A Brief History of the Approximation of π

 Ashlie Gaines
 8 months ago
 Views:
Transcription
1 A Brief History of the Approximation of π Communication Presentation Nicole Clizzie Orion ThompsonVogel April 3, 2017
2 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
3 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
4 Archimedes (287 BC212 BC) Archimedes uses Euclid s Theorem Theorem Euclid s Theorem If a straight line bisects an angle of a triangle and cuts the base then the resulting segments of the base have the same ratio as the remaining sides of the triangle. Let s now prove this theorem.
5 Archimedes (287 BC212 BC) Archimedes uses Euclid s Theorem Proof. If we have AOC and OD bisects AOC, then OC OA = CD DA. So the first step is to draw AOC and subsequently draw in the angle bisector OD. Afterwards, we may add a point E such that it forms a line EC, which is parallel to OD, and a line AE, where AE is an extension of AO. Here we can reason a few things:
6 Archimedes (287 BC212 BC) Archimedes uses Euclid s Theorem Proof. m DOA=m CEA since they re corresponding angles m OCE+m CEO+m EOC=180 as these form a triangle m AOD+m DOC+m EOC=180 since they re supplementary m DOC+m AOD=m OCE+m CEO with substitution COE is isosceles by definition and using AA similarity theorem, DOA is similar to CEA so
7 Archimedes (287 BC212 BC) Archimedes uses Euclid s Theorem Proof. CA AE = DA AO CA DA = AE AO AO+OE AO = CD+DA DA OE AO = CD DA CO AO = CD DA and since CA=CD+DA and AE=AO+OE
8 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
9 Archimedes (287 BC212 BC) Let O be the center and AB be the diameter while AC is tangent at A. Let m AOC be 30 degrees or π 6 radians. So note CF = 2 CA where CF is a side of a circumscribed hexagon. Also note that where n is the number of sides, n AC Perimeter of the polygon OA Diameter = π. Now let s bisect m AOC, and using Euclid s theorem, CD DA = CO OA we see CD+DA DA = CO+OA OA = CA DA CO+OA CA = OA DA and so (i) CO CA + OA CA = OA AD.
10 Archimedes (287 BC212 BC) Let O be the center and AB be the diameter while AC is tangent at A. Let m AOC be 30 degrees or π 6 radians. So note CF = 2 CA where CF is a side of a circumscribed hexagon. Also note that where n is the number of sides, n AC Perimeter of the polygon OA Diameter = π. Now let s bisect m AOC, and using Euclid s theorem, CD DA = CO OA we see CD+DA DA = CO+OA OA = CA DA CO+OA CA = OA DA and so (i) CO CA + OA CA = OA AD.
11 Archimedes (287 BC212 BC) Now using Pythagorean Theorem: OA 2 + AD 2 = OD 2 OA2 + 1 = AD OD2 2 DA 2 OA (ii) = OD AD 2 DA. Archimedes then approximates 3 so OA AC = 3 since OA AC = cot( π 6 ) so OA AC >
12 Archimedes (287 BC212 BC) Now using Pythagorean Theorem: OA 2 + AD 2 = OD 2 OA2 + 1 = AD OD2 2 DA 2 OA (ii) = OD AD 2 DA. Archimedes then approximates 3 so OA AC = 3 since OA AC = cot( π 6 ) so OA AC > also note that OC CA = 2 1 = as for us this is 1 sin( π 6 (i) we have < OA AD < OA AD ) and using
13 Archimedes (287 BC212 BC) Let s find OD DA using (ii). So since we know something about OA AD we can plug this in knowing it is less than the actual value so we get OA2 + 1 > AD and OA > AD which Archimedes reduces to OD AD =
14 Archimedes (287 BC212 BC) And now the next step is to bisect AOD by OE, and using the same approach as before, we have OA and so OA AE > = AE = OD DA + OA DA 153 and so this gives us a 12gon s approximation. [4] We can continue this until we have a 96gon, which is what Archimedes did, where he achieved an upperbound of and a lower bound of [8]
15 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
16 Francois Viete ( ) First Infinite Series for π Lawyer in France who worked for King Henry III and Henry IV Dabbled in quite a bit of mathematics, as in 1593 Adriaan van Roomen, a mathematician from the Netherlands, challenged french mathematicians with a problem posed about a 45 degree polynomial. An ambassador from the Netherlands commented on the poor quality of french mathematicians as none could solve this problem, however, this prompted King Henry to present the problem to Viete, who found a solution on the first day and 22 more the next day. First infinite expansion of π 2 π = A variation of Archimedes but instead of using polygons perimeters he used the areas. [5]
