Inverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4
|
|
- Leonard Bennett
- 5 years ago
- Views:
Transcription
1 Inverse Functions
2 Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) 1 E: f 5 and f 5 are inverses because their input and output are switched. For instance: f 4 f f f ' 5 4
3 Tables and Graphs of Inverses Orginal Switch and y Inverse (0,5) (,16) (6,4) (10,0) (0,5) (18,16) (14,4) X Y X Y (4,14) (0,10) (4,6) Line of Symmetry: y = Switch and y (16,18) (16,) Although transformed, the graphs are identical
4 Inverse and Compositions In order for two functions to be inverses: f g AND g f
5 One-to-One Functions A function f() is one-to-one on a domain D if, for every value c, the equation f() = c has at most one solution for every in D. Or, for every a and b in D: f a f b a b unless Theorems: 1. A function has an inverse function if and only if it is one-to-one.. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.
6 The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
7 The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
8 Eample Without graphing, decide if the function below has an inverse function. f 6 3 If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See if the derivative is always one sign: f ' 6 Since the derivative is always negative, the inverse of f is a function.
9 Find the Inverse of a Function 1. Switch the and y of the function whose inverse you desire.. Solve for y to get the Inverse function 3. Make sure that the domains and ranges of your inverse and original function match up.
10 Eample Find the inverse of the following: d 4 3 Switch and y Only Half Parabola Solve for y 34 4 y 3 y Really y = Restrict the Domain! 3 Full Parabola y (too much) 4 d 3 y when 3 =3 Make sure to check with a table and graph on the calculator.
11 Logarithms v Eponentials
12 Definition of Logarithm The logarithm base a of b is the eponent you put on a to get b: lo g a b a > 0 if and only a i.e. Logs give you eponents! b if and b > 0 The logarithm to the base e, denoted ln, is called the natural logarithm.
13 Logarithm and Eponential Forms Logarithm Form 5 = log (3) Logs Give you Eponents Input Becomes Output 5 = 3 Base Stays the Base Eponential Form
14 Eamples Write each equation in eponential form 1.log 15 (5) = /3.Log 8 () = 1/3 15 /3 = 5 8 1/3 = Write each equation in logarithmic form 1.If 64 = 4 3.If 1/7 = 3 log 4 (64) = 3 Log 3 (1/7) =
15 Eample Complete the table if a is a positive real number and: f a Domain Range Continuous? One-to-One? Concavity Left End Behavior Right End Behavior 1 f a f All Reals All Positive Reals Yes Yes Always Up lim a 0 lim a log a All Positive Reals All Reals Yes Yes Always Down 0 lim log a lim log a
16 The Change of Base Formula The following formula allows you to evaluate any valid logarithm statement: log b a log log c c a b For a and b greater than 0 AND b 1. Eample: Evaluate log1.04 ln ln
17 Solve: Solving Equations with the Change of Base Formula Isolate the base and power Change the eponential equation to an logarithm equation Use the Change of Base Formula log log 454 log
18 Properties of Logarithms For a>0, b>0, m>0, m 1, and any real number n. Logarithm of 1: log 1 0 m Logarithm of the base: log 1 m m Power Property: log m a n nlog a m Product Property: ab a b log log log m m m Quotient Property: a log log a log b m b m m
19 Eample 1 Condense the epression: 1 3 log log 10 log log log 10 log log 10 log log
20 Eample Epand the epression: ln 3y ln 3 ln ln y ln 3 ln ln y
21 Eample 3 Solve the equation: 4 4 log log log 3 3
22 AP Reminders Do not forget the following relationships: ln e e ln e a e b e a b e e a b e ab
23 Inverse Trigonometry
24 Tangent Cosine Sine Trigonometric Functions Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Secant Cotangent
25 Tangent Cosine Sine Trigonometric Functions with Restricted Domains D :, D : 0, D :, In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become oneto-one. D :,0 0, D : 0,, D : 0, Cosecant Secant Cotangent
26 Trigonometric Functions with Restricted Domains Function Domain Range f () = sin f () = cos f () = tan f () = csc f () = sec f () = cot, 0,,,0 0, 0,, 0, 1,1 1,1,,, 1 1,, 1 1,
27 Tan -1 Cos -1 Sin -1 Inverse Trigonometric Functions Csc -1 Sec -1 Cot -1
28 Inverse Trigonometric Functions Function Domain Range f () = sin -1 f () = cos -1 f () = tan -1 f () = csc -1 f () = sec -1 f () = cot -1 1,1 1,1,,, 1 1,, 1 1,, 0,,,0 0, 0,, 0,
29 Alternate Names/Defintions for Inverse Trigonometric Functions Familiar Alternate Calculator f () = sin -1 f () = arcsin f () = sin -1 f () = cos -1 f () = arccos f () = tan -1 f () = arctan f () = cos -1 f () = tan -1 f () = csc -1 f () = arccsc f () = sin -1 1/ f () = sec -1 f () = arcsec f () = cos -1 1/ f () = cot -1 f () = arccot f () = -tan -1 + Arccot is different because it is always positive but tan can be negative.
