixing some values of the shape parameters. Without solving a linear system, the spline curves can be also used to interpolate sets of points with C2 continuity partly or entirely.

Wilker- and Huygens-type inequalities and solution to Oppenheim's problem

Abstract: We establish Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions. We also provide a laconic proof to Oppenheim’s problem associated with inequalities involving the sine and cosine functions.

Electronic Scheme for Computing Inverse-Cosine and its Application to a GMR Based Angle Sensor

Abstract: This paper presents a new, simple but effective, electronic method to obtain inverse-cosine of an electrical variable. The proposed method is very useful for linearization of sensors whose output is a cosine or sine function of the physical quantity being sensed. The inverse-cosine of the variable is obtained by comparing it with a reference sinusoidal wave. The proposed method is easy to implement using electronic components. Using the new technique, a giant magneto-resistance-based angle sensor that provides an output linearly proportional to the angle being sensed has been developed, and the details are reported in this paper. A prototype of the angle sensor has been built, and the practicality of the new method has been tested successfully. The developed sensor provides a linear output for a range of 0° to 180°. The worst case error was found to be less than 0.35° for a range of 10°-170°.

Transcendental Representations with Applications to Solids and Fluids

Abstract: Sequences of Fractions or Products Power Series, Singularities, and Functions Series of Fractions for Meromorphic Functions (Mittag-Leffler 1876, 1884) Meromorphic Function as a Ratio of Two Integral Functions Factorization with Infinite Number of Zeros Infinite Products for Circular Functions Recurrence Formulas and Continued Fractions (Wallis 1656 Euler 1737) Optimal and Doubly Bounded Sharpening Approximations Transformation of Series and Products into Fractions (Euler 1785) Continued Fraction for the Ratio of Two Series (Lambert 1770) Conclusion Compressible and Rotational Flows Source, Sink, and Vortex in a Compressible Flow Potential Vortex with Rotational Core (Rankine Hallock and Burnham 1997) Minimum Energy (Thomson 1849) and Intrinsic Equations of Motion Laplace/Poisson Equations in Complex Conjugate Coordinates Second Forces/Moment and Circle Theorems Cylinder in a Unidirectional Shear Flow Monopole Interactions and Equilibrium Positions Cylinder in a Stream with Two Trailing Monopoles (Foppl 1913) Reciprocity Theorem (Green 1828) and Path Function (Routh 1881) Conclusion Exponential and Logarithmic Functions Derivation Property, Series, and Rational Limit Continued Fractions and Computation of the Number e Transformation of Sums to Products and Powers Limits, Period, and Absence of Zeros Logarithm as the Function Inverse to Exponential Series Expansions and Continued Fractions Exponential and Logarithm with Complex Base Tables of Natural (Napier 1614) and Decimal (Briggs 1624) Logarithms Gaussian