COMPLEX NUMBERS. Powers of j. Consider the quadratic equation; It has no solutions in the real number system since. j 1 ie. j 2 1
|
|
- Kelly Higgins
- 5 years ago
- Views:
Transcription
1 COMPLEX NUMBERS Consider the quadratic equation; x 0 It has no solutions in the real number system since x or x j j ie. j Similarly x 6 0 gives x 6 or x 6 j Powers of j j j j j. j j j j.j jj j j 5 j j j j 6 j. j j j 7 j. j j Clearly all the powers of j can be described by j, j, j, j. e.g. j 07 j 0. j j 576. j j
2 General form of complex numbers Consider now the quadratic equation x x 0 0 b ac 0 x 6 6 j j This leads us t oa general form for complex numbers z a bj a and b are real a real part of z Rez b imaginary part of z Imz In the previous example; roots z j, hence a, b For complex numbers : j,, 7j, 6j are all special cases a bj is purely real if b 0, a. a bj is purely imaginary if a 0, bj. Basic Rules of algebra: Consider : z a b j, z a b j Sum : z z a a b b j Difference : z z a a b b j Example : for : z j and : z 7j Product : z z 7j j z z 7j 7 j z. z a b ja b j a a a b j b a j b b j a a b b a b a b j Example : if z j and z j find z z z z z z z j j 5j j 7j
3 Complex Conjugate: For any complex number a bj there corresponds a complex number a bj obtained by changing the sign of the "imarigary part". a bj is the Complex Conjugate of a bj. Notation : If z a bj then z a bj Clearly and z z _ a z z bj, hence a z z_ Rez b z z_ Imz Also zz a bja bj a j b, hence zz a b, Quotient : To find z z where z a b j, z a b j z z z z z z z z z z Complex number in general form NB : to find the quotient z ie. by z z a b j a b j a b j a b j a a b b b a a b j a b j a a b b a b a b j a b a b in general a bj form we multiply above and below by the complex conjugate of the denominator
4 Equality Let and z a b j z a b j then z z when a a and b b ie and Rez Rez Imz Imz Complex Number zero : 0 0j a 0 and b 0 ie. both real part and imaginary part are zero. Example : find the real numbers Α and Β such that z z 0, given that z Αj and z Β 5j. z z Β Α 5j 0 if Β 0 or Β and Α 5 0 or Α 5 For complex numbers, z, z and z : Commutative property : z z z z and z z z z Associative property: z z z z z z and z z z z z z Distributive property : z z z z z z z Geometrical Representation: A complex number a b j can be represented either as a point a, b or as a position vector of a, b in the x y plane complex plane or z plane. x axis is called real axis y axis is called imaginary axis. This geographical representation is called Argand Diagram Example :
5 Modulus of a complex number Let z a bj, the modulus of z is denoted by z and z a b Example: For z j z z length of the vector for z and is always 0 Also zz a bja bj a b z Polar form of z Point x, y represents z x yj, where x r cos Θ, y r sin Θ thus z r cos Θ j r sin Θ r cos Θ j sin Θ This is called the polar form of z Here r x y z Θ is called the "argument of z" denoted by Θ arg z The Θ arg z is not unique since Θ k k an integer produces another value of arg z. However Θ gives the principle value of arg z. Clearly r x y and Θ tan x y Also cos Θ x r sin Θ y r x x y y x y
6 Determination of pricipal Argument of z : Θ. z x jy in first quadrant x 0, Y 0 Θ argz tan x y 0 Θ Example : determine the modulus and argument of z j r z j Θ arg z tan 0.588radians. z x zy in second quadrant x 0, y 0 Θ argz tan x y, Θ Example : Determine z and arg z for z j r z j Since tan tan thus Θ argz rad.
7 . z x zy in third quadrant x 0, y 0 : Θ argz tan x, y Θ Example Determine the polar form of complex number r z z 6 j 6 j 6 Also tan x 8 y tan 6 tan 6 rad.
8 Example: Verify the inequailities z z z z Triangle inequality Also z z 5j 9 5 Polar form of z : 5 z z 9 z 8 cos 5 5 j sin cos 5 5 j sin 6 6. z x jy in fourth quadrant x, y 0 : Θ arg z tan x y Θ 0 Example : find the priciple argument of z j. Θ arg z tan x y tan tan rad.