30 Evaluate: Eample 1 sin 1 1 This epression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is,. 1 Since sin and 6, 6 sin 6 1 1
31 Eample Evaluate: csc 1 1 This epression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is,0 0,. Since csc 1 and 0, 1 csc 1
32 Eample 3 Evaluate: tan arcsin 1 3 The embedded epression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): 3 a Find the missing side length(s) a a 8 It does not even matter what the angle is, we only need to find: tan opp adj 1 4 Is the result positive or negative? Since arcsin 0, 1 3 tan 0
33 Eample 4 Evaluate: tan cos 1 ( 1 ) 6 The embedded epression asks us to find the angle whose cosine is -1/6. Draw a picture (There are infinite varieties): Ignore the negative for now. 6 1 Find the missing side length(s) o 1 6 o 35 o tan It does not even matter what the angle is, we only need to find: opp adj Is the result positive or negative? 1 1 Since cos, tan 0 6
34 Eample 3 Evaluate: costan 1 The embedded epression asks us to find the angle whose tangent is. Draw a possible picture (There are infinite varieties): h 1 Find the missing side length(s) 1 h h 1 It does not even matter what the angle is, we only need to find: cos adj hyp 1 1 Is the result positive or negative? Since - tan, 1 cos 0
35 White Board Challenge Evaluate without a calculator: sec 1 4
36 White Board Challenge Evaluate without a calculator: cot
37 White Board Challenge Evaluate without a calculator: arccos 3
38 White Board Challenge Evaluate without a calculator: cot csc 1 5 6
39 White Board Challenge Evaluate without a calculator: tan sin 1 1
THEOREM: THE CONSTANT RULE
MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More informationName: Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions
Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following:
More informationSection Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
More informationSummer 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 informationA 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(ii) y = ln 1 ] t 3 t x x2 9
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
More informationCALCULUS: Graphical,Numerical,Algebraic by Finney,Demana,Watts and Kennedy Chapter 3: Derivatives 3.3: Derivative of a function pg.
CALCULUS: Graphical,Numerical,Algebraic b Finne,Demana,Watts and Kenned Chapter : Derivatives.: Derivative of a function pg. 116-16 What ou'll Learn About How to find the derivative of: Functions with
More informationTrigonometric 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 informationCHINO 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 information90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
More informationCore Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description
Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation;
More informationName: Top Ten Things You ve Learned #10: Graphing Lines, Parabolas, and other Functions I. No Calculator: Sketch a graph of each equation
Name: #: Graphing Lines, Parabolas, and other Functions I. No Calculator: Sketch a graph of each equation... - - - - - -.. 6. - - - - - - 7. 8. 9. - - - - - - ... a f - - - - - - II. Use a graphing calculator
More informationMath 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 informationTrigonometry Outline
Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the
More informationDerivatives of Inverse Functions
Derivatives of Inverse Functions Implicit differentiation enables us to determine the derivatives of inverse functions. determine the derivatives of arcsin, arccos, arctan, and ln. In this lecture, we
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationNext, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
More informationAP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student
AP Calculus AB SUMMER ASSIGNMENT Dear future Calculus AB student We are ecited to work with you net year in Calculus AB. In order to help you be prepared for this class, please complete the summer assignment.
More informationMath 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 informationTRIG 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 informationMath 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 information5.3 Properties of Trigonometric Functions Objectives
Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.
More informationMath 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 informationWest 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 informationMath 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 informationPrecalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.