and Related Hypergeometric Functions Conclusion Plane Elasticity and Multiharmonic Functions Displacement Vector and Deformation and Strain Tensors Stress Vector, Tensor, and Function Elastic Energy and Moduli of a Material (Hooke 1678 Poisson 1829a Lame 1852) Momentum Equation for Isotropic Elasticity Cavities and Static and Rotating Cylinders Multiharmonic Equation and Fluid Loading on a Dam Forces and Moments on a Wedge Elastic Potential and Stresses in an Infinite Medium Driven Loaded Wheel with Traction or Braking Conclusion Circular and Hyperbolic Functions Sine/Cosine Representations on the Ellipse/Hyperbola Secant, Cosecant, Tangent, and Cotangent Formulas of Addition of Several Variables Formulas for Multiple, Double, and Half Variable Powers, Products, and Sums of the Functions Chebychev (1859) Polynomials of Two Kinds Orthogonal and Normalized Trigonometric Functions Relations between Complex and Real Functions Periods, Symmetries, Values, and Limits Conclusion Membranes, Capillarity, and Torsion Linear and Nonlinear Deflection of a Membrane Large Deflection of a Membrane by Weight or Pressure Boundary Condition with Surface Tension Wetting Angle and Capillary Rise Warping, Stress, and Displacement Functions Torsional Stiffness of a Multiply Connected Section Hollow Elliptical or Thin or Cut Cross Sections Torsion of Prisms with Triangular Cross Section (Saint-Venant 1885 Campos and Cunha 2010) Trajectories of Fluid Particles in a Rotating Vessel Conclusion Infinite and Cyclometric Representations Power Series and Euler (1755)/Bernoulli (1713) Numbers Branch Points and Branch Cuts for Cyclometric Functions Derivatives/Primitives of Direct/Inverse Functions Power Series for Cyclometric Functions Slopes at Zeros and Residues at Singularities Series of Fractions for Meromorphic Functions Relation with Factorization in Infinite Products Continued Fractions for Direct/Inverse Functions Gregory, Leibnitz, Brouncker, and Wallis Quadratures Summary Confined and Unsteady Flows Cylinder Moving in Large Cavity Two Cylinders in Relative Motion Eccentric, Biconcave/Biconvex, and Semi-Recessed Cylinders Ramp/Step in a Wall and Thick Pointed/Blunt Plate Channel with Contraction/Expansion and a Thick Plate Flat Plate with Partially Separated Flow Airfoil with Plain/Slotted Flap/Slat Flow Past Tandem Airfoils and Cascades Instability of a Plane Vortex Sheet (Helmholtz 1868 Kelvin 1871) Conclusion Infinite Processes and Summability Bounds for Integrals of Monotonic Functions Genus of an Infinite Canonical Product (Weierstrass 1876) Convergent and Periodic Continued Fractions Derangement of Conditionally Convergent Series Summation of Series of Rational Functions Extension of Convergence to Summability (Euler 1755 Cesaro 1890) Cardinals of Enumerable, Continuum, and Discontinuous (Cantor 1874, 1878) Transfinite Cardinal and Ordinal Numbers (Cantor 1883a,b Hardy 1903) Three Antinomies and the Axiom of Selection (Burali-Forti 1897 Russell 1903 Zermelo 1908) Conclusion Twenty Examples Examples 10.1 through 10.20 Conclusion Bibliography Index