17 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
18 Gottfried Leibniz ( ), James Gregory ( ), and Nilakantha LiebnizGregory Series LiebnizGregory Series π = 4( ) So we may quickly prove this case using the Taylor series. The Taylor series is the following: 1 1 y = 1 + y + y and substituting in y = x 2 we get 1 1+x 2 = 1 x 2 + x 4... and note that the left hand side is the same as d dx tan(x) 1 and so if we integrate both sides we have tan(x) 1 = x x3 3 + x and if we plugin x=1, and multiply the right side by 4, we have our result. [10]
19 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
20 John Machin ( ) Machin Series π 4 = artan 1 5 arctan so we can derive this as follows: let tan(α)=1/5 where alpha is some angle and now we see using the double angle formula tan(2α)= 2 tan(α) 1 tan 2 (α) = = 5 12 and repeating again tan(4α)= 2 tan(2α) 1 tan 2 (2α) = 5 6 = Now note that this differs from 1 by only a tiny amount, and so we can examine the following: tan(4α π 4 ) = tan(4α) tan( π 4 tan(4α)+tan( π 4 taking the arctan of both sides gives 4α pi 4 = arctan = tan(4α) 1 tan(4α+1) = = and which simplified to the result above. [3]
21 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha LeibnizGregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π
22 Srinivasa Ramanujan ( ) Ramanujan s Approximation of π Ben Lynn from Stanford University stated Its my favourite formula for pi. I have no idea how it works. 1 π = 8 (4n)! 9801 n= n+1103 (n!) n The formula involved modular equations. [2]
23 Summary A good approximation of π was first done by Archimedes, and all that have followed suit have involved infinite series. Some approximations are better than others due to their convergence rate. The mysterious Ramanujan crafted the essence of the best π approximation formula to date.
24 Bibliography I Ben, Lynn. The GregoryLeibniz Series, Standford. Web 2 Apr Ben, Lynn. Ramanujans Formula for Pi, Standford. Web 2 Apr Derivation of the Machin Formula. Arctan Formulae for Computing Pi. Southwestern Adventist University. Web. 2 Apr Han, Kyutae Paul.Pi and Archimedes Polygon Method, Dartmouth College. Web 28 Apr Hartshorne, Robin. Francois VieteLife. Mathematicans. Web 2 Apr O Connor, J. J., and E. F. Robertson, John Machin, Historical Topics, School of Mathematics and Statistics University of St. Andrews, April Web 2 Apr
25 Bibliography II O Connor, J. J., and E. F. Robertson, Gottfried Wilhelm von Liebniz, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Oct Web 2 Apr O Connor, J. J., and E. F. Robertson, Archimedes of Syracuse, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Jan Web 2 Apr O Connor, J. J., and E. F. Robertson, Srinivasa Aiyangar Ramanujan, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Jun Web 2 Apr Roy, Ranjan. The Discovery of the Series Formula of Pi by Leibniz, Gregory, and Nilakatha. Mathematics Magazine Dec. 1990: Web 2 Apr
Properties of the Circle
9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference
More informationGeometry Facts Circles & Cyclic Quadrilaterals
Geometry Facts Circles & Cyclic Quadrilaterals Circles, chords, secants and tangents combine to give us many relationships that are useful in solving problems. Power of a Point Theorem: The simplest of
More information1 Solution of Final. Dr. Franz Rothe December 25, Figure 1: Dissection proof of the Pythagorean theorem in a special case
Math 3181 Dr. Franz Rothe December 25, 2012 Name: 1 Solution of Final Figure 1: Dissection proof of the Pythagorean theorem in a special case 10 Problem 1. Given is a right triangle ABC with angle α =
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationHigher Geometry Problems
Higher Geometry Problems (1) Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationChapter (Circle) * Circle  circle is locus of such points which are at equidistant from a fixed point in
Chapter  10 (Circle) Key Concept * Circle  circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle  Circle having same centre called concentric circle.
