Cambridge Year 12 3u

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YEAR

12 BILL PENDER

DAVID SADLER

JULIA SHEA

DEREK WARD

CAMBRIDGE

Enhanced

3 Unit

Extension 1

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Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

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iv Contents

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Chapter Five—The Binomial Theorem . . . 173

5A The Pascal Triangle . . . 173

5B Further Work with the Pascal Triangle . . . 179

5C Factorial Notation . . . 185

5D The Binomial Theorem . . . 189

5E Greatest Coeﬃcient and Greatest Term . . . 197

5F Identities on the Binomial Coeﬃcients . . . 201

Chapter Six—Further Calculus . . . 208

6A Diﬀerentiation of the Six Trigonometric Functions . . . 208

6B Integration Using the Six Trigonometric Functions . . . 213

6C Integration by Substitution . . . 218

6D Further Integration by Substitution . . . 222

6E Approximate Solutions and Newton’s Method . . . 226

6F Inequalities and Limits Revisited . . . 233

Chapter Seven—Rates and Finance . . . 240

7A Applications of APs and GPs . . . 240

7B Simple and Compound Interest . . . 248

7C Investing Money by Regular Instalments . . . 253

7D Paying Oﬀ a Loan . . . 258

7E Rates of Change — Diﬀerentiating . . . 262

7F Rates of Change — Integrating . . . 267

7G Natural Growth and Decay . . . 270

7H Modiﬁed Natural Growth and Decay . . . 277

Chapter Eight—Euclidean Geometry . . . 282

8A Points, Lines, Parallels and Angles . . . 283

8B Angles in Triangles and Polygons . . . 292

8C Congruence and Special Triangles . . . 300

8D Trapezia and Parallelograms . . . 310

8E Rhombuses, Rectangles and Squares . . . 314

8F Areas of Plane Figures . . . 321

8G Pythagoras’ Theorem and its Converse . . . 325

8H Similarity . . . 329

8I . . . 338

Chapter Nine—Circle Geometry . . . 344

9A Circles, Chords and Arcs . . . 344

9B Angles at the Centre and Circumference . . . 352

9C Angles on the Same and Opposite Arcs . . . 358

9D Concyclic Points . . . 364

9E Tangents and Radii . . . 369

9F The Alternate Segment Theorem . . . 377

9G Similarity and Circles . . . 382 Intercepts on Transversals

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Contents v

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Chapter Ten—Probability and Counting . . . 389

10A Probability and Sample Spaces . . . 389

10B Probability and Venn Diagrams . . . 398

10C Multi-Stage Experiments . . . 403

10D Probability Tree Diagrams . . . 409

10E Counting Ordered Selections . . . 414

10F Counting with Identical Elements, and Cases . . . 421

10G Counting Unordered Selections . . . 425

10H Using Counting in Probability . . . 432

10I Arrangements in a Circle . . . 438

10J Binomial Probability . . . 442

Answers to Exercises . . . 450

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Preface

This textbook has been written for students in Years 11 and 12 taking the course previously known as ‘3 Unit Mathematics’, but renamed in the new HSC as two courses, ‘Mathematics’ (previously called ‘2 Unit Mathematics’) and ‘Mathemat-ics, Extension 1’. The book develops the content at the level required for the 2 and 3 Unit HSC examinations. There are two volumes — the present volume is roughly intended for Year 12, and the previous volume for Year 11. Schools will, however, diﬀer in their choices of order of topics and in their rates of progress. Although these Syllabuses have not been rewritten for the new HSC, there has been a gradual shift of emphasis in recent examination papers.

• The interdependence of the course content has been emphasised. • Graphs have been used much more freely in argument.

• Structured problem solving has been expanded. • There has been more stress on explanation and proof.

This text addresses these new emphases, and the exercises contain a wide variety of diﬀerent types of questions.

