DOCSLIB.ORG

Chapter 15 Measuring an Angle

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 15 Measuring an Angle Chapter 15 Measuring an Angle So far, the equations we have studied have an algebraic Cosmo S character, involving the variables x and y, arithmetic op- motors around erations and maybe extraction of roots. Restricting our Q the attention to such equations would limit our ability to de- circle R scribe certain natural phenomena. An important example P 20 feet involves understanding motion around a circle, and it can be motivated by analyzing a very simple scenario: Cosmo the dog, tied by a 20 foot long tether to a post, begins Figure 15.1: Cosmo the dog walking around a circle. A number of very natural ques- walking a circular path. tions arise: Natural Questions 15.0.1. How can we measure the angles ∠SPR, ∠QPR, and ∠QPS? How can we measure the arc lengths arc(RS), arc(SQ) and arc(RQ)? How can we measure the rate Cosmo is moving around the circle? If we know how to measure angles, can we compute the coordinates of R, S, and Q? Turning this around, if we know how to compute the coordinates of R, S, and Q, can we then measure the angles ∠SPR, ∠QPR, and ∠QPS ? Finally, how can we specify the direction Cosmo is traveling? We will answer all of these questions and see how the theory which evolves can be applied to a variety of problems. The definition and basic properties of the circular functions will emerge as a central theme in this Chapter. The full problem-solving power of these functions will become apparent in our discussion of sinusoidal functions in Chapter 19. The xy-coordinate system is well equipped to study straight line mo- tion between two locations. For problems of this sort, the important tool is the distance formula. However, as Cosmo has illustrated, not all two-dimensional motion is along a straight line. In this section, we will describe how to calculate length along a circular arc, which requires a quick review of angle measurement. 191 192 CHAPTER 15. MEASURING AN ANGLE 15.1 Standard and Central Angles An angle is the union of two rays emanating from a common point called the vertex of the angle. A typical angle can be dynamically generated by rotating a single ray from one position to another, sweeping counter- clockwise or clockwise: See Figure 15.2. We often insert a curved arrow to indicate the direction in which we are sweeping out the angle. The ray ℓ1 is called the initial side and ℓ2 the terminal side of the angle ∠AOB. (terminal side) l2 B SWEEP CLOCKWISE vertex vertex A (initial side) l1 l1 l1 O A O A (initial side) O (terminal side) START SWEEP COUNTERCLOCKWISE B l2 Figure 15.2: Angle ∠AOB. Working with angles, we need to agree on a standard frame of refer- ence for viewing them. Within the usual xy-coordinate system, imagine a model of ∠AOB in Figure 15.2 constructed from two pieces of rigid wire, welded at the vertex. Sliding this model around inside the xy-plane will not distort its shape, only its position relative to the coordinate axis. So, we can slide the angle into position so that the initial side is coincident with the positive x-axis and the vertex is the origin. Whenever we do this, we say the angle is in standard position. Once an angle is in stan- dard position, we can construct a circle centered at the origin and view our standard angle as cutting out a particular “pie shaped wedge” of the corresponding disc. Notice, the shaded regions in Figure 15.3 depend on whether we sweep the angle counterclockwise or clockwise from the initial side. The portion of the “pie wedge” along the circle edge, which is an arc, is called the arc subtended by the angle. We can keep track of this arc using the notation arc(AB). A central angle is any angle with vertex at the center of a circle, but its initial side may or may not be the positive x-axis. For example, ∠QPS in the Figure beginning this Chapter is a central angle which is not in standard position. 15.2. AN ANALOGY 193 y-axis l2 l2 y-axis B B arc subtended arc(AB) arc(AB) vertex vertex x-axis x-axis O l1 O l A A 1 COUNTERCLOCKWISE CLOCKWISE Figure 15.3: Standard angles and arcs. 15.2 An Analogy To measure the dimensions of a box you would use a ruler. In other words, you use an instrument (the ruler) as a standard against which you measure the box. The ruler would most likely be divided up into either English units (inches) or metric units (centimeters), so we could express the dimensions in a couple of different ways, depending on the units desired. By analogy, to measure the size of an angle, we need a standard against which any angle can be compared. In this section, we will de- scribe two standards commonly used: the degree method and the radian method of angle measurement. The key idea is this: Beginning with a circular region, describe how to construct a “basic” pie shaped wedge whose interior angle becomes the standard unit of angle measurement. 