# Week 13 Trigonometric Form of Complex Numbers

Size: px
Start display at page:

Transcription

1 Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working with complex numbers. We will have to go through some basics to get to the point where trigonometric form is most helpful raising complex numbers to powers or finding roots of complex numbers. I will not go into as much detail as a textbook, but try to make sense out of what we are using. We will: Convert standard form of a complex number to trigonometric form. Convert trigonometric form to standard form. Multiply and divide complex numbers. Raise complex numbers to a power using DeMoivre s Theorem. Find all roots to a complex number. Geometry knowledge needed: Basic properties of circles Algebra skills needed: Complex number form and graphing in complex number plane. Multiplying and dividing complex numbers (for checking purposes). Rationaliing a denominator using conjugates Simplifying a radical. Laws of exponents. Trigonometric Form of a Complex Number Recall the basics of complex numbers, where we use the letter. o The standard form of a complex number is a bi, where a and b are real numbers, e.g., +i, -4+i, 0+i, -i, etc. o A complex number has a real part, a, and a pure imaginary part, bi. In the number i, is the real part and -i is the pure imaginary part. o We can graph a complex number in the complex number plane, which has a horiontal real axis and a vertical pure imaginary axis. See the points in the plane in the figure.

2 You can see that this is very much like plotting corresponding points in the rectangular coordinate system. We can work with these complex numbers in much the same way. Let s focus on a general complex number a + bi and bring up some of the ideas that we have been using in this course - r and angle θ. Notice that the complex number is at a distance of r from the origin. Using the right triangle, we know r a b. We use θ and its functions to get cos a b a r cos while sin b r sin r r. Using these relations, we write the standard form of a complex number in trigonometric form. This substitution yields ( r cos ) ( r sin ) i. When we remove parentheses, there can be a little confusion with the imaginary part. The i is not multiplying θ. The definition will rewrite this term and factor the r from both parts. The trigonometric form of the complex number a bi is r(cos i sin ), b where a r cos, b r sin, r a b, and tan. a The r is called the modulus of, and θ is called the argument of.

4 4 Example : Write the complex number 7 4i in trigonometric form. Make a sketch of. We keep in mind that is in quadrant III. r r a b ( 7) ( 4) 6 which cannot be simplified. b 4 4 tan a tan 9.7 as the reference angle. 7 Since is in QIII, Therefore, 74i 6(cos09.7 i sin09.7 ) or 6cis09.7. Example : Write 4(cos0 isin0 ) in standard form. Make a sketch of. We do what the trigonometric form indicates and multiply. Since the reference angle is a special angle of 60, we use exact values. This number is definitely in QII. 4(cos0 isin0 ) 4 i 4 i 4 i The answer above also illustrates what I mentioned about being careful with the radical sign. The i is not under the radical sign. It might be safer to leave a little space after the radical before the i, e.g. i. You can also put the i in front of the term, but that is not usual for standard form. When I write on the board in class, I usually just add a little hook at the end of the radical sign top, as a separator like this: á Now work the problem set for Trigonometric Form of a Complex Number.

5 Problems for Trigonometric Form of a Complex Number. Convert the complex number i to trigonometric form. Make a sketch of.. Write the complex number i in trigonometric form. Make a sketch of.. Write (cos0 isin0 ) in standard form. Make a sketch of.

6 6 Answers to Problems for Trigonometric Form of a Complex Number. The sketch is a reminder that is in quadrant II. r a b r 9 ( ) 8 b tan a tan ( ) Reference angle is 4. Therefore, i (cos i sin ). If radian measure is used, goes in 4 place of.. The sketch is a reminder that is in QIII. r ( ) ( ) 4 r 4 It cannot be simplified. tan = tan 9.0 Reference angle in QI. Therefore, i 4(cos9 i sin9 ).. The sketch reminds us that is in QIV. We will be exact with this special angle. (cos0 i sin0 ) Reference angle i s 0 i( ) i in standard form.