There is an abundance of questions in each exercise — too many for any one student — carefully grouped in three graded sets, so that with proper selection the book can be used at all levels of ability. In particular, both those who subse-quently drop to 2 Units of Mathematics, and those who in Year 12 take 4 Units of Mathematics, will ﬁnd an appropriate level of challenge. We have written a separate book, also in two volumes, for the 2 Unit ‘Mathematics’ course alone. We would like to thank our colleagues at Sydney Grammar School and Newington College for their invaluable help in advising us and commenting on the successive drafts, and for their patience in the face of some diﬃculties in earlier drafts. We would also like to thank the Headmasters of Sydney Grammar School and Newington College for their encouragement of this project, and Peter Cribb and the team at Cambridge University Press, Melbourne, for their support and help in discussions. Finally, our thanks go to our families for encouraging us, despite the distractions it has caused to family life.

Preface to the enhanced version

To provide students with practice for the new objective response (multiple choice) questions to be included in HSC examinations, online self-marking quizzes have been provided for each chapter, on Cambridge GO (access details can be found in the following pages). In addition, an interactive textbook version is available through the same website.

Dr Bill Pender

Subject Master in Mathematics Sydney Grammar School College Street

Darlinghurst NSW 2010 David Sadler

Mathematics

Sydney Grammar School

Julia Shea Head of Mathematics Newington College 200 Stanmore Road Stanmore NSW 2048 Derek Ward Mathematics

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How to Use This Book

This book has been written so that it is suitable for the full range of 3 Unit students, whatever their abilities and ambitions. The book covers the 2 Unit and 3 Unit content without distinction, because 3 Unit students need to study the 2 Unit content in more depth than is possible in a 2 Unit text. Nevertheless, students who subsequently move to the 2 Unit course should ﬁnd plenty of work here at a level appropriate for them.

The Exercises: No-one should try to do all the questions! We have written long exercises so that everyone will ﬁnd enough questions of a suitable standard — each student will need to select from them, and there should be plenty left for revision. The book provides a great variety of questions, and representatives of all types should be selected.

Each chapter is divided into a number of sections. Each of these sections has its own substantial exercise, subdivided into three groups of questions:

Foundation: _{These questions are intended to drill the new content of the} sec-tion at a reasonably straightforward level. There is little point in proceeding without mastery of this group.

Development: _{This group is usually the longest. It contains more substantial} questions, questions requiring proof or explanation, problems where the new content can be applied, and problems involving content from other sections and chapters to put the new ideas in a wider context. Later questions here can be very demanding, and Groups 1 and 2 should be suﬃcient to meet the demands of all but exceptionally diﬃcult problems in 3 Unit HSC papers. Extension: _{These questions are quite hard, and are intended principally for}

those taking the 4 Unit course. Some are algebraically challenging, some establish a general result beyond the theory of the course, some make diﬃcult connections between topics or give an alternative approach, some deal with logical problems unsuitable for the text of a 3 Unit book. Students taking the 4 Unit course should attempt some of these.

The Theory and the Worked Exercises: The theory has been developed with as much rigour as is appropriate at school, even for those taking the 4 Unit course. This leaves students and their teachers free to choose how thoroughly the theory is presented in a particular class. It can often be helpful to learn a method ﬁrst and then return to the details of the proof and explanation when the point of it all has become clear.

The main formulae, methods, deﬁnitions and results have been boxed and num-bered consecutively through each chapter. They provide a summary only, and

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x How to Use This Book

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represent an absolute minimum of what should be known. The worked examples have been chosen to illustrate the new methods introduced in the section, and should be suﬃcient preparation for the questions of the following exercise.

The Order of the Topics: We have presented the topics in the order we have found most satisfactory in our own teaching. There are, however, many effective or-derings of the topics, and the book allows all the flexibility needed in the many different situations that apply in different schools (apart from the few questions that provide links between topics).

The time needed for the work on polynomials in Chapter Four, on Euclidean geometry in Chapters Eight and Nine, and on the ﬁrst few sections of probability in Chapter Ten, will depend on students’ experiences in Years 9 and 10. The Study Notes at the start of each chapter make further speciﬁc remarks about each topic.