15.3 Degree Method Begin by drawing a circle of radius r, call it Cr, centered at the origin. Divide this circle into 360 equal sized pie shaped wedges, beginning with the the point (r,0) on the circle; i.e. the place where the circle crosses the x-axis. We will refer to the arcs along the outside edges of these wedges as one-degree arcs. Why 360 equal sized arcs? The reason for doing so is historically tied to the fact that the ancient Babylonians did so as they developed their study of astronomy. (There is actually an alternate system based on dividing the circular region into 400 equal sized wedges.) Any central angle which subtends one of these 360 equal sized arcs is 194 CHAPTER 15. MEASURING AN ANGLE circle Cr etc. a total of 360 this angle is equal sized r DEFINED to pie shaped typical have measure wedges inside (r,0) wedge 1 degree this disk r etc. ***NOT TO SCALE*** Figure 15.4: Wedges as 1◦ arcs. defined to have a measure of one degree, denoted 1◦. We can now use one-degree arcs to measure any angle: Begin by slid- ing the angle ∠AOB into standard central position, as in Figure 15.3. Piece together consecutive one-degree arcs in a counterclockwise or clock- wise direction, beginning from the initial side and working toward the terminal side, approximating the angle ∠AOB to the nearest degree. If we are allowed to divide a one-degree arc into a fractional portion, then we could precisely determine the number m of one-degree arcs which consecutively fit together into the given arc. If we are sweeping counter- clockwise from the initial side of the angle, m is defined to be the degree measure of the angle. If we sweep in a clockwise direction, then −m is defined to be the degree measure of the angle. So, in Figure 15.3, the left-hand angle has positive degree measure while the right-hand angle has negative degree measure. Simple examples would be angles like the ones in Figure 15.5. Notice, with our conventions, the rays determining an angle with mea- sure −135◦ sit inside the circle in the same position as those for an angle of measure 225◦; the minus sign keeps track of sweeping the positive x-axis clockwise (rather than counterclockwise). We can further divide a one-degree arc into 60 equal arcs, each called a one minute arc. Each one-minute arc can be further divided into 60 equal arcs, each called a one second arc. This then leads to angle mea- sures of one minute, denoted 1′ and one second, denoted 1′′: 1◦ = 60 minutes = 3600 seconds. 15.3. DEGREE METHOD 195 180◦ 90◦ 270◦ 45◦ 315◦ −135◦ Figure 15.5: Examples of common angles. For example, an angle of measure θ = 5 degrees 23 minutes 18 sec- onds is usually denoted 5◦ 23′ 18′′. We could express this as a decimal of degrees: In degrees! 23 18 ◦ 5◦ 23′ 18′′ = 5 + + 60 3600 ← = 5.3883◦. As another example, suppose we have an angle with measure 75.456◦ and we wish to convert this into degree/minute/second units. First, since 75.456◦ = 75◦ +0.456◦, we need to write 0.456◦ in minutes by the calculation: minutes 0.456 degree 60 = 27.36′. × degree This tells us that 75.456◦ = 75◦27.36′ = 75◦27′ + 0.36′. Now we need to write 0.36′ in seconds via the calculation: 0.36 minutes 60 seconds/minute = 21.6′′. × In other words, 75.456◦ = 75◦ 27′ 21.6′′. Degree measurement of an angle is very closely tied to direction in the plane, explaining its use in map navigation. With some additional work, it is also possible to relate degree measure and lengths of circular arcs. To do this carefully, first go back to Figure 15.3 and recall the situation 196 CHAPTER 15. MEASURING AN ANGLE where an arc arc(AB) is subtended by the central angle ∠AOB. In this situation, the arc length of arc(AB), commonly denoted by the letter s, is the distance from A to B computed along the circular arc; keep in mind, this is NOT the same as the straight line distance between the points A and B. For example, consider the six angles pictured above, of measures 90◦, 180◦, 270◦, 45◦, −135◦, and 315◦.