7 7 Multiplication, Division, and Powers of Complex Numbers In algebra, multiplying complex numbers involved the same procedure as multiplying binomials. For example: ( i )( 4 i ) 8 0i i i 8 i ( ) 7i This is not a lengthy process, so using trigonometric form is not necessarily better. It is a step towards where we are heading, however. We can also check some answers this way. Developing the rule for multiplying complex numbers in trigonometric form uses good algebra and past identities: r (cos i sin ) r (cos i sin ) r r ( cos cos i cos sin i sin cos i sin sin ) r r ( cos cos sin sin ) i(cos sin sin cos r r cos( ) i sn i ( In words, the rule tells us to multiply the moduli (plural of modulus) and add the arguments. That requires each complex number to be in trigonometric form. Example : Find the product of (cos60 i sin60 ) and 4(cos0 i sin0 ). This is where the abbreviation is particularly handy. Abbreviate both numbers and multiply. cis60 4cis0 4cis(600 ) 8cis0 Reference angle is 0. 8(cos0i sin0 ) 8( i( )) 4 4i Sometimes you may have to convert each number to trigonometric form first. Also, the sum of the angles or arguments is not always a special angle. When that happens you will have to use your calculator in the appropriate mode and round off as specified. Dividing complex numbers algebraically took considerably more work (Doesn t division always require more effort?). We will look at one example as a reminder. For example, let us divide two complex numbers such as i and i. The process was as follows.

8 8 i i i i i i 6 9i 0i 4 i 6 9i 0( ) 4 ( ) 4 9i 4 9 i Multiply by to rationalie the denominator. Multiplying binomials. Developing the rule for dividing complex numbers in trigonometric form uses this same process along with past identities. The rule: r cos( ) i sin ( ), 0. r In words, to divide two complex numbers in trigonometric form, divide the moduli and subtract the arguments. That certainly seems shorter than algebra, but it does require the trigonometric form first. Example : Find the quotient if (cos0 i sin0 ) and 4(cos70 i sin70 ). Since these are not special angles, to do this problem algebraically would require a lot of decimals. We use the rule to divide (and the handy abbreviation again). cis0 cis(0 70 ) 4cis70 4 cis40 (cos40 i sin40 ) i( ) i 4 4 The real purpose of this section is to get to powers of complex numbers. A power of a number is a repeated multiplication. For example, is a multiplication problem with factors (but only four multiplications). If we want, using the multiplication rule 4 times, then we would have r as a factor times and we would add arguments: r r r r r i cos( ) sin( ) r (cos i sin )

9 9 In general, the power of a complex number is found by: n n r (c osn i s inn), n a positive integer. This is called DeMoivre s Theorem, named after a French mathematician. Example : Find if (cos4 i sin4 ). Using the abbreviation will save a lot of writing and worrying about enough parentheses. (cis 4 ) Using the abbreviation. cis( 4 ) Applying the power rule. 4cis 0 Reference angle is 60 in QII. 4 cos0 i sin0 4 i 4 4 i There is no advantage in writing with mixed numbers. Example 4: Find if i. We will keep in mind that is in quadrant IV. We need to first find the trigonometric form. r tan 00 ( ) 4 tan ( ) Reference angle is 60 in QIV. The trigonometric form of is (cos00 i sin00 ) or cis 00. Raising to a power: (cis00 ) cis(00 ) 8cis 900 Find an equivalent smaller angle. 8cis 80 Subtracted 60 twice. 8(cos80 i sin80) 8( i(0)) 8 0i

10 0 You may have thought of a couple of questions while reading through these examples. Are all of these steps necessary? No. Since I cannot anticipate where someone might have any trouble or a question, I write out every step. The last example, for instance, could be shortened to only the critical steps. r ( ) tan 60 in QI, 00 in QIV cis 00 cis 900 8cis 80 8( i(0)) 8 0i This procedure is still less work than doing it algebraically. What if the angle is not a special angle? In reality, that happens frequently. If directions do not specify, carry the angle you find for trigonometric form out to one decimal place. Since no other calculation is done until the final product or quotient or power, you can then carry out the cosine and sine values to or 4 decimal places, unless otherwise specified. For example, suppose your tangent was a number such that θ =. so that the final trigonometric form for is 4cis.. Then you write out the form and use your calculator. 4cis. 4(cos. i sin. ) 4(.48 i(.9048)).70.69i Depending on the objective, a textbook author may leave the trigonometric form when numbers are like this. If you are raising a complex number to a power, and the parts involve integers only, then the power (final answer) should also be rounded to integers, if that does not automatically happen. á Now work the problem set for Multiplication, Division, and Powers of Complex Numbers.