We have left Euclidean geometry, polynomials and elementary probability until Year 12 for two reasons. First, we believe as much calculus as possible should be developed in Year 11, ideally including the logarithmic and exponential functions and the trigonometric functions. These are the fundamental ideas in the course, and it is best if Year 12 is used then to consolidate and extend them (and students subsequently taking the 4 Unit course particularly need this material early). Sec-ondly, the Years 9 and 10 Advanced Course already develops elementary probility in the Core, and much of the work on polynomials and Euclidean geometry in Options recommended for those proceeding to 3 Unit, so that revisiting them in Year 12 with the extensions and greater sophistication required seems an ideal arrangement.

The Structure of the Course: Recent examination papers have included longer ques-tions combining ideas from different topics, thus making clear the strong inter-connections amongst the various topics. Calculus is the backbone of the course, and the two processes of differentiation and integration, inverses of each other, dominate most of the topics. We have introduced both processes using geomet-rical ideas, basing differentiation on tangents and integration on areas, but the subsequent discussions, applications and exercises give many other ways of un-derstanding them. For example, questions about rates are prominent from an early stage.

Besides linear functions, three groups of functions dominate the course:

The Quadratic Functions: _{These functions are known from earlier years.} They are algebraic representations of the parabola, and arise naturally in situations where areas are being considered or where a constant acceleration is being applied. They can be studied without calculus, but calculus provides an alternative and sometimes quicker approach.

The Exponential and Logarithmic Functions: _{Calculus is essential for} the study of these functions. We have chosen to introduce the logarithmic function ﬁrst, using deﬁnite integrals of the reciprocal functiony = 1/x. This approach is more satisfying because it makes clear the relationship between these functions and the rectangular hyperbolay = 1/x, and because it gives a clear picture of the new numbere. It is also more rigorous. Later, however, one can never overemphasise the fundamental property that the exponential

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How to Use This Book xi

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function with basee is its own derivative — this is the reason why these func-tions are essential for the study of natural growth and decay, and therefore occur in almost every application of mathematics.

Arithmetic and geometric sequences arise naturally throughout the course. They are the values, respectively, of linear and exponential functions at in-tegers, and these interrelationships need to be developed, particularly in the context of applications to ﬁnance.

The Trigonometric Functions: _{Again, calculus is essential for the study} of these functions, whose deﬁnition, like the associated deﬁnition of π, is based on the circle. The graphs of the sine and cosine functions are waves, and they are essential for the study of all periodic phenomena — hence the detailed study of simple harmonic motion in Year 12.

Thus the three basic functions of the course —x2,exand sinx — and the related numberse and π are developed from the three most basic degree 2 curves — the parabola, the rectangular hyperbola and the circle. In this way, everything in the course, whether in calculus, geometry, trigonometry, coordinate geometry or algebra, is easily related to everything else.

The geometry of the circle is mostly studied using Euclidean methods, and the highly structured arguments used here contrast with the algebraic arguments used in the coordinate geometry approach to the parabola. In the 4 Unit course, the geometry of the rectangular hyperbola is given special consideration in the context of a coordinate geometry treatment of general conics.

Polynomials constitute a generalisation of quadratics, and move the course a little beyond the degree 2 phenomena described above. The particular case of the binomial theorem then becomes the bridge from elementary probability us-ing tree diagrams to the binomial distribution with all its practical applications. Unfortunately, the power series that link polynomials with the exponential and trigonometric functions are too sophisticated for a school course. Projective ge-ometry and calculus with complex numbers are even further removed, so it is not really possible to explain that exponential and trigonometric functions are the same thing, although there are many clues.

Algebra, Graphs and Language: One of the chief purposes of the course, stressed in recent examinations, is to encourage arguments that relate a curve to its equation. Being able to predict the behaviour of a curve given only its equation is a constant concern of the exercises. Conversely, the behaviour of a graph can often be used to solve an algebraic problem. We have drawn as many sketches in the book as space allowed, but as a matter of routine, students should draw diagrams for almost every problem they attempt. It is because sketches can so easily be drawn that this type of mathematics is so satisfactory for study at school.

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xii How to Use This Book

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This course is intended to develop simultaneously algebraic agility, geometric intuition, and rigorous language and logic. Ideally then, any solution should display elegant and error-free algebra, diagrams to display the situation, and clarity of language and logic in argument.

