Lesson 6: Trigonometric Identities
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Trigonometric Functions
Trigonometric Functions This worksheet covers the basic characteristics of the sine, cosine, tangent, cotangent, secant, and cosecant trigonometric functions. Sine Function: f(x) = sin (x) • Graph • Domain: all real numbers • Range: [-1 , 1] • Period = 2π • x intercepts: x = kπ , where k is an integer. • y intercepts: y = 0 • Maximum points: (π/2 + 2kπ, 1), where k is an integer. • Minimum points: (3π/2 + 2kπ, -1), where k is an integer. • Symmetry: since sin (–x) = –sin (x) then sin(x) is an odd function and its graph is symmetric with respect to the origin (0, 0). • Intervals of increase/decrease: over one period and from 0 to 2π, sin (x) is increasing on the intervals (0, π/2) and (3π/2 , 2π), and decreasing on the interval (π/2 , 3π/2). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Cosine Function: f(x) = cos (x) • Graph • Domain: all real numbers • Range: [–1 , 1] • Period = 2π • x intercepts: x = π/2 + k π , where k is an integer. • y intercepts: y = 1 • Maximum points: (2 k π , 1) , where k is an integer. • Minimum points: (π + 2 k π , –1) , where k is an integer. • Symmetry: since cos(–x) = cos(x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. • Intervals of increase/decrease: over one period and from 0 to 2π, cos (x) is decreasing on (0 , π) increasing on (π , 2π). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Tangent Function : f(x) = tan (x) • Graph • Domain: all real numbers except π/2 + k π, k is an integer. -
CHAPTER 8. COMPLEX NUMBERS Why Do We Need Complex Numbers? First of All, a Simple Algebraic Equation Like X2 = −1 May Not Have
CHAPTER 8. COMPLEX NUMBERS Why do we need complex numbers? First of all, a simple algebraic equation like x2 = 1 may not have a real solution. − Introducing complex numbers validates the so called fundamental theorem of algebra: every polynomial with a positive degree has a root. However, the usefulness of complex numbers is much beyond such simple applications. Nowadays, complex numbers and complex functions have been developed into a rich theory called complex analysis and be- come a power tool for answering many extremely difficult questions in mathematics and theoretical physics, and also finds its usefulness in many areas of engineering and com- munication technology. For example, a famous result called the prime number theorem, which was conjectured by Gauss in 1849, and defied efforts of many great mathematicians, was finally proven by Hadamard and de la Vall´ee Poussin in 1896 by using the complex theory developed at that time. A widely quoted statement by Jacques Hadamard says: “The shortest path between two truths in the real domain passes through the complex domain”. The basic idea for complex numbers is to introduce a symbol i, called the imaginary unit, which satisfies i2 = 1. − In doing so, x2 = 1 turns out to have a solution, namely x = i; (actually, there − is another solution, namely x = i). We remark that, sometimes in the mathematical − literature, for convenience or merely following tradition, an incorrect expression with correct understanding is used, such as writing √ 1 for i so that we can reserve the − letter i for other purposes. But we try to avoid incorrect usage as much as possible. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Derivation of Sum and Difference Identities for Sine and Cosine
Derivation of sum and difference identities for sine and cosine John Kerl January 2, 2012 The authors of your trigonometry textbook give a geometric derivation of the sum and difference identities for sine and cosine. I find this argument unwieldy | I don't expect you to remember it; in fact, I don't remember it. There's a standard algebraic derivation which is far simpler. The only catch is that you need to use complex arithmetic, which we don't cover in Math 111. Nonetheless, I will present the derivation so that you will have seen how simple the truth can be, and so that you may come to understand it after you've had a few more math courses. And in fact, all you need are the following facts: • Complex numbers are of the form a+bi, where a and b are real numbers and i is defined to be a square root of −1. That is, i2 = −1. (Of course, (−i)2 = −1 as well, so −i is the other square root of −1.) • The number a is called the real part of a + bi; the number b is called the imaginary part of a + bi. All the real numbers you're used to working with are already complex numbers | they simply have zero imaginary part. • To add or subtract complex numbers, add the corresponding real and imaginary parts. For example, 2 + 3i plus 4 + 5i is 6 + 8i. • To multiply two complex numbers a + bi and c + di, just FOIL out the product (a + bi)(c + di) and use the fact that i2 = −1. -
