Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.

Size: px
Start display at page:

Download "Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear."

Transcription

1 Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain for all polynomial functions is ALWAYS - The degree of a polynomial is - Graph behavior based on its degree: End behavior: Number of turning points A function is said to be even if A function is said to be odd if

2 2. Rational Functions (a.k.a. Fractional Functions) - General form: - Domain: - Unique Graph attributes: Vertical asymptotes: Horizontal or Slanted Asymptotes: 1. degree numerator < degree denominator 2. degree numerator = degree denominator 3. degree numerator > degree denominator Ex) For the function Domain: x + 3 y = determine the following: x 5 Vertical Asymptote Horizontal or Slanted Asymptote Also helps to plot the intercepts: x-intercept y-intercept Ex) Determine the asymptotes (vertical, horizontal/slant) for 2 x x 6x + 7 a) f( x) = b) g( x) = 2 x 9 x 5

3 3. Inverse Functions In order for a function f( x ) to have an inverse, it must be which means The inverse of the function f( x ) is denoted as Ex) f( x) x 3 = Ex) h( x) = x 2 The purpose of an inverse function is to Properties of inverse functions: - Domain and Range: - Graph symmetry Ex) Sketch the graph of the inverse of the function f( x ) on the blank axes. Domain: Range: Domain: Range:

4 4. Exponential and Logarithmic Functions Exponential functions of base a and Logarithmic functions of base a are inverses of each other. General Exp. Function: x y = a General Log. Function: y = log ( x) a Domain: Domain: Range: Range: Intercept: Intercept: Asymptote: Asymptote: Graph: Graph: Most frequently used base is whose log inverse is Approximate value:

5 Properties of Exponentials and Logarithms you ll need in calculus: Rewrite between exponential form and logarithmic form: x a = b can be rewritten as can be rewritten as ln( b) = x Cancellation Properties (VERY handy when solving equations) Solving equations: Solve the equation x = 45 Solve the equation 6ln(15 7 x) + 20 = 38 Base Change Formula The Laws of Logarithms These are handy when you need to expand or condense logarithmic expressions. I. ln( UV ) = U II. ln( V ) = M III. ln( U ) =

6 5. Trigonometric Functions Trigonometric functions were defined in several ways: -Right Triangle Definitions: The main three and their reciprocals sinθ = cosθ = tanθ = REMEMBER: The roles of OPPOSITE and ADJACENT depend on which acute angle you re calling θ. - Unit Circle Definitions: Let t be a radian angle measure and ( x, y) represents the point on the unit circle paired with the angle t sin( t ) = cos( t ) = tan( t ) =

7 FOR REFERENCE ONLY!!! THIS IS PREREQUISITE MATERIAL!!! YOU WILL NOT BE ALLOWED TO USE THIS ON THE TEST!!!

8 You ll need the unit circle for various reasons this semester: 7π Ex) Evaluate csc( ) 4 Ex) Solve the equation 2 3tan ( x ) = 1 on the interval [0,2 π ). Ex) What interval (or intervals) make 2cost on the interval [0,2 π )? - Trigonometric Function Graphs y = sin( x) y = cos( x) Domain of Sine and Cosine: Range of Sine and Cosine: Graph is periodic with a cycle repeating every interval of length.

9 Ex) Sketch two full periods of the graph of y = 8cos(10 x). How does the 8 affect the graph? How does the 10 affect the graph? The other trigonometric function graphs for reference (again prerequisite material won t be given on the test) Graph of y = csc( x) Graph of y = sec( x)

10 Graph of y = tan( x) Graph of y = cot( x) y = tan x y = cot x Cosecant and Cotangent have vertical asymptotes at every multiple of π (where Sine has x- intercepts) Secant and Tangent have vertical asymptotes at every ODD multiple of 2 π (where Cosine has x- intercepts) Trigonometric Identities we will need in Calculus (KNOW THEM!) Reciprocal Identities sinu = 1 csc u cos u = 1 secu tan u = 1 cot u csc u = 1 sin u sec u = 1 cosu cot u = 1 tanu Pythagorean Identities also cos also tan also cot sin u + cos u = 1 u = 1 sin u and sin u = 1 cos 1+ tan u = sec u u = sec u 1 and 1 = sec u tan 1+ cot u = csc u u = csc u 1 and 1 = csc u cot u u u Quotient Identities sin u tan u = cosu cosu cot u = sinu Even / Odd Identities ODDS sin( u) = sin u csc( u) = cscu tan( u) = tanu cot( u) = cot u EVENS cos( u) = cosu sec( u) = secu

