Lesson 73 Polar Form of Complex Numbers. Relationships Among x, y, r, and. Polar Form of a Complex Number. x r cos y r sin. r x 2 y 2.

Size: px
Start display at page:

Download "Lesson 73 Polar Form of Complex Numbers. Relationships Among x, y, r, and. Polar Form of a Complex Number. x r cos y r sin. r x 2 y 2."

Transcription

1 Lesson 7 Polar Form of Complex Numbers HL Math - Santowski Relationships Among x, y, r, and x r cos y r sin r x y tan y x, if x 0 Polar Form of a Complex Number The expression r(cos isin ) is called the polar form (or trigonometric form) of the complex number x + yi. The expression cos + i sin is sometimes abbreviated cis. Using this notation r(cos isin ) is written r cis. 1

2 Example Express (cos 10 + i sin 10) in rectangular form. Example Express (cos 10 + i sin 10) in rectangular form. 1 1 cos10 (cos10 isin10 ), i 1 i sin10 Notice that the real part is negative and the imaginary part is positive, this is consistent with 10 degrees being a quadrant II angle. Converting from Rectangular to Polar Form Step 1 Sketch a graph of the number x + yi in the complex plane. Step Find r by using the equation r x y. Step y Find by using the equation tan, x 0 x choosing the quadrant indicated in Step 1.

3 Example Example: Find trigonometric notation for 1 i. Example Example: Find trigonometric notation for 1 i. First, find r. r a b r ( 1) ( 1) r 1 1 sin cos Thus, 1 i cos isin or cis Product of Complex Numbers Find the product of 4(cos50 isin50 ) and (cos10 isin10 ).

4 Product Theorem If r1 cos1 isin 1 and r cos isin, are any two complex numbers, then cos sin cos sin cos isin. r i r i rr In compact form, this is written r r rr cis cis cis Example: Product Find the product of 4(cos50 isin50 ) and (cos10 isin10 ). Example: Product Find the product of 4(cos50 isin50 ) and (cos10 isin10 ). 4(cos50 isin50 ) (cos10 isin10 ) 4 cos(50 10 ) isin(50 10 ) 8(cos60 isin60 ) 8 1 i 4 4i 4

5 Quotient of Complex Numbers Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) Quotient Theorem If r cos isin and r cos isin are any two complex numbers, where r cos isin, r 0, then r1 cos1 isin1 r1 cos1 isin 1. r cos isin r In compact form, this is written r1 cis 1 r1 cis1 r cis r Example: Quotient Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) 5

6 Example: Quotient Find the quotient. 16(cos70 isin70 ) and 4(cos40 isin 40 ) 16(cos70 isin70 ) 4(cos40 isin 40 ) =16 4 cos(70 40 ) isin(70 40 ) 4cos0 isin0 4 1 i i cos 40 i sin 40 and w 6cos10 i sin10 If z 4, find : (a) zw (b) z w If z 4cos 40 o isin40 o and w 6cos10 o isin10 o, find: (a) zw (b) z w zw 4 cos 40 i sin 40 6 cos10 i sin cos40 10 i sin40 10 multiply the moduli i 4 cos160 i sin 160 add the arguments (the i sine term will have same argument) 0.40i If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 4 through. 6

7 z w 4 cos 40 o isin40 o 6 cos10 o isin10 o 4 6 cos 40o 10 o isin 40 o 10 o divide the moduli cos 80o isin 80 o In rectangular coordinates: cos 80oisin 80 o subtract the arguments In polar form we want an angle between 0 and 60 so add 60 to the i i z1 divide z Example Where z1 i 6cos45 isin45 z i 4cos0 isin0 Rectangular form Trig form z1 divide z Example Where z1 i 6cos45 isin45 Rectangular form i i i i i i i 6 6i 6 i 1 4i i 6 6i i z i 4cos0 isin0 Trig form 6cos45 isin45 4cos0 isin0 6 cos 45 0 i sin cos15 i sin r tan r r

