Solving Trigonometric Equations with Identities

Solving Trigonometric Equations with Identities tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về...

[...]... for the Reader (ii) Engineers, quantitative analysts and others with a more technical background in mathematical and quantitative methods who are interested in applying stochastic differential equations with jumps, and in implementing efficient simulation methods or developing new schemes could use the book according to the following suggested flowchart Without too much emphasis on proofs the selected material... larger class of SDEs driven by fairly general semimartingales The class of SDEs driven by Wiener processes and Poisson jump measures with finite intensity appears to be large enough for realistic modeling of the dynamics of quantities in finance Here continuous trading noise and a few single events model the typical sources of uncertainty Furthermore, stochastic jump sizes and stochastic intensities,... weighted index ODE ordinary differential equation SDE stochastic differential equation PDE partial differential equation PIDE partial integro differential equation Iν (·) modified Bessel function of the first kind with index ν modified Bessel function of the third kind with index λ time step size of a time discretization Kλ (·) Δ i l = i! l!(i−l)! combinatorial coefficient C k (Rd , R) set of k times continuously... determinant of a matrix A A−1 inverse of a matrix A (x, y) inner product of vectors x and y N = {1, 2, } set of natural numbers 1 2 d d with ith component xi XX Basic Notation ∞ in nity (a, b) open interval a < x < b in [a, b] closed interval a ≤ x ≤ b in = (−∞, ∞) + = [0, ∞) d set of real numbers set of nonnegative real numbers d-dimensional Euclidean space Ω sample space ∅ empty set A∪B the union of. .. uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic Processes Stochastic Process If not otherwise stated, throughout the book we shall assume that there exists a common underlying probability space (Ω, A, P ) consisting of the sample space Ω, the sigma-algebra or collection of events... Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability 64, DOI 10.1007/978-3-642-13694-8 1, © Springer-Verlag Berlin Heidelberg 2010 1 2 1 SDEs with Jumps We set the time set to the interval T = [0, ∞) if not otherwise stated On some occasions the time set may become the bounded interval [0, T ] for T ∈ (0, ∞) or a set of discrete time points {t0 , t1 , t2... Solving Trigonometric Equations with Identities Solving Trigonometric Equations with Identities By: OpenStaxCollege International passports and travel documents In espionage movies, we see international spies with multiple passports, each claiming a different identity However, we know that each of those passports represents the same person The trigonometric identities act in a similar manner to multiple passports—there are many ways to represent the same trigonometric expression Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions 1/20 Solving Trigonometric Equations with Identities Verifying the Fundamental Trigonometric Identities Identities enable us to simplify complicated expressions They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations In fact, we use algebraic techniques constantly to simplify trigonometric expressions Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle Consequently, any trigonometric identity can be written in many ways To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result In this first section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities We will begin with the Pythagorean identities (see [link]), which are equations involving trigonometric functions based on the properties of a right triangle We have already seen and used the first of these identifies, but now we will also use additional identities Pythagorean Identities sin2θ + cos2θ = 1 + cot2θ = csc2θ + tan2θ = sec2θ The second and third identities can be obtained by manipulating the first The identity + cot2θ = csc2θ is found by rewriting the left side of the equation in terms of sine and cosine Prove: + cot2θ = csc2θ 2/20 Solving Trigonometric Equations with Identities ( ) ( )( ) + cot2θ = + cos2θ sin2θ Rewrite the left side sin2θ cos2θ = + sin2θ sin2θ Write both terms with the common denominator sin2θ + cos2θ = sin2θ = sin2θ = csc2θ Similarly, + tan2θ = sec2θ can be obtained by rewriting the left side of this identity in terms of sine and cosine This gives ( ) cos θ sin θ =( +( ) cos θ cos θ ) + tan2θ = + sin θ cos θ = cos2 θ + sin2 θ cos2 θ = cos2 θ Rewrite left side Write both terms with the common denominator = sec2 θ The next set of fundamental identities is the set of even-odd identities The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even (See [link]) Even-Odd Identities tan( − θ) = − tan θ sin( − θ) = − sin θ cos( − θ) = cos θ cot( − θ) = − cot θ csc( − θ) = − csc θ sec( − θ) = sec θ Recall that an odd function is one in which f (− x) = − f(x) for all x in the domain of f The sine function is an odd function because sin( − θ) = − sin θ The graph of an odd 3/20 Solving Trigonometric Equations with Identities function is symmetric about the origin For example, consider corresponding inputs of π π π π and − The output of sin is opposite the output of sin − Thus, () sin ( ) ( π2 ) = and ( π2 ) = − sin( π2 ) sin − = −1 This is shown in [link] Graph of y = sin θ Recall that an even function is one in which f( − x) = f(x) for all x in the domain of f The graph of an even function is symmetric about the y-axis The cosine function is an π even function because cos( − θ) = cos θ For example, consider corresponding inputs π and − The output of cos ( cos − π ( π4 ) is the same as the output of cos( − π4 ) Thus, ) = cos( π4 ) ≈ 0.707 See [link] 4/20 Solving Trigonometric Equations with Identities Graph of y = cos θ For all θ in the domain of the sine and cosine functions, respectively, we can state the following: • Since sin (− θ) = −sin θ, sine is an odd function • Since, cos (− θ) = cos θ, cosine is an even function The other even-odd identities follow from the even and odd nature of the sine and cosine functions For example, consider the tangent identity, ... Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn 1 MỤC LỤC Trang Mở đầu 2 Chƣơng I Một số khái niệm về hệ phƣơng trình vi phân đại số 5 1.1 Phép chiếu - Chỉ số của cặp ma trận 5 1.2 Hệ phương trình vi phân đại số tuyến tính với hệ số hằng 7 1.3 Phân rã hệ phương trình vi phân đại số thành hệ phương trình vi phân thường và hệ phương trình đại số 10 1.4 Sự ổn định (Lyapunov) của hệ phương trình vi phân đại số 13 Chƣơng II Bán kinh ổn định của hệ phƣơng trình vi phân đại số tuyến tính với ma trận hệ số hằng 15 2.1 Bán kính ổn định phức của hệ phương trình vi phân đại số 15 2.2 Liên hệ giữa bán kính ổn định thực và bán kính ổn định phức của hệ phương trình vi phân đại số 24 Chƣơng III Bán kính ổn định của hệ phƣơng trình vi phân đại số tuyến tính với nhiễu động 34 3.1 Hệ phương trình vi phân đại số tuyến tính với hệ số biến thiên 35 3.2 Nghiệm yếu và các khái niệm ổn định 37 3.3 Công thức bán kính ổn định 44 3.4 Các trường hợp đặc biệt 55 Kết luận 59 Tài liệu tham khảo 60 Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn 2 MỞ ĐẦU Từ cuối thế kỷ XIX nhiều nhà khoa học đã quan tâm tìm lời giải cho bài toán ổn định của chuyển động. Ở thời điểm đó, người ta đã đưa ra nhiều định nghĩa khác nhau về khái niệm này, chẳng hạn như định nghĩa của A.Poincaré, V.Rumyantsev, Chỉ từ khi A.M. Lyapunov (1857-1918) công bố công trình “Bài toán tổng quát về tính ổn định của chuyển động” vào năm 1892 ở Nga và dịch sang tiếng Pháp (Problème général de la stabilité du mouvement) năm 1907, lý thuyết ổn định mới được nghiên cứu một cách có hệ thống và trở thành một bộ phận quan trọng trong lý thuyết định tính phương trình vi phân. Kể từ đó, lý thuyết ổn định đã được nhiều nhà khoa học trên khắp thế giới quan tâm nghiên cứu. Đến nay, đã hơn một thế kỷ trôi qua, lý thuyết ổn định vẫn là một lĩnh vực toán học được nghiên cứu sôi nổi và đã thu được nhiều thành tựu rực rỡ, sâu sắc, như: vật lý, khoa học kỹ thuật công nghệ, sinh thái học, Lyapunov đã giải quyết bài toán ổn định bằng cả hai phương pháp, đó là phương pháp số mũ đặc trưng Lyapunov (còn gọi là phương pháp phổ hay phương pháp thứ nhất của Lyapunov) và phương pháp hàm Lyapunov (còn gọi là phương pháp thứ hai của Lyapunov). Vào những năm 70 của thế kỷ trước, một số bài toán có liên quan đến phương trình vi phân dạng:     '( ) + ( ) 0ttA x t B x t  ở đó,           , , , : , , , nn A B C I L x I I a     RR a là hằng số,   det 0 A t t I   . Đây chính là một dạng đặc biệt của phương trình vi phân đại số (differential algebraic equation-DAE). Ngay sau đó, loại phương trình vi phân này được nhiều nhà toán học đi sâu nghiên cứu. Để nghiên cứu DAE người ta thường làm như sau: phân rã chúng nhờ các phép chiếu để được một hệ phương trình vi phân thường và một hệ phương trình đại số. Ngoài ra, Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn 3 cũng còn một vài phương pháp khác. Đến nay người ta cũng đã tìm ra khá nhiều kết quả cho phương trình vi phân đại số tương tự như ở phương trình vi phân thường chẳng hạn như lý thuyết Floquet, tính ổn định tiệm cận của nghiệm của phương trình với ma trận hệ số hằng. Trong hơn hai thập kỷ qua, từ khái niệm bán kính ổn định mà D.Hinrichsen và A.J.Pritchard đưa ra, hai ông đã hình thành một hướng nghiên cứu mới là nghiên cứu tính ổn định vững của các hệ động lực dựa trên khái niệm bán kính ổn định. Hướng nghiên cứu này đã thu hút sự chú ý và tâm huyết của nhiều nhà toán học vì tính hiệu quả và tính thời sự của nó cũng như những ứng dụng trong các bài toán kỹ thuật. Nhóm tác giả Nguyễn Hữu Dư, Vũ Hoàng Linh đã nghiên cứu sự ổn định của hệ phương trình vi phân đại số với ma trận hệ số phụ thuộc tham số thời gian và đưa ra công thức bán kính ổn định trong bài báo “Stability radii for linear time - VNU Journal of Science, Mathematics - Physics 26 (2010) 175-184 Stability Radii for Difference Equations with Time-varying Coefficients Le Hong Lan* Department of Basic Sciences, University of Transport and Communication, Hanoi, Vietnam Received 10 August 2010 Abstract. This paper deals with a formula of stability radii for an linear difference equation (LDEs for short) with the coefficients varying in time under structured parameter perturbations. It is shown that the l p − real and complex stability radii of these systems coincide and they are given by a formula of input-output operator. The result is considered as an discrete version of a previous result for time-varying ordinary differential equations [1]. Keywords: Robust stability, Linear difference equation, Input-output operator, Stability ra- dius 1. Introduction Many control systems are subject to perturbations in terms of uncertain parameters. An important quantitative measure of stability robustness of a system to such perturbations is called the stability radius. The concept of stability radii was introduced by Hinrichsen and Pritchard 1986 for time- invariant differential (or difference) systems (see [2, 3]). It is defined as the smallest value ρ of the norm of real or complex perturbations destabilizing the system. If complex perturbations are allowed, ρ is called the complex stability radius. If only real perturbations are considered, the real radius is obtained. The computation of a stability radius is a subject which has attracted a lot of interest over recent decades, see e.g. [2, 3, 4, 5]. For further considerations in abstract spaces, see [6] and the references therein. Earlier results for time-varying systems can be found, e.g., in [1, 7]. The most successful attempt for finding a formula of the stability radius was an elegant result given by Jacob [1]. In that paper, it has been given by virtue of output-input operator a formula for L p − stability for time-varying system subjected to additive structured perturbations of the form ˙x(t) = B(t)x(t) + E(t)∆(F (·)x(·))(t), t  0, x(0) = x 0 , where E(t) and F (t) are given scaling matrices defining the structure of the perturbation and ∆ is an unknown disturbance. We now want to study a discrete version of this work by considering a difference equation with coefficients varying in time x(n + 1) = (A n + E n ∆F n )x(n), n ∈ N. (1) ∗ E-mail: honglanle229@gmail.com This work was supported by the project B2010 - 04. 175 176 L.H. Lan / VNU Journal of Science, Mathematics - Physics 26 (2010) 175-184 This problem has been studied by F. Wirth [8]. However, in this work, he has just given an estimate for stability radius. Following the idea in [1], we set up a formula for stability radius in the space l p and show that when p = 2 and A, E, F are constant matrix, we obtain the result dealt with in [5] The technique we use in this paper is somewhat similar to one in [1]. However, in applying the main idea of Jacob in [1] to the difference equations, we need some improvements. Many steps of the proofs in the paper [1] are considerably reduced and this reduction is valid not only in discrete case but also in continuous time one. An outline of the remainder of the paper is as follows: the next section introduces the concept of Stability radius for difference equation in l p . In Section 3 we prove a formula for computing the l p − stability radius. 