More informationChapter 3 Cumulative Review Answers
Chapter 3 Cumulative Review Answers 1a. The triangle inequality is violated. 1b. The sum of the angles is not 180º. 1c. Two angles are equal, but the sides opposite those angles are not equal. 1d. The
More informationClass : IX(CBSE) Worksheet  1 Sub : Mathematics Topic : Number system
Class : IX(CBSE) Worksheet  Sub : Mathematics Topic : Number system I. Solve the following:. Insert rational numbers between. Epress 57 65 in the decimal form. 8 and.. Epress. as a fraction in the simplest
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &3D) AND CALCULUS. TIME : 3hrs Ma. Marks.75 Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.
More informationKENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 01 (201718) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationConstructing Trig Values: The Golden Triangle and the Mathematical Magic of the Pentagram
Constructing Trig Values: The Golden Triangle and the Mathematical Magic of the Pentagram A Play in Five Acts 1 "Mathematicians always strive to confuse their audiences; where there is no confusion there
More informationThe CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Wednesday, April 15, 2015
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 015 Euclid Contest Wednesday, April 15, 015 (in North America and South America) Thursday, April 16, 015 (outside of North America
More informationPractice Test Geometry 1. Which of the following points is the greatest distance from the yaxis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E.
April 9, 01 Standards: MM1Ga, MM1G1b Practice Test Geometry 1. Which of the following points is the greatest distance from the yaxis? (1,10) B. (,7) C. (,) (,) (,1). Points P, Q, R, and S lie on a line
More informationarxiv: v1 [math.ho] 29 Nov 2017
The Two Incenters of the Arbitrary Convex Quadrilateral Nikolaos Dergiades and Dimitris M. Christodoulou ABSTRACT arxiv:1712.02207v1 [math.ho] 29 Nov 2017 For an arbitrary convex quadrilateral ABCD with
More informationINMO2001 Problems and Solutions
INMO2001 Problems and Solutions 1. Let ABC be a triangle in which no angle is 90. For any point P in the plane of the triangle, let A 1,B 1,C 1 denote the reflections of P in the sides BC,CA,AB respectively.
More informationChapter 06: Analytic Trigonometry
Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is onetoone. Are any of our trigonometric
More informationAbhilasha Classses. Class X (IX to X Moving) Date: MM 150 Mob no (SetAAA) Sol: Sol: Sol: Sol:
Class X (IX to X Moving) Date: 06 MM 0 Mob no. 97967 Student Name... School.. Roll No... Contact No....... If = y = 8 z and + + =, then the y z value of is (a) 7 6 (c) 7 8 [A] (b) 7 3 (d) none of these
More informationPage 1 of 15. Website: Mobile:
Exercise 10.2 Question 1: From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5
More informationπ is a mathematical constant that symbolizes the ratio of a circle s circumference to its
Ziya Chen Math 4388 Shanyu Ji Origin of π π is a mathematical constant that symbolizes the ratio of a circle s circumference to its diameter, which is approximately 3.14159265 We have been using this symbol
More informationMath 9 Chapter 8 Practice Test
Name: Class: Date: ID: A Math 9 Chapter 8 Practice Test Short Answer 1. O is the centre of this circle and point Q is a point of tangency. Determine the value of t. If necessary, give your answer to the
More informationGeometry: Introduction, Circle Geometry (Grade 12)
OpenStaxCNX module: m39327 1 Geometry: Introduction, Circle Geometry (Grade 12) Free High School Science Texts Project This work is produced by OpenStaxCNX and licensed under the Creative Commons Attribution
More informationEuclidean Geometry Proofs
Euclidean Geometry Proofs History Thales (600 BC) First to turn geometry into a logical discipline. Described as the first Greek philosopher and the father of geometry as a deductive study. Relied on rational
More informationConcurrency and Collinearity
Concurrency and Collinearity Victoria Krakovna vkrakovna@gmail.com 1 Elementary Tools Here are some tips for concurrency and collinearity questions: 1. You can often restate a concurrency question as a