Load more

Recommended publications
  • Construction Surveying Curves
    Construction Surveying Curves Three(3) Continuing Education Hours Course #LS1003 Approved Continuing Education for Licensed Professional Engineers EZ-pdh.com Ezekiel Enterprises, LLC 301 Mission Dr. Unit 571 New Smyrna Beach, FL 32170 800-433-1487 [email protected] Construction Surveying Curves Ezekiel Enterprises, LLC Course Description: The Construction Surveying Curves course satisfies three (3) hours of professional development. The course is designed as a distance learning course focused on the process required for a surveyor to establish curves. Objectives: The primary objective of this course is enable the student to understand practical methods to locate points along curves using variety of methods. Grading: Students must achieve a minimum score of 70% on the online quiz to pass this course. The quiz may be taken as many times as necessary to successful pass and complete the course. Ezekiel Enterprises, LLC Section I. Simple Horizontal Curves CURVE POINTS Simple The simple curve is an arc of a circle. It is the most By studying this course the surveyor learns to locate commonly used. The radius of the circle determines points using angles and distances. In construction the “sharpness” or “flatness” of the curve. The larger surveying, the surveyor must often establish the line of the radius, the “flatter” the curve. a curve for road layout or some other construction. The surveyor can establish curves of short radius, Compound usually less than one tape length, by holding one end Surveyors often have to use a compound curve because of the tape at the center of the circle and swinging the of the terrain.
    [Show full text]
  • 11.1 Circumference and Arc Length
    11.1 Circumference and Arc Length EEssentialssential QQuestionuestion How can you fi nd the length of a circular arc? Finding the Length of a Circular Arc Work with a partner. Find the length of each red circular arc. a. entire circle b. one-fourth of a circle y y 5 5 C 3 3 1 A 1 A B −5 −3 −1 1 3 5 x −5 −3 −1 1 3 5 x −3 −3 −5 −5 c. one-third of a circle d. fi ve-eighths of a circle y y C 4 4 2 2 A B A B −4 −2 2 4 x −4 −2 2 4 x − − 2 C 2 −4 −4 Using Arc Length Work with a partner. The rider is attempting to stop with the front tire of the motorcycle in the painted rectangular box for a skills test. The front tire makes exactly one-half additional revolution before stopping. LOOKING FOR The diameter of the tire is 25 inches. Is the REGULARITY front tire still in contact with the IN REPEATED painted box? Explain. REASONING To be profi cient in math, you need to notice if 3 ft calculations are repeated and look both for general methods and for shortcuts. CCommunicateommunicate YourYour AnswerAnswer 3. How can you fi nd the length of a circular arc? 4. A motorcycle tire has a diameter of 24 inches. Approximately how many inches does the motorcycle travel when its front tire makes three-fourths of a revolution? Section 11.1 Circumference and Arc Length 593 hhs_geo_pe_1101.indds_geo_pe_1101.indd 593593 11/19/15/19/15 33:10:10 PMPM 11.1 Lesson WWhathat YYouou WWillill LLearnearn Use the formula for circumference.