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
More informationCalculus with business applications, Lehigh U, Lecture 05 notes Summer
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often
More informationAP 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 informationThe Chain Rule. This is a generalization of the (general) power rule which we have already met in the form: then f (x) = r [g(x)] r 1 g (x).
The Chain Rule This is a generalization of the general) power rule which we have already met in the form: If f) = g)] r then f ) = r g)] r g ). Here, g) is any differentiable function and r is any real
More informationReview of elements of Calculus (functions in one variable)
Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints
More information4-3 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 informationAlgebra/Pre-calc Review
Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge
More information7.3 Inverse Trigonometric Functions
58 transcendental functions 73 Inverse Trigonometric Functions We now turn our attention to the inverse trigonometric functions, their properties and their graphs, focusing on properties and techniques
More informationAP Calculus AB Summer Assignment
Name: AP Calculus AB Summer Assignment Due Date: The beginning of class on the last class day of the first week of school. The purpose of this assignment is to have you practice the mathematical skills
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More informationC. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2
AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are
More informationUsing 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 informationUsing 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 informationLesson 11 Inverse Trig Functions
Unit : Trig Equations & Graphs Student ID #: Lesson 11 Inverse Trig Functions Goal: IX. use inverse trig to calculate an angle measure given a (special) ratio of sides Opener: Determine which values of
More informationUnit 10 Prerequisites for Next Year (Calculus)
Unit 0 Prerequisites for Net Year (Calculus) The following Study Guide is the required INDEPENDENT review for you to work through for your final unit. You WILL have a test that covers this material after
More informationUnit 2 - The Trigonometric Functions - Classwork
Unit 2 - The Trigonometric Functions - Classwork Given a right triangle with one of the angles named ", and the sides of the triangle relative to " named opposite, adjacent, and hypotenuse (picture on
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationNOTES ON INVERSE TRIGONOMETRIC FUNCTIONS
NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS MATH 5 (S). Definitions of Inverse Trigonometric Functions () y = sin or y = arcsin is the inverse function of y = sin on [, ]. The omain of y = sin = arcsin is
More informationChapter 5 Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5 Logarithmic, Exponential, an Other Transcenental Functions 5.1 The Natural Logarithmic Function: Differentiation 5.2 The Natural Logarithmic Function: Integration 5.3 Inverse Functions 5.4 Exponential
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment
More informationTRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
12 TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. 12.2 The Trigonometric Functions Copyright Cengage Learning. All rights reserved. The Trigonometric Functions and Their Graphs
More informationAlgebra/Trigonometry Review Notes
Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More informationAnalytic 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 information2. 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 informationTrigonometric Functions. Section 1.6
Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian
More information1 Functions and Inverses
October, 08 MAT86 Week Justin Ko Functions and Inverses Definition. A function f : D R is a rule that assigns each element in a set D to eactly one element f() in R. The set D is called the domain of f.
More informationAP Calculus BC Summer Review
AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it
More informationMethods of Integration
Methods of Integration Essential Formulas k d = k +C sind = cos +C n d = n+ n + +C cosd = sin +C e d = e +C tand = ln sec +C d = ln +C cotd = ln sin +C + d = tan +C lnd = ln +C secd = ln sec + tan +C cscd
More informationAP CALCULUS AB - SUMMER ASSIGNMENT 2018
Name AP CALCULUS AB - SUMMER ASSIGNMENT 08 This packet is designed to help you review and build upon some of the important mathematical concepts and skills that you have learned in your previous mathematics
More informationPre-Calculus (#9400)
AASD MATHEMATICS CURRICULUM Pre-Calculus (#9400) Description This course is a foundation course for college-level mathematics classes. The topics covered include functions and their graphs; the circular
More informationSect 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 informationCalculus Summer TUTORIAL
Calculus Summer TUTORIAL The purpose of this tutorial is to have you practice the mathematical skills necessary to be successful in Calculus. All of the skills covered in this tutorial are from Pre-Calculus,
More informationJune 9 Math 1113 sec 002 Summer 2014
June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the
More informationOctober 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 informationSection 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 informationChapter 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 informationAs 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 informationAP Calculus AB Summer Assignment
AP Calculus AB 07-08 Summer Assignment Welcome to AP Calculus AB! You are epected to complete the attached homework assignment during the summer. This is because of class time constraints and the amount
More informationTrigonometry 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 informationFox Lane High School Department of Mathematics
Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationFUNDAMENTAL 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 informationCK- 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 informationAP Calculus AB Summer Assignment School Year
AP Calculus AB Summer Assignment School Year 018-019 Objective of the summer assignment: The AP Calculus summer assignment is designed to serve as a review for many of the prerequisite math skills required
More informationSummer Mathematics Prep
Summer Mathematics Prep Entering Calculus Chesterfield County Public Schools Department of Mathematics SOLUTIONS Domain and Range Domain: All Real Numbers Range: {y: y } Domain: { : } Range:{ y : y 0}
More information6.1 Antiderivatives and Slope Fields Calculus
6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.