University Calculus, Early Transcendentals

Abstract: 1. Functions 1.1 Functions and Their Graphs 1.2 Combining Functions Shifting and Scaling Graphs 1.3 Trigonometric Functions 1.4 Graphing with Calculators and Computers 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms 2. Limits and Continuity 2.1 Rates of Change and Tangents to Curves 2.2 Limit of a Function and Limit Laws 2.3 The Precise Definition of a Limit 2.4 One-Sided Limits 2.5 Continuity 2.6 Limits Involving Infinity Asymptotes of Graphs 3. Differentiation 3.1 Tangents and the Derivative at a Point 3.2 The Derivative as a Function 3.3 Differentiation Rules 3.4 The Derivative as a Rate of Change 3.5 Derivatives of Trigonometric Functions 3.6 The Chain Rule 3.7 Implicit Differentiation 3.8 Derivatives of Inverse Functions and Logarithms 3.9 Inverse Trigonometric Functions 3.10 Related Rates 3.11 Linearization and Differentials 4. Applications of Derivatives 4.1 Extreme Values of Functions 4.2 The Mean Value Theorem 4.3 Monotonic Functions and the First Derivative Test 4.4 Concavity and Curve Sketching 4.5 Indeterminate Forms and L'Hi?½pital's Rule 4.6 Applied Optimization 4.7 Newton's Method 4.8 Antiderivatives 5. Integration 5.1 Area and Estimating with Finite Sums 5.2 Sigma Notation and Limits of Finite Sums 5.3 The Definite Integral 5.4 The Fundamental Theorem of Calculus 5.5 Indefinite Integrals and the Substitution Rule 5.6 Substitution and Area Between Curves 6. Applications of Definite Integrals 6.1 Volumes Using Cross-Sections 6.2 Volumes Using Cylindrical Shells 6.3 Arc Length 6.4 Areas of Surfaces of Revolution 6.5 Work 6.6 Moments and Centers of Mass 7. Integrals and Transcendental Functions 7.1 The Logarithm Defined as an Integral 7.2 Exponential Change and Separable Differential Equations 7.3 Hyperbolic Functions 8. Techniques of Integration 8.1 Integration by Parts 8.2 Trigonometric Integrals 8.3 Trigonometric Substitutions 8.4 Integration of Rational Functions by Partial Fractions 8.5 Integral Tables and Computer Algebra Systems 8.6 Numerical Integration 8.7 Improper Integrals 9. Infinite Sequences and Series 9.1 Sequences 9.2 Infinite Series 9.3 The Integral Test 9.4 Comparison Tests 9.5 The Ratio and Root Tests 9.6 Alternating Series, Absolute and Conditional Convergence 9.7 Power Series 9.8 Taylor and Maclaurin Series 9.9 Convergence of Taylor Series 9.10 The Binomial Series and Applications of Taylor Series 10. Parametric Equations and Polar Coordinates 10.1 Parametrizations of Plane Curves 10.2 Calculus with Parametric Curves 10.3 Polar Coordinates 10.4 Graphing in Polar Coordinates 10.5 Areas and Lengths in Polar Coordinates 10.6 Conics in Polar Coordinates 11. Vectors and the Geometry of Space 11.1 Three-Dimensional Coordinate Systems 11.2 Vectors 11.3 The Dot Product 11.4 The Cross Product 11.5 Lines and Planes in Space 11.6 Cylinders and Quadric Surfaces 12. Vector-Valued Functions and Motion in Space 12.1 Curves in Space and Their Tangents 12.2 Integrals of Vector Functions Projectile Motion 12.3 Arc Length in Space 12.4 Curvature and Normal Vectors of a Curve 12.5 Tangential and Normal Components of Acceleration 12.6 Velocity and Acceleration in Polar Coordinates 13. Partial Derivatives 13.1 Functions of Several Variables 13.2 Limits and Continuity in Higher Dimensions 13.3 Partial Derivatives 13.4 The Chain Rule 13.5 Directional Derivatives and Gradient Vectors 13.6 Tangent Planes and Differentials 13.7 Extreme Values and Saddle Points 13.8 Lagrange Multipliers 14. Multiple Integrals 14.1 Double and Iterated Integrals over Rectangles 14.2 Double Integrals over General Regions 14.3 Area by Double Integration 14.4 Double Integrals in Polar Form 14.5 Triple Integrals in Rectangular Coordinates 14.6 Moments and Centers of Mass 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 14.8 Substitutions in Multiple Integrals 15. Integration in Vector Fields 15.1 Line Integrals 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 15.3 Path Independence, Conservative Fields, and Potential Functions 15.4 Green's Theorem in the Plane 15.5 Surfaces and Area 15.6 Surface Integrals 15.7 Stokes' Theorem 15.8 The Divergence Theorem and a Unified Theory 16. First-Order Differential Equations (Online) 16.1 Solutions, Slope Fields, and Euler's Method 16.2 First-Order Linear Equations 16.3 Applications 16.4 Graphical Solutions of Autonomous Equations 16.5 Systems of Equations and Phase Planes 17. Second-Order Differential Equations (Online) 17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 Power Series Solutions Appendices 1. Real Numbers and the Real Line 2. Mathematical Induction 3. Lines, Circles, and Parabolas 4. Conic Sections 5. Proofs of Limit Theorems 6. Commonly Occurring Limits 7. Theory of the Real Numbers 8. Complex Numbers 9. The Distributive Law for Vector Cross Products 10. The Mixed Derivative Theorem and the Increment Theorem 11. Taylor's Formula for Two Variables