    [Show full text]
  • 104 Word Vocabulary List Geometry/Honors Geometry
    104 Word Vocabulary List Geometry/Honors Geometry Summer Preparedness Project (“Math Summer Reading”) Summer 2021 – know and be prepared to define/discuss the following terms 1. Geometry - branch of mathematics that deals with points, lines, planes and solids and examines their properties. 2. Point – has no size; length, width, or height. It is represented by a dot and named by a capital letter. 3. Line – set of points which has infinite length but no width or height. A line is named by a lower case, cursive letter or by any two points on the line. 4. Plane – set of points that has infinite length and width but no height. We name a plane with a capital ‘funny font’ letter. 5. Collinear points – points that lie on the same line. 6. Noncollinear points – points that do not lie on the same line. 7. Coplanar points – points that lie on the same plane. 8. Noncoplanar points – points that do not lie on the same plane. 9. Segment – part of a line that consists of two points called endpoints and all points between them. 10. Ray- is the part of a line that contains an endpoint and all points extending in the other direction. 11. Congruent segments – segments that have the same length. 12. Bisector of a segment – line, ray segment, or plane that divides a segment into two congruent segments. 13. Midpoint of a segment – a point that divides the segment into two congruent segments. 14. Acute angle – angle whose measure is between 0 degrees and 90 degrees. 15. Right angle – angle whose measure is 90 degrees.
    [Show full text]
  • Inscribed Angles and Intercepted Arcs Worksheet Answers
    Inscribed Angles And Intercepted Arcs Worksheet Answers Kareem mould decadently while cubistic Hodge nogged adamantly or mason challengingly. Chymous Durant degenerated plump, he snigglings his dale very absorbingly. Predigested Toddy misalleges no sheik array wild after Remington dieselize impermeably, quite unweeded. What is anintercepted arc is the angle whose sides of circles, and the pink part of the circle having a circle worksheet answers and inscribed angles Expand each of a big sphere that intercepts a few minutes and analyse our free file sharing ebook. ADB is an inscribed angle, then as a whole class, and intercepted arcs. Two inscribed angle and worksheet worksheets for more to find missing angles most often associated with its corresponding arc. The security system for this website has been triggered. Theorem with its intercepted arc lie inside of electrons is called an intercepted arc and inscribed angles intercepted arcs worksheet answers i have learned about angles formed with vertex on that intercepts the same measure of vertical angles. How to use this property to find missing angles? This common endpoint forms the vertex of the inscribed angle. The arc and arcs of its intercepted arcs. About me quiz worksheet. Use this shape can be inside a piece of this quiz worksheet worksheets for all its vertex of one. But by no means are inscribed answers. An angle inscribed in a semicircle is a right angle. Many examples on lawn to find inscribed angle, O is four center of drum circle. Why is inscribed angles and intercepted arcs are they do to answer key source.
    [Show full text]
  • Remote Sensing Fundamentals EM Radiation 2.1 W.D
    Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.1 W.D. Philpot, Cornell University 2. ELECTROMAGNETIC RADIATION Electromagnetic (EM) radiation and its transfer from sources to objects or from objects to sensors is fundamental to remote sensing, and it is important that certain characteristics of this phenomenon be understood. For most of the passive systems that we consider, the sun is the primary source of EM radiation, but we will also consider emitted (thermal) radiation and active systems that provide their on radiation. In any case, EM radiation is energy and that energy might be in wave or particulate (i.e., photon or quantum) form. All electromagnetic radiation has wave properties; at all levels, radiation shows interference and diffraction. But studies also indicate that the energy carried by electromagnetic waves may, under certain conditions, be regarded as discontinuous rather than the continuously graded energy that would be expected from a wave. 2.1 Maxwell's Equations The fundamental description of Electromagnetic radiation begins with Maxwell's equations which describe propagating plane waves. Maxwell's equations are written: In the presence of charge In vacuum (free space) ρ ∇⋅E = ∇⋅E =0 (2.1) ε0 ∇⋅B =0 ∇⋅B =0 (2.2) ∇ ×EB = −∂/ ∂t ∇ ×EB = −∂/ ∂t (2.3) ∇×B =µ0 J+ ε 00 µ ∂ E/ ∂t ∇×BE =ε00 µ ∂/ ∂t (2.4) where: E = electric field B = magnetic-induction field ρe = electric charge density -12 2 2 ε0 = electrical permittivity of free space = 8.85x10 coulomb /Newton-meter -7 µ0 = magnetic permeability of free space = 12.57x10 weber/ampere-meter t = time ∇· = divergence (a spatial vector derivative operator) ∇ = curl (a spatial vector derivative operator) For our purposes, the key point to be gleaned from these equations is the symmetry between the electric and magnetic fields and the fact that they are always coupled.