Section 5.4 the Other Trigonometric Functions 333
Section 5.4 The Other Trigonometric Functions 333 Section 5.4 The Other Trigonometric Functions In the previous section, we defined the sine and cosine functions as ratios of the sides of a right triangle in a circle. Since the triangle has 3 sides there are 6 possible combinations of ratios. While the sine and cosine are the two prominent ratios that can be formed, there are four others, and together they define the 6 trigonometric functions. Tangent, Secant, Cosecant, and Cotangent Functions For the point (x, y) on a circle of radius r at an angle of θ , we can define four additional important functions as the ratios of the (x, y) sides of the corresponding triangle: y r The tangent function: tan(θ ) = y x θ r The secant function: sec(θ ) = x x r The cosecant function: csc(θ ) = y x The cotangent function: cot(θ ) = y Geometrically, notice that the definition of tangent corresponds with the slope of the line segment between the origin (0, 0) and the point (x, y). This relationship can be very helpful in thinking about tangent values. You may also notice that the ratios defining the secant, cosecant, and cotangent are the reciprocals of the ratios defining the cosine, sine, and tangent functions, respectively. Additionally, notice that using our results from the last section, y r sin(θ ) sin(θ ) tan(θ ) = = = x r cos(θ ) cos(θ ) Applying this concept to the other trig functions we can state the other reciprocal identities. Identities The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships: sin(θ ) 1 1 1 cos(θ ) tan(θ ) = sec(θ ) = csc(θ ) = cot(θ ) = = cos(θ ) cos(θ ) sin(θ ) tan(θθ ) sin( ) 334 Chapter 5 These relationships are called identities. -
Complex Numbers and Functions
Complex Numbers and Functions Richard Crew January 20, 2018 This is a brief review of the basic facts of complex numbers, intended for students in my section of MAP 4305/5304. I will discuss basic facts of com- plex arithmetic, limits and derivatives of complex functions, power series and functions like the complex exponential, sine and cosine which can be defined by convergent power series. This is a preliminary version and will be added to later. 1 Complex Numbers 1.1 Arithmetic. A complex number is an expression a + bi where i2 = −1. Here the real number a is the real part of the complex number and bi is the imaginary part. If z is a complex number we write <(z) and =(z) for the real and imaginary parts respectively. Two complex numbers are equal if and only if their real and imaginary parts are equal. In particular a + bi = 0 only when a = b = 0. The set of complex numbers is denoted by C. Complex numbers are added, subtracted and multiplied according to the usual rules of algebra: (a + bi) + (c + di) = (a + c) + (b + di) (1.1) (a + bi) − (c + di) = (a − c) + (b − di) (1.2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i (1.3) (note how i2 = −1 has been used in the last equation). Division performed by rationalizing the denominator: a + bi (a + bi)(c − di) (ac − bd) + (bc − ad)i = = (1.4) c + di (c + di)(c − di) c2 + d2 Note that denominator only vanishes if c + di = 0, so that a complex number can be divided by any nonzero complex number. -
Complex Numbers Euler's Identity