Phase extraction method for phase-shifting interferometric fringe

Abstract: The invention relates to a phase extraction method for a phase-shifting interferometric fringe. Phase measurement error is in sine periodic distribution, while each phase of a trigonometric function has opposite positive and negative amplitudes every pi, namely the error distribution is in revert phase. Therefore, a basic algorithm can be expanded, numerators and denominators in arc tangent formulae in multiple measurement groups are subjected to overlapping average respectively, and errors are reduced by utilizing mutual cancellation of error phase distribution. On the basis of the idea, a plurality of average compensation algorithms of different situations can be deduced. The algorithm is particularly insensitive to phase displacement errors, and the phase measurement accuracy can be remarkably improved.

Transient-Time fractional-space trigonometry and application

Abstract: In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time of a non-integer-order system have significantly different values from their steady-state values. Identities such as sin2(t)+cos2(t)=1 are shown to be invalid in the transient-time of a fractional-order system. Some generalized hyperbolic functions and identities are also given in this work. Application to the evaluation of the step-response of a non-integer-order system is given.

Inequalities for the generalized trigonometric and hyperbolic functions

Abstract: The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovic-Adamovic inequality, Lazarevic's inequality, Huygens-type inequalities, Wilker-type inequalities, and Cuza-Huygens-type inequalities, are generalized to the case of generalized functions.

Calculus for Scientists and Engineers

Abstract: 1. Functions 1.1 Review of functions 1.2 Representing functions 1.3 Trigonometric functions and their inverses Review 2. Limits 2.1 The idea of limits 2.2 Definitions of limits 2.3 Techniques for computing limits 2.4 Infinite limits 2.5 Limits at infinity 2.6 Continuity 2.7 Precise definitions of limits Review 3. Derivatives 3.1 Introducing the derivative 3.2 Rules of differentiation 3.3 The product and quotient rules 3.4 Derivatives of trigonometric functions 3.5 Derivatives as rates of change 3.6 The Chain Rule 3.7 Implicit differentiation 3.8 Derivatives of inverse trigonometric functions 3.9 Related rates Review 4. Applications of the Derivative 4.1 Maxima and minima 4.2 What derivatives tell us 4.3 Graphing functions 4.4 Optimization problems 4.5 Linear approximation and differentials 4.6 Mean Value Theorem 4.7 L'Hopital's Rule 4.8 Newton's method 4.9 Antiderivatives Review 5. Integration 5.1 Approximating areas under curves 5.2 Definite integrals 5.3 Fundamental Theorem of Calculus 5.4 Working with integrals 5.5 Substitution rule Review 6. Applications of Integration 6.1 Velocity and net change 6.2 Regions between curves 6.3 Volume by slicing 6.4 Volume by shells 6.5 Length of curves 6.6 Surface area 6.7 Physical applications 6.8 Hyperbolic functions Review 7. Logarithmic and Exponential Functions 7.1 Inverse functions 7.2 The natural logarithm and exponential functions 7.3 Logarithmic and exponential functions with general bases 7.4 Exponential models 7.5 Inverse trigonometric functions 7.6 L'Hopital's rule and growth rates of functions Review 8. Integration Techniques 8.1 Basic approaches 8.2 Integration by parts 8.3 Trigonometric integrals 8.4 Trigonometric substitutions 8.5 Partial fractions 8.6 Other integration strategies 8.7 Numerical integration 8.8 Improper integrals Review 9. Differential Equations 9.1 Basic ideas 9.2 Direction fields and Euler's method 9.3 Separable differential equations 9.4 Special first-order differential equations 9.5 Modeling with differential equations Review 10. Sequences and Infinite Series 10.1 An overview 10.2 Sequences 10.3 Infinite series 10.4 The Divergence and Integral Tests 10.5 The Ratio, Root, and Comparison Tests 10.6 Alternating series Review 11. Power Series 11.1 Approximating functions with polynomials 11.2 Properties of power series 11.3 Taylor series 11.4 Working with Taylor series Review 12. Parametric and Polar Curves 12.1 Parametric equations 12.2 Polar coordinates 12.3 Calculus in polar coordinates 12.4 Conic sections Review 13. Vectors and Vector-Valued Functions 13.1 Vectors in the plane 13.2 Vectors in three dimensions 13.3 Dot products 13.4 Cross products 13.5 Lines and curves in space 13.6 Calculus of vector-valued functions 13.7 Motion in space 13.8 Length of curves 13.9 Curvature and normal vectors Review 14. Functions of Several Variables 14.1 Planes and surfaces 14.2 Graphs and level curves 14.3 Limits and continuity 14.4 Partial derivatives 14.5 The Chain Rule 14.6 Directional derivatives and the gradient 14.7 Tangent planes and linear approximation 14.8 Maximum/minimum problems 14.9 Lagrange multipliers Review 15. Multiple Integration 15.1 Double integrals over rectangular regions 15.2 Double integrals over general regions 15.3 Double integrals in polar coordinates 15.4 Triple integrals 15.5 Triple integrals in cylindrical and spherical coordinates 15.6 Integrals for mass calculations 15.7 Change of variables in multiple integrals Review 16. Vector Calculus 16.1 Vector fields 16.2 Line integrals 16.3 Conservative vector fields 16.4 Green's theorem 16.5. Divergence and curl 16.6 Surface integrals 16.6 Stokes' theorem 16.8 Divergence theorem Review