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters

Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,

More information

Math Analysis Chapter 5 Notes: Analytic Trigonometric

Math Analysis Chapter 5 Notes: Analytic Trigonometric Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot

More information

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

AP Calculus Summer Packet

AP Calculus Summer Packet AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept

More information

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2 29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with

More information

Using the Definitions of the Trigonometric Functions

Using the Definitions of the Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective

More information

Analytic Trigonometry

Analytic Trigonometry Chapter 5 Analytic Trigonometry Course Number Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions

More information

Precalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring

Precalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring Unit 1 : Algebra Review Factoring Review Factoring Using the Distributive Laws Factoring Trinomials Factoring the Difference of Two Squares Factoring Perfect Square Trinomials Factoring the Sum and Difference

More information

A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.

A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1. Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,

More information

Section 6.2 Trigonometric Functions: Unit Circle Approach

Section 6.2 Trigonometric Functions: Unit Circle Approach Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal

More information

NAME DATE PERIOD. Trigonometric Identities. Review Vocabulary Complete each identity. (Lesson 4-1) 1 csc θ = 1. 1 tan θ = cos θ sin θ = 1

NAME DATE PERIOD. Trigonometric Identities. Review Vocabulary Complete each identity. (Lesson 4-1) 1 csc θ = 1. 1 tan θ = cos θ sin θ = 1 5-1 Trigonometric Identities What You ll Learn Scan the text under the Now heading. List two things that you will learn in the lesson. 1. 2. Lesson 5-1 Active Vocabulary Review Vocabulary Complete each

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

Chapter 4 Trigonometric Functions

Chapter 4 Trigonometric Functions SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios

More information

Crash Course in Trigonometry

Crash Course in Trigonometry Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................

More information

Math Section 4.3 Unit Circle Trigonometry

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

More information

Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B

Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in set B.

More information

CHAPTERS 5-7 TRIG. FORMULAS PACKET

CHAPTERS 5-7 TRIG. FORMULAS PACKET CHAPTERS 5-7 TRIG. FORMULAS PACKET PRE-CALCULUS SECTION 5-2 IDENTITIES Reciprocal Identities sin x = ( 1 / csc x ) csc x = ( 1 / sin x ) cos x = ( 1 / sec x ) sec x = ( 1 / cos x ) tan x = ( 1 / cot x

More information

Lone Star College-CyFair Formula Sheet

Lone Star College-CyFair Formula Sheet Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST

More information

Honors Algebra 2 Chapter 14 Page 1

Honors Algebra 2 Chapter 14 Page 1 Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0

More information

October 15 MATH 1113 sec. 51 Fall 2018

October 15 MATH 1113 sec. 51 Fall 2018 October 15 MATH 1113 sec. 51 Fall 2018 Section 5.5: Solving Exponential and Logarithmic Equations Base-Exponent Equality For any a > 0 with a 1, and for any real numbers x and y a x = a y if and only if

More information

More with Angles Reference Angles

More with Angles Reference Angles More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o

More information

PreCalculus First Semester Exam Review

PreCalculus First Semester Exam Review PreCalculus First Semester Eam Review Name You may turn in this eam review for % bonus on your eam if all work is shown (correctly) and answers are correct. Please show work NEATLY and bo in or circle

More information

A. Incorrect! This equality is true for all values of x. Therefore, this is an identity and not a conditional equation.

A. Incorrect! This equality is true for all values of x. Therefore, this is an identity and not a conditional equation. CLEP-Precalculus - Problem Drill : Trigonometric Identities No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as. Which of the following equalities is

More information

secθ 1 cosθ The pythagorean identities can also be expressed as radicals

secθ 1 cosθ The pythagorean identities can also be expressed as radicals Basic Identities Section Objectives: Students will know how to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. We use trig. identities