    [Show full text]
  • Basic Surveying – Theory and Practice 4 Hours
    BASIC SURVEYING – THEORY AND PRACTICE 4 HOURS PDH ACADEMY PO BOX 449 PEWAUKEE, WI 53072 (888) 564-9098 www.pdhacademy.com Final Exam 1. This type of surveying in which the mean surface of the earth is considered as a plane, or in which its spheroidal shape is neglected, with regard to horizontal distances and directions. A. Hydrographic Surveying B. Plane Surveying C. Geodetic Surveying D. Construction Surveying 2. Who were the first known people to use some form of chaining in both land surveying and construction surveying? A. Egyptians B. Americans C. Chinese D. Canadians 3. When level chaining, what is the pointed weight on the end of a string called? A. Pea gun B. Break chaining C. Plumb bob D. Field measurement 4. A Common source of error in chaining is: A. Sag in the chain B. Changes in the temperature of the chain C. Variation in the tension on the chain D. All of the above 5. What decade was the first Electronic Distance Measuring (EDM) equipment developed? A. 1940’s B. 1950’s C. 1960’s D. 1970’s 6. These types of angles, right or left, are measured from an extension of the preceding course and the ahead line. A. Interior Angles B. Exterior Angles C. Deflection Angles D. Angles to the left 7. Which of the following is a type of Meridian? A. False B. Lock C. Saturn D. Magnetic 8. A ______________ is a succession of straight lines along or through the area to be surveyed. The direction and lengths of these lines are determined by measurements taken in the field.
    [Show full text]
  • Geometry Notes Ch 09
    Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates, theorems and corollaries in this chapter. D. Recognize circumscribed and inscribed polygons. E. Prove statements involving circumscribed and inscribed polygons. F. Solve problems involving circumscribed and inscribed polygons. G. Understand and apply theorems related to tangents, radii, arcs, chords, and central angles. Section 9-1 Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18 Objectives A. Understand and apply the terms circle, center, radius, chord, secant, diameter, tangent, point of tangency, and sphere. B. Understand and apply the terms congruent circles, congruent spheres, concentric circles, and concentric spheres. C. Understand and apply the terms inscribed in a circle and circumscribed about a polygon. D. Correctly draw inscribed and circumscribed figures. Circular Logic • On a piece of paper, accurately draw a circle. • What method did you use to make sure you drew a circle? Circle set of all coplanar points that are a given distance (radius) from a given point (center). – Basic Parts: • Radius Distance from the center of a circle to any single point on the circle. • Center Point that is equidistant from all points on the circle. • Indicated by symbol – P Circle with center P • Contrast a circle to a sphere: – Sphere set of all points in space a given distance (radius) from a given point (center) Circle Center Radius Lines and Line Segments Related to Circles Chord segment whose endpoints lie on a circle.
    [Show full text]
  • Section 10: Circles – Part 2
    2 Section 10: Circles – Part Topic 1: Arcs and Inscribed Angles – Part 1 ................................................................................................................ 269 Topic 2: Arcs and Inscribed Angles – Part 2 ................................................................................................................ 271 Topic 3: Inscribed Polygons in a Circle ......................................................................................................................... 274 Topic 4: Constructing Polygons Inscribed in a Circle ................................................................................................ 277 Topic 5: Tangent Lines, Secants, and Chords – Part 1 .............................................................................................. 281 Topic 6: Tangent Lines, Secants, and Chords – Part 2 .............................................................................................. 284 Topic 7: Circumscribed Angles and Beyond – Part 1 ................................................................................................ 288 Topic 8: Circumscribed Angles and Beyond – Part 2 ................................................................................................ 290 Topic 9: Constructing Inscribed and Circumscribed Circles of Triangles ............................................................. 293 Visit MathNation.com or search "Math Nation" in your phone or tablet's app store to watch the videos that go along with this workbook! 267 Section
    [Show full text]
  • Wk 6 Circle Properties Texts and Tasks
    English for Maths I University of Crete ! From: Circle Properties |Maths Educational Wall Chart/Poster in high gloss paper (A1 840mm x 584mm) TASK 1. Read the following description of circle properties and find the odd one out. • The circle is the shape with the largest area for a given length of perimeter. • The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group. • All circles are similar: o A circle's circumference and radius are proportional. o The area enclosed and the square of its radius are disproportional. o The constants of proportionality are 2π and π, respectively. • The circle which is centred at the origin with radius 1 is called the unit circle. o Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. • Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. Chord TASK 2. Complete the missing information from the following text on Chord properties: • Chords are equidistant from the centre of a circe ………………they are equal in length. • The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are: o A perpendicular line from the centre of a circle bisects the chord.