11 Problems for Multiplication, Division, and Powers of Complex Numbers. Find the product if (cos7 i sin7 ) and (cos i sin ).. Find the quotient if (cos i sin ) and 6(cos i sin ).. Find if (cos0 isin0 ). 4. Find 6 if i.

12 Answers to Multiplying, Dividing, and Powers of a Complex Number cis 7 and cis (cis7 )(cis ) cis(7) 6cis 0 6(cos0 i sin0 ) Reference angle is 0 in QII. 6 i( ) i cis and 6cis cis cis( ) 6cis 6 cis 80 (cos80 i sin80 ) ( i(0)) 0i (cos0 i sin0 ) cis0 cis(0 ) cis 600 Find an equivalent angle less than 60. cis 40 Subtracted 60. (cos40 i sin40 ) Reference angle is 60 in QIII. i( ) 6 6 i i r ( ) ( ) 4 tan cis 4 (cis 4 ) tan 4 and is in QI cis( 64 ) 64cis 70 64(cos70 i sin70 ) 64(0 i( )) 064i

13 Roots of a Complex Number In algebra, you were able to find roots to polynomial equations such as 4 x 0, x 6 0, 8x 0 by factoring, the quadratic formula, or a combination of these methods. You may have also learned how to use theorems to find accurate guesses for roots of higher degree polynomial equations. However, algebra methods failed for finding all solutions to some polynomial equations, such as 7 x 0, a simple equation but not easy to solve. Trigonometry can help find roots to some equations that algebra cannot, or uses an alternate method to solve some of those equations Recall two notations for the n th root of a number a: Using a radical, the notation is n a. n Using exponential form: a. Trigonometry has a method for finding all roots of a simple polynomial equation. It is a result of DeMoivre s Theorem applied in reverse (plus algebra techniques). By definition, a complex number, say w, is the nth root of another complex number, say, if w How do we find the roots? The complex number r(cosi sin ) has exactly n (a positive integer) distinct roots given by: n. cos k n r i sin k where k 0,,,..., n. n n This can look intimidating, so let s look at it in pieces. First, for all roots we take the nth root of the complex number s modulus r. The first root s argument comes from the first π or 60, then dividing by n. The remaining arguments for other roots come from going around the circle (adding π) one more time, before dividing by n, for each argument. Since k starts at 0, k will only go up to n- to get the last root. (Roots repeat after that.)

14 4 Example : Find all the cube roots of. (Let = + 0i.) Write the number in trigonometric form, noting that is on the positive x-axis, and then apply the rule. 0 i r cis 0 cis cis0 (cos0 i sin0 ) ( i 0) 0 60 cis cis 0 i( ) i 0, 0 and tan (60 ) cis cis 40 i( ) i This problem is the equivalent of solving the equation x 0. You can check this by factoring and using the quadratic formula (if you remember how to factor the difference of two cubes). Some textbook authors focus on a shortcut for finding the argument. If you want to do that, it is fine. I think that working neatly and orderly where you can see the thinking process, as in the example, is the best thing to do, plus it is easy to forget shortcuts. Before we go through another example, it is useful to note the geometric significance of the roots. They are spaced equally around the circle. (This example involved special angles, which is not always the case, nor do the roots always start at 0.) While this method is most useful for higher degree (power), I am keeping the example function degrees smaller so as not to overwhelm you at the start. Once you get used to the method, you can (and should) try problems that have more roots.

15 Example : Find all the fourth roots of -4. We first change to trigonometric form, noticing that = i is on the negative x-axis, and then use the rule i r ( 4) 0 4 and tan 80 this time. 4 4cis 80 n r (4) ( ) 80 cis cis 4 i i 4 Simplify the radical cis cis i i 4 80 (60 ) cis cis i( ) i 4 80 (60 ) 4 cis cis i( i 4 Starting at 4 for the first root, we see that the other roots are spaced equally around the circle of radius. See the next figure. This problem is the equivalent of solving the equation x Also, notice that a theorem from algebra that talks about complex roots to a real number occurring in conjugate pairs holds true here. Okay, what happens when there are angles in the roots that are not special? (It is also possible that the trigonometric form of will involve an angle that is not special.)

