Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES XXXIII April 20 (Mon), 2015 Yasuyuki Kachi. 33. Trigonometry III.
|
|
|
- Aubrey Tucker
- 5 years ago
- Views:
Transcription
1 Instructor: Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES XXXIII April 20 Mon, 2015 Yasuyuki Kachi Line #: Trigonometry III In the previous lecture we calculated the distance between P π 4, π 4, Q Now, we can more generally consider the following question: π 6, π 6 Question 1 P Find the distance between,, Q φ, φ where and φ are arbitrary [ Solution PQ ] : φ 2 φ φ φ φ φ φ 2 2 φ 2 φ 2 φ 1
2 Now, exactly as we previously did in the case π 4 and φ π 6, we pose: Question 2 R Find the distance between φ, φ, S 1, 0 Explain why PQ and RS are equal Then use that fact and the result of Question 1 above to write up φ in terms of,, φ and φ [ Solution ] : P and Q are both lying in the unit circle such that the angle POQ equals φ R and S are also both lying in the unit circle such that the angle ROS equals φ Hence PQ and RS are naturally equal Now, RS φ 2 1 φ 2 φ 2 2 φ φ φ φ 2 This equals PQ 2 2 φ 2 φ 2
3 So 2 2 φ 2 2 φ 2 φ Solve this for φ : φ φ φ Let s highlight this result: Axiom 1 φ φ φ Now, we shoot for pulling an analogous identity for φ Toward that goal, working on the following two questions Question 1, Question 2 is the correct step to take: Question 1 P Find the distance between,, Q φ, φ where and φ are arbitrary The answer is, P Q 2 2 φ 2 φ The calculation is similar to Question 1 3
4 Question 2 R Find the distance between φ, φ, S 1, 0 Explain why P Q and R S are equal Then use that fact and the result of Question 1 above to write up φ in terms of,, φ and φ [ Solution ] : You can solve this exactly the same way as Question 2: P and Q are both lying in the unit circle such that the angle P OQ equals π φ R and S are also both lying in the unit circle such that the angle 2 R π OS equals 2 φ Hence P Q and R S are naturally equal Now, R S 2 2 φ The calculation is similar to the calculation of RS This equals P Q 2 2 φ 2 φ So 2 2 φ 2 2 φ 2 φ Solve this for φ : φ φ φ 4
5 Let s pair Axiom 1 which we deduced earlier with the above new result, and highlight them together: Axiom 1 φ φ φ Axiom 2 φ φ φ How about analogs for φ and for φ? For that matter, we rely on Axiom 1 and Axiom 2, and proceed as follows: First, set φ α Thus α φ Then the above two identities are φ αφ φ αφ φ αφ α, φ αφ α These two equations are regarded as ax by p, cx dy q, where a φ, b φ, p α, c φ, d φ, q α, x φ, and y φ Clearly ad bc 1 Thus we rely on the result in Supplement : x dp bq, y cp aq 5
6 That is: αφ αφ φ α φ α φ α, φ α At this point, we are free to replace the letter α with Let s go ahead and make that replacement, and highlight: Axiom 3 φ φ φ Axiom 4 φ φ φ In Axiom 1 and Axiom 2, set φ and we get Double angle formula for Double angle formula for version Double angle formula for version Double angle formula for 2 2 6
7 From Axiom 1 and Axiom 3, we have Formula A 2 φ Formula B 2 φ φ φ φ φ, Similarly, from Axiom 2 and Axiom 4, we have Formula C 2 φ φ φ Today s final topic is lim 0 This is somehow very important Don t ask me yet how so As for this, consider P,, P 0 1, 0, T 1, Assume > 0 T is the unique point lying in the extension of the segment OP whose x-coordinate reading is 1 By drawing the figure, we have the area of POP 0 < < the area of circle sector POP 0 the area of TOP 0 7
8 This translates into 2 < 2 < # 2 Here, we use the fact that the area of the entire disc of radius 1 equals π the middle term is 2π times the area of the entire unit disc From we have < 1, whereas from # we have < In short, < < 1 From this it is inferred that lim 0 1 Indeed, lim To highlight: Formula D lim 0 1 8
9 Technically, this last implication is due to the so-called squeeze theorem The squeeze theorem asserts that, if f x < g x < h x, for a < x < b where a is some negative number, a is some positive number, and moreover if lim f x lim h x, x 0 x 0 then these limits further equal lim g x x 0 9
Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A.