8 DeMoivre s Theorem Taking Complex Numbers to Higher Powers z r cos i sin cos sin cos sin z?? r i r i r cos icos sin i sin r cos r z r sin isin cos cos isin cos i sin cos sin cos sin cos sin z r i z r i r i r cos icos sin i sin r r cos sin isin cos cos isin z r cos i sin Nice 8

9 What about z? cos isin, cos sin z z r cos i sin r cos i sin z r z r i z z r i i r r r cos cos sin sin cos sin isin cos cos isin cos isin cos sin cos sin r i i r cos cos i cos sin i sin cos i sin sin r i n isin cos i s cos sin sin cos sin cos cos sin sin cos si r cos cos sin sin sin co r cos i r i z r cos isin Hooray!! 9

10 We saw z r cos isin and z r cos i Similarly. 4 4 z r cos4 isin z r i cos5 sin 5 sin n n z r cosn isin n We saw z r cos isin and z r cos i Similarly. 4 4 z r cos4 isin z r i sin cos5 sin 5 n n z r cosn isin n Hooray DeMoivre and his incredible theorem Powers of Complex Numbers This is horrible in rectangular form. n a bi a bia bia bi... a bi The best way to expand one of these is using Pascal s triangle and binomial expansion. You d need to use an i-chart to simplify. It s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent. z r cos i sin n n z r cosn i sinn Example z 5cos0 isin0 z 5 cos0 isin0 z 15cos60 isin60 10

11 De Moivre s Theorem If r cos isin is a complex number, and if n is any real number, then n n r cos 1 isin 1 r cos n isin n. In compact form, this is written n n r r n cis cis. Example: Find (1 i) 5 and express the result in rectangular form. Example: Find (1 i) 5 and express the result in rectangular form. First, find trigonometric notation for 1 i 1 i cos 5 isin 5 Theorem 5 1 i cos5 isin 5 5 cos( 5 5 ) isin( 5 5 ) 4 cos115 i sin115 4 i 44i 5 11

12 Example z 1 i Let. Find z 10 Example Further Examples 1

13 nth Roots For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if a bi x yi. n nth Root Theorem If n is any positive integer, r is a positive real number, and is in degrees, then the nonzero complex number r(cos + i sin ) has exactly n distinct nth roots, given by n where n r cos isin or r cis, 60 k 60 k or =, k 0,1,,..., n1. n n n Example: Square Roots i Find the square roots of 1 1

14 Example: Square Roots Find the square roots of 1 Polar notation: For k = 0, root is For k = 1, root is 1 i i cos60 isin cos60 isin60 cos 60 k 60 isin 60 k 60 cos 0 k 180 isin 0 k 180 cos0 isin0 cos10 isin10 Example: Fourth Root Find all fourth roots of 8 8i. Example: Fourth Root Find all fourth roots of 8 8i. roots in rectangular form. Write the Write in polar form. 8 8i 16 cis 10 Here r = 16 and = 10. The fourth roots of this number have absolute value k 0 90 k

15 Example: Fourth Root continued There are four fourth roots, let k = 0, 1, and. k k k k Using these angles, the fourth roots are cis 0, cis 10, cis 10, cis 00 Example: Fourth Root continued Written in rectangular form i 1i i 1 i The graphs of the roots are all on a circle that has center at the origin and radius. Find the complex fifth roots of 15

16 Find the complex fifth roots of The five complex roots are: for k = 0, 1,,,4. Example Find all the complex fifth roots of 16

17 0i r a b tan tan i sin cos k 0 60k cos i sin cos 7k i sin 7k k 0 cos 0 i sin0 1 0i k 1 cos 7k i sin 7k i sin cos i k cos 144 i sin i k cos 16 i sin i k 4 cos 88 i sin i Example Find i You may assume it is the principle root you are seeking unless specifically stated otherwise. First express i as a complex number in standard form. 0 i Then change to polar form r a b r 1 b tan a 1 tan 0 i sin 1cos tan

18 i sin 1cos Since we are looking for the cube root, use DeMoivre s Theorem 1 and raise it to the power cos 90 i sin 90 i sin 1 cos i 1 i Example: Find the 4 th root of i i 1 4 Change to polar form 5 cos i sin Apply DeMoivre s Theorem cos i sin cos 8. 4 i sin i 18

Polar Form of Complex Numbers

Polar Form of Complex Numbers OpenStax-CNX module: m49408 1 Polar Form of Complex Numbers OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:

More information

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10.