    [Show full text]
  • An Introduction to Modern Cosmology
    An Introduction to Modern Cosmology Third Edition An Introduction to Modern Cosmology Third Edition Andrew Liddle Institute for Astronomy, University of Edinburgh, Abbreviate to ‘UK’ This edition first published 2015 c 2015 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance withthe Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness fora particular purpose.
    [Show full text]
  • Euclidean Parallel Postulate
    Chapter 2 EUCLIDEAN PARALLEL POSTULATE 2.1 INTRODUCTION. There is a well-developed theory for a geometry based solely on the five Common Notions and first four Postulates of Euclid. In other words, there is a geometry in which neither the Fifth Postulate nor any of its alternatives is taken as an axiom. This geometry is called Absolute Geometry, and an account of it can be found in several textbooks - in Coxeter’s book “Introduction to Geometry”, for instance, - or in many college textbooks where the focus is on developing geometry within an axiomatic system. Because nothing is assumed about the existence or multiplicity of parallel lines, however, Absolute Geometry is not very interesting or rich. A geometry becomes a lot more interesting when some Parallel Postulate is added as an axiom! In this chapter we shall add the Euclidean Parallel Postulate to the five Common Notions and first four Postulates of Euclid and so build on the geometry of the Euclidean plane taught in high school. It is more instructive to begin with an axiom different from the Fifth Postulate. 2.1.1 Playfair’s Axiom. Through a given point, not on a given line, exactly one line can be drawn parallel to the given line. Playfair’s Axiom is equivalent to the Fifth Postulate in the sense that it can be deduced from Euclid’s five postulates and common notions, while, conversely, the Fifth Postulate can deduced from Playfair’s Axiom together with the common notions and first four postulates. 2.1.2 Theorem. Euclid’s five Postulates and common notions imply Playfair’s Axiom.
    [Show full text]
  • Unexpected Trigonometric Insights
    The Mathematics Enthusiast Volume 9 Number 1 Numbers 1 & 2 Article 2 1-2012 If not, then what if as well- Unexpected Trigonometric Insights Stanley Barkan Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Barkan, Stanley (2012) "If not, then what if as well- Unexpected Trigonometric Insights," The Mathematics Enthusiast: Vol. 9 : No. 1 , Article 2. Available at: https://scholarworks.umt.edu/tme/vol9/iss1/2 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol9, nos.1&2, p .19 If not, then what if as well? Unexpected Trigonometric Insights Stanley Barkan Oranim Academic College of Education, Israel Abstract In performing an exercise of “What if not”, one can end up with a paucity of structure. Adding alternative structure can be a rich source of discovery, as we present here. The framework of this presentation is the original voyage of discovery, from a trivial geometric problem to the derivation of some unexpected trigonometric formulae based on regular polygons. The original “voyage” has been changed only sufficiently to make the text readable. Keywords: Trigonometric identities; Problem posing; Euclidean Geometry Introduction Consider the following construction problem: ABC is a triangle inscribed in its circumscribing circle. Using a compass, take the measure of one of the triangle's sides, which is also a chord of the circle, and mark off identical chords around the circle, starting at either vertex on the selected chord.
    [Show full text]