9 Exponential form of z : Recall Let x jθ refer to "Tables of mathmatical formulas" e x x x j x x... e jθ jθ jθ cos Θ jsin Θ Θ jθ jθ Θ... j Θ... Θ Θ Thus z r cos Θ jsin Θ has a new form z r e jθ the exponential form Important Points :. For the correct value in exponential form Θ must be in radians not in degrees.. Negative Θ is measured in the clockwise direction. Replacing Θ by Θ in e jθ cos Θ jsin Θ we get e jθ cos Θ sinθ, or e jθ cos Θ jsin Θ. e jθ and e jθ are complex cojugates hence z re jθ and z re jθ Consider e jθ cos Θ sin Θ and e jθ cos Θ sinθ Adding and substracting e jθ e jθ j sin Θ gives two important results : cos Θ ejθ e jθ sin Θ ejθ e jθ
10 Log of complex numbers : Let z re jθk k 0,,,... ln z ln re jθk ln r jθ k ln e ln r jθ k The principal value of ln z is ln z ln r jθ k 0 Example : Principal value of ln e j k is ln j j
11 Multiplication in polar form Let z r cos Θ j sin Θ z r cos Θ j sin Θ then z z r r cos Θ j sin Θ cos Θ j sin Θ r r cos Θ cos Θ sin Θ sin Θ j sin Θ cos Θ r r cosθ Θ jsin Θ Θ r r e jθ Θ Similarly z r cosθ Θ jsin Θ Θ z r cos Θ sin Θ r r e jθ Θ Clearly z z r r z z z z r r z z and arg z z Θ Θ arg z arg z arg z z Θ Θ arg z arg z Example : z z cos for z cos and z cos j sin j sin j sin cos 7 7 j sin and z z cos j sin
12 De Moivre s Theorem : If z r e jθ z r e jθ z n r n e jθ n then z z z n r r r n e j Θ Θ..Θ n r r r n cos Θ Θ.Θ n j sinθ Θ.Θ n Set then, z z... z n z r e jθ r cos Θ j sin Θ z n r n cos Θ j sin Θ n r n e jnθ r n cos nθ j sin nθ Set r to give De Moivre s Theorem cos Θ j sin Θ n cos nθ j sin nθ Application: If z then z cos Θ j sin Θ e jθ z cos Θ j sin Θ cos Θ j sin Θ cos Θ j sin Θ cos Θ j sin Θ cos Θ sin Θ cos Θ j sin Θ e jθ Hence and z n e jnθ cos nθ j sin nθ z n e jnθ cos nθ j sin nθ Adding and subtracting gives cos nθ z n z n and j sin nθ z n z n
13 Example : Evaluate i cos 6 ii j 5, j sin 6 iii j. 8 Solution : By De Moivre s Theorem i cos 6 j sin 6 cos 6 j sin 6 cos j sin j 0 j ii Let z j, then r z and Θ arg z tan Thus z 5 cos j sin 5 5 cos 5 j sin 5 5 cos 6 j sin 6 cos j sin j j iii Set z j thus from ii r z and Θ arg z tanh Thus z 8 z8 cos j sin 8 8 cos 8 j sin 8 cos j sin 6
14 Powers of cos Θ and sin Θ : Example : cos 6 Θ Binomial expansion gives z z cos 6 Θ 6 z6 6z 5z 0 5z 6z z 6 cos6θ z6 6 z 6 z 5 z z 0 z cos 6Θ cos Θ 0 cos Θ 0 6 cos Θ 5 cos Θ 5 6 Example : Express cos 6Θ and sin 6Θ in terms of cos Θ and sin Θ Example : Sketch z j on an Argand Diagram Polar form of z is and evaluate z z arg z 6 z cos j sin 6 6 cos 6 e j 6 j sin 6 Hence z e j 6 e j cos j sin
15 Roots of complex numbers: Recall de Moivre s theorem; For z rcos Θ j sin Θ z n r n cos nθ j sin nθ ffor any value of n integer or fraction, positive or negative This is a very important result. When n is a fraction we are finding roots of complex numbers Consider w n z integer The n different solutions w 0, w, w,,..., w n of this equation are the n th roots of z denoted by n z or z n Let z rcos Θ j sin Θ and w Ρcos Φ j sin Φ Equation w n z gives Ρ n cos nφ j sin nφ rcos Θ j sin Θ Equality of two complex numbers in polar form means that; i their modulus are equal i.e. Ρ n r or Ρ r n ii their arguments, arg w n and arg z may differ by a multiple of, say k, k 0,,... Thus nφ Θ k or Θ k Φ, k 0,,,..., n n all other choicesof k duplicate the values of Φ Thus n th roots of z, w 0, w, w,,..., w n are given by w k r n cos Θ k n j sin Θ k n r n exp Θk j n k 0,,..., n
16 Example : Find all the cube roots of 8 Set z 8, then r 8 8, arg 8 Hence 8 8cos j sin and 8 8 exp 8 j k There are cube roots of w z 8 Thus the cube roots of z 8 are w 0 cos cos k j sin k j j k 0,, since n j sin w cos j sin 0 w cos 5 j j j sin 5 w 0, w, w are equally spaced radians 0 apart on circle of radius 8, centered at 0, 0