1 Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. Required : To construct an angle at A equal to POQ. 1.
Question Bank Tangent Properties of a Circle
Question Bank Tangent Properties of a Circle 1. In quadrilateral ABCD, D = 90, BC = 38 cm and DC = 5 cm. A circle is inscribed in this quadrilateral which touches AB at point Q such that QB = 7 cm. Find
Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS
Mathematics Revision Guides Vectors Page of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Version:.4 Date: 05-0-05 Mathematics Revision Guides Vectors Page of 9 VECTORS
Class IX Chapter 5 Introduction to Euclid's Geometry Maths
Class IX Chapter 5 Introduction to Euclid's Geometry Maths Exercise 5.1 Question 1: Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can
CIRCLES MODULE - 3 OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE. Circles. Geometry. Notes
Circles MODULE - 3 15 CIRCLES You are already familiar with geometrical figures such as a line segment, an angle, a triangle, a quadrilateral and a circle. Common examples of a circle are a wheel, a bangle,
Free Sample set of investigative questions
Free Sample set of investigative questions SPECIALIST MATHEMATICS: Year Contents Unit Topic : Combinatorics Task : Extended investigation: Relationships in Pascal s Triangle 3 Task : In-class investigation:
Geometry beyond Euclid
Geometry beyond Euclid M.S. Narasimhan IISc & TIFR, Bangalore narasim@math.tifrbng.res.in 1 Outline: Aspects of Geometry which have gone beyond Euclid Two topics which have played important role in development
Full Question Paper Maths, X Class
Full Question Paper Maths, X Class Time: 3 Hrs MM: 80 Instructions: 1. All questions are compulsory. 2. The questions paper consists of 34 questions divided into four sections A,B,C and D. Section A comprises
H2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
Page 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
1 / 24
CBSE-XII-017 EXAMINATION CBSE-X-01 EXAMINATION MATHEMATICS Paper & Solution Time: 3 Hrs. Max. Marks: 90 General Instuctions : 1. All questions are compulsory.. The question paper consists of 34 questions
All E Maths Formulas for O levels E Maths by Ethan Wu
All E Maths Formulas for O levels E Maths by Ethan Wu Chapter 1: Indices a 5 = a x a x a x a x a a m a n = a m + n a m a n = a m n (a m ) n = a m n (ab) n = a n b n ( a b )n = an b n a 0 = 1 a -n = 1 a
Logic, Proof, Axiom Systems
Logic, Proof, Axiom Systems MA 341 Topics in Geometry Lecture 03 29-Aug-2011 MA 341 001 2 Rules of Reasoning A tautology is a sentence which is true no matter what the truth value of its constituent parts.
C-1. Snezana Lawrence
C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary
(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
AEA 2007 Extended Solutions
AEA 7 Extended Solutions These extended solutions for Advanced Extension Awards in Mathematics are intended to supplement the original mark schemes, which are available on the Edexcel website.. (a The
Year 12 into 13 Maths Bridging Tasks
Year 1 into 13 Maths Bridging Tasks Topics covered: Surds Indices Curve sketching Linear equations Quadratics o Factorising o Completing the square Differentiation Factor theorem Circle equations Trigonometry
Regent College. Maths Department. Core Mathematics 4. Vectors
Regent College Maths Department Core Mathematics 4 Vectors Page 1 Vectors By the end of this unit you should be able to find: a unit vector in the direction of a. the distance between two points (x 1,
MA 460 Supplement: Analytic geometry
M 460 Supplement: nalytic geometry Donu rapura In the 1600 s Descartes introduced cartesian coordinates which changed the way we now do geometry. This also paved for subsequent developments such as calculus.
P1 Chapter 11 :: Vectors
P1 Chapter 11 :: Vectors jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 21 st August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
Answer all the questions
SECTION A ( 38 marks) Answer all the questions 1 The following information refer to the set A and set B. Set A = { -3, -2, 2, 3 } Set B = { 4, 9 } The relations between set A and set B is defined by the
SAMPLE QUESTION PAPER 11 Class-X ( ) Mathematics
SAMPLE QUESTION PAPER 11 Class-X (2017 18) Mathematics GENERAL INSTRUCTIONS (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B,C and D. (iii)
Welcome Accelerated Algebra 2!