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10. Complete Solutions to Examination Questions 0 Complete Solutions to Examination Questions 0. (i We need to determine + given + j, j: + + j + j (ii The product ( ( + j6 + 6 j 8 + j is given by ( + j( j

More information

Chapter 9: Complex Numbers

Chapter 9: Complex Numbers Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number - definition - argand diagram - equality of comple number 9.3 Algebraic operations on comple number - addition and subtraction - multiplication

More information

The modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then.

The modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then. COMPLEX NUMBERS _+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

More information

Chapter 6 Additional Topics in Trigonometry, Part II

Chapter 6 Additional Topics in Trigonometry, Part II Chapter 6 Additional Topics in Trigonometry, Part II Section 3 Section 4 Section 5 Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Vocabulary Directed line segment

More information

Module 10 Polar Form of Complex Numbers

Module 10 Polar Form of Complex Numbers MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

More information

Complex Numbers, Polar Coordinates, and Parametric Equations

Complex Numbers, Polar Coordinates, and Parametric Equations 8 Complex Numbers, Polar Coordinates, and Parametric Equations If a golfer tees off with an initial velocity of v 0 feet per second and an initial angle of trajectory u, we can describe the position of

More information

1,cost 1 1,tant 0 1,cott ,cost 0 1,tant 0. 1,cott 1 0. ,cost 5 6,tant ,cott x 2 1 x. 1 x 2. Name: Class: Date:

1,cost 1 1,tant 0 1,cott ,cost 0 1,tant 0. 1,cott 1 0. ,cost 5 6,tant ,cott x 2 1 x. 1 x 2. Name: Class: Date: Class: Date: Practice Test (Trigonometry) Instructor: Koshal Dahal Multiple Choice Questions SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE CREDIT. 1. Find the values of the trigonometric

More information

Complex Numbers and Polar Coordinates

Complex Numbers and Polar Coordinates Chapter 5 Complex Numbers and Polar Coordinates One of the goals of algebra is to find solutions to polynomial equations. You have probably done this many times in the past, solving equations like x 1

More information

AH Complex Numbers.notebook October 12, 2016

AH Complex Numbers.notebook October 12, 2016 Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does

More information

Complex Numbers. Introduction

Complex Numbers. Introduction 10 Assessment statements 1.5 Complex numbers: the number i 5 1 ; the term s real part, imaginary part, conjugate, modulus and argument. Cartesian form z 5 a 1 ib. Sums, products and quotients of complex

More information

Chapter 8: Polar Coordinates and Vectors

Chapter 8: Polar Coordinates and Vectors Chapter 8: Polar Coordinates and Vectors 8.1 Polar Coordinates This is another way (in addition to the x-y system) of specifying the position of a point in the plane. We give the distance r of the point

More information

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i}

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i} Chapter 8 Complex Numbers, Polar Equations, and Parametric Equations 6. { ± 6i} Section 8.1: Complex Numbers 1. true. true. true 4. true 5. false (Every real number is a complex number. 6. true 7. 4 is

More information

Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Trigonometry Capacity Matrix Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their

More information

College Trigonometry

College Trigonometry College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 131 George Voutsadakis (LSSU) Trigonometry January 2015 1 / 25 Outline 1 Functions

More information

Find all solutions cos 6. Find all solutions. 7sin 3t Find all solutions on the interval [0, 2 ) sin t 15cos t sin.