16 6 Example : Find all fifth roots of = - + i. We notice that is in quadrant II when we write it in trigonometric form. n i r r ( ) and tan. 0 0 Be careful with the double radical. 0 0 cis cis 7.078cis i cis cis cis i 0 (60 ) 0 cis cis 7.078cis i 0 (60 ) 0 4 cis cis 4.078cis i 0 4( 60 ) 0 cis cis.078cis i The roots are equally spaced around a circle of radius 0.078, starting at 7. If you are haphaard about the way you do things, you are apt to make quite a few mistakes. Instead of being in a hurry, go at a steady and orderly pace. á Now work the problem set for Roots of a Complex Number.

17 7 Problems for Roots of a Complex Number. Find all fourth roots of 6. ( = 6 + 0i). Mark the roots on the appropriate circle.. Find all cube roots of -7. ( = i) Mark the roots on the appropriate circle.. Find all cube roots of i. Mark the roots on the appropriate circle.

18 8 Answers to Problems for Roots of Complex Numbers. n 0 i r 6 6cis and tan 0 (on positive x-axis) r cis cis 0 ( i 0) cis cis 90 (0 i()) 0 i 4 0 (60 ) cis cis 80 ( i 0) 4 0 (60 ) 4 cis cis 70 (0 i( ) 0 i 4 You can leave the 0 out in answers and 4.. n 0 i r 7 7cis ( 7) 0 7 and tan 80 ( is on negative x-axis) r 7 (I know it's a lot of 's. 80 cis cis 60 i( ) i cis cis 80 ( i 0) 80 (60 ) cis cis 00 i( ) i )

19 9. n i r ( ) 8 and tan ( is in QIV) 8cis r 8 ( 8) ( ) ( ) ( ) cis cis0 (cos0 i sin0 ) i ( 60 ) cis cis (cos i sin ) i (60 ) cis cis 4 (cos4 i sin4 ) i I hope that this course has been helpful, and that you understand what Trigonometry does. Thank you for giving it a try. You can still contact me if you need help. A Note about the Polar Coordinate System If you are taking Trigonometry in a Pre-Calculus course, the Polar Coordinate System will be a topic, probably a chapter, in your textbook. That is because the Polar Coordinate System is used at times in Calculus. If you are headed toward Calculus, you should do next week s lesson on the Polar Coordinate System.

### MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

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

### COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS _4+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

### Trigonometric Functions and Equations

Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

### Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### Unit 6 Trigonometric Identities, Equations, and Applications

Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

### Trigonometry Hard Problems

Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.

### Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

### Evaluating trigonometric functions

MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### Math Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.

Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Section 6-3 Double-Angle and Half-Angle Identities

6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Euler s Formula Math 220

Euler s Formula Math 0 last change: Sept 3, 05 Complex numbers A complex number is an expression of the form x+iy where x and y are real numbers and i is the imaginary square root of. For example, + 3i

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### 2312 test 2 Fall 2010 Form B

2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

### Tips for Solving Mathematical Problems

Tips for Solving Mathematical Problems Don Byrd Revised late April 2011 The tips below are based primarily on my experience teaching precalculus to high-school students, and to a lesser extent on my other

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### GRE Prep: Precalculus

GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### How to Graph Trigonometric Functions

How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

### 2.5 Zeros of a Polynomial Functions

.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### X On record with the USOE.

Textbook Alignment to the Utah Core Algebra 2 Name of Company and Individual Conducting Alignment: Chris McHugh, McHugh Inc. A Credential Sheet has been completed on the above company/evaluator and is

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### DRAFT. Further mathematics. GCE AS and A level subject content

Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

### One advantage of this algebraic approach is that we can write down

. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### CAMI Education linked to CAPS: Mathematics

- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Unit 11 Additional Topics in Trigonometry - Classwork

Unit 11 Additional Topics in Trigonometry - Classwork In geometry and physics, concepts such as temperature, mass, time, length, area, and volume can be quantified with a single real number. These are

### Trigonometry for AC circuits

Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### Homework 2 Solutions

Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### 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.

### 1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what

### Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### ACT Math Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Complex Algebra. What is the identity, the number such that it times any number leaves that number alone?

Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession

### Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts