Section 63 DoubleAngle and HalfAngle Identities


 Baldric Merritt
 5 years ago
 Views:
Transcription
1 63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities called doubleangle and halfangle identities. We can derive these identities directly from the sum and difference identities given in Section 6. Even though the names use the word angle, the new identities hold for real numbers as well. DoubleAngle Identities Start with the sum identity for sine, sin ( y) sin cos y cos sin y and replace y with to obtain sin ( ) sin cos cos sin On simplification, this gives sin sin cos Doubleangle identity for sine () If we start with the sum identity for cosine, cos ( y) cos cos y sin sin y and replace y with, we obtain cos ( ) cos cos sin sin On simplification, this gives cos cos sin First doubleangle identity for cosine () Now, using the Pythagorean identity sin cos (3) in the form cos sin (4) and substituting it into equation (), we get cos sin sin On simplification, this gives cos sin Second doubleangle identity for cosine (5)
2 47 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Or, if we use equation (3) in the form sin cos and substitute it into equation (), we get cos cos ( cos ) On simplification, this gives cos cos Third doubleangle identity for cosine (6) Doubleangle identities can be established for the tangent function in the same way by starting with the sum formula for tangent (a good eercise for you). We list the doubleangle identities below for convenient reference. DOUBLEANGLE IDENTITIES sin sin cos cos cos sin sin cos tan tan tan cot cot cot tan The identities in the second row can be solved for sin and cos to obtain the identities sin cos cos cos These are useful in calculus to transform a power form to a nonpower form. Eplore/Discuss (A) Discuss how you would show that, in general, sin sin cos cos tan tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A. Identity Verification Verify the identity cos tan. tan
3 63 DoubleAngle and HalfAngle Identities 473 Verification We start with the right side: tan tan sin cos sin cos cos sin cos sin cos sin cos Quotient identities Algebra Pythagorean identity Doubleangle identity Key Algebraic Steps in Eample a b b a b a b b a b b a b a Solution Verify the identity sin Finding Eact Values tan tan. Find the eact values, without using a calculator, of sin and cos if tan 3 4 and is a quadrant IV angle. First draw the reference triangle for and find any unknown sides: r 4 3 r (3) 4 5 sin 3 5 cos 4 5 Now use doubleangle identities for sine and cosine: sin sin cos ( 3 5 )( 4 5 ) 4 5 cos cos ( 4 5 ) 7 5 Find the eact values, without using a calculator, of cos and tan if sin and is a quadrant II angle. 4 5
4 474 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS HalfAngle Identities Halfangle identities are simply doubleangle identities stated in an alternate form. Let s start with the doubleangle identity for cosine in the form cos m sin m Now replace m with / and solve for sin (/) [if m is twice m, then m is half of m think about this]: cos sin sin cos sin cos Halfangle identity for sine (7) where the choice of the sign is determined by the quadrant in which / lies. To obtain a halfangle identity for cosine, start with the doubleangle identity for cosine in the form cos m cos m and let m / to obtain cos cos Halfangle identity for cosine (8) where the sign is determined by the quadrant in which / lies. To obtain a halfangle identity for tangent, use the quotient identity and the halfangle formulas for sine and cosine: Thus, tan sin cos cos cos cos cos tan cos cos Halfangle identity for tangent (9) where the sign is determined by the quadrant in which / lies. Simpler versions of equation (9) can be obtained as follows: tan cos cos cos cos cos cos (0)
5 cos ( cos ) sin ( cos ) sin ( cos ) sin cos 63 DoubleAngle and HalfAngle Identities 475 sin sin and ( cos ) cos, since cos is never negative. All absolute value signs can be dropped, since it can be shown that tan (/) and sin always have the same sign (a good eercise for you). Thus, tan sin cos Halfangle identity for tangent () By multiplying the numerator and the denominator in the radicand in equation (0) by cos and reasoning as before, we also can obtain tan cos sin Halfangle identity for tangent () We now list all the halfangle identities for convenient reference. HALFANGLE IDENTITIES sin cos cos cos tan cos cos sin cos cos sin where the sign is determined by the quadrant in which / lies. Eplore/Discuss (A) Discuss how you would show that, in general, sin cos tan sin cos tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A.
6 476 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 3 Finding Eact Values Compute the eact value of sin 65 without a calculator using a halfangle identity. Solution sin 65 sin 330 cos 330 (3/) 3 Use halfangle identity for sine with a positive radical, since sin 65 is positive. 3 4 Solution Compute the eact value of tan 05 without a calculator using a halfangle identity. Finding Eact Values Find the eact values of cos (/) and cot (/) without using a calculator if sin 3 5, 3/. Draw a reference triangle in the third quadrant, and find cos. Then use appropriate halfangle identities. a a 5 (3) 4 cos (a, 3) If 3/, then 3 4 Divide each member of 3/ by. Thus, / is an angle in the second quadrant where cosine and cotangent are negative, and cos cos (4 5 ) 0 or 0 0 cot tan (/) sin cos 3 5 ( 4 5 ) 3
7 63 DoubleAngle and HalfAngle Identities Verification 5 Find the eact values of sin (/) and tan (/) without using a calculator if cot 4 3, /. Identity Verification Verify the identity: sin sin cos sin cos tan tan cos tan tan cos tan tan sin tan Verify the identity cos tan sin tan Halfangle identity for sine Square both sides. Algebra Algebra tan sin. tan Quotient identity Answers to Matched Problems tan tan sin cos cos sin cos sin cos sin cos sin cos sin. sin cos cos sin cos. cos cos 5, tan 4 3 sin (/) 30/0, tan (/) 3 cos tan 7 tan cos tan tan cos tan sin tan tan EXERCISE 63 A 5. sin cos, (Choose the correct sign.) In Problems 6, verify each identity for the values indicated.. cos cos sin, 30. sin sin cos, tan cot tan, 3 4. tan tan tan, 6 6. cos cos, (Choose the correct sign.) In Problems 7 0, find the eact value without a calculator using doubleangle and halfangle identities. 7. sin.5 8. tan cos tan 5
8 478 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS In Problems 4, graph y and y in the same viewing window for. Use TRACE to compare the two graphs.. y cos, y cos sin. y sin, y sin cos B Verify the identities in Problems (sin cos ) sin 6. sin (tan )( cos ) 7. sin ( cos ) 8. cos (cos ) 9. cos tan sin 0. sin t (sin t cos t). sin. cos 3. cot tan 4. cot tan 5. cot 6. sin cos 7. cos u tan u 8. tan u sec sec csc tan tan sec cot tan 3. cos tan (/) 3. cos tan (/) cot tan Compute the eact values of sin, cos, and tan using the information given in Problems and appropriate identities. Do not use a calculator. 33. sin 3 5, / In Problems 37 40, compute the eact values of sin (/), cos (/), and tan (/) using the information given and appropriate identities. Do not use a calculator y tan, y y tan, y sin cos tan tan tan 5, / 0 cot 5, / 0 sin 3, 3/ cos 4, 3/ 39. cot 3 4, / 40. tan 3 4, / cos cos cot tan cot cos sin cos u sin u tan u tan u 4 cos 5, / Suppose you are tutoring a student who is having difficulties in finding the eact values of sin and cos from the information given in Problems 4 and 4. Assuming you have worked through each problem and have identified the key steps in the solution process, proceed with your tutoring by guiding the student through the solution process using the following questions. Record the epected correct responses from the student. (A) The angle is in what quadrant and how do you know? (B) How can you find sin and cos? Find each. (C) What identities relate sin and cos with either sin or cos? (D) How would you use the identities in part C to find sin and cos eactly, including the correct sign? (E) What are the eact values for sin and cos? 4. Find the eact values of sin and cos, given tan, Find the eact values of sin and cos, given sec, Verify each of the following identities for the value of indicated in Problems Compute values to five significant digits using a calculator. (A) tan tan (B) cos cos tan (Choose the correct sign.) In Problems 47 50, graph y and y in the same viewing window for, and state the intervals for which the equation y y is an identity C y cos (/), y cos y cos (/), y cos y sin (/), y y sin (/), y Verify the identities in Problems cos 3 4 cos 3 3 cos 5. sin 3 3 sin 4 sin 3 cos cos 53. cos 4 8 cos 4 8 cos 54. sin 4 (cos )(4 sin 8 sin 3 )
9 63 DoubleAngle and HalfAngle Identities 479 In Problems 55 60, find the eact value of each without using a calculator. 3 5 )] 55. cos [ cos ( 3 5 )] 56. sin [ cos ( 57. tan [ cos ( 4 5 )] 58. tan [ tan ( 59. cos [ cos ( 3 5 )] 60. sin [ tan ( 4 3 )] 3 4 )] (B) Using the resulting equation in part A, determine the angle that will produce the maimum distance d for a given initial speed v 0. This result is an important consideration for shotputters, javelin throwers, and discus throwers. In Problems 6 66, graph f() in a graphing utility, find a simpler function g() that has the same graph as f(), and verify the identity f() g(). [Assume g() k A T(B) where T() is one of the si trigonometric functions.] 6. f() csc cot 6. f() csc cot cos cos 63. f() 64. f() sin cos 65. f() 66. f() cot cot sin cos APPLICATIONS 70. Geometry. In part (a) of the figure, M and N are the midpoints of the sides of a square. Find the eact value of cos. [Hint: The solution uses the Pythagorean theorem, the definition of sine and cosine, a halfangle identity, and some auiliary lines as drawn in part (b) of the figure.] M M 67. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use cos cos.] s N s / / N 68. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use tan ( tan )/( tan ).] 69. Sports Physics. The theoretical distance d that a shotputter, discus thrower, or javelin thrower can achieve on a given throw is found in physics to be given approimately by d 4 feet v 0 sin cos 3 feet per second per second 8 m where v 0 is the initial speed of the object thrown (in feet per second) and is the angle above the horizontal at which the object leaves the hand (see the figure). (A) Write the formula in terms of sin by using a suitable identity. feet 7 m s (a) s (b) 7. Area. An nsided regular polygon is inscribed in a circle of radius R. (A) Show that the area of the nsided polygon is given by A n nr sin n [Hint: (Area of a triangle) ( )(base)(altitude). Also, a doubleangle identity is useful.] (B) For a circle of radius, complete Table, to five decimal places, using the formula in part A: T A B L E n 0 00,000 0,000 A n (C) What number does A n seem to approach as n increases without bound? (What is the area of a circle of radius?) (D) Will A n eactly equal the area of the circumscribed circle for some sufficiently large n? How close can A n be made to get to the area of the circumscribed circle? [In calculus, the area of the circumscribed circle is called the limit of A n as n increases without bound. In symbols, for a circle of radius, we would write lim A n n. The limit concept is the cornerstone on which calculus is constructed.]
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
More informationHomework 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
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 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 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 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 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 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 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 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 informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
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 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 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 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 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 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 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 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 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 informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 13 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1 Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
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 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 informationParallel 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
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 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 informationSAT 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
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
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 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 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 informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationSelfPaced Study Guide in Trigonometry. March 31, 2011
SelfPaced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the SelfPaced Review Module Trigonometry SelfPaced Review Module 4.1 Right Triangles..........................
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 informationTechniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an artform than a collection of algorithms. Many problems in applied mathematics involve the integration
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationAdditional Topics in Math
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
More informationMATH 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: Prealgebra Algebra Precalculus Calculus Statistics
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More information(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
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 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 informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationPrentice 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
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 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 informationBiggar 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
More information1. Introduction circular deﬁnition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationTrigonometric 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
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 informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationHow 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
More informationTrigonometry LESSON ONE  Degrees and Radians Lesson Notes
210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationSolutions 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
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationTrigonometry 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.
More informationMath 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
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 informationCOMPLEX 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
More informationLesson Plan. Students will be able to define sine and cosine functions based on a right triangle
Lesson Plan Header: Name: Unit Title: Right Triangle Trig without the Unit Circle (Unit in 007860867) Lesson title: Solving Right Triangles Date: Duration of Lesson: 90 min. Day Number: Grade Level: 11th/1th
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
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 informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationLIES MY CALCULATOR AND COMPUTER TOLD ME
LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocketsize calculating devices are currently
More informationSOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions
More informationChapter 5 Resource Masters
Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois StudentWorks TM This CDROM includes the entire Student Edition along with the Study Guide, Practice,
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
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 informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationTHE 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
More informationHigh School Geometry Test Sampler Math Common Core Sampler Test
High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slopeintercept 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
More informationSolving Quadratic Equations
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
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 informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationCourse outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems)
Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems) Functions, domain and range Domain and range of rational and algebraic
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given
More informationChapter 5: Trigonometric Functions of Angles
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationSample Problems. Practice Problems
Lecture Notes Circles  Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
More informationUse finite approximations to estimate the area under the graph of the function. f(x) = x 3
5.1: 6 Use finite approximations to estimate the area under the graph of the function f(x) = x 3 between x = 0 and x = 1 using (a) a lower sum with two rectangles of equal width (b) a lower sum with four
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfdprep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationModuMath Basic Math Basic Math 1.1  Naming Whole Numbers Basic Math 1.2  The Number Line Basic Math 1.3  Addition of Whole Numbers, Part I
ModuMath Basic Math Basic Math 1.1  Naming Whole Numbers 1) Read whole numbers. 2) Write whole numbers in words. 3) Change whole numbers stated in words into decimal numeral form. 4) Write numerals in
More information