Find all solutions cos 6. Find all solutions. 7sin 3t Find all solutions on the interval [0, 2 ) sin t 15cos t sin. 7.1 Solving Trigonometric Equations with Identities In this section, we explore the techniques needed to solve more complex trig equations: By Factoring Using the Quadratic Formula Utilizing Trig Identities

More information

PreCalculus: Chapter 9 Test Review

PreCalculus: Chapter 9 Test Review Name: Class: Date: ID: A PreCalculus: Chapter 9 Test Review Short Answer 1. Plot the point given in polar coordinates. 3. Plot the point given in polar coordinates. (-4, -225 ) 2. Plot the point given

More information

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( ) Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.

More information

CHAPTER 1 COMPLEX NUMBER

CHAPTER 1 COMPLEX NUMBER BA0 ENGINEERING MATHEMATICS 0 CHAPTER COMPLEX NUMBER. INTRODUCTION TO COMPLEX NUMBERS.. Quadratic Equations Examples of quadratic equations:. x + 3x 5 = 0. x x 6 = 0 3. x = 4 i The roots of an equation

More information

1! i 3$ (( )( x! 1+ i 3)

1! i 3$ (( )( x! 1+ i 3) Math 4C Fall 2008 Final Exam (Name) (PID) (Section) Read each question carefully; answer each question completely. Show all work: no credit for unsupported answers. Attach additional sheets if necessary.

More information

1 Review of complex numbers

1 Review of complex numbers 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers Syllabus Objectives: 5.1 The student will eplore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Math Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2.

Math Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2. Math 70300 Homework 1 September 1, 006 The homework consists mostly of a selection of problems from the suggested books. 1. (a) Find the value of (1 + i) n + (1 i) n for every n N. We will use the polar

More information

12) y = -2 sin 1 2 x - 2

12) y = -2 sin 1 2 x - 2 Review -Test 1 - Unit 1 and - Math 41 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find and simplify the difference quotient f(x + h) - f(x),

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

in Trigonometry Name Section 6.1 Law of Sines Important Vocabulary

in Trigonometry Name Section 6.1 Law of Sines Important Vocabulary Name Chapter 6 Additional Topics in Trigonometry Section 6.1 Law of Sines Objective: In this lesson you learned how to use the Law of Sines to solve oblique triangles and how to find the areas of oblique

More information

Trigonometry and Analysis Mathematics

Trigonometry and Analysis Mathematics Scope And Sequence Timeframe Unit Instructional Topics 6 Week(s) 3 Week(s) 5 Week(s) 4 Week(s) 5 Week(s) Equations, Inequalties, Functions 1. Basic Equations (1.1) 2. Modeling with Equations (1.2) 3. Quadratic

More information

Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.

Unit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount. Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that

More information

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t

More information

Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Trigonometry Capacity Matrix Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions

More information

MATH 130 FINAL REVIEW

MATH 130 FINAL REVIEW MATH 130 FINAL REVIEW Problems 1 5 refer to triangle ABC, with C=90º. Solve for the missing information. 1. A = 40, c = 36m. B = 53 30', b = 75mm 3. a = 91 ft, b = 85 ft 4. B = 1, c = 4. ft 5. A = 66 54',

More information

Complex Practice Exam 1

Complex Practice Exam 1 Complex Practice Exam This practice exam contains sample questions. The actual exam will have fewer questions, and may contain questions not listed here.. Be prepared to explain the following concepts,

More information

In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.

In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q. THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various

More information

Unit Activity Correlations to Common Core State Standards. Precalculus. Table of Contents

Unit Activity Correlations to Common Core State Standards. Precalculus. Table of Contents Unit Activity Correlations to Common Core State Standards Precalculus Table of Contents Number and Quantity 1 Algebra 3 Functions 3 Geometry 5 Statistics and Probability 6 Number and Quantity The Complex

More information

Unit 3 Specialist Maths

Unit 3 Specialist Maths Unit 3 Specialist Maths succeeding in the vce, 017 extract from the master class teaching materials Our Master Classes form a component of a highly specialised weekly program, which is designed to ensure

More information

Radical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots

Radical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots 8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions

More information

Secondary Honors Algebra II Objectives

Secondary Honors Algebra II Objectives Secondary Honors Algebra II Objectives Chapter 1 Equations and Inequalities Students will learn to evaluate and simplify numerical and algebraic expressions, to solve linear and absolute value equations

More information

PreCalculus Honors Curriculum Pacing Guide First Half of Semester

PreCalculus Honors Curriculum Pacing Guide First Half of Semester Unit 1 Introduction to Trigonometry (9 days) First Half of PC.FT.1 PC.FT.2 PC.FT.2a PC.FT.2b PC.FT.3 PC.FT.4 PC.FT.8 PC.GCI.5 Understand that the radian measure of an angle is the length of the arc on

More information

Complex Numbers CK-12. Say Thanks to the Authors Click (No sign in required)

Complex Numbers CK-12. Say Thanks to the Authors Click  (No sign in required) Complex Numbers CK-12 Say Thanks to the Authors Click http://www.ck12.org/saythanks No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

More information

List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015)

List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) MAT 155P MAT 155 1 Absolute Value Equations P 7 P 3 2 Absolute Value Inequalities P 9 P 4 3 Algebraic Expressions:

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

Math 005A Prerequisite Material Answer Key

Math 005A Prerequisite Material Answer Key Math 005A Prerequisite Material Answer Key 1. a) P = 4s (definition of perimeter and square) b) P = l + w (definition of perimeter and rectangle) c) P = a + b + c (definition of perimeter and triangle)

More information

Pre-Calculus & Trigonometry Scope and Sequence

Pre-Calculus & Trigonometry Scope and Sequence WHCSD Scope and Sequence Pre-Calculus/ 2017-2018 Pre-Calculus & Scope and Sequence Course Overview and Timing This section is to help you see the flow of the unit/topics across the entire school year.

More information

Chetek-Weyerhaeuser High School

Chetek-Weyerhaeuser High School Chetek-Weyerhaeuser High School Advanced Math A Units and s Advanced Math A Unit 1 Functions and Math Models (7 days) 10% of grade s 1. I can make connections between the algebraic equation or description

More information

MATH 135: COMPLEX NUMBERS

MATH 135: COMPLEX NUMBERS MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex

More information

An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.

An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis. Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise

More information

Some commonly encountered sets and their notations

Some commonly encountered sets and their notations NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their

More information

MAT01A1: Complex Numbers (Appendix H)

MAT01A1: Complex Numbers (Appendix H) MAT01A1: Complex Numbers (Appendix H) Dr Craig 14 February 2018 Announcements: e-quiz 1 is live. Deadline is Wed 21 Feb at 23h59. e-quiz 2 (App. A, D, E, H) opens tonight at 19h00. Deadline is Thu 22 Feb

More information

Math 370 Semester Review Name

Math 370 Semester Review Name Math 370 Semester Review Name 1) State the following theorems: (a) Remainder Theorem (b) Factor Theorem (c) Rational Root Theorem (d) Fundamental Theorem of Algebra (a) If a polynomial f(x) is divided

More information

NFC ACADEMY COURSE OVERVIEW

NFC ACADEMY COURSE OVERVIEW NFC ACADEMY COURSE OVERVIEW Algebra II Honors is a full-year, high school math course intended for the student who has successfully completed the prerequisite course Algebra I. This course focuses on algebraic

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler

More information

Secondary Math GRAPHING TANGENT AND RECIPROCAL TRIG FUNCTIONS/SYMMETRY AND PERIODICITY

Secondary Math GRAPHING TANGENT AND RECIPROCAL TRIG FUNCTIONS/SYMMETRY AND PERIODICITY Secondary Math 3 7-5 GRAPHING TANGENT AND RECIPROCAL TRIG FUNCTIONS/SYMMETRY AND PERIODICITY Warm Up Factor completely, include the imaginary numbers if any. (Go to your notes for Unit 2) 1. 16 +120 +225

More information

Complex Numbers Class Work. Complex Numbers Homework. Pre-Calc Polar & Complex #s ~1~ NJCTL.org. Simplify using i b 4 3.