17 For w n z roots are spaced by n n rad and lie on a circle of radius r Example : Find the values of j There are two ways depending upon how you express j i j j ii j j i.e find cube roots of i.e find the cube roots of j and square them. Here it appears sensible to use method i i polar form of : r and Θ arg w k cos k j sin k k 0,, w 0 cos j sin j w cos j sin w cos 5 5 j sin j Exercise : Solve the equation w j
Simplifying Radical Expressions and the Distance Formula
1 RD. Simplifying Radical Expressions and the Distance Formula In the previous section, we simplified some radical expressions by replacing radical signs with rational exponents, applying the rules of
More informationMath 4. Unit 1: Conic Sections Lesson 1.1: What Is a Conic Section?
Unit 1: Conic Sections Lesson 1.1: What Is a Conic Section? 1.1.1: Study - What is a Conic Section? Duration: 50 min 1.1.2: Quiz - What is a Conic Section? Duration: 25 min / 18 Lesson 1.2: Geometry of
More informationAlgebra I: A Fresh Approach. By Christy Walters
Algebra I: A Fresh Approach By Christy Walters 2016 A+ Education Services All rights reserved. No part of this publication may be reproduced, distributed, stored in a retrieval system, or transmitted,
More informationMathematics. Leaving Certificate Examination Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30
2016. M29 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2016 Mathematics Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30 300 marks Examination number
More informationAlgebra I: A Fresh Approach. By Christy Walters
Algebra I: A Fresh Approach By Christy Walters 2005 A+ Education Services All rights reserved. No part of this publication may be reproduced, distributed, stored in a retrieval system, or transmitted,
More informationBesides the reported poor performance of the candidates there were a number of mistakes observed on the assessment tool itself outlined as follows:
MATHEMATICS (309/1) REPORT The 2013 Mathematics (309/1) paper was of average standard. The paper covered a wide range of the syllabus. It was neither gender bias nor culture bias. It did not have language
More informationBASICS OF TRIGONOMETRY
Mathematics Revision Guides Basics of Trigonometry Page 1 of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF TRIGONOMETRY Version: 1. Date: 09-10-015 Mathematics Revision
More informationAP Physics 1 Summer Packet Review of Trigonometry used in Physics
AP Physics 1 Summer Packet Review of Trigonometry used in Physics For some of you this material will seem pretty familiar and you will complete it quickly. For others, you may not have had much or any
More informationVectors in the City Learning Task
Vectors in the City Learning Task Amy is spending some time in a city that is laid out in square blocks. The blocks make it very easy to get around so most directions are given in terms of the number of
More informationContent Design Structure, Scope & Sequence of Mathematics Content Addressed Within Numbers Up! Volcanic Panic
7-8 9-10 8-9 11-12 9 13-14 Division facts 2x, 5x, 10x Recognises division as repeated subtraction Recognises division as grouping Recognises division as sharing equally Recognises numbers that are divisible
More informationOperations on Radical Expressions; Rationalization of Denominators
0 RD. 1 2 2 2 2 2 2 2 Operations on Radical Expressions; Rationalization of Denominators Unlike operations on fractions or decimals, sums and differences of many radicals cannot be simplified. For instance,
More informationName Date. 5. In each pair, which rational number is greater? Explain how you know.