A DSP-based resolver-to-digital conversion using pulse excitation

Abstract: Resolvers are absolute angle measurement and are mounted on the motor shaft to get the motor's absolute angular position. This study proposes a digital signal processor (DSP) based resolver-to-digital (R/D) conversion using pulse excitation. A pulse signal generated by a DSP is utilized as a reference signal coupled into the resolver's rotor winding and provides primary excitation. The output signals of the resolver are two orthogonal stator coils modulated with the sine and cosine of the motor shaft angle. The sine and cosine modulated output signals are sampled by the analog-to-digital (A/D) converter of the DSP. The sampled frequency of output signals are the same frequency as the pulse signal. The angular position is derived by the inverse tangent of the quotient of the demodulated sine and cosine samples. Test results are presented to validate the performances of the proposed scheme.

Compute mode of multifunctional computer

Abstract: The invention provides a compute mode of a multifunctional computer, wherein operations are performed according to the internal functions of the computer; some mathematical symbols are automatically recognized in the operation process; an expression is automatically computed according to an arithmetical operation rule; some trigonometric functions in mathematics such as sine function (sin), cosine function (cos), tangent (tg), cotangent (ctg), exponential function (exp), square root function (sqr), square root function (sqrt), logarithmic function (log), arc-sin function (asin), arc-cosine function (acos), arc tangent (atg), arc cotangent (actg) and the like are supported, and simultaneously operators +, -, *, / and ^ (power) also are supported; an absolute value is in the form of |expresion|; and a decimal point is rounded off by means of a round function; and a function for rounding behind the decimal point is Trunc. The mode is capable of simultaneously performing hybrid operations of mathematical formulas, and capable of obtaining a correct value by inputting the expression expresion in an input box.

Some Nonlinear Model for Solving the Nonlinear Partial Differential Equations in Mathematical Physics

Abstract: The solitary waves are derived from the traveling waves. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. MSC 2000: 35Dxx

Hardware performance analysis of a parametric CORDIC IP

Abstract: In this paper, a parametric CORDIC IP is presented, implementing fixed point arithmetic (sine, cosine, and arctangent) trigonometric functions evaluation. The VHDL design uses parameters to define the bitwidth and the precision of the input and output signals. Important design options for hardware implementation include iterative, unrolled and unrolled pipelined architectures of the CORDIC module. The design uses the VHDL '93 backwards-compatible version of the fixed point package, as defined according to the VHDL 2008 standard. Hardware performance analysis results are presented in this paper for more than 75 circuit variations implemented on a Virtex-5 Xilinx FPGA, each for different parameter values of the proposed CORDIC module. A maximum 47% increase of speed, and 57% area reduction are accomplished, in comparison with other designs.