2.003 Fall 2003 Complex Exponentials Complex Numbers ² Complex numbers have both real and imaginary components. A complex number r may be expressed in Cartesian or Polar forms: r = a + jb (cartesian) = jrjeÁ (polar) The following relationships convert from cartesian to polar forms: p Magnitude jrj = a2 + b2 ( ¡1 b tan a a > 0 Angle Á = ¡1 b tan a § ¼ a < 0 ² Complex numbers can be plotted on the complex plane in either Cartesian or Polar forms Fig.1. Figure 1: Complex plane plots: Cartesian and Polar forms Euler's Identity Euler's Identity states that ejÁ = cos Á + j sin Á 1 2.003 Fall 2003 Complex Exponentials This can be shown by taking the series expansion of sin, cos, and e. Á3 Á5 Á7 sin Á = Á ¡ + ¡ + ::: 3! 5! 7! Á2 Á4 Á6 cos Á = 1 ¡ + ¡ + ::: 2! 4! 6! Á2 Á3 Á4 Á5 ejÁ = 1 + jÁ ¡ ¡ j + + j + ::: 2! 3! 4! 5! Combining (Á)2 Á3 Á4 Á5 cos Á + j sin Á = 1 + jÁ ¡ ¡ j + + j + ::: 2! 3! 4! 5! = ejÁ Complex Exponentials ² Consider the case where Á becomes a function of time increasing at a constant rate ! Á(t) = !t: then r(t) becomes r(t) = ej!t Plotting r(t) on the complex plane traces out a circle with a constant radius = 1 (Fig. 2 ). Plotting the real and imaginary components of r(t) vs time (Fig. 3 ), we see that the real component is Refr(t)g = cos !t while the imaginary component is Imfr(t)g = sin !t. ² Consider the variable r(t) which is de¯ned as follows: r(t) = est where s is a complex number s = σ + j! 2 2.003 Fall 2003 Complex Exponentials t Figure 2: Complex plane plots: r(t) = ej!t t 0 t Re[ r(t) ]=cos t 0 t Im[ r(t) ]=sin t Figure 3: Real and imaginary components of r(t) vs time ² What path does r(t) trace out in the complex plane ? Consider r(t) = est = e(σ+j!)t = eσt ¢ ej!t One can look at this as a time varying magnitude (eσt) multiplying a point rotating on the unit circle at frequency ! via the function ej!t. -
Inverse Trig Functions Summary
Inverse trigonometric functions 1 1 z = sec ϑ The inverse sine function. The sine function restricted to [ π, π ] is one-to-one, and its inverse on this 3 − 2 2 ϑ = π ϑ = arcsec z interval is called the arcsine (arcsin) function. The domain of arcsin is [ 1, 1] and the range of arcsin is 2 1 1 1 −1 [ π, π ]. Below is a graph of y = sin ϑ, with the the part over [ π, π ] emphasized, and the graph − 2 2 − 2 2 of ϑ y. π 1 = arcsin ϑ = 2 π y = sin ϑ ϑ = arcsin y 1 π π 2 π − 1 ϑ 1 1 z − 1 π 1 π ϑ 1 1 y − 2 2 − 1 − 1 1 3 π ϑ = 2 π ϑ = 2 π − 2 By definition, By definition, 1 1 1 3 ϑ = arcsin y means that sin ϑ = y and π 6 ϑ 6 π. ϑ = arcsec z means that sec ϑ = t and 0 6 ϑ< 2 π or π 6 ϑ< 2 π. − 2 2 Differentiating the equation on the right implicitly with respect to y, gives Again, differentiating the equation on the right implicitly with respect to z, and using the restriction in ϑ, dϑ dϑ 1 1 1 one computes the derivative cos ϑ = 1, or = , provided π<ϑ< π. dy dy cos ϑ − 2 2 d 1 (arcsec z)= 2 , for z > 1. 1 1 2 2 dz z√z 1 | | Since cos ϑ> 0 on ( 2 π, 2 π ), it follows that cos ϑ = p1 sin ϑ = p1 y . -
Intervening in Student Identity in Mathematics Education: an Attempt to Increase Motivation to Learn Mathematics
INTERNATIONAL ELECTRONIC JOURNAL OF MATHEMATICS EDUCATION e-ISSN: 1306-3030. 2020, Vol. 15, No. 3, em0597 OPEN ACCESS https://doi.org/10.29333/iejme/8326 Intervening in Student Identity in Mathematics Education: An Attempt to Increase Motivation to Learn Mathematics Kayla Heffernan 1*, Steven Peterson 2, Avi Kaplan 3, Kristie J. Newton 4 1 Department of Mathematics, University of Pittsburgh at Greensburg, 150 Finoli Drive, Greensburg, PA 15601, USA 2 Haverford High School, 200 Mill Road, Havertown, PA 19083, USA 3 Department of Psychological Studies in Education, Temple University, 1301 Cecil B. Moore Avenue, Philadelphia, PA 19122, USA 4 Department of Teaching and Learning, Temple University, 1301 Cecil B. Moore Avenue, Philadelphia, PA 19122, USA * CORRESPONDENCE: [email protected] ABSTRACT Students’ relationships with mathematics continuously remain problematic, and researchers have begun to look at this issue through the lens of identity. In this article, the researchers discuss identity in education research, specifically in mathematics classrooms, and break down the various perspective on identity. A review of recent literature that explicitly invokes identity as a construct in intervention studies is presented, with a devoted attention to research on identity interventions in mathematics classrooms categorized based on the various perspectives of identity. Across perspectives, the review demonstrates that mathematics identities motivate action and that mathematics educators can influence students’ mathematical identities. The purpose of this paper is to help readers, researchers, and educators understand the various perspectives on identity, understand that identity can be influenced, and learn how researchers and educators have thus far, and continue to study identity interventions in mathematics classrooms. -
On Orders of Elements in Quasigroups