Master 3.18 Extra Practice 1 Lesson 3.1: What Is a Rational Number? 1. Which of the following numbers are equal to? 2. Write the rational number represented by each letter as a decimal. 3. Write the rational
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (Right-Angled Triangles)
More informationMathematics (Project Maths Phase 3)
L.17 NAME SCHOOL TEACHER Pre-Leaving Certificate Examination, 2014 Mathematics (Project Maths Phase 3) Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks For examiner Question 1 2 School stamp 3
More informationB.U.G. Newsletter. Full Steam Ahead! September Dr. Brown
B.U.G. Newsletter September 2014 THIS NEWSLETTER IS A SERVICE THAT WAS FUNDED BY "NO CHILD LEFT BEHIND" TITLE II PART A HIGHER EDUCATION IMPROVING TEACHER QUALITY HIGHER EDUCATION GRANT ADMINISTERED THROUGH
More informationStudent Resource / Program Workbook INTEGERS
INTEGERS Integers are whole numbers. They can be positive, negative or zero. They cannot be decimals or most fractions. Let us look at some examples: Examples of integers: +4 0 9-302 Careful! This is a
More information8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
8-1 The Pythagorean Theorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number 9 Square Positive Square Root 1 4 1 16 Vocabulary Builder leg (noun)
More informationTrigonometric Functions
Trigonometric Functions (Chapters 6 & 7, 10.1, 10.2) E. Law of Sines/Cosines May 21-12:26 AM May 22-9:52 AM 1 degree measure May 22-9:52 AM Measuring in Degrees (360 degrees) is the angle obtained when
More informationUnit 2 Day 4 Notes Law of Sines
AFM Unit 2 Day 4 Notes Law of Sines Name Date Introduction: When you see the triangle below on the left and someone asks you to find the value of x, you immediately know how to proceed. You call upon your
More informationAlgebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task 3.1.2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task 3.. TASK 3..: 30-60 RIGHT TRIANGLES Solutions. Shown here is a 30-60 right triangle that has one leg of length and
More information8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
8-1 he Pythagorean heorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root 9 81 3 1 4 1 16 1 2 Vocabulary Builder leg
More informationNCERT solution Decimals-2
NCERT solution Decimals-2 1 Exercise 8.2 Question 1 Complete the table with the help of these boxes and use decimals to write the number. (a) (b) (c) Ones Tenths Hundredths Number (a) - - - - (b) - - -
More informationMATHEMATICS: PAPER II
CLUSTER PAPER 2016 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 28 pages and an Information Sheet of 2 pages(i-ii).
More informationCMIMC 2018 Official Contest Information
CMIMC 2018 Official Contest Information CMIMC Staff Latest version: January 14, 2018 1 Introduction 1. This document is the official contest information packet for the 2018 Carnegie Mellon Informatics
More informationThe Bruins I.C.E. School
The Bruins I.C.E. School Lesson 1: Decimal Place Value Lesson 2: Measurement and Data Lesson 3: Coordinate Graphing Lesson 4: Adding and Subtracting Fractions Lesson 5: Order of Operations Worksheets Included:
More informationMORE TRIGONOMETRY
MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram
More information13.7 Quadratic Equations and Problem Solving
13.7 Quadratic Equations and Problem Solving Learning Objectives: A. Solve problems that can be modeled by quadratic equations. Key Vocabulary: Pythagorean Theorem, right triangle, hypotenuse, leg, sum,
More informationSolving Quadratic Equations (FAL)
Objective: Students will be able to (SWBAT) solve quadratic equations with real coefficient that have complex solutions, in order to (IOT) make sense of a real life situation and interpret the results
More informationSummer Work. 6 th Grade Enriched Math to 7 th Grade Pre-Algebra
Summer Work 6 th Grade Enriched Math to 7 th Grade Pre-Algebra Attached is a packet for Summer 2017. Take your time. Do not wait until the weeks right before school to begin. The benefit of summer work
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (Right-Angled Triangles)
More informationCleveland State University MCE441: Intr. Linear Control Systems. Lecture 6: The Transfer Function Poles and Zeros Partial Fraction Expansions
Cleveland State University MCE441: Intr. Linear Control Systems Lecture 6: The and Zeros Partial Fraction Expansions Prof. Richter 1 / 12 System and Zeros and Zeros in Decomposition- Obtaining the - in
More informationBishop Kelley High School Summer Math Program Course: Trigonometry and Trigonometry with Pre-Calculus
015 01 Summer Math Program Course: Trigonometr and Trigonometr with Pre-Calculus NAME: DIRECTIONS: Show all work on loose-leaf paper, which ou will turn in with the packet. (NO WORK IN PACKET!) Put final
More informationApplication of Geometric Mean
Section 8-1: Geometric Means SOL: None Objective: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse
More informationAP Physics B Summer Homework (Show work)
#1 NAME: AP Physics B Summer Homework (Show work) #2 Fill in the radian conversion of each angle and the trigonometric value at each angle on the chart. Degree 0 o 30 o 45 o 60 o 90 o 180 o 270 o 360 o
More informationA COURSE OUTLINE (September 2001)
189-265A COURSE OUTLINE (September 2001) 1 Topic I. Line integrals: 2 1 2 weeks 1.1 Parametric curves Review of parametrization for lines and circles. Paths and curves. Differentiation and integration
More information11.4 Apply the Pythagorean
11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse Legs of a right triangle Pythagorean theorem THE PYTHAGOREAN THEOREM Words If a triangle is
More informationSQUARE ROOTS. Pythagoras theorem has been a perennially interesting. Drawing A SPIRAL OF. ClassRoom
Drawing A SPIRAL OF SQUARE ROOTS ClassRoom KHUSHBOO AWASTHI "Mathematics possesses a beauty cold and austere, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
More informationWrite these equations in your notes if they re not already there. You will want them for Exam 1 & the Final.