Applied Calculus: For Business, Economics, and the Social and Life Sciences

Abstract: Chapter 1: Functions, Graphs, and Limits 1.1 Functions 1.2 The Graph of a Function 1.3 Lines and Linear Functions 1.4 Functional Models 1.5 Limits 1.6 One-Sided Limits and Continuity Chapter 2: Differentiation: Basic Concepts 2.1 The Derivative 2.2 Techniques of Differentiation 2.3 Product and Quotient Rules Higher-Order Derivatives 2.4 The Chain Rule 2.5 Marginal Analysis and Approximations Using Increments 2.6 Implicit Differentiation and Related Rates Chapter 3: Additional Applications of the Derivative 3.1 Increasing and Decreasing Functions Relative Extrema 3.2 Concavity and Points of Inflection 3.3 Curve Sketching 3.4 Optimization Elasticity of Demand 3.5 Additional Applied Optimization Chapter 4: Exponential and Logarithmic Functions 4.1 Exponential Functions Continuous Compounding 4.2 Logarithmic Functions 4.3 Differentiation of Exponential and Logarithmic Functions 4.4 Additional Applications Exponential Models Chapter 5: Integration 5.1 Indefinite Integration and Differential Equations 5.2 Integration by Substitution 5.3 The Definite Integral and the Fundamental Theorem of Calculus 5.4 Applying Definite Integration: Distribution of Wealth and Average Value 5.5 Additional Applications to Business and Economics 5.6 Additional Applications to the Life and Social Sciences Chapter 6: Additional Topics in Integration 6.1 Integration by Parts Integral Tables 6.2 Numerical Integration 6.3 Improper Integrals Chapter 7: Calculus of Several Variables 7.1 Functions of Several Variables 7.2 Partial Derivatives 7.3 Optimizing Functions of Two Variables 7.4 The Method of Least-Squares 7.5 Constrained Optimization: The Method of Lagrange Multipliers 7.6 Double Integrals Chapter 8: Trigonometric Functions 8.1 Angle Measurement Trigonometric Functions 8.2 Trigonometric Applications Involving Differentiation 8.3 Trigonometric Applications Involving Integration Chapter 9: Differential Equations 9.1 Modeling with Differential Equations 9.2 First-Order Linear Differential Equations 9.3 Additional Applications of Differential Equations 9.4 Approximate Solutions of Differential Equations 9.5 Difference Equations The Cobweb Model Chapter 10: Infinite Series and Taylor Series Approximations 10.1 Infinite Series Geometric Series 10.2 Tests for Convergence 10.3 Functions as Power Series Taylor Series Chapter 11: Probability and Calculus 11.1 Introduction to Probability Discrete Random Variables 11.2 Continuous Probability Distributions 11.3 Expected Value and Variance of Continuous Random Variables 10.4 Normal and Poisson Probability Distributions Appendix A: Algebra Review A.1 A Brief Review of Algebra A.2 Factoring Polynomials and Solving Systems of Equations A.3 Evaluating Limits with L'Hopital's Rule A.4 The Summation Notation

Probabilistic Nodes Combination: Modeling and Interpolation of 2D Curve

Abstract: Mathematics and computer science are interested in methods of 2D curve interpolation and extrapolation using the set of key points knots or nodes. Proposed method, called by author Probabilistic Nodes Combination PNC, is such a method. This novel PNC method is introduced in the case of Hurwitz-Radon Matrices MHR. MHR method is based on the family of Hurwitz-Radon HR matrices which possess columns composed of orthogonal vectors. Two-dimensional curve is modeled and interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions. It is shown how to build the orthogonal matrix operator and how to use it in a process of curve reconstruction.