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationHigh School Math Contest
High School Math Contest University of South Carolina February 4th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) 4 (e) 48 Problem. Suppose the function g(x) = f(x)
More informationMath 4388 Amber Pham 1. The Birth of Calculus. for counting. There are two major interrelated topics in calculus known as differential and
Math 4388 Amber Pham 1 The Birth of Calculus The literal meaning of calculus originated from Latin, which means a small stone used for counting. There are two major interrelated topics in calculus known
More informationLabel carefully each of the following:
Label carefully each of the following: Circle Geometry labelling activity radius arc diameter centre chord sector major segment tangent circumference minor segment Board of Studies 1 These are the terms
More informationAnalysis and synthesis (and other peculiarities): Euclid, Apollonius. 2 th March 2014
Analysis and synthesis (and other peculiarities): Euclid, Apollonius 2 th March 2014 What is algebra? Algebra (today): Advanced level : Groups, rings,..., structures; Elementary level : equations. The
More information4 Arithmetic of Segments Hilbert s Road from Geometry
4 Arithmetic of Segments Hilbert s Road from Geometry to Algebra In this section, we explain Hilbert s procedure to construct an arithmetic of segments, also called Streckenrechnung. Hilbert constructs
More informationMODEL QUESTION PAPERS WITH ANSWERS SET 1
MTHEMTICS MODEL QUESTION PPERS WITH NSWERS SET 1 Finish Line & Beyond CLSS X Time llowed: 3 Hrs Max. Marks : 80 General Instructions: (1) ll questions are compulsory. (2) The question paper consists of
More informationSlopes, Derivatives, and Tangents. Matt Riley, Kyle Mitchell, Jacob Shaw, Patrick Lane
Slopes, Derivatives, and Tangents Matt Riley, Kyle Mitchell, Jacob Shaw, Patrick Lane S Introduction Definition of a tangent line: The tangent line at a point on a curve is a straight line that just touches
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationSUMMATIVE ASSESSMENT I, IX / Class IX
I, 0 SUMMATIVE ASSESSMENT I, 0 0 MATHEMATICS / MATHEMATICS MATHEMATICS CLASS CLASS  IX  IX IX / Class IX MA0 90 Time allowed : hours Maximum Marks : 90 (i) (ii) 8 6 0 0 (iii) 8 (iv) (v) General Instructions:
More information1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle?
1 For all problems, NOTA stands for None of the Above. 1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle? (A) 40 (B) 60 (C) 80 (D) Cannot be determined
More informationMathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions
Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions Quiz #1. Tuesday, 17 January, 2012. [10 minutes] 1. Given a line segment AB, use (some of) Postulates I V,
More informationMarking Scheme. Mathematics Class X ( ) Section A
Marking Scheme Mathematics Class X (01718) Section A S.No. Answer Marks 1. Non terminating repeating decimal expansion.. k ±4 3. a 11 5 4. (0, 5) 5. 9 : 49 6. 5 Section B 7. LCM (p, q) a 3 b 3 HCF (p,
More information6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities
Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of
More informationCircle Theorems Standard Questions (G10)
Circle Theorems Standard Questions (G10) Page 1 Q1.(a) A, B and C are points on the circumference of a circle with centre O. Not drawn accurately Work out the size of angle x. (1) Page 2 (b) P, Q and R
More information2001 Solutions Euclid Contest (Grade 12)
Canadian Mathematics Competition An activity of The Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 001 s Euclid Contest (Grade 1) for The CENTRE for EDUCATION
More informationFROSHSOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2017 STATE DIVISION AA 1. Determine the sum of all distinct positive integers between 8 and 16 inclusive that can be expressed in one and only
More informationStarting with the base and moving counterclockwise, the measured side lengths are 5.5 cm, 2.4 cm, 2.9 cm, 2.5 cm, 1.3 cm, and 2.7 cm.
