Spherical Trigonometry After the Cesaro Method
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
1 Chapter 3 Plane and Spherical Trigonometry 3.1
1 CHAPTER 3 PLANE AND SPHERICAL TRIGONOMETRY 3.1 Introduction It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls. This is followed by a quick review of spherical coordinates and direction cosines in three- dimensional geometry. The formulas for the velocity and acceleration components in two- dimensional polar coordinates and three-dimensional spherical coordinates are developed in section 3.4. Section 3.5 deals with the trigonometric formulas for solving spherical triangles. This is a fairly long section, and it will be essential reading for those who are contemplating making a start on celestial mechanics. Sections 3.6 and 3.7 deal with the rotation of axes in two and three dimensions, including Eulerian angles and the rotation matrix of direction cosines. Finally, in section 3.8, a number of commonly encountered trigonometric formulas are gathered for reference. 3.2 Plane Triangles . This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles. Conventionally, a plane triangle is described by its three angles A, B, C and three sides a, b, c, with a being opposite to A, b opposite to B, and c opposite to C. -
Geodesy. Application of the Theory of Least Squares to the Adjustment Of
SI No Serial ^) DEPARTME]S'T OF COMMERCE 0. S. COAST AND GEODETIC SURVEY it E. LESTER JONES, Supei^intendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION RY OSCAR S. ADAM^ COMPUTER UNITED STATES COAST AND GEODETIC SURJ^Y No. 28 Special Publication { WASHINGTON GOVERNMENT I'RINTING OFFICE 1915 Serial No. 9 DEPARTMEISTT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, Superintendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION BY OSCAR S. ADAMS COMPUTER UNITED STATES COAST AND GEODETIC SURVEY Special Publication No. 28 WASHINGTON GOVERNMENT PRINTING OFFICE 1815 ADDITIONAL COPIES OF THIS PUBLICATION MAY BE PROCURED FROM THE SUPERINTENDENT OF DOCUMENTS GOVERNMENT PRINTING OFFICE WASHINGTON, D. C. AT 25 CENTS PER COPY QB 32/ CONTENTS Page. General statement 7 Station adjustment 7 Observed angles 8 List of directions 8 Condition equations , 9 Formation of normal equations by differentiation 9 Correlate equations 11 Formation of normal equations 11 Normal equations 12 Discussion of method of solution of normal equations 12 Solution of normal equations 13 Back solution 13 Computation of corrections 13 Adjustment of a quadrilateral 14 General statement 14 Lists of directions 16 Figure 16 Angle equations 17 Side equation 17 Formation of normal equations by differentiation 17 Correlate equations 18 Normal equations 18 Solution of normal equations 19 Back solution 19 Computation of corrections 19 Adjustment of a quadrilateral by the use of two angle and two -
Hyperbolic Trigonometry in Two-Dimensional Space-Time Geometry
Hyperbolic trigonometry in two-dimensional space-time geometry F. Catoni, R. Cannata, V. Catoni, P. Zampetti ENEA; Centro Ricerche Casaccia; Via Anguillarese, 301; 00060 S.Maria di Galeria; Roma; Italy January 22, 2003 Summary.- By analogy with complex numbers, a system of hyperbolic numbers can be intro- duced in the same way: z = x + hy; h2 = 1 x, y R . As complex numbers are linked to the { ∈ } Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representa- tion, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalise the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry. PACS 03 30 - Special Relativity PACS 02.20. Hj - Classical groups and Geometries 1 Introduction Complex numbers are strictly related to the Euclidean geometry: indeed their invariant (the module) arXiv:math-ph/0508011v1 3 Aug 2005 is the same as the Pythagoric distance (Euclidean invariant) and their unimodular multiplicative group is the Euclidean rotation group. It is well known that these properties allow to use complex numbers for representing plane vectors. In the same way hyperbolic numbers, an extension of complex numbers [1, 2] defined as z = x + hy; h2 =1 x, y R , { ∈ } are strictly related to space-time geometry [2, 3, 4]. Indeed their square module given by1 z 2 = 2 2 | | zz˜ x y is the Lorentz invariant of two dimensional special relativity, and their unimodular multiplicative≡ − group is the special relativity Lorentz group [2]. -
Spherical Trigonometry