Hardware implementation of coordinate rotation digital computer in field programmable gate array

Abstract: Trigonometry is of great importance in mathematics as well as in physics, engineering, and chemistry. Astronomy, geography, navigation, study of optics and acoustics, oceanography, architecture, calculus, etc. are just several examples where trigonometry is significantly practiced. Historical figures like Pythagoras and Columbus used trigonometric tables in their careers. The birth of software has empowered relatively faster trigonometric functions performed by processors. In real-time applications though, such as trajectory calculations in military or space exploration, or in biomedical authentication system for fast access or rejection decision, trigonometric computation by software is a considerably time-consuming process. Coordinate Rotation Digital Computer (CORDIC) is an algorithm developed for hardware implementation as a real-time solution to trigonometric computation. This report presents a design approach to realize the CORDIC algorithm, prototyped as an embedded system in an Altera Field Programmable Gate Array (FPGA) development board running at 100 MHz clock frequency. The design flow applies the systematic Register Transfer Level (RTL) methodology, partitioning the design into a Datapath Unit (DU) for computation tasks, and a Control Unit (CU) for controlling the operation flow. Experimental results show that a high accuracy was obtained, with mean computation errors between 0.0014% and 0.0023% with respect to a software implementation on the same platform. The speed up in the execution time is about 89 times for the computation of cosine and sine functions, and 69 times for the arctangent. The work demonstrates the power of the CORDIC algorithm, and presents a methodology for an efficient complex hardware design.

Mathematics Solutions for Inverse Tangents

Abstract: Sinusoidal circuit analysis is very co mmon in Alternating Current (AC) electrical circu it analysis and so close to the term of phasor concept using complex frequency, which is mainly use trigonometry figures such as sin, cos, and tangent, for calculations. One of the most important tasks that we will meet e.g. finding phase angles using inverse tangent formu la. The author presents this paper to introduce a mathematic solution derived fro m basic concept of trigonometry, to solve circuit analysis problems related to phase angles and inverse tangents.

The Deformed Trigonometric Functions of two Variables

Abstract: Recently, various generalizations and deformations of the elementary functions were introduced. Since a lot of natural phenomena have both discrete and continual aspects, deformations which are able to express both of them are of particular interest. In this paper, we consider the trigonometry induced by one parameter deformation of the exponential function of two variables eh(x, y) = (1 + hx) y/h (h ∈ R \ {0}, x ∈ C \ {−1/h}, y ∈ R). In this manner, we define deformed sine and cosine functions and analyze their various properties. We give series expansions of these functions, formulas which have their similar counterparts in the classical trigonometry, and interesting difference and differential properties.

integrals connecting with sums related to inverse tangent function

Abstract: Some denite integrals are evaluated through sums related to inverse tangent function that found in chapter 2 of Ramanujan notebooks, Part-I. In addition, integrals involving complicated arguments of inverse tangent function are evaluated through innite products.

Simulation study of asynchronous motor VC system based on the simplified SVPWM algorithm

Abstract: Simplified SVPWM algorithm is an algorithm, which chooses the space voltage vector according to the one-to-one relationships between the motor line voltage and the sector of space voltage vector. This algorithm can overcome many disadvantages in the traditional algorithm, for example the complex operation of coordinates rotation, trigonometric function, inverse trigonometric function etc, and make it easy to realize digital control. In the paper, the simulation model of vector Control system is built by using the simplified SVPWM algorithm and is simulated. Simulation results show that the asynchronous motors VC system based on the simplified SVPWM algorithm has better static and dynamic performance, and verify that the algorithm is feasible.

Integrability of trigonometric series with generalized semi-convex coefficients

Abstract: In this paper we deal with cosine and sine trigonometric series with generalized semi-convex coefficients. Integrability conditions for them are obtained.