Tuesday January 30 Assignment 3: Due Friday, 11:59pm.like every Friday Pre-Class Assignment: 15min before class like every class Office Hours: Wed. 10-11am, 204 EAL Help Room: Wed. & Thurs. 6-9pm, here
More information2.6 Related Rates Worksheet Calculus AB. dy /dt!when!x=8
Two Rates That Are Related(1-7) In exercises 1-2, assume that x and y are both differentiable functions of t and find the required dy /dt and dx /dt. Equation Find Given 1. dx /dt = 10 y = x (a) dy /dt
More information2013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:
013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and
More informationPacific Charter Institute Pacing Guide Grade(s): _5 Subject Area: _Math in Focus grade 5 CP: yes _X no
Week 1 Pre-requisite skills review Chapter 1 pre-test Assessment 5 chapter pre-test pp. 1-4 or MIF Student book 5A pp. 1-4 Reteach 4A (1.1, 1.2, 2.1, 3.2, 3.4) NWEA Kids start Wed. Week 2 1.1 Writing numbers
More information77.1 Apply the Pythagorean Theorem
Right Triangles and Trigonometry 77.1 Apply the Pythagorean Theorem 7.2 Use the Converse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 Apply the Tangent Ratio
More informationSection 2C Formulas with Dividing Decimals
Section 2C Formulas with Dividing Decimals x Look at the following z-score formula again from Statistics. z. Suppose we want to calculate the z-score if x 17.6 pounds, 13.8 pounds, and 2.5 pounds. Not
More informationLesson 5. Section 2.2: Trigonometric Functions of an Acute Angle 1 = 1
Lesson 5 Diana Pell March 6, 2014 Section 2.2: Trigonometric Functions of an Acute Angle 1 = 1 360 We can divide 1 into 60 equal parts, where each part is called 1 minute, denoted 1 (so that 1 minute is
More informationGary Delia AMS151 Fall 2009 Homework Set 3 due 10/07/2010 at 11:59pm EDT
Gary Delia AMS151 Fall 2009 Homework Set 3 due 10/07/2010 at 11:59pm EDT 1. (2 pts) The angle of elevation to the top of a building is found to be 9 from the ground at a distance of 6000 feet from the
More informationREAL LIFE GRAPHS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Real Life Graphs Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier REAL LIFE GRAPHS Version: 2.1 Date: 20-10-2015 Mathematics Revision Guides
More informationThe Diver Returns Circular Functions, Vector Components, and Complex Numbers
Co n t e n t s The Diver Returns Circular Functions, Vector Components, and Complex Numbers Back to the Circus 3 The Circus Act 4 As the Ferris Wheel Turns 6 Graphing the Ferris Wheel 8 Distance with Changing
More informationMinimum Mean-Square Error (MMSE) and Linear MMSE (LMMSE) Estimation
Minimum Mean-Square Error (MMSE) and Linear MMSE (LMMSE) Estimation Outline: MMSE estimation, Linear MMSE (LMMSE) estimation, Geometric formulation of LMMSE estimation and orthogonality principle. Reading:
More informationChapter 10. Right Triangles
Chapter 10 Right Triangles If we looked at enough right triangles and experimented a little, we might eventually begin to notice some relationships developing. For instance, if I were to construct squares
More informationCIRCLE AREA Lesson 3: Round Peg in a Square Hole
CIRCLE AREA Lesson 3: Round Peg in a Square Hole Australian Curriculum: Mathematics (Year 8) ACMMG197: Investigate the relationship between features of circles such as circumference, area, radius and diameter.