2. Stability radius for difference equation We now establish a formulation for stability radius of the varying in times system  x(n + 1) = B n x(n), n ∈ N, n > m x(m) = x 0 ) ∈ R d . (2) It is easy to see that the equation (2) has Using the Explorer IDE After a plot has been selected, you can change its properties via the Plot Properties dialog accessible from the right-mouse button within the plot window. The Plot Prop- erties dialog allows you to set generic, axis, and signal attributes. Generic plot attributes include the plot type as well as display of titles and/or subti- tles. Axis properties of the plot allow you to set the axis styles for both the X- and Y-axis, including labels and tic-mark styles. Verilog-A Explorer IDE 197 Verilog-A Explorer IDE Signal properties allow you to edit the description of the signals displayed in the leg- end box, as well as the data format and drawing attributes. D.3.2 Creating a New Designs Starting a new design follows essentially the same procedure as previously outlined, but with the addition to creating a new circuit and/or Verilog-A file(s). From the main Explorer menu, select File->New, which raises the following dialog box: If you select a circuit file, the workspace will be cleared of any open files. If you select a Verilog-A file, it is assumed that it is associated with any existing circuit design open within the workspace. In both cases, a new file is created and initialized with a template file of the appropriate type. If you prefer your own template files, change the path of the template via the respective Editor Properties dialog accessible via the right mouse button. 198 Verilog-A HDL Appendix E Spice Quick Reference E.1 Introduction Spice is a general-purpose circuit simulation program for nonlinear DC, nonlinear transient, and linear AC analysis. Originating from the University of California at Berkeley, is by far the best known and most widely used circuit simulator. It is availa- ble in for a wide variety of computer platforms, in both commercial and proprietary derivatives of the original version. Newer versions of Spice offer many extensions, but the input format for circuit descriptions reflect the original batch-oriented program architecture. This appendix overviews the Spice input format, or netlist files including the fundamental types and analyses supported. Omitted for brevity are details regarding semiconductor device models and the various Spice options. Spice Quick Reference 199 Spice Quick Reference E.2 Circuit Netlist Description The netlist (also referred to as the input deck) consists of element lines which describes both the circuit topology and element values and control lines which describe analyses to be performed for Spice. The first card in the input deck must be a title card, and the last card must be the .END control line. The order of the remaining element and control lines is arbitrary. The input format is free format. Fields on an element or control line are separated by one or more blanks, commas, equal (=) sign, or a left or right parenthesis. A element or control line may be continued by placing a (+) in column 1 on the following line. Spice will continue reading beginning with column 2. Name fields must begin with a letter [a–z] and cannot contain any delimeters. Names within Spice netlists are considered case-insensitive 1 . An integer or a floating point number can be followed by one of the following scale factors: G = 1.0e9 MEG = 1.0e6 K = 1.0e3 MIL = 25.4e-4 M = 1.0e-3 U = 1.0e-6 N = l.0e-9 P = 1.0e-12 1. Names in Verilog are case-sensitive requiring a certain level of awareness for modelers in developing Verilog-A models that are case-independent for use within Spice netlists. 200 Verilog-A HDL Spice Quick Reference 201 E.3 Components Letters immediately following a Modeling with Trigonometric Equations Modeling with Trigonometric Equations By: OpenStaxCollege The hands on a clock are periodic: they repeat positions every twelve hours (credit: “zoutedrop”/Flickr) Suppose we charted the average daily temperatures in New York City over the course of one year We would expect to find the lowest temperatures in January and February and highest in July and ... and practice with the fundamental trigonometric identities • Fundamental Trigonometric Identities • Verifying Trigonometric Identities Key Equations sin2θ + cos2θ = Pythagorean identities + cot2θ... will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities We will begin with the Pythagorean identities. .. sin2x =1 12/20 Solving Trigonometric Equations with Identities Using Algebra to Simplify Trigonometric Expressions We have seen that algebra is very important in verifying trigonometric identities,