More informationPrentice Hall: Algebra 2 with Trigonometry 2006 Correlated to: California Mathematics Content Standards for Trigonometry (Grades 9-12)
California Mathematics Content Standards for Trigonometry (Grades 9-12) Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric
More information2. Pythagorean Theorem:
Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle
More informationChapter 5 The Next Wave: MORE MODELING AND TRIGONOMETRY
ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter The Next Wave: MORE MODELING AND TRIGONOMETRY NW-1. TI-8, points; Casio, points a) An infinite number of them. b) 17p, - 7p c) Add p n to p, p
More informationA. 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 informationPrecalculus Midterm Review
Precalculus Midterm Review Date: Time: Length of exam: 2 hours Type of questions: Multiple choice (4 choices) Number of questions: 50 Format of exam: 30 questions no calculator allowed, then 20 questions
More informationInverse Trigonometric Functions. September 5, 2018
Inverse Trigonometric Functions September 5, 08 / 7 Restricted Sine Function. The trigonometric function sin x is not a one-to-one functions..0 0.5 Π 6, 5Π 6, Π Π Π Π 0.5 We still want an inverse, so what
More informationTrigonometric substitutions (8.3).
Review for Eam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Eam covers: 7.4, 7.6, 7.7, 8-IT, 8., 8.2. Solving differential equations
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationExercise Set 4.3: Unit Circle Trigonometry
Eercise Set.: Unit Circle Trigonometr Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle. Sketch each of the following angles in
More informationExercise 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 informationDifferential Calculus
Differential Calculus. Compute the derivatives of the following functions a() = 4 3 7 + 4 + 5 b() = 3 + + c() = 3 + d() = sin cos e() = sin f() = log g() = tan h() = 3 6e 5 4 i() = + tan 3 j() = e k()
More informationChapter 8: Trig Equations and Inverse Trig Functions
Haberman MTH Section I: The Trigonometric Functions Chapter 8: Trig Equations and Inverse Trig Functions EXAMPLE : Solve the equations below: a sin( t) b sin( t) 0 sin a Based on our experience with the
More informationCHAPTER 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 informationCalculus 1 (AP, Honors, Academic) Summer Assignment 2018
Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have
More informationCHAPTER 3 DERIVATIVES (continued)
CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationTrigonometry Exam 2 Review: Chapters 4, 5, 6
Trig Exam Review F07 O Brien Trigonometry Exam Review: Chapters,, 0% of the questions on Exam will come from Chapters through. The other 70 7% of the exam will come from Chapters through. There may be
More informationBasic Math Formulas. Unit circle. and. Arithmetic operations (ab means a b) Powers and roots. a(b + c)= ab + ac
Basic Math Formulas Arithmetic operations (ab means ab) Powers and roots a(b + c)= ab + ac a+b c = a b c + c a b + c d = ad+bc bd a b = a c d b d c a c = ac b d bd a b = a+b ( a ) b = ab (y) a = a y a
More informationSummer AP Assignment Coversheet Falls Church High School
Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose:
More informationSANDERSON 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 informationTroy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
More informationChapter 7, Continued
Math 150, Fall 008, c Benjamin Aurispa Chapter 7, Continued 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-Angle Formulas Formula for Sine: Formulas for Cosine: Formula for Tangent: sin
More informationPre Calc. Trigonometry.
1 Pre Calc Trigonometry 2015 03 24 www.njctl.org 2 Table of Contents Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Power Reducing
More informationPre-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 informationTransition to College Math
Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain
More informationMTH 112: Elementary Functions
1/19 MTH 11: Elementary Functions Section 6.6 6.6:Inverse Trigonometric functions /19 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line crosses the graph of a 1
More informationCore Mathematics 2 Unit C2 AS
Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences
More information