More informationGeom- Chpt. 8 Algebra Review Before the Chapter
Geom- Chpt. 8 Algebra Review Before the Chapter Solving Quadratics- Using factoring and the Quadratic Formula Solve: 1. 2n 2 + 3n - 2 = 0 2. (3y + 2) (y + 3) = y + 14 3. x 2 13x = 32 1 Working with Radicals-
More informationParking Lot HW? Joke of the Day: What do you call a leg that is perpendicular to a foot? Goals:
Parking Lot Joke of the Day: HW? What do you call a leg that is perpendicular to a foot? a right ankle Goals: Agenda 1 19 hw? Course Recommendations Simplify Radicals skill practice L8 2 Special Right
More informationMath 3 Plane Geometry Review Special Triangles
Name: 1 Date: Math 3 Plane Geometry Review Special Triangles Special right triangles. When using the Pythagorean theorem, we often get answers with square roots or long decimals. There are a few special
More informationAPPROVED FACILITY SCHOOLS CURRICULUM DOCUMENT SUBJECT: Mathematics GRADE: 6. TIMELINE: Quarter 1. Student Friendly Learning Objective
TIMELINE: Quarter 1 i-ready lesson: Rational Numbers and Absolute Value i-ready lesson: Numerical Expressions and Order of Operations 6/16/15 1 i-ready lesson (2a, 2b and 2c): Algebraic Expressions 6/16/15
More informationArithmetic with Units of Measure
Arithmetic with Units of Measure Reteaching 81 Math Course 1, Lesson 81 If units are not the same, convert first. Example: 2 ft + 12 in. 24 in. + 12 in. or 2 ft + 1 ft To add or subtract measures, keep
More informationETA Cuisenaire The Super Source Grades: 7, 8 States: Texas Essential Knowledge and Skills (TEKS) Subjects: Mathematics
ETA Cuisenaire The Super Source Grades: 7, 8 States: Texas Essential Knowledge and Skills () Subjects: Mathematics TX.111.23 (7.1) Texas Essential Knowledge and Skills () Mathematics Grade 7 Number, operation,
More informationSimilar Right Triangles
MATH 1204 UNIT 5: GEOMETRY AND TRIGONOMETRY Assumed Prior Knowledge Similar Right Triangles Recall that a Right Triangle is a triangle containing one 90 and two acute angles. Right triangles will be similar
More informationThe Pythagorean Theorem Diamond in the Rough
The Pythagorean Theorem SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Visualization, Interactive Word Wall Cameron is a catcher trying out for the school baseball team. He
More informationChapter 12 Practice Test
Chapter 12 Practice Test 1. Which of the following is not one of the conditions that must be satisfied in order to perform inference about the slope of a least-squares regression line? (a) For each value
More information2011 Canadian Intermediate Mathematics Contest
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING www.cemc.uwaterloo.ca 011 Canadian Intermediate Mathematics Contest Tuesday, November, 011 (in North America and South America) Wednesday, November
More informationAlgebra Date Lesson Independent Work Computer Tuesday, Introduction (whole class) Problem with Dice
Tuesday, Introduction (whole class) Problem with Dice Critical Thinking Puzzles September 3 Station expectations Count the Squares Math Riddles Wednesday, Computer expectations (whole class) Tangrams Read
More informationConcave and Flat Expander Anti-Wrinkle Rollers:
Concave and Flat Expander Anti-Wrinkle Rollers: For Better or For Worse? Timothy J. Walker tjwalker@tjwa.com (651) 686-5400 Office (651) 249-1121 Mobile (866) 572-3139 Fax NEW WWW.TJWA.COM 164 Stonebridge
More informationPractice 9-1. The Real Numbers. Write all names that apply to each number
Chapter 9 Practice 9-1 The Real Numbers Write all names that apply to each number. 1. 3.2 2. 2 5 3. 12 4. 4 2 5. 20 6. 16 7. 7 8 8. 0.15 9. 18 2 10. 45 11. 25 12. 6.75 State if the number is rational,
More informationTwo Special Right Triangles
Page 1 of 7 L E S S O N 9.3 In an isosceles triangle, the sum of the square roots of the two equal sides is equal to the square root of the third side. Two Special Right Triangles In this lesson you will
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (20 minutes)
Student Outcomes Students explain a proof of the converse of the Pythagorean Theorem. Students apply the theorem and its converse to solve problems. Lesson Notes Students had their first experience with
More informationTutorial 5 Relative equilibrium
Tutorial 5 Relative equilibrium 1. n open rectangular tank 3m long and 2m wide is filled with water to a depth of 1.5m. Find the slope of the water surface when the tank moves with an acceleration of 5m/s
More information2.5. All games and sports have specific rules and regulations. There are rules about. Play Ball! Absolute Value Equations and Inequalities
Play Ball! Absolute Value Equations and Inequalities.5 LEARNING GOALS In this lesson, you will: Understand and solve absolute values. Solve linear absolute value equations. Solve and graph linear absolute