Complex Numbers Class Work. Complex Numbers Homework. Pre-Calc Polar & Complex #s ~1~ NJCTL.org. Simplify using i b 4 3. Complex Numbers Class Work Simplify using i. 1. 16 2. 36b 4 3. 8a 2 4. 32x 6 y 7 5. 16 25 6. 8 10 7. 3i 4i 5i 8. 2i 4i 6i 8i 9. i 9 10. i 22 11. i 75 Complex Numbers Homework Simplify using i. 12. 81 13.

More information

Reteach Simplifying Algebraic Expressions

Reteach Simplifying Algebraic Expressions 1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order

More information

Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number

Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number Complex Numbers December, 206 Introduction 2 Imaginary Number In your study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions. For example, try as

More information

College Algebra & Trig w Apps

College Algebra & Trig w Apps WTCS Repository 10-804-197 College Algebra & Trig w Apps Course Outcome Summary Course Information Description Total Credits 5.00 This course covers those skills needed for success in Calculus and many

More information

UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS

UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS Revised Dec 10, 02 38 SCO: By the end of grade 12, students will be expected to: C97 construct and examine graphs in the polar plane Elaborations

More information

Units. Year 1. Unit 1: Course Overview

Units. Year 1. Unit 1: Course Overview Mathematics HL Units All Pamoja courses are written by experienced subject matter experts and integrate the principles of TOK and the approaches to learning of the IB learner profile. This course has been

More information

Catholic Central High School

Catholic Central High School Catholic Central High School Course: Basic Algebra 2 Department: Mathematics Length: One year Credit: 1 Prerequisite: Completion of Basic Algebra 1 or Algebra 1, Basic Plane Geometry or Plane Geometry,

More information

Chapter 7 PHASORS ALGEBRA

Chapter 7 PHASORS ALGEBRA 164 Chapter 7 PHASORS ALGEBRA Vectors, in general, may be located anywhere in space. We have restricted ourselves thus for to vectors which are all located in one plane (co planar vectors), but they may

More information

Math 370 Exam 3 Review Name

Math 370 Exam 3 Review Name Math 370 Exam 3 Review Name The following problems will give you an idea of the concepts covered on the exam. Note that the review questions may not be formatted like those on the exam. You should complete

More information

C) ) cos (cos-1 0.4) 5) A) 0.4 B) 2.7 C) 0.9 D) 3.5 C) - 4 5

C) ) cos (cos-1 0.4) 5) A) 0.4 B) 2.7 C) 0.9 D) 3.5 C) - 4 5 Precalculus B Name Please do NOT write on this packet. Put all work and answers on a separate piece of paper. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the

More information

CHAPTER 6: ADDITIONAL TOPICS IN TRIG

CHAPTER 6: ADDITIONAL TOPICS IN TRIG (Section 6.1: The Law of Sines) 6.01 CHAPTER 6: ADDITIONAL TOPICS IN TRIG SECTION 6.1: THE LAW OF SINES PART A: THE SETUP AND THE LAW The Law of Sines and the Law of Cosines will allow us to analyze and

More information

Curriculum Mapper - Complete Curriculum Maps CONTENT. 1.2 Evaluate expressions (p.18 Activity 1.2).

Curriculum Mapper - Complete Curriculum Maps CONTENT. 1.2 Evaluate expressions (p.18 Activity 1.2). Page 1 of 9 Close Window Print Page Layout Show Standards View Paragraph Format View Course Description MATH 3 (MASTER MAP) School: Binghamton High School Teacher: Master Map Email: Course #: 203 Grade

More information

Glossary Common Core Curriculum Maps Math/Grade 9 Grade 12

Glossary Common Core Curriculum Maps Math/Grade 9 Grade 12 Glossary Common Core Curriculum Maps Math/Grade 9 Grade 12 Grade 9 Grade 12 AA similarity Angle-angle similarity. When twotriangles have corresponding angles that are congruent, the triangles are similar.

More information

Objectives List. Important Students should expect test questions that require a synthesis of these objectives.