BULETINUL ACADEMIEI DE S¸TIINT¸E A REPUBLICII MOLDOVA. MATEMATICA Number 2(45), 2004, Pages 49–54 ISSN 1024–7696 On orders of elements in quasigroups Victor Shcherbacov Abstract. We study the connection between the existence in a quasigroupof(m, n)- elements for some natural numbers m, n and properties of this quasigroup. The special attention is given for case of (m, n)-linear quasigroups and (m, n)-T-quasigroups. Mathematics subject classification: 20N05. Keywords and phrases: Quasigroup, medial quasigroup, T-quasigroup, order of an element of a quasigroup. 1 Introduction We shall use basic terms and concepts from books [1, 2, 11]. We recall that a binary groupoid (Q, A) with n-ary operation A such that in the equality A(x1,x2)= x3 knowledge of any two elements of x1,x2,x3 the uniquely specifies the remaining one is called a binary quasigroup [3]. It is possible to define a binary quasigroup also as follows. Definition 1. A binary groupoid (Q, ◦) is called a quasigroup if for any element (a, b) of the set Q2 there exist unique solutions x,y ∈ Q to the equations x ◦ a = b and a ◦ y = b [1]. An element f(b) of a quasigroup (Q, ·) is called a left local identity element of an element b ∈ Q, if f(b) · b = b. An element e(b) of a quasigroup (Q, ·) is called a right local identity element of an element b ∈ Q, if b · e(b)= b. The fact that an element e is a left (right) identity element of a quasigroup (Q, ·) means that e = f(x) for all x ∈ Q (respectively, e = e(x) for all x ∈ Q). -
Euler's Formula: 1 Powers of E: First Pass
Euler’s Formula: The purpose of these notes is to explain Euler’s famous formula eiθ = cos(θ)+ i sin(θ). (1) 1 Powers of e: First Pass Euler’s equation is complicated because it involves raising a number to an imaginary power. Let’s build up to this slowly. Integer Powers: It’s pretty clear that e2 = e e and e3 = e e e, × × × and so on. For any positive integer p, we have ep = e ... e, a total of p times. For negative integers, the definition is also pretty× clear.× For instance e−2 =1/(e e) and e−3 =1/(e e e). And so on. × × × Fractional Powers: How would you define e2/3. This really means the cube root of e e. So e2/3 is the number y such that y y y = e e. Using this system, it× is pretty clear that you can make good sense× × of ep/q×where p/q is any positive fraction. You can make sense of negative fractions using the formula e−p/q =1/ep/q. Real Powers: If a is a positive real number, then you could define ea to pn/qn be the limit of expressions of the form e , where pn/qn is a sequence of rational numbers converging to a. To give an example of what I’m talking about, let a = √2. 17/12 is close to √2 and e17/12 =4.1233529... • 41/29 is closer to √2 and e41/29 =4.111521... • 577/408 is closer to √2 and e577/408 =4.113259 • 1393/985 is closer to √2 and e1393/985 =4.1132488. -
American Protestantism and the Kyrias School for Girls, Albania By
Of Women, Faith, and Nation: American Protestantism and the Kyrias School For Girls, Albania by Nevila Pahumi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (History) in the University of Michigan 2016 Doctoral Committee: Professor Pamela Ballinger, Co-Chair Professor John V.A. Fine, Co-Chair Professor Fatma Müge Göçek Professor Mary Kelley Professor Rudi Lindner Barbara Reeves-Ellington, University of Oxford © Nevila Pahumi 2016 For my family ii Acknowledgements This project has come to life thanks to the support of people on both sides of the Atlantic. It is now the time and my great pleasure to acknowledge each of them and their efforts here. My long-time advisor John Fine set me on this path. John’s recovery, ten years ago, was instrumental in directing my plans for doctoral study. My parents, like many well-intended first generation immigrants before and after them, wanted me to become a different kind of doctor. Indeed, I made a now-broken promise to my father that I would follow in my mother’s footsteps, and study medicine. But then, I was his daughter, and like him, I followed my own dream. When made, the choice was not easy. But I will always be grateful to John for the years of unmatched guidance and support. In graduate school, I had the great fortune to study with outstanding teacher-scholars. It is my committee members whom I thank first and foremost: Pamela Ballinger, John Fine, Rudi Lindner, Müge Göcek, Mary Kelley, and Barbara Reeves-Ellington.