More informationMATH 110 Sample 02 Exam 1
1. Evaluate: log 4 64 [A] 3 [B] 1 12 [C] 12 [D] 1 3 2. Given log 5 = T and log 2 = U, find log. [A] T + U [B] TU [C] T + U [D] TU. 3. Find x if e x 52 8 =, and ou are given ln 8 = 2.0794. [A] 0.3999 [B]
More informationPART 3 MODULE 6 GEOMETRY: UNITS OF GEOMETRIC MEASURE
PART 3 MODULE 6 GEOMETRY: UNITS OF GEOMETRIC MEASURE LINEAR MEASURE In geometry, linear measure is the measure of distance. For instance, lengths, heights, and widths of geometric figures are distances,
More informationCenter Distance Change of Silent Chain Drive Effect on Sprocket Tooth Profile Modification Wei Sun 1,a, Xiaolun Liu 1,b, Jiajun Liu 1 and Wei Zhang 1
Key Engineering Materials Online: 2012-08-24 ISSN: 1662-9795, Vol. 522, pp 574-577 doi:10.4028/www.scientific.net/kem.522.574 2012 Trans Tech Publications, Switzerland Center Distance Change of Silent
More informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationUNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios Instruction
Prerequisite Skills This lesson requires the use of the following skills: defining and calculating sine, cosine, and tangent setting up and solving problems using the Pythagorean Theorem identifying the
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above)
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Solutions Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas
More informationMixture Models & EM. Nicholas Ruozzi University of Texas at Dallas. based on the slides of Vibhav Gogate
Mixture Models & EM Nicholas Ruozzi University of Texas at Dallas based on the slides of Vibhav Gogate Previously We looked at -means and hierarchical clustering as mechanisms for unsupervised learning
More informationUAB MATH-BY-MAIL CONTEST, 2004
UAB MATH-BY-MAIL CONTEST, 2004 ELIGIBILITY. Math-by-Mail competition is an individual contest run by the UAB Department of Mathematics and designed to test logical thinking and depth of understanding of
More information0607 CAMBRIDGE INTERNATIONAL MATHEMATICS
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 0 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL
More informationLearning Goal: I can explain when to use the Sine, Cosine and Tangent ratios and use the functions to determine the missing side or angle.
MFM2P Trigonometry Checklist 1 Goals for this unit: I can solve problems involving right triangles using the primary trig ratios and the Pythagorean Theorem. U1L4 The Pythagorean Theorem Learning Goal:
More information1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely.
9.7 Warmup 1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely. 2. A right triangle has a leg length of 7 in. and a hypotenuse length of 14 in. Solve the triangle
More informationTitle: Direction and Displacement
Title: Direction and Displacement Subject: Mathematics Grade Level: 10 th 12 th Rational or Purpose: This activity will explore students knowledge on directionality and displacement. With the use angle
More informationLesson 1: Decimal Place Value. Concept/Topic to Teach: Students use Bruins statistical data to order and compare decimals to the thousandths.
Math Lesson 1: Decimal Place Value Concept/Topic to Teach: Students use Bruins statistical data to order and compare decimals to the thousandths. Standards Addressed: Standard 1: 5.NBT.3 Read, write, and
More informationMark Scheme (Results) Summer 2009
Mark Scheme (Results) Summer 2009 GCSE GCSE Mathematics (Linear) - 1380 Paper: 1380_2F 2 NOTES ON MARKING PRINCIPLES 1 Types of mark M marks: method marks A marks: accuracy marks B marks: unconditional
More informationAddition and Subtraction of Rational Expressions
RT.3 Addition and Subtraction of Rational Expressions Many real-world applications involve adding or subtracting algebraic fractions. Similarly as in the case of common fractions, to add or subtract algebraic
More informationBIOMECHANICAL MOVEMENT
SECTION PART 5 5 CHAPTER 12 13 CHAPTER 12: Biomechanical movement Practice questions - text book pages 169-172 1) For which of the following is the athlete s centre of mass most likely to lie outside of
More informationPort Jefferson-Bridgeport Ferry Project: Wind Correction
Port Jefferson-Bridgeport Ferry Project: Wind Correction Background Joseph Giannotti SUNY-Stony Brook Dr. Duane Waliser Marine Sciences Research Center SUNY-Stony Brook Jeffrey Tongue National Weather
More informationMath 154 Chapter 7.7: Applications of Quadratic Equations Objectives:
Math 154 Chapter 7.7: Applications of Quadratic Equations Objectives: Products of numbers Areas of rectangles Falling objects Cost/Profit formulas Products of Numbers Finding legs of right triangles Finding
More informationChapter 11 Waves. Waves transport energy without transporting matter. The intensity is the average power per unit area. It is measured in W/m 2.