Chapter 6 Geometric Vectors Chapter 6 Prerequisite Skills Chapter 6 Prerequisite Skills Question 1 Page 302 Starting with the base and moving counterclockwise, the measured side lengths are 5.5 cm, 2.4
More informationThe Computation of π by Archimedes. Bill McKeeman Dartmouth College
The Computation of π by Archimedes Bill McKeeman Dartmouth College 2012.02.15 Abstract It is famously known that Archimedes approximated π by computing the perimeters of manysided regular polygons, one
More informationCCE RR. ( / English Version ) ( / New Syllabus ) ( / Regular Repeater )
CCE RR 1 81E : 81E Code No. : 81E CCE RR Subject : MATHEMATICS ( / English Version ) ( / New Syllabus ) ( / Regular Repeater ) General Instructions : i) The QuestioncumAnswer Booklet consists of objective
More informationEuclidian Geometry Grade 10 to 12 (CAPS)
Euclidian Geometry Grade 10 to 12 (CAPS) Compiled by Marlene Malan marlene.mcubed@gmail.com Prepared by Marlene Malan CAPS DOCUMENT (Paper 2) Grade 10 Grade 11 Grade 12 (a) Revise basic results established
More informationIt is known that the length of the tangents drawn from an external point to a circle is equal.
CBSE MATHSSET 12014 Q1. The first three terms of an AP are 3y1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More informationGEOMETRICAL DEFINITION OF π AND ITS APPROXIMATIONS BY NESTED RADICALS. Milan Janjić. 1. Introduction
THE TECHING OF THETICS 008, Vol. XI,, pp. 5 34 GEOETRICL DEFINITION OF π ND ITS PPROXITIONS Y NESTED RDICLS ilan Janjić bstract. In this paper the length of the arc of a circle and the area of a circular
More informationy= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.
. (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the
More informationSquaring of circle and arbelos and the judgment of arbelos in choosing the real Pi value (Bhagavan Kaasi Visweswar method)
IOSR Journal of Engineering (IOSRJEN) ISSN (e): 50301, ISSN (p): 788719 Vol. 0, Issue 07 (July. 01), V3 PP 6370 www.iosrjen.org Squaring of circle and arbelos and the judgment of arbelos in choosing
More informationEXERCISE 10.1 EXERCISE 10.2
NCERT Class 9 Solved Questions for Chapter: Circle 10 NCERT 10 Class CIRCLES 9 Solved Questions for Chapter: Circle EXERCISE 10.1 Q.1. Fill in the blanks : (i) The centre of a circle lies in of the circle.
More informationPOLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL. A real number α is a root or zero of polynomial f(x) = x + a x + a x +...
POLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL n n 1 n an n 1 n 1 + 0 A real number α is a root or zero of polynomial f(x) = x + a x + a x +... + a x a, n n an n 1 n 1 0 = if f (α) = 0. i.e. α + a + a
More informationCore Mathematics 3 Trigonometry
Edexcel past paper questions Core Mathematics 3 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure
More informationHigh School Math Contest
High School Math Contest University of South Carolina February th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) (e) 8 Solution: Note that xy = (x + y) (x y) = 169
More information2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016
6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge
More informationweebly.com/ Core Mathematics 3 Trigonometry
http://kumarmaths. weebly.com/ Core Mathematics 3 Trigonometry Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure and had to find areas of sectors and segments.
More informationMATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 TaRight Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
More informationCLASS X FORMULAE MATHS
Real numbers: Euclid s division lemma Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 r < b. Euclid s division algorithm: This is based on Euclid s division
More informationTRIANGLES CHAPTER 7. (A) Main Concepts and Results. (B) Multiple Choice Questions
CHAPTER 7 TRIANGLES (A) Main Concepts and Results Triangles and their parts, Congruence of triangles, Congruence and correspondence of vertices, Criteria for Congruence of triangles: (i) SAS (ii) ASA (iii)
More informationProblem 1. Solve the equation 3 x + 9 x = 27 x. Solution: 3 x + 3 2x = 3 3x. Denote: y = 3 x, then. y + y 2 = y 3. y 3 y 2 y = 0. y(y 2 y 1) = 0.
Problem 1. Solve the equation 3 x + 9 x = 7 x. Solution: 3 x + 3 x = 3 3x. Denote: y = 3 x, then Therefore, y + y = y 3. y 3 y y = 0. y(y y 1) = 0. y = 0 or y = 1 ± 5. i) 3 x = 0 has no solutions, ii)
More informationConstructing Taylor Series
Constructing Taylor Series 88200 The Taylor series for fx at x = c is fc + f cx c + f c 2! x c 2 + f c x c 3 + = 3! f n c x c n. By convention, f 0 = f. When c = 0, the series is called a Maclaurin series.
More information2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is
. If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time
More informationMathematics, Algebra, and Geometry
Mathematics, Algebra, and Geometry by Satya http://www.thesatya.com/ Contents 1 Algebra 1 1.1 Logarithms............................................ 1. Complex numbers........................................
More informationTOPIC1 REAL NUMBERS. Rational Numbers WORKSHEET1 WORKSHEET2 P1 SECTION. \ x = [CBSE Marking Scheme, 2012] S O L U T I O N S CHAPTER
SECTION CHAPTER REAL NUMERS TOPIC Rational Numbers WORKSHEET.. 98 98 9 So, it is a rational number. 9 9.. Yes, zero is a rational number. Zero can be epressed as 0 0, 0 6, 00 etc, which are in the form
More informationChapter 8 Similar Triangles
Chapter 8 Similar Triangles Key Concepts:.A polygon in which all sides and angles are equal is called a regular polygon.. Properties of similar Triangles: a) Corresponding sides are in the same ratio b)
More informationLLT Education Services
8. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. (a) 4 cm (b) 3 cm (c) 6 cm (d) 5 cm 9. From a point P, 10 cm away from the
More informationUnderstand and Apply Theorems about Circles
UNIT 4: CIRCLES AND VOLUME This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,
More informationChapter 7 Trigonometric Identities and Equations 71 Basic Trigonometric Identities Pages
Trigonometric Identities and Equations 7 Basic Trigonometric Identities Pages 47 430. Sample answer: 45 3. tan, cot, cot tan cos cot, cot csc 5. Rosalinda is correct; there may be other values for which
More informationSUMMATIVE ASSESSMENT  I (2012) MATHEMATICS CLASS IX. Time allowed : 3 hours Maximum Marks :90
SUMMATIVE ASSESSMENT  I (2012) MATHEMATICS CLASS IX Time allowed : 3 hours Maximum Marks :90 General Instructions: i. All questions are compulsory. ii. The question paper consists of 34 questions divided
More informationLesson 2B: Thales Theorem
Lesson 2B: Thales Theorem Learning Targets o I can identify radius, diameter, chords, central circles, inscribed circles and semicircles o I can explain that an ABC is a right triangle, then A, B, and
More information( 3x 2 y) 6 (6x 3 y 2 ) x 4 y 4 b.
1. Simplify 3 x 5 4 64x Algebra Practice Problems for MDPT Pre Calculus a. 1 18x 10 b. 7 18x 7 c. x 6 3x d. 8x 1 x 4. Solve 1 (x 3) + x 3 = 3 4 (x 1) + 1 9 a. 77 51 b. 3 17 c. 3 17 d. 3 51 3. Simplify
More informationClass 9 Quadrilaterals
ID : in9quadrilaterals [1] Class 9 Quadrilaterals For more such worksheets visit www.edugain.com Answer t he quest ions (1) The diameter of circumcircle of a rectangle is 13 cm and rectangle's width
More informationCOORDINATE GEOMETRY LOCUS EXERCISE 1. The locus of P(x,y) such that its distance from A(0,0) is less than 5 units is x y 5 ) x y 10 x y 5 4) x y 0. The equation of the locus of the point whose distance
More informationMath 3 Unit Skills Checklist
Unit 1 Modeling with Statistics Use Normal Distributions Math 3 Unit Skills Checklist Describe the characteristics of a standard normal curve. Use the mean and standard deviation of a data set to fit it
More informationSURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS
SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH  017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION 
More informationBOARD QUESTION PAPER : MARCH 2016 GEOMETRY
BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential
More information17. The length of a diagonal of a square is 16 inches. What is its perimeter? a. 8 2 in. b in. c in. d in. e in.
