M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.


 Michael Lucas
 2 years ago
 Views:
Transcription
1 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides of sin θ cos θ + = to see another form of this identity. b. Subtract sin θ from both sides of sin θ cos θ + = to see another form of this identity.. There are two more variations of the Pythagorean Identity, involving the other trigonometric functions. a. To find one of the variations, divide both sides of simplify each term. sin θ cos θ + = by cos θ, and b. Subtract from both sides of your equation in part (a) to write this identity in a different form. c. To find another variation of the Pythagorean Identity, divide both sides of sin θ + cos θ = by sin θ, and simplify each term. d. Subtract from both sides of your equation in part (c) to write this identity in a different form.
2 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Part 3: Simplifying Trigonometric Epressions General guidelines for simplifying trig epressions: More simplified means fewer operations and fewer fractions. Positive angles are simpler than negative angles. For eample, tan is more simplified tan. than ( ) Things to try: Substitute one side of a trig identity in place of the other side. For eample, replaced by ( sin θ ). cos θ can be Factor or multiply. Use a common denominator to add or subtract fractions. Write all parts in terms of sines and cosines. When you see sums or differences of squares and s, substitute with a Pythagorean identity. 5. Before we begin simplifying, let s write all of our trig identities together in one place: a. Reciprocal identities: cscθ = secθ = cotθ = b. Quotient identities: tanθ = cotθ = c. Pythagorean identities: (see page ) d. Cofunction identities: (see page of packet from unit 3) π sin θ = π cos θ = π tan θ = π csc θ = π sec θ = π cot θ = e. Even/odd identities: sin csc ( θ ) = cos ( θ ) = tan ( θ ) ( θ ) = sec ( θ ) = cot ( θ ) = =
3 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Simplify these trigonometric epressions. A hint is given for each. a. csc tan Write in terms of sines and cosines. b. + cot cos Write in terms of sines and cosines, and get a common denominator. c. ( secθ )( secθ ) + Multiply. Then use a Pythagorean identity to substitute. d. + cot sec Use a Pythagorean identity, or write in terms of sines and cosines. e. + cos + cos Get a common denominator, and use a Pythagorean identity to substitute. f. sin cos ( ) ( ) Use even/odd identities. 3
4 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO VERIFYING TRIGONOMETRIC IDENTITIES 5. Part : The Basics A trigonometric identity is a statement about trig functions that is true for all values in their domain. For eample, the identity tan cos To verify an identity, use algebra to prove the two sides equal. = is true for all values at which tan eists. Because you are proving equality, you are not allowed to use properties of equality in your proof. Thus, adding, subtracting, multiplying, or dividing the same number to both sides is not allowed. In fact, you cannot do anything that changes the value of either side, such as squaring, taking the square root, etc You must either work with one side until you get it to match the other, or work on each side separately until you get them to match.. One technique is to work with only the more complicatedlooking side. Try simplifying the right side until you get it to match the left side: cotθ = cosθcscθ. If you see a sum or difference of fractions, use a common denominator to combine the fractions first. Try this one, only working on the left side: cos cos + = + cos 4
5 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Try simplifying the left side by multiplying: ( ) sinθ + cosθ tanθ = tanθ + sin θ 4. As with simplifying, when you see sums or differences of squares and s, it may be helpful to substitute with Pythagorean identities on one or both sides: = sec Part : More Advanced Techniques 5. This identity is already written in terms of sine and cosine and does not contain squares. By our guidelines on page, you d be stuck! A more advanced technique is to multiply the numerator and denominator by a conjugate to create a difference of squares. Multiply : the numerator and denominator of the left side by ( + ), the conjugate of ( ) cos + = cos 5
6 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Factoring can also be helpful. Factor the numerator of the left side: = cos 4 sin cos 7. To simplify a comple fraction (fractions within a fraction), multiply the numerator and denominator by an LCD to eliminate denominators within the fraction. Here, convert the left side to sines and cosines, and simplify the comple fraction: cos tan + cot = 8. Look for features that are easy to match. In this eample, the left side has sec in the numerator, and the right side has cos in the denominator. Make one of these match the other, and then see what you can do with the rest: ( sec )( sec ) + = + cos sec tan tan 6
7 5D M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 PRACTICE VERIFYING TRIGONOMETRIC IDENTITIES 5D Verify each trigonometric identity. Use the hints on p. 46 of this packet to help you start.. csc = csc csc. sec β + tan β = tan β tan β 3. cos y sec y+ tan y = sin y 4. cos sin tan = cos 5. cos + = 0 cos + 7
8 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO SOLVING TRIGONOMETRIC EQUATIONS 5.3 Part : Finding solutions in [ ) 0,π. Label the radian measures of the angles in the other 3 quadrants with the given reference angle. π π 3 π 4 6. To solve a trig equation, use inverse operations to isolate the trig function. a. In the equation + 3 = 0, first isolate. b. Now list all angles in [ 0,π ) that satisfy the resulting statement. 3. Sometimes, you need to factor to isolate the trig function. a. Factor the equation 0 =. (If you can t factor as is, let u sin = first.) b. Set each factor equal to 0, and isolate in each. c. List all angles in [ 0,π ) that satisfy the resulting statements. 8
9 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Do you see more than one trig function? Use an identity to rewrite in terms of only one. Here, convert to all cosines, and then find all solutions in [ 0,π ) : 3cos + 3 = sin 5. In this equation, remember the ± after you take the square root of both sides: Find all solutions in [ 0,π ) for 4cos 3 = 0. Part : Finding a General Formula for Solutions 6. So far, we ve only found solutions in the interval [ 0,π ), but all of these equations really have infinite solutions. Why is that? 7. a. Isolate in + = 0. b. In the interval [ 0,π ), there are solutions. Fill them in the blanks below: = + nπ or = + nπ, where n is any integer The + nπ makes this a general solution. It means that you can take the angle measures you found and add π ( ), π ( ), ( ) makes the equation true. 3π, etc to find another angle that For sine and cosine equations, we use + nπ in our general formulas because the period of the graph is π. 9
10 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO a. Isolate tan in tan =, and fill in the blanks with solutions from the interval [ ) 3 0,π : = + nπ or = + nπ, where n is any integer b. The + nπ means that we can add π, π, 3π, etc to find another angle that solves the equation. Why do we use nπ here instead of nπ? 9. Solve these, giving general formulas for your answers. a. + = b. = 3( cos ) cos cos 0 Part 3: Solving Equations with b 0. Suppose cos = : a. Fill in the blanks with angles in [ 0,π ) that have cosine value = + nπ or = + nπ, where n is any integer : b. Divide both sides of each equation by to finish solving for. c. Now use your formulas from part (b) to find all angles in the interval [ 0,π ) that solve the original equation. 0
11 . Consider the equation ( ) a. Isolate sin( 3 ). M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 sin 3 = 0. b. Fill in the blanks with angle measures in [ 0,π ) that have the sine value shown in your isolated equation: 3 = + nπ or 3 = + nπ or 3 = + nπ or 3 = + nπ, n is any integer c. Divide by 3 to finish solving for. d. Use your formulas to list all angles in [ 0,π ) that solve the original equation.. Find a general formula and all solutions in [ 0, π ) for the equation ( ) tan =. 3. Find a general formula and all solutions in [ 0,π ) for the equation 3 sin =.
12 Review M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 REVIEW OF 5. TO 5.3 Review Identities to memorize: = csc csc = cos = sec sec = cos tan = cot cot = tan tan = cos cos cot = Pythagorean Identity: + cos = π Cofunction Identities: sin = cos π cos = π csc = sec π sec = csc cos cos Odd/Even Identities: ( ) = ( ) = ( ) csc( ) = csc sec( ) = sec cot ( ) π tan = cot π cot = tan tan = tan = cot. Derive all other forms of the Pythagorean Identity, and write them below. Simplify.. sec + + tan cos cos cos cos cos tan tan + sec sec + 6. cot cos cos + 7. ( sec + tan )( sec tan )
13 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Verify each identity. 8. tan sec cos = 9. ( )( ) sec + tan = cos sec tan = ( ) sin + cos + = csc + cos. csc + sec = csc sec 3. cos cos + = sec csc cos 3
14 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Find a general formula and all solutions in [ 0,π ). 4. cos = = 6. sin 3sin 0 3sec = sec Find a general formula and all solutions in [ 0,π ). 7. sin = 3cos 8. cos = 9. sin3 = 0. tan 4 =. 4 tan sec + 3 = 0 Answers:. csc 3. cos 4. cot 5. csc 6. tan = nπ; { 0, π} π 5π π π 5π π 5. = + nπ, + nπ, + nπ;,, , 6. no solution π 5π π 5π 7. = + nπ, + nπ;, π π 3π 8. nπ ;, π nπ 5π nπ π 3π 5π 5π 7π 9π = + 9. = +, + ;,,,,, π nπ 3π 7π π 5π 9π 3π 7π 3π 0. = + ;,,,,,,, π nπ π 3π 5π 7π. = + ;,,,
PreCalculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.
PreCalculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationAlgebra. 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
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
More informationTrigonometric Identities and Equations
LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now
More informationComplex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers
Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension
More informationSection 63 DoubleAngle and HalfAngle Identities
63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
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
More informationClass XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34
Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in
More informationTrigonometric 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
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationMath 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
More informationIntegration Involving Trigonometric Functions and Trigonometric Substitution
Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationALGEBRA 2/ TRIGONOMETRY
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/ TRIGONOMETRY Friday, June 14, 2013 1:15 4:15 p.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 28.......................
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More information2. Right Triangle Trigonometry
2. Right Triangle Trigonometry 2.1 Definition II: Right Triangle Trigonometry 2.2 Calculators and Trigonometric Functions of an Acute Angle 2.3 Solving Right Triangles 2.4 Applications 2.5 Vectors: A Geometric
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More information1 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 informationSample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x
Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More informationGeometry 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
More informationTRIGONOMETRY 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
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, June 14, 013 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationASA Angle Side Angle SAA Side Angle Angle SSA Side Side Angle. B a C
8.2 The Law of Sines Section 8.2 Notes Page 1 The law of sines is used to solve for missing sides or angles of triangles when we have the following three cases: S ngle Side ngle S Side ngle ngle SS Side
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus 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
More informationTrigonometry 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:
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationHCC COLLEGE PLACEMENT TEST MATHEMATICS
HCC COLLEGE PLACEMENT TEST MATHEMATICS PLEASE READ BEFORE TAKING THE MATHEMATICS PORTION OF THE COLLEGE PLACEMENT TEST It is highl recommended that ou review this packet BEFORE ou take the College Placement
More informationEvaluating trigonometric functions
MATH 1110 0090906 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationTrigonometry Chapter 3 Lecture Notes
Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More informationAngles 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
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationDr. Foote s Algebra & Calculus Tips or How to make life easier for yourself, make fewer errors, and get more points
Dr. Foote s Algebra & Calculus Tips or How to make life easier for yourself, make fewer errors, and get more points This handout includes several tips that will help you streamline your computations. Most
More informationINVERSE TRIGONOMETRIC FUNCTIONS
Mathematics, in general, is fundamentally the science of selfevident things. FELIX KLEIN. Introduction In Chapter, we have studied that the inverse of a function f, denoted by f, eists if f is oneone
More informationHigher 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
More informationD.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
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationWeek 13 Trigonometric Form of Complex Numbers
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
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More informationAlgebra 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
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationTrigonometric 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
More informationThis 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
More informationSection 64 Product Sum and Sum Product Identities
480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 64 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
More informationPreCalculus Review Problems Solutions
MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry PreCalculus Review Problems Solutions Problem 1. Give equations for the following lines in both pointslope and slopeintercept form. (a) The
More informationUnit 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
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationRight Triangle Trigonometry
Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned
More informationThnkwell 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
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationSection 5.2 Trigonometric Functions of Real Numbers
Section. Trigonometric Functions of Real Numbers The Trigonometric Functions EXAMPLE: Use the Table below to find the six trigonometric functions of each given real number t. a) t = π b) t = π 1 EXAMPLE:
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
More informationRational Expressions and Rational Equations
Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple
More information5.2 Unit Circle: Sine and Cosine Functions
Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and
More informationThe 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
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationTOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM
Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 34 weeks Trigonometry 3 weeks Graphs of Trig Functions 34 weeks Analytic Trigonometry 56 weeks
More informationPOLYNOMIAL 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
More informationAlgebra 2/Trigonometry Practice Test
Algebra 2/Trigonometry Practice Test Part I Answer all 27 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. For each question, write on the separate
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
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
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationALGEBRA 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
More informationG. GRAPHING FUNCTIONS
G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More informationFriday, 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
More informationy 1 x dx ln x y a x dx 3. y e x dx e x 15. y sinh x dx cosh x y cos x dx sin x y csc 2 x dx cot x 7. y sec 2 x dx tan x 9. y sec x tan x dx sec x
Strateg for Integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should appl. But it
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are welldefined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
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
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationWho uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)
1 The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview
More informationn th roots of complex numbers
n th roots of complex numbers Nathan Pflueger 1 October 014 This note describes how to solve equations of the form z n = c, where c is a complex number. These problems serve to illustrate the use of polar
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationSolutions 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
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationUsing Trigonometry to Find Missing Sides of Right Triangles
Using Trigonometry to Find Missing Sides of Right Triangles A. Using a Calculator to Compute Trigonometric Ratios 1. Introduction: Find the following trigonometric ratios by using the definitions of sin(),
More informationTrigonometry LESSON TWO  The Unit Circle Lesson Notes
(cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each
More informationActivity 1: Using base ten blocks to model operations on decimals
Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More information