Energy can be transported by particles or waves: Chapter 11 Waves A wave is characterized as some sort of disturbance that travels away from a source. The key difference between particles and waves is
More informationMarch Madness Basketball Tournament
March Madness Basketball Tournament Math Project COMMON Core Aligned Decimals, Fractions, Percents, Probability, Rates, Algebra, Word Problems, and more! To Use: -Print out all the worksheets. -Introduce
More informationNote! In this lab when you measure, round all measurements to the nearest meter!
Distance and Displacement Lab Note! In this lab when you measure, round all measurements to the nearest meter! 1. Place a piece of tape where you will begin your walk outside. This tape marks the origin.
More informationECO 199 GAMES OF STRATEGY Spring Term 2004 Precept Materials for Week 3 February 16, 17
ECO 199 GAMES OF STRATEGY Spring Term 2004 Precept Materials for Week 3 February 16, 17 Illustration of Rollback in a Decision Problem, and Dynamic Games of Competition Here we discuss an example whose
More informationSHOT ON GOAL. Name: Football scoring a goal and trigonometry Ian Edwards Luther College Teachers Teaching with Technology
SHOT ON GOAL Name: Football scoring a goal and trigonometry 2006 Ian Edwards Luther College Teachers Teaching with Technology Shot on Goal Trigonometry page 2 THE TASKS You are an assistant coach with
More informationThe statements of the Law of Cosines
MSLC Workshop Series: Math 1149 and 1150 Law of Sines & Law of Cosines Workshop There are four tools that you have at your disposal for finding the length of each side and the measure of each angle of
More informationWeek of July 2 nd - July 6 th. The denominator is 0. Friday
Week of July 2 nd - July 6 th The temperature is F at 7 A.M. During the next three hours, the temperature increases 1 F. What is the temperature at 10 A.M.? A quotient is undefined. What does this mean?
More informationFractions Print Activity. Use the Explore It mode to answer the following questions:
Name: Fractions Print Activity Use the Explore It mode to answer the following questions:. Use the fraction strips below to shade in the following values: a. b. 5 c. d. 4. a. Select The multiples of that
More information2017 AMC 12B. 2. Real numbers,, and satisfy the inequalities,, and. Which of the following numbers is necessarily positive?
2017 AMC 12B 1. Kymbrea's comic book collection currently has 30 comic books in it, and she is adding to her collection at the rate of 2 comic books per month. LaShawn's comic book collection currently
More information*Definition of Cosine
Vetors - Unit 3.3A - Problem 3.5A 3 49 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? ()
More informationFOURTH GRADE MATHEMATICS UNIT 4 STANDARDS. MGSE.4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Dear Parents, FOURTH GRADE MATHEMATICS UNIT 4 STANDARDS We want to make sure that you have an understanding of the mathematics your child will be learning this year. Below you will find the standards we
More informationMarch Madness Basketball Tournament
March Madness Basketball Tournament Math Project COMMON Core Aligned Decimals, Fractions, Percents, Probability, Rates, Algebra, Word Problems, and more! To Use: -Print out all the worksheets. -Introduce
More informationParallel Lines Cut by a Transversal
Name Date Class 11-1 Parallel Lines Cut by a Transversal Parallel Lines Parallel Lines Cut by a Transversal A line that crosses parallel lines is a transversal. Parallel lines never meet. Eight angles
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and
More informationCONTENTS III CUMULATIVE REVIEW Copyright by Phoenix Learning Resources. Inc. All Rights Reserved.
CONTENTS Chapter 1 WHOLE NUMBERS Pretest.............................. 1 Adding Whole Numbers.................. 2 Subtracting Whole Numbers.............. 4 Adding and Subtracting Whole Numbers..... 7 Using
More information