Welcome Accelerated Algebra 2! U7H3: Worksheet 10.3 #15-22, 24-25, 27-30, 33 Complete on graph paper Updates: U7Q1 will be March 23 rd U7T will be April 3 rd Agenda (1) Warm-Up! (2) Review U7H1 + U7H2
Models of Hyperbolic Geometry
October 23, 2011 Poincaré s Disk Model C O A B N l M Begin with a circle C in the Euclidean plane, and its interior H, as shown in th figure above. The components of this geometry are as follows: Point:
Shape Booster 6 Similar Shapes
Shape Booster 6 Similar Shapes Check: 85T) The two triangles are similar. 5cm y x 37.8cm 8cm 43.2cm a) Work out the size of x. b) Work out the size of y. a) x = 27cm b) y = 7cm Learn: Maths Watch Reference
Conic Sections Session 3: Hyperbola
Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct 2017 1 / 16 Problem 3.1 1 Recall that an ellipse is defined as the locus of points P such that
2012 Canadian Senior Mathematics Contest
The CENTRE for EDUCATION in ATHEATICS and COPUTING cemc.uwaterloo.ca 01 Canadian Senior athematics Contest Tuesday, November 0, 01 (in North America and South America) Wednesday, November 1, 01 (outside
INVERSION IN THE PLANE BERKELEY MATH CIRCLE
INVERSION IN THE PLANE BERKELEY MATH CIRCLE ZVEZDELINA STANKOVA MILLS COLLEGE/UC BERKELEY SEPTEMBER 26TH 2004 Contents 1. Definition of Inversion in the Plane 1 Properties of Inversion 2 Problems 2 2.
5. Introduction to Euclid s Geometry
5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +
Successful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
CBSE Sample Paper-05 (Solved) SUMMATIVE ASSESSMENT I MATHEMATICS Class IX. Time allowed: 3 hours Maximum Marks: 90. Section A.
CBSE Sample Paper-05 (Solved) SUMMATIVE ASSESSMENT I MATHEMATICS Class IX Time allowed: hours Maximum Marks: 90 General Instructions: a) All questions are compulsory. b) The question paper consists of
b) The trend is for the average slope at x = 1 to decrease. The slope at x = 1 is 1.
Chapters 1 to 8 Course Review Chapters 1 to 8 Course Review Question 1 Page 509 a) i) ii) [2(16) 12 + 4][2 3+ 4] 4 1 [2(2.25) 4.5+ 4][2 3+ 4] 1.51 = 21 3 = 7 = 1 0.5 = 2 [2(1.21) 3.3+ 4][2 3+ 4] iii) =
Module 7 : Applications of Integration - I. Lecture 21 : Relative rate of growth of functions [Section 21.1] Objectives
![Module 7 : Applications of Integration - I. Lecture 21 : Relative rate of growth of functions [Section 21.1] Objectives](/thumbs/89/99619893.jpg "Module 7 : Applications of Integration - I. Lecture 21 : Relative rate of growth of functions [Section 21.1] Objectives")
Module 7 : Applications of Integration - I Lecture 21 : Relative rate of growth of functions [Section 211] Objectives In this section you will learn the following : How to compare the rate of growth of
Sum and difference formulae for sine and cosine. Elementary Functions. Consider angles α and β with α > β. These angles identify points on the
Consider angles α and β with α > β. These angles identify points on the unit circle, P (cos α, sin α) and Q(cos β, sin β). Part 5, Trigonometry Lecture 5.1a, Sum and Difference Formulas Dr. Ken W. Smith
Exercise 5.1: Introduction To Euclid s Geometry
Exercise 5.1: Introduction To Euclid s Geometry Email: info@mywayteaching.com Q1. Which of the following statements are true and which are false? Give reasons for your answers. (i)only one line can pass
5.7 Justifying the Laws
SECONDARY MATH III // MODULE 5 The Pythagorean theorem makes a claim about the relationship between the areas of the three squares drawn on the sides of a right triangle: the sum of the area of the squares
MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation
MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between
E Math (4016/01) Total marks : 80. x 1. Solve Answer x = [1]
![E Math (4016/01) Total marks : 80. x 1. Solve Answer x = [1]](/thumbs/87/96795538.jpg "E Math (4016/01) Total marks : 80. x 1. Solve Answer x = [1]")
Requirement : Answer all questions Total marks : 80 Duration : hours x 1. Solve 14 8. 5 8 14 30 x 5 Answer x = [1]. Frank bought an antique vase for $345. One year later he sold it for a profit of 180%
For the intersections: cos x = 0 or sin x = 1 2
Chapter 6 Set-up examples The purpose of this document is to demonstrate the work that will be required if you are asked to set-up integrals on an exam and/or quiz.. Areas () Set up, do not evaluate, any
2. 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
DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS
NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric
CLASS-IX MATHEMATICS. For. Pre-Foundation Course CAREER POINT
CLASS-IX MATHEMATICS For Pre-Foundation Course CAREER POINT CONTENTS S. No. CHAPTERS PAGE NO. 0. Number System... 0 3 0. Polynomials... 39 53 03. Co-ordinate Geometry... 54 04. Introduction to Euclid's
= ( +1) BP AC = AP + (1+ )BP Proved UNIT-9 CIRCLES 1. Prove that the parallelogram circumscribing a circle is rhombus. Ans Given : ABCD is a parallelogram circumscribing a circle. To prove : - ABCD is
Spring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam.