Spherical trigonometry 1 The spherical Pythagorean theorem Proposition 1.1 On a sphere of radius R, any right triangle 4ABC with \C being the right angle satisfies cos(c=R) = cos(a=R) cos(b=R): (1) Proof: Let O be the center of the sphere, we may assume its coordinates are (0; 0; 0). We may rotate −! the sphere so that A has coordinates OA = (R; 0; 0) and C lies in the xy-plane, see Figure 1. Rotating z B y O(0; 0; 0) C A(R; 0; 0) x Figure 1: The Pythagorean theorem for a spherical right triangle around the z axis by β := AOC takes A into C. The edge OA moves in the xy-plane, by β, thus −−!\ the coordinates of C are OC = (R cos(β);R sin(β); 0). Since we have a right angle at C, the plane of 4OBC is perpendicular to the plane of 4OAC and it contains the z axis. An orthonormal basis −−! −! of the plane of 4OBC is given by 1=R · OC = (cos(β); sin(β); 0) and the vector OZ := (0; 0; 1). A rotation around O in this plane by α := \BOC takes C into B: −−! −−! −! OB = cos(α) · OC + sin(α) · R · OZ = (R cos(β) cos(α);R sin(β) cos(α);R sin(α)): Introducing γ := \AOB, we have −! −−! OA · OB R2 cos(α) cos(β) cos(γ) = = : R2 R2 The statement now follows from α = a=R, β = b=R and γ = c=R. ♦ To prove the rest of the formulas of spherical trigonometry, we need to show the following. -
Elementary Functions Triangulation and the Law of Sines Three Bits of A
Triangulation and the Law of Sines A triangle has three sides and three angles; their values provide six pieces of information about the triangle. In ordinary geometry, most of the time three pieces of information are Elementary Functions sufficient to give us the other three pieces of information. Part 5, Trigonometry Lecture 5.3a, The Laws of Sines and Triangulation In order to more easily discuss the angles and sides of a triangle, we will label the angles by capital letters (such as A; B; C) and label the sides by small letters (such as a; b; c.) Dr. Ken W. Smith We will assume that a side labeled with a small letter is the side opposite Sam Houston State University the angle with the same, but capitalized, letter. For example, the side a is opposite the angle A. 2013 We will also use these letters for the values or magnitudes of these sides and angles, writing a = 3 feet or A = 14◦: Smith (SHSU) Elementary Functions 2013 1 / 26 Smith (SHSU) Elementary Functions 2013 2 / 26 Three Bits of a Triangle AA and AAA Suppose we have three bits of information about a triangle. Can we recover all the information? AAA If the 3 bits of information are the magnitudes of the 3 angles, and if we If we know the values of two angles then since the angles sum to π have no information about lengths of sides, then the answer is No. (= 180◦), we really know all 3 angles. Given a triangle with angles A; B; C (in ordinary Euclidean geometry) we can always expand (or contract) the triangle to a similar triangle which So AAA is really no better than AA! has the same angles but whose sides have all been stretched (or shrunk) by some constant factor. -
7 TRIGONOMETRY Objectives in This Chapter You Will Learn About • Revision of Solution of Triangles
7 TRIGONOMETRY Objectives In this chapter you will learn about • Revision of solution of triangles. • Solving problems in 2–dimensions and in 3-dimensions by constructing and interpreting models. 1 | P a g e 7.1 Revision In previous grades, we dealt with the types of triangles named according to side and angles. In trigonometry, we are able to find the unknown sides and angles when given the magnitudes of the minimum required number of angles and sides. We looked at two main types of triangles: • Right angled triangles. • Non-right angled triangles Right angled triangles When a right angled triangle is given, the following may be useful: • Pythagoras theorem • Definitions of Trigonometric ratios Non-right angled triangles • Sine rule • Cosine rule • Area rule NOTE: • Not all triangles that you need to solve are right-angled triangles. In grade 11 you learned about rules which can be used to solve non-right-angled triangles. • Theorems/axioms involving lines, triangles, quadrilateral and circles are also useful. 7.1.1 The Sine Rule The sine rule expresses the relationship between the two of the sides of the triangles and two of the angles. In any triangle, a is the side opposite angle A, b is the side opposite angle B and c is the side opposite angle C, then the lengthof any side the length of any other side = the sine of the angle opposite that side the sine of the angle opposite that side That is: 2 | P a g e In ABC, a b c = = sin A sinB sinC This can also be written as: sin A sinB sinC = = a b c The sine rule can be used when the following information about a triangle is given: • Two sides and an angle opposite to one of the two sides • One side and any two angles Worked Example 1 Solve the triangle given below. -
How to Learn Trigonometry Intuitively | Betterexplained 9/26/15, 12:19 AM
How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (/) How To Learn Trigonometry Intuitively by Kalid Azad · 101 comments Tweet 73 Trig mnemonics like SOH-CAH-TOA (http://mathworld.wolfram.com/SOHCAHTOA.html) focus on computations, not concepts: TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half- dedicated to vision processing, seem to enjoy imagery. And “TOA” evokes the stunning beauty of an abstract ratio. I think you deserve better, and here’s what made trig click for me. Visualize a dome, a wall, and a ceiling Trig functions are percentages to the three shapes http://betterexplained.com/articles/intuitive-trigonometry/ Page 1 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM Motivation: Trig Is Anatomy Imagine Bob The Alien visits Earth to study our species. Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”. After creating specific terms for anatomy, Bob might jot down typical body proportions (http://en.wikipedia.org/wiki/Body_proportions): The armspan (fingertip to fingertip) is approximately the height A head is 5 eye-widths wide Adults are 8 head-heights tall http://betterexplained.com/articles/intuitive-trigonometry/ Page 2 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (http://en.wikipedia.org/wiki/Vitruvian_Man) How is this helpful? Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate the owner’s height. -
The Sphere of the Earth Description and Technical Manual
The Sphere of the Earth Description and Technical Manual December 20, 2012 Daniel Ramos ∗ MMACA (Museu de Matem`atiquesde Catalunya) 1 The exhibit The exhibit we are presenting explores the science of cartography and the geometry of the sphere. Althoug this subject has been widely treated on science exhibitions and fairs, our proposal aims to be a more comprehensive and complete treatment on the subject, as well as to be a fully open source, documented material that both museographers and public can experience, learn and modify. The key concept comes from the problem of representing the spherical surface of the Earth onto a flat map. Studying this problem is the subject of cartography, and has been an important mathematical problem in History (navigation, position, frontiers, land ownership...). An essential theorem in Geometry (Gauss' Egregium theorem) ensures that there is no perfect map, that is, there is no way of repre- senting the Earth keeping distances at scale. However, this is exactly what makes cartography a discipline: developing several different maps that try to solve well enough the problem of representing the Earth. A common activity is comparing an Earth globe with a (Mercator) map, observ- ing that distances are not preserved. Our module explores far beyond that. We present: • Several different maps, currently 6 map projections are used, each one featuring different special properties. Printed at nominal scale 1:1 of a model globe. • The script programs that generate these pictures. Generating a map takes less than 50 lines of code, anyone could generate his own. ∗E-mail: [email protected] 1 • A collection of tools and models: a flexible ruler to be used over the globe, plane and spherical protractors, models showing longitude and latitude.. -
Spherical Trigonometry Through the Ages
School of Mathematics and Statistics Newcastle University MMath Mathematics Report Spherical Trigonometry Through the Ages Author: Supervisor: Lauren Roberts Dr. Andrew Fletcher May 1, 2015 Contents 1 Introduction 1 2 Geometry on the Sphere Part I 4 2.1 Cross-sections and Measurements . .4 2.2 Triangles on the Sphere . .7 2.2.1 What is the Largest Possible Spherical Triangle? . .7 2.2.2 The Sum of Spherical Angles . .9 3 Menelaus of Alexandria 12 3.1 Menelaus' Theorem . 13 3.2 Astronomy and the Menelaus Configuration: The Celestial Sphere . 16 4 Medieval Islamic Mathematics 18 4.1 The Rule of Four Quantities . 18 4.2 Finding the Qibla . 20 4.3 The Spherical Law of Sines . 22 5 Right-Angled Triangles 25 5.1 Pythagoras' Theorem on the Sphere . 25 5.2 The Spherical Law of Cosines . 27 6 Geometry on the Sphere Part II 30 6.1 Areas of Spherical Polygons . 30 6.2 Euler's Polyhedral Formula . 33 7 Quaternions 35 7.1 Some Useful Properties . 35 7.1.1 The Quaternion Product . 36 7.1.2 The Complex Conjugate and Inverse Quaternions . 36 1 CONTENTS 2 7.2 Rotation Sequences . 37 7.2.1 3-Dimensional Representation . 39 7.3 Deriving the Spherical Law of Sines . 39 8 Conclusions 42 Bibliography 44 Abstract The art of applying trigonometry to triangles constructed on the surface of the sphere is the branch of mathematics known as spherical trigonometry. It allows new theorems relating the sides and angles of spherical triangles to be derived. The history and development of spherical trigonometry throughout the ages is discussed. -
Arxiv:1409.4736V1 [Math.HO] 16 Sep 2014
ON THE WORKS OF EULER AND HIS FOLLOWERS ON SPHERICAL GEOMETRY ATHANASE PAPADOPOULOS Abstract. We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A05 ; 53A35. Keywords: spherical trigonometry, spherical geometry, Euler, Lexell the- orem, cartography, calculus of variations. -
Trigonometric Computation David Altizio, Andrew Kwon
Trig Computation Western PA ARML 2016-2017 Page 1 Trigonometric Computation David Altizio, Andrew Kwon 0.1 Lecture • There are many different formulas for the area of a triangle. The ones that you'll probably need to know the most are summarized below: 1 1 abc A = bh = ab sin C = rs = = ps(s − a)(s − b)(s − c) = r (s − a): 2 2 4R a Make sure you know how to derive these! • Law of Sines: In any triangle ABC, we have a b c = = = 2R: sin A sin B sin C • Law of Cosines: In any triangle ABC, we have c2 = a2 + b2 − 2ab cos C: • Ratio Lemma: In any triangle ABC, if D is a point on BC, then BD BA sin BAD = \ : DC AC sin \DAC Note that this is a generalization of the Angle Bisector Theorem. Note further that this does not necessarily require that D lie on the segment BC. • Make sure you know your trig identities! Here are a few ones to keep in mind: { Negation: We have sin(−α) = − sin α and cos(−α) = cos α for all angles α. { Pythagorean Identity: For any angle α, sin2 α + cos2 α = 1: Be sure to memorize this one! { Sum and Difference Identities: For any angles α and β, sin(α + β) = sin α cos β + sin β cos α; cos(α + β) = cos α cos β − sin α sin β: { Double Angle Identities: We have 2 tan A sin(2A) = 2 sin A cos A; cos(2A) = cos2 A − sin2 A; tan(2A) = : 1 − tan2 A { Half Angle Identities: We have r r A 1 − cos A A 1 + cos A A r1 − cos A sin = ; cos = ; tan = : 2 2 2 2 2 1 + cos A Trig Computation Western PA ARML 2016-2017 Page 2 0.2 Problems 1. -
4 Trigonometry and Complex Numbers
4 Trigonometry and Complex Numbers Trigonometry developed from the study of triangles, particularly right triangles, and the relations between the lengths of their sides and the sizes of their angles. The trigono- metric functions that measure the relationships between the sides of similar triangles have far-reaching applications that extend far beyond their use in the study of triangles. Complex numbers were developed, in part, because they complete, in a useful and ele- gant fashion, the study of the solutions of polynomial equations. Complex numbers are useful not only in mathematics, but in the other sciences as well. Trigonometry Most of the trigonometric computations in this chapter use six basic trigonometric func- tions The two fundamental trigonometric functions, sine and cosine, can be de¿ned in terms of the unit circle—the set of points in the Euclidean plane of distance one from the origin. A point on this circle has coordinates +frv w> vlq w,,wherew is a measure (in radians) of the angle at the origin between the positive {-axis and the ray from the ori- gin through the point measured in the counterclockwise direction. The other four basic trigonometric functions can be de¿ned in terms of these two—namely, vlq { 4 wdq { @ vhf { @ frv { frv { frv { 4 frw { @ fvf { @ vlq { vlq { 3 ?w? For 5 , these functions can be found as a ratio of certain sides of a right triangle that has one angle of radian measure w. Trigonometric Functions The symbols used for the six basic trigonometric functions—vlq, frv, wdq, frw, vhf, fvf—are abbreviations for the words cosine, sine, tangent, cotangent, secant, and cose- cant, respectively.You can enter these trigonometric functions and many other functions either from the keyboard in mathematics mode or from the dialog box that drops down when you click or choose Insert + Math Name.