More information

Pre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives

Pre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives Pre-Calculus MATH 119 Fall 2013 Learning Objectives Section 1.1 1. Use the Distance Formula 2. Use the Midpoint Formula 4. Graph Equations Using a Graphing Utility 5. Use a Graphing Utility to Create Tables

More information

Summer Packet Greetings Future AP Calculus Scholar,

Summer Packet Greetings Future AP Calculus Scholar, Summer Packet 2017 Greetings Future AP Calculus Scholar, I am excited about the work that we will do together during the 2016-17 school year. I do not yet know what your math capability is, but I can assure

More information

4-3 Trigonometric Functions on the Unit Circle

4-3 Trigonometric Functions on the Unit Circle Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on

More information

MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically

MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically 1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram

More information

A Library of Functions

A Library of Functions LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us

More information

1 Chapter 2 Perform arithmetic operations with polynomial expressions containing rational coefficients 2-2, 2-3, 2-4

1 Chapter 2 Perform arithmetic operations with polynomial expressions containing rational coefficients 2-2, 2-3, 2-4 NYS Performance Indicators Chapter Learning Objectives Text Sections Days A.N. Perform arithmetic operations with polynomial expressions containing rational coefficients. -, -5 A.A. Solve absolute value

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

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

More information

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities

6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of

More information

Functions and their Graphs

Functions and their Graphs Chapter One Due Monday, December 12 Functions and their Graphs Functions Domain and Range Composition and Inverses Calculator Input and Output Transformations Quadratics Functions A function yields a specific

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

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

More information

The American School of Marrakesh. AP Calculus AB Summer Preparation Packet

The American School of Marrakesh. AP Calculus AB Summer Preparation Packet The American School of Marrakesh AP Calculus AB Summer Preparation Packet Summer 2016 SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents)

More information

Analytic Trigonometry

Analytic Trigonometry 0 Analytic Trigonometry In this chapter, you will study analytic trigonometry. Analytic trigonometry is used to simplify trigonometric epressions and solve trigonometric equations. In this chapter, you

More information

FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant

FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant NOTES 8: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 8.1 TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sinθ 1 cscθ cosθ 1 secθ tanθ 1

More information

NYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order)

NYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order) 1 of 6 UNIT P.I. 1 - INTEGERS 1 A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable 1 A2.A.4 * Solve quadratic inequalities in one and two variables, algebraically

More information

Find: sinθ. Name: Date:

Find: sinθ. Name: Date: Name: Date: 1. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: sinθ c a θ a a =

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.

More information

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.1.1 Solve Simple Equations Involving Absolute Value 0.2 Solving Quadratic Equations 0.2.1 Use the

More information

Sect 7.4 Trigonometric Functions of Any Angles

Sect 7.4 Trigonometric Functions of Any Angles Sect 7.4 Trigonometric Functions of Any Angles Objective #: Extending the definition to find the trigonometric function of any angle. Before we can extend the definition our trigonometric functions, we

More information

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics

More information

Chapter 2 Overview: Anti-Derivatives. As noted in the introduction, Calculus is essentially comprised of four operations.

Chapter 2 Overview: Anti-Derivatives. As noted in the introduction, Calculus is essentially comprised of four operations. Chapter Overview: Anti-Derivatives As noted in the introduction, Calculus is essentially comprised of four operations. Limits Derivatives Indefinite Integrals (or Anti-Derivatives) Definite Integrals There

More information

Milford Public Schools Curriculum. Department: Mathematics Course Name: Precalculus Level 1

Milford Public Schools Curriculum. Department: Mathematics Course Name: Precalculus Level 1 Milford Public Schools Curriculum Department: Mathematics Course Name: Precalculus Level 1 UNIT 1 Unit Description: Students will construct polynomial graphs with zeros and end behavior, and apply limit

More information

CALCULUS ASSESSMENT REVIEW

CALCULUS ASSESSMENT REVIEW CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness

More information

NOTES 10: ANALYTIC TRIGONOMETRY

NOTES 10: ANALYTIC TRIGONOMETRY NOTES 0: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 0. USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sin csc cos sec

More information

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra

More information

(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think:

(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think: PART F: EVALUATING INVERSE TRIG FUNCTIONS Think: (Section 4.7: Inverse Trig Functions) 4.82 A trig function such as sin takes in angles (i.e., real numbers in its domain) as inputs and spits out outputs

More information

Welcome to AP Calculus!!!

Welcome to AP Calculus!!! Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you

More information

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained. Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive

More information

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

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

More information

Notes on Radian Measure

Notes on Radian Measure MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius

More information

Chapter 1. Functions 1.3. Trigonometric Functions

Chapter 1. Functions 1.3. Trigonometric Functions 1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius

More information

CHAPTER 5: Analytic Trigonometry

CHAPTER 5: Analytic Trigonometry ) (Answers for Chapter 5: Analytic Trigonometry) A.5. CHAPTER 5: Analytic Trigonometry SECTION 5.: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) csc( x) sin x Reciprocal

More information

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT P r e c a l c u l u s ( A ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS and Graphs Polynomial, Power,

More information

A List of Definitions and Theorems

A List of Definitions and Theorems Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One

More information

A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019

A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019 A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.

More information

West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12

West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12 West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12 Unit 1: Linear Relationships & Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, 11

More information

Trigonometric Functions. Copyright Cengage Learning. All rights reserved.

Trigonometric Functions. Copyright Cengage Learning. All rights reserved. 4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.3 Right Triangle Trigonometry Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate trigonometric

More information

MATH 2412 Sections Fundamental Identities. Reciprocal. Quotient. Pythagorean

MATH 2412 Sections Fundamental Identities. Reciprocal. Quotient. Pythagorean MATH 41 Sections 5.1-5.4 Fundamental Identities Reciprocal Quotient Pythagorean 5 Example: If tanθ = and θ is in quadrant II, find the exact values of the other 1 trigonometric functions using only fundamental

More information

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Winter 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Winter 01/01! This problems in this packet are designed to help you review the topics from

More information

Numbers Content Points. Reference sheet (1 pt. each) 1-7 Linear Equations (1 pt. each) / Factoring (2 pt. each) /28

Numbers Content Points. Reference sheet (1 pt. each) 1-7 Linear Equations (1 pt. each) / Factoring (2 pt. each) /28 Summer Packet 2015 Your summer packet will be a major test grade for the first nine weeks. It is due the first day of school. You must show all necessary solutions. You will be tested on ALL material;

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f

( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,

More information

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

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

More information

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

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

More information

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)

TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal) TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents

More information

Polynomials and Rational Functions. Quadratic Equations and Inequalities. Remainder and Factor Theorems. Rational Root Theorem

Polynomials and Rational Functions. Quadratic Equations and Inequalities. Remainder and Factor Theorems. Rational Root Theorem Pre-Calculus Pre-AP Scope and Sequence - Year at a Glance Pre-Calculus Pre-AP - First Semester Pre-calculus with Limits; Larson/Hostetler Three Weeks 1 st 3 weeks 2 nd 3 weeks 3 rd 3 weeks 4 th 3 weeks

More information

SECTION 3.6: CHAIN RULE

SECTION 3.6: CHAIN RULE SECTION 3.6: CHAIN RULE (Section 3.6: Chain Rule) 3.6.1 LEARNING OBJECTIVES Understand the Chain Rule and use it to differentiate composite functions. Know when and how to apply the Generalized Power Rule

More information

Chapter 8: Techniques of Integration

Chapter 8: Techniques of Integration Chapter 8: Techniques of Integration Section 8.1 Integral Tables and Review a. Important Integrals b. Example c. Integral Tables Section 8.2 Integration by Parts a. Formulas for Integration by Parts b.