Objectives List. Important Students should expect test questions that require a synthesis of these objectives. MATH 1040 - of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List

More information

Notice that we are switching from the subtraction to adding the negative of the following term

Notice that we are switching from the subtraction to adding the negative of the following term MTH95 Day 6 Sections 5.3 & 7.1 Section 5.3 Polynomials and Polynomial Functions Definitions: Term Constant Factor Coefficient Polynomial Monomial Binomial Trinomial Degree of a term Degree of a Polynomial

More information

Name. Use Two-Color Counters to model each addition problem. Make pairs of red and yellow counters. Find the sum.

Name. Use Two-Color Counters to model each addition problem. Make pairs of red and yellow counters. Find the sum. Lesson 1 The Number System Name Use Two-Color Counters to model each addition problem. Make pairs of red and yellow counters. Find the sum. 1. 2. 9 + ( 10) 18 + 9 Using Two-Color Counters, model each addition

More information

CURRICULUM GUIDE. Honors Algebra II / Trigonometry

CURRICULUM GUIDE. Honors Algebra II / Trigonometry CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed

More information

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT Algebra 2B KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS RATIONAL AND RADICAL FUNCTIONS PROPERTIES AND

More information

Overall Description of Course Trigonometry is a College Preparatory level course.

Overall Description of Course Trigonometry is a College Preparatory level course. Radnor High School Course Syllabus Modified 9/1/2011 Trigonometry 444 Credits: 1 Grades: 11-12 Unweighted Prerequisite: Length: Year Algebra 2 Format: Meets Daily or teacher recommendation Overall Description

More information

Lesson 25 Solving Linear Trigonometric Equations

Lesson 25 Solving Linear Trigonometric Equations Lesson 25 Solving Linear Trigonometric Equations IB Math HL - Santowski EXPLAIN the difference between the following 2 equations: (a) Solve sin(x) = 0.75 (b) Solve sin -1 (0.75) = x Now, use you calculator

More information

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

More information

Pre-Calculus EOC Review 2016

Pre-Calculus EOC Review 2016 Pre-Calculus EOC Review 2016 Name The Exam 50 questions, multiple choice, paper and pencil. I. Limits 8 questions a. (1) decide if a function is continuous at a point b. (1) understand continuity in terms

More information

Math Precalculus Blueprint Assessed Quarter 1

Math Precalculus Blueprint Assessed Quarter 1 PO 11. Find approximate solutions for polynomial equations with or without graphing technology. MCWR-S3C2-06 Graphing polynomial functions. MCWR-S3C2-12 Theorems of polynomial functions. MCWR-S3C3-08 Polynomial

More information

C. Complex Numbers. 1. Complex arithmetic.

C. Complex Numbers. 1. Complex arithmetic. C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.

More information

Lesson 22 - Trigonometric Identities

Lesson 22 - Trigonometric Identities POP QUIZ Lesson - Trigonometric Identities IB Math HL () Solve 5 = x 3 () Solve 0 = x x 6 (3) Solve = /x (4) Solve 4 = x (5) Solve sin(θ) = (6) Solve x x x x (6) Solve x + = (x + ) (7) Solve 4(x ) = (x

More information

Discrete mathematics I - Complex numbers

Discrete mathematics I - Complex numbers Discrete mathematics I - Emil Vatai (based on hungarian slides by László Mérai) 1 January 31, 018 1 Financed from the financial support ELTE won from the Higher Education Restructuring

More information

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards Core: Operations on Numbers and Expressions Priority: 15% Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions,

More information

Common Core State Standards. What is the goal of the Common Core State Standards?

Common Core State Standards. What is the goal of the Common Core State Standards? SECTION 1 Trusted Content Common Core State W ith American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. Common Core State

More information

MTH 122: Section 204. Plane Trigonometry. Test 1

MTH 122: Section 204. Plane Trigonometry. Test 1 MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π

More information

Unit 13: Polynomials and Exponents

Unit 13: Polynomials and Exponents Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:

More information

Pre-Calculus and Trigonometry Capacity Matrix

Pre-Calculus and Trigonometry Capacity Matrix Information Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving

More information

Unit 1. Revisiting Parent Functions and Graphing

Unit 1. Revisiting Parent Functions and Graphing Unit 1 Revisiting Parent Functions and Graphing Precalculus Analysis Pacing Guide First Nine Weeks Understand how the algebraic properties of an equation transform the geometric properties of its graph.

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area

More information

A basic trigonometric equation asks what values of the trig function have a specific value.

A basic trigonometric equation asks what values of the trig function have a specific value. Lecture 3A: Solving Basic Trig Equations A basic trigonometric equation asks what values of the trig function have a specific value. The equation sinθ = 1 asks for what vales of θ is the equation true.

More information

Lecture 3f Polar Form (pages )

Lecture 3f Polar Form (pages ) Lecture 3f Polar Form (pages 399-402) In the previous lecture, we saw that we can visualize a complex number as a point in the complex plane. This turns out to be remarkable useful, but we need to think

More information

Curriculum Catalog

Curriculum Catalog 2017-2018 Curriculum Catalog 2017 Glynlyon, Inc. Table of Contents PRE-CALCULUS COURSE OVERVIEW...1 UNIT 1: RELATIONS AND FUNCTIONS... 1 UNIT 2: FUNCTIONS... 1 UNIT 3: TRIGONOMETRIC FUNCTIONS... 2 UNIT

More information

Math Analysis Curriculum Map Kennett High School

Math Analysis Curriculum Map Kennett High School Section Topic Specific Concept Standard Assessment mp assignments 1.1 Coordinate geometry distance formula, midpoint fomrula G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles

More information

Chapter 5 Trigonometric Functions of Angles

Chapter 5 Trigonometric Functions of Angles Chapter 5 Trigonometric Functions of Angles Section 3 Points on Circles Using Sine and Cosine Signs Signs I Signs (+, +) I Signs II (+, +) I Signs II (, +) (+, +) I Signs II (, +) (+, +) I III Signs II

More information

Upon completion of this course, the student should be able to satisfy the following objectives.

Upon completion of this course, the student should be able to satisfy the following objectives. Homework: Chapter 6: o 6.1. #1, 2, 5, 9, 11, 17, 19, 23, 27, 41. o 6.2: 1, 5, 9, 11, 15, 17, 49. o 6.3: 1, 5, 9, 15, 17, 21, 23. o 6.4: 1, 3, 7, 9. o 6.5: 5, 9, 13, 17. Chapter 7: o 7.2: 1, 5, 15, 17,

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

Lesson 28 Working with Special Triangles

Lesson 28 Working with Special Triangles Lesson 28 Working with Special Triangles Pre-Calculus 3/3/14 Pre-Calculus 1 Review Where We ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane

More information

Math 144 Activity #7 Trigonometric Identities

Math 144 Activity #7 Trigonometric Identities 144 p 1 Math 144 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value

More information

0615a2. Algebra 2/Trigonometry Regents Exam x 2 y? 4 x. y 2. x 3 y

0615a2. Algebra 2/Trigonometry Regents Exam x 2 y? 4 x. y 2. x 3 y Algebra /Trigonometry Regents Exam 065 www.jmap.org 065a Which list of ordered pairs does not represent a one-to-one function? ) (, ),(,0),(,),(4,) ) (,),(,),(,4),(4,6) ) (,),(,4),(,),(4,) 4) (,5),(,4),(,),(4,0)

More information

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations

sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on

More information

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS

CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS Course Number 5121 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra

More information

Math 1316 Exam 3. if u = 4, c. ÄuÄ = isin π Ë 5 34, , 5 34, 3

Math 1316 Exam 3. if u = 4, c. ÄuÄ = isin π Ë 5 34, , 5 34, 3 Math 36 Exam 3 Multiple Choice Identify the choice that best completes the statement or answers the question.. Find the component form of v if ÄÄ= v 0 and the angle it makes with the x-axis is 50. 0,0

More information