Geometry 2 nd Semester Final Review Name: 1. Pentagon FGHIJ pentagon. 2. Find the scale factor of FGHIJ to KLMNO. 3. Find x. 4. Find y. 5. Find z. 6. Find the scale factor of ABCD to EFGD. 7. Find the
More informationThe Golden Ratio and Viète s Formula
/ (04), 43 54 The Golden Ratio and Viète s Formula Esther M. García Caballero, Samuel G. Moreno and Michael P. Prophet Abstract. Viète s formula uses an infinite product to express π. In this paper we
More informationThe analysis method for construction problems in the dynamic geometry
The analysis method for construction problems in the dynamic geometry Heechan Lew Korea National University of Education SEMEORECSAM University of Tsukuba of Tsukuba Joint Seminar Feb. 15, 2016, Tokyo
More informationSolving equations UNCORRECTED PAGE PROOFS
1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1.3 Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal equations and simultaneous equations 1.6 Review
More informationRitangle  an A Level Maths Competition 2016
Ritangle  an A Level Maths Competition 2016 Questions and Answers  121216 A. The Taster Questions Answer: this sequence cycles. The first eight terms are, r, i, t, a, n, g, l, e, 1 while the ninth
More informationBowling Balls and Binary Switches
Andrew York April 11, 2016 Back in the old programming days, my favorite line of code was "ifthen." If some condition is met, then do something specific  incredibly useful for making things happen exactly
More information0609ge. Geometry Regents Exam AB DE, A D, and B E.
0609ge 1 Juliann plans on drawing ABC, where the measure of A can range from 50 to 60 and the measure of B can range from 90 to 100. Given these conditions, what is the correct range of measures possible
More informationGeometry 3 SIMILARITY & CONGRUENCY Congruency: When two figures have same shape and size, then they are said to be congruent figure. The phenomena between these two figures is said to be congruency. CONDITIONS
More informationCollege Algebra with Trigonometry
College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (556 topics + 614 additional
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationMathematics Project. Class:10 Date of submission :
Mathematics Project Class:10 Date of submission : 090711 General Instructions:  The project should be hand written in about 58 A4 size sheets  Credit will be given to original and creative use of
More informationMaharashtra State Board Class X Mathematics Geometry Board Paper 2015 Solution. Time: 2 hours Total Marks: 40
Maharashtra State Board Class X Mathematics Geometry Board Paper 05 Solution Time: hours Total Marks: 40 Note: () Solve all questions. Draw diagrams wherever necessary. ()Use of calculator is not allowed.
More informationEdexcel GCSE Mathematics (1387) Higher Tier Model Answers
Edexcel GCSE Mathematics (387) Higher Tier 003 Model Answers In general, the number of significant figures in an answer should not exceed the number of significant figures in the input data, or if this
More informationDESIGN OF THE QUESTION PAPER
SETII DESIGN OF THE QUESTION PAPER MATHEMATICS CLASS IX Time : 3 Hours Maximum Marks : 80 The weightage or the distribution of marks over different dimensions of the question paper shall be as follows:
More information2012 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (21) AADBD (26) BCDCD SOLUTIONS
01 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (1) AADBD (6) BCDCD SOLUTIONS 1. Noting that x = ( x + )( x ), we have circle is π( x +
More informationThe number π. Rajendra Bhatia. December 20, Indian Statistical Institute, Delhi India 1 / 78
1 / 78 The number π Rajendra Bhatia Indian Statistical Institute, Delhi India December 20, 2012 2 / 78 CORE MATHEMATICS Nature of Mathematics Reasoning and Proofs in Mathematics History of Mathematics
More informationSMT 2018 Geometry Test Solutions February 17, 2018
SMT 018 Geometry Test Solutions February 17, 018 1. Consider a semicircle with diameter AB. Let points C and D be on diameter AB such that CD forms the base of a square inscribed in the semicircle. Given
More informationGeometry Advanced Fall Semester Exam Review Packet  CHAPTER 1
Name: Class: Date: Geometry Advanced Fall Semester Exam Review Packet  CHAPTER 1 Multiple Choice. Identify the choice that best completes the statement or answers the question. 1. Which statement(s)
More informationClassical Theorems in Plane Geometry 1
BERKELEY MATH CIRCLE 1999 2000 Classical Theorems in Plane Geometry 1 Zvezdelina StankovaFrenkel UC Berkeley and Mills College Note: All objects in this handout are planar  i.e. they lie in the usual
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment II. Revision CLASS X Prepared by
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationSuggested problems  solutions
Suggested problems  solutions Parallel lines Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 4.1, pp 219223.
More information