More information

2.1 Limits, Rates of Change and Slopes of Tangent Lines

2.1 Limits, Rates of Change and Slopes of Tangent Lines 2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0

More information

3.1 Fundamental Identities

3.1 Fundamental Identities www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,

More information

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

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

More information

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities Fundamental Trigonometric Identities MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: recognize and write the fundamental trigonometric

More information

Information Page. Desmos Graphing Calculator IXL Username: Password: gothunder

Information Page. Desmos Graphing Calculator  IXL  Username: Password: gothunder i Information Page Class Website: http://padlet.com/william_kaden/algebra2stem Has PDFs of assignments. Additionally they can sometimes be found on Skyward. Desmos Graphing Calculator http://www.desmos.com/

More information

Math RE - Calculus I Trigonometry Limits & Derivatives Page 1 of 8. x = 1 cos x. cos x 1 = lim

Math RE - Calculus I Trigonometry Limits & Derivatives Page 1 of 8. x = 1 cos x. cos x 1 = lim Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page of 8 Trigonometric Limits It has been shown in class that: lim 0 sin lim 0 sin lim 0 cos cos 0 lim 0 cos lim 0 + cos + To evaluate trigonometric

More information

Course Outline and Objectives. MA 1453 Precalculus with Graphing Calculators

Course Outline and Objectives. MA 1453 Precalculus with Graphing Calculators Effective Fall 2011 Course Outline and Objectives MA 1453 Precalculus with Graphing Calculators TEXT: Precalculus, 5 th Edition, by Faires and DeFranza ISBN 978-0-8400-6862-0 NOTE: A graphing calculator

More information

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

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

More information

One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle.

One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the unit circle. 2.24 Tanz and the Reciprocals Derivatives of Other Trigonometric Functions One of the powerful themes in trigonometry is that the entire subject emanates from a very simple idea: locating a point on the

More information

CURRICULUM GUIDE. Honors Algebra II / Trigonometry

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

More information

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles

More information

Algebra II B Review 5

Algebra II B Review 5 Algebra II B Review 5 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measure of the angle below. y x 40 ο a. 135º b. 50º c. 310º d. 270º Sketch

More information

Grade 11 or 12 Pre-Calculus

Grade 11 or 12 Pre-Calculus Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard

More information

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach

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

More information

West Essex Regional School District. AP Calculus AB. Summer Packet

West Essex Regional School District. AP Calculus AB. Summer Packet West Esse Regional School District AP Calculus AB Summer Packet 05-06 Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral

More information

Topic Outline for Algebra 2 & and Trigonometry One Year Program

Topic Outline for Algebra 2 & and Trigonometry One Year Program Topic Outline for Algebra 2 & and Trigonometry One Year Program Algebra 2 & and Trigonometry - N - Semester 1 1. Rational Expressions 17 Days A. Factoring A2.A.7 B. Rationals A2.N.3 A2.A.17 A2.A.16 A2.A.23

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

Hello Future Calculus Level One Student,

Hello Future Calculus Level One Student, Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will

More information

The Other Trigonometric

The Other Trigonometric The Other Trigonometric Functions By: OpenStaxCollege A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is or less, regardless

More information

Summer Assignment MAT 414: Calculus

Summer Assignment MAT 414: Calculus Summer Assignment MAT 414: Calculus Calculus - Math 414 Summer Assignment Due first day of school in September Name: 1. If f ( x) = x + 1, g( x) = 3x 5 and h( x) A. f ( a+ ) x+ 1, x 1 = then find: x+ 7,

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340 ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

More information

Trigonometric Ratios. θ + k 360

Trigonometric Ratios. θ + k 360 Trigonometric Ratios These notes are intended as a summary of section 6.1 (p. 466 474) in your workbook. You should also read the section for more complete explanations and additional examples. Coterminal

More information

Chapter 4 Trigonometric Functions

Chapter 4 Trigonometric Functions Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs

More information

SET 1. (1) Solve for x: (a) e 2x = 5 3x

SET 1. (1) Solve for x: (a) e 2x = 5 3x () Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x

More information

Honors Pre-calculus Midterm Review

Honors Pre-calculus Midterm Review Honors Pre-calculus Midterm Review Name: Chapter 1: Functions and Their Graphs 1. Evaluate the function f(x) = x 2 + 1 at each specified value of the independent variable and simplify. a. f( 3) b. f(x

More information

As we know, the three basic trigonometric functions are as follows: Figure 1

As we know, the three basic trigonometric functions are as follows: Figure 1 Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an

More information

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014)

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014) WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM PRE-CALCULUS (June 2014) COURSE NAME: Pre-Calculus UNIT: Chapter 1 NO. OF DAYS: KEY LEARNING (S): UNIT ESSENTIAL QUESTIONS: What methods are used to solve equations

More information