MTH 282 final Spring 2010 Exam 2 Time Limit: 110 Minutes Name (Print): Instructor: Prof. Houhong Fan This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages are
P-CEP. Mathematics. From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet
P-CEP Mathematics From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet Dear future AP Calculus BC student: There are certain math skills that have been
Mathematics 123.3: Solutions to Lab Assignment #1
Mathematics 123.3: Solutions to Lab Assignment #1 2x 2 1 if x= 1/2 (A12) 1 2x 2 = 1 2x 2 if 1/2
Math 142, Final Exam. 12/7/10.
Math 4, Final Exam. /7/0. No notes, calculator, or text. There are 00 points total. Partial credit may be given. Write your full name in the upper right corner of page. Number the pages in the upper right
Definition: A vector is a directed line segment which represents a displacement from one point P to another point Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have
Edexcel GCSE (9-1) Maths for post-16
Edexcel GCSE (9-1) Maths for post-16 Fiona Mapp with Su Nicholson 227227_Front_EDEXCEL.indd 1 02/06/2017 11:34 Contents Introduction 5 How to use this book 13 1 Numbers 14 1.1 Positive and negative numbers
Rectangular Hyperbola Conics HSC Maths Extension 2
Rectangular Hyperbola HSC Maths Etension Question For the y, find: (c) (d) (e) the eccentricity the coordinates of the foci the equations of the directrices the equations of the asymptotes sketch the hyperbola.
Correct substitution. cos = (A1) For substituting correctly sin 55.8 A1
Circular Functions and Trig - Practice Problems (to 07) MarkScheme 1. (a) Evidence of using the cosine rule eg cos = cos Correct substitution eg cos = = 55.8 (0.973 radians) N2 (b) Area = sin For substituting
ieducation.com Tangents given as follows. the circle. contact. There are c) Secant:
h Tangents and Secants to the Circle A Line and a circle: let us consider a circle and line say AB. There can be three possibilities given as follows. a) Non-intersecting line: The line AB and the circle
RMT 2016 Calculus Test Solutions February 20, 2016
RMT 6 Calculus Test Solutions February, 6. Find the volume common to 3 cylinders of radius that lie along the three coordinate axes. Answer: ( Solution: There are many ways to do this problem. Using multi-variable
Complex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
Math 3c Solutions: Exam 2 Fall 2017
Math 3c Solutions: Exam Fall 07. 0 points) The graph of a smooth vector-valued function is shown below except that your irresponsible teacher forgot to include the orientation!) Several points are indicated
Math 152: Affine Geometry
Math 152: Affine Geometry Christopher Eur October 21, 2014 This document summarizes results in Bennett s Affine and Projective Geometry by more or less following and rephrasing Faculty Senate Affine Geometry
1. Number a. Using a calculator or otherwise 1 3 1 5 i. 3 1 4 18 5 ii. 0.1014 5.47 1.5 5.47 0.6 5.1 b. Bus tour tickets . Algebra a. Write as a single fraction 3 4 11 3 4 1 b. 1 5 c. Factorize completely
6.3 More Sine Language
6.3 More Sine Language A Solidify Understanding Task Clarita is helping Carlos calculate his height at different locations around a Ferris wheel. They have noticed when they use their formula h(t) = 30
Cambridge International Examinations Cambridge International General Certificate of Secondary Education
Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 9 1 1 9 9 9 9 1 * ADDITIONAL MATHEMATICS 0606/1 Paper May/June 016 hours Candidates answer on the
Honors 213 / Math 300. Second Hour Exam. Name
Honors 213 / Math 300 Second Hour Exam Name Monday, March 6, 2006 95 points (will be adjusted to 100 pts in the gradebook) Page 1 I. Some definitions (5 points each). Give formal definitions of the following:
Section A.6. Solving Equations. Math Precalculus I. Solving Equations Section A.6
Section A.6 Solving Equations Math 1051 - Precalculus I A.6 Solving Equations Simplify 1 2x 6 x + 2 x 2 9 A.6 Solving Equations Simplify 1 2x 6 x + 2 x 2 9 Factor: 2x 6 = 2(x 3) x 2 9 = (x 3)(x + 3) LCD
Lecture 6 : Induction DRAFT
CS/Math 40: Introduction to Discrete Mathematics /8/011 Lecture 6 : Induction Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we began discussing proofs. We mentioned some proof
1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
CHAPTER 9. Conformal Mapping and Bilinear Transformation. Dr. Pulak Sahoo
CHAPTER 9 Conformal Mapping and Bilinear Transformation BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University of Kalyani West Bengal, India E-mail : sahoopulak@gmail.com Module-3:
Q1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. askiitians
Class: VII Subject: Math s Topic: Properties of triangle No. of Questions: 20 Q1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. Greater
MATHEMATICS Unit Pure Core 2
General Certificate of Education June 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Thursday 15 May 2008 9.00 am to 10.30 am For this paper you must have: an 8-page answer book
Exam Question 10: Differential Equations. June 19, Applied Mathematics: Lecture 6. Brendan Williamson. Introduction.
Exam Question 10: June 19, 2016 In this lecture we will study differential equations, which pertains to Q. 10 of the Higher Level paper. It s arguably more theoretical than other topics on the syllabus,
Mathematics Extension 2 SOLUTIONS
3 HSC Examiatio Mathematics Extesio SOLUIONS Writte by Carrotstics. Multiple Choice. B 6. D. A 7. C 3. D 8. C 4. A 9. B 5. B. A Brief Explaatios Questio Questio Basic itegral. Maipulate ad calculate as
A Pair in a Crowd of Unit Balls
Europ. J. Combinatorics (2001) 22, 1083 1092 doi:10.1006/eujc.2001.0547 Available online at http://www.idealibrary.com on A Pair in a Crowd of Unit Balls K. HOSONO, H. MAEHARA AND K. MATSUDA Let F be a
4 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
Euclidean Geometry Rediscovered
Euclidean Geometry Rediscovered Presenter: John C. Mayer Assistants: William Bond & David Cosper University of Alabama at Birmingham Greater Birmingham Mathematics Partnership Supported by NSF EHR-0632522
So, eqn. to the bisector containing (-1, 4) is = x + 27y = 0
Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y
Pure Core 2. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic
7 Asymptotics for Meromorphic Functions
Lecture G jacques@ucsd.edu 7 Asymptotics for Meromorphic Functions Hadamard s Theorem gives a broad description of the exponential growth of coefficients in power series, but the notion of exponential
Exercise 2. Prove that [ 1, 1] is the set of all the limit points of ( 1, 1] = {x R : 1 <
![Exercise 2. Prove that [ 1, 1] is the set of all the limit points of ( 1, 1] = {x R : 1 <](/thumbs/79/80157367.jpg "Exercise 2. Prove that [ 1, 1] is the set of all the limit points of ( 1, 1] = {x R : 1 <")
Math 316, Intro to Analysis Limits of functions We are experts at taking limits of sequences as the indexing parameter gets close to infinity. What about limits of functions as the independent variable
Notes on Complex Numbers
Notes on Complex Numbers Math 70: Ideas in Mathematics (Section 00) Imaginary Numbers University of Pennsylvania. October 7, 04. Instructor: Subhrajit Bhattacharya The set of real algebraic numbers, A,
Have both a magnitude and direction Examples: Position, force, moment
Force Vectors Vectors Vector Quantities Have both a magnitude and direction Examples: Position, force, moment Vector Notation Vectors are given a variable, such as A or B Handwritten notation usually includes
SULIT 47/ The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. x b ± b 4ac a ALGEBRA 8 log
SULIT 47/ 47/ Matematik Tambahan Kertas ½ jam 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kertas Dua jam tiga puluh minit JANGAN BUKA
Udaan School Of Mathematics Class X Chapter 10 Circles Maths
Exercise 10.1 1. Fill in the blanks (i) The common point of tangent and the circle is called point of contact. (ii) A circle may have two parallel tangents. (iii) A tangent to a circle intersects it in
Vectors. Advanced Math Circle. October 18, 2018
Vectors Advanced Math Circle October 18, 2018 Today we are going to spend some time studying vectors. Vectors are not only ubiquitous in math, and also used a lot in physics and engineering. Vectors are
Mathematics 5 Worksheet 14 The Horizon
Mathematics 5 Worksheet 14 The Horizon For the problems below, we will assume that the Earth is a sphere whose radius is 4,000 miles. Note that there are 5,280 feet in one mile. Problem 1. If a line intersects
Edexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.
Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line
DISCRIMINANT EXAM QUESTIONS
DISCRIMINANT EXAM QUESTIONS Question 1 (**) Show by using the discriminant that the graph of the curve with equation y = x 4x + 10, does not cross the x axis. proof Question (**) Show that the quadratic
Adam Blank Spring 2017 CSE 311. Foundations of Computing I
Adam Blank Spring 2017 CSE 311 Foundations of Computing I Pre-Lecture Problem Suppose that p, and p (q r) are true. Is q true? Can you prove it with equivalences? CSE 311: Foundations of Computing Lecture
Integrated Mathematics I, II, III 2016 Scope and Sequence
Mathematics I, II, III 2016 Scope and Sequence I Big Ideas Math 2016 Mathematics I, II, and III Scope and Sequence Number and Quantity The Real Number System (N-RN) Properties of exponents to rational
Math 142, Final Exam, Fall 2006, Solutions
Math 4, Final Exam, Fall 6, Solutions There are problems. Each problem is worth points. SHOW your wor. Mae your wor be coherent and clear. Write in complete sentences whenever this is possible. CIRCLE
Strand 2 of 5. 6 th Year Maths Ordinary Level. Topics: Trigonometry Co-ordinate Geometry of the Line Co-ordinate Geometry of the Circle Geometry
6 th Year Maths Ordinary Level Strand 2 of 5 Topics: Trigonometry Co-ordinate Geometry of the Line Co-ordinate Geometry of the Circle Geometry No part of this publication may be copied, reproduced or transmitted
MATH 2200 Final Review
MATH 00 Final Review Thomas Goller December 7, 01 1 Exam Format The final exam will consist of 8-10 proofs It will take place on Tuesday, December 11, from 10:30 AM - 1:30 PM, in the usual room Topics
Course 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
Sec 4 Maths. SET A PAPER 2 Question
S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark
G-C, G-CO Tangent Lines and the Radius of a Circle
G-C, G-CO Tangent ines and the Radius of a Circle Task O P P P Consider a circle with center and let be a point on the circle. Suppose is a tangent line to the circle at, that is meets the circle only
cbse.jagranjosh.com CBSE Class 10 th Mathematics Solved Guess Paper
Solved Guess Paper 6 Mathematics Class X SA II SET I Time Allowed: hours Maximum Marks: 9 General Instructions:. All questions are compulsory..the question paper consists of questions divided into four
Paper reference. 5505/05 Edexcel GCSE Mathematics A 1387 Paper 5 (Non-Calculator) Friday 5 November 2004 Morning Time: 2 hours
Centre No Paper reference Candidate No 5 5 0 5 / 0 5 Surname Signature Initial(s) Paper References(s) 5505/05 Edexcel GCSE Mathematics A 1387 Paper 5 (Non-Calculator) Higher Tier Friday 5 November 2004
National 5 Course Notes. Scientific Notation (or Standard Form) This is a different way of writing very large and very small numbers in the form:-
National 5 Course Notes Scientific Notation (or Standard Form) This is a different way of writing very large and very small numbers in the form:- a x 10 n where a is between 1 and 10 and n is an integer
Maharashtra State Board Class IX Mathematics Geometry Board Paper 1 Solution
Maharashtra State Board Class IX Mathematics Geometry Board Paper Solution Time: hours Total Marks: 40. i. Let the measure of each interior opposite angle be x. Since, Sum of two interior opposite angles
Math 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
Math 164-1: Optimization Instructor: Alpár R. Mészáros
Math 164-1: Optimization Instructor: Alpár R. Mészáros First Midterm, April 20, 2016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By writing
Math Review for AP Calculus
Math Review for AP Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
MATH20411 PDEs and Vector Calculus B
MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411