Theory and Applications: Elegance of argument and perfection of structure are fun-damental in mathematics. We have kept to these values as far as is reasonable in the development of the theory and in the exercises. The application of math-ematics to the world around us is equally fundamental, and we have given many examples of the usefulness of everything in the course. Calculus is particularly suitable for presenting this double view of mathematics.

We would therefore urge the reader sometimes to pay attention to the details of argument in proofs and to the abstract structures and their interrelationships, and at other times to become involved in the interpretation provided by the applications.

Limits, Continuity and the Real Numbers: This is a ﬁrst course in calculus, geometri-cally and intuitively developed. It is not a course in analysis, and any attempt to provide a rigorous treatment of limits, continuity or the real numbers would be quite inappropriate. We believe that the limits required in this course present little diﬃculty to intuitive understanding — really little more is needed than

lim

x→∞1/x = 0 and the occasional use of the sandwich principle in proofs.

Char-acterising the tangent as the limit of the secant is a dramatic new idea, clearly marking the beginning of calculus, and quite accessible. Continuity and differ-entiability need only occasional attention, given the well-behaved functions that occur in the course. The real numbers are defined geometrically as points on the number line, and provided that intuitive ideas about lines are accepted, ev-erything needed about them can be justified from this definition. In particular, the intermediate value theorem, which states that a continuous function can only change sign at a zero, is taken to be obvious.

These unavoidable gaps concern only very subtle issues of ‘foundations’, and we are fortunate that everything else in the course can be developed rigorously so that students are given that characteristic mathematical experience of certainty and total understanding. This is the great contribution that mathematics brings to all our education.

Technology: There is much discussion, but little agreement yet, about what role tech-nology should play in the mathematics classroom and which calculators or soft-ware may be eﬀective. This is a time for experimentation and diversity. We have therefore given only a few speciﬁc recommendations about technology, but we encourage such investigation, and to this version we have added some optional technology resources that can be accessed via the Cambridge GO website. The graphs of functions are at the centre of the course, and the more experience and intuitive understanding students have, the better able they are to interpret the mathematics correctly. A warning here is appropriate — any machine drawing of a curve should be accompanied by a clear understanding of why such a curve arises from the particular equation or situation.

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About the Authors

Dr Bill Pender is Subject Master in Mathematics at Sydney Grammar School, where he has taught since 1975. He has an MSc and PhD in Pure Mathematics from Sydney University and a BA (Hons) in Early English from Macquarie Uni-versity. In 1973–4, he studied at Bonn University in Germany and he has lectured and tutored at Sydney University and at the University of NSW, where he was a Visiting Fellow in 1989. He was a member of the NSW Syllabus Committee in Mathematics for two years and subsequently of the Review Committee for the Years 9–10 Advanced Syllabus. He is a regular presenter of inservice courses for AIS and MANSW, and plays piano and harpsichord.

David Sadler is Second Master in Mathematics and Master in Charge of Statistics at Sydney Grammar School, where he has taught since 1980. He has a BSc from the University of NSW and an MA in Pure Mathematics and a DipEd from Sydney University. In 1979, he taught at Sydney Boys’ High School, and he was a Visiting Fellow at the University of NSW in 1991.

Julia Shea is Head of Mathematics at Newington College, with a BSc and DipEd from the University of Tasmania. She taught for six years at Rosny College, a State Senior College in Hobart, and then for ﬁve years at Sydney Grammar School. She was a member of the Executive Committee of the Mathematics Association of Tasmania for ﬁve years.

Derek Ward has taught Mathematics at Sydney Grammar School since 1991, and is Master in Charge of Database Administration. He has an MSc in Applied Mathematics and a BScDipEd, both from the University of NSW, where he was subsequently Senior Tutor for three years. He has an AMusA in Flute, and sings in the Choir of Christ Church St Laurence.

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The Book of Nature is written in the language of Mathematics.

— The seventeenth century Italian scientist Galileo

It is more important to have beauty in one’s equations than to have them ﬁt experiment.

— The twentieth century English physicist Dirac

Even if there is only one possible uniﬁed theory, it is just a set of rules and equations. What is it that breathes ﬁre into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe.

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CHAPTER ONE

The Inverse

Trigonometric Functions

A proper understanding of how to solve trigonometric equations requires a theory of inverse trigonometric functions. This theory is complicated by the fact that the trigonometric functions are periodic functions — they therefore fail the horizontal line test quite seriously, in that some horizontal lines cross their graphs inﬁnitely many times. Understanding inverse trigonometric functions therefore requires further discussion of the procedures for restricting the domain of a function so that the inverse relation is also a function. Once the functions are established, the usual methods of diﬀerential and integral calculus can be applied to them. This theory gives rise to primitives of two purely algebraic functions

1

√

1− x2 dx = sin

−1_{x (or − cos}−1_x) _and 1

1 +x2 dx = tan−1x,

which are similar to the earlier primitive

1

xdx = log x in that in all three cases,

a purely algebraic function has a primitive which is non-algebraic.

Study Notes: _{Inverse relations and functions were ﬁrst introduced in} Sec-tion 2H of the Year 11 volume. That material is summarised in SecSec-tion 1A in preparation for more detail about restricted functions, but some further revision may be necessary. Sections 1B–1E then develop the standard theory of inverse trigonometric functions and their graphs, and the associated derivatives and in-tegrals. In Section 1F these functions are used to establish some formulae for the general solutions of trigonometric equations.

1 A

Restricting the Domain

Section 2H of the Year 11 volume discussed how the inverse relation of a function may or may not be a function, and brieﬂy mentioned that if the inverse is not a function, then the domain can be restricted so that the inverse of this restricted function is a function. This section revisits those ideas and develops a more systematic approach to restricting the domain.

Inverse Relations and Inverse Functions: First, here is a summary of the basic theory of inverse functions and relations. The examples given later will illustrate the various points. Suppose that f(x) is a function whose inverse relation is being considered.

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2 CHAPTER1: The Inverse Trigonometric Functions CAMBRIDGEMATHEMATICS3 UNITYEAR12

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1

INVERSE FUNCTIONS AND RELATIONS:

• The graph of the inverse relation is obtained by reﬂecting the original graph

in the diagonal liney = x.

• The inverse relation of a given relation is a function if and only if no horizontal

line crosses the original graph more than once.

• The domain and range of the inverse relation are the range and domain

re-spectively of the original function.

• To ﬁnd the equations and conditions of the inverse relation, write x for y and y for x every time each variable occurs.

• If the inverse relation is also a function, the inverse function is written as f−1₍_{x). Then the composition of the function and its inverse, in either order,}

leaves every number unchanged:

f−1_f(x)₌_x _and _f_f−1₍_x)₌_x.

• If the inverse is not a function, then the domain of the original function can

be restricted so that the inverse of the restricted function is a function. The following worked exercise illustrates the fourth and ﬁfth points above.

WORKEDEXERCISE: _{Find the inverse function of} f(x) = x − 2

x + 2. Then show

di-rectly thatf−1f(x)=x and ff−1(x)=x. SOLUTION: Let y = x − 2

x + 2.

Then the inverse relation is x = y − 2

y + 2 (writingy for x and x for y) xy + 2x = y − 2

y(x − 1) = −2x − 2 y = 2 + 2x

1− x .

Since there is only one solution fory, the inverse relation is a function, and f−1(x) = 2 + 2x 1− x . Thenff−1(x)=f 2 + 2x 1− x = 2+ 2x 1−x − 2 2+ 2x 1−x + 2 ×1− x 1− x = (2 + 2x) − 2(1 − x) (2 + 2x) + 2(1 − x) = 4x 4 =x, as required. and f−1f(x) =f−1 x − 2 x + 2 = 2 + 2(x−2) x+2 1−x−2_x+2 × x + 2 x + 2 = 2(x + 2) + 2(x − 2) (x + 2) − (x − 2) = 4x 4 =x as required.

Increasing and Decreasing Functions: Increasing means getting bigger, and we say that a functionf(x) is an increasing function if f(x) increases as x increases:

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CHAPTER1: The Inverse Trigonometric Functions 1A Restricting the Domain 3

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For example, if f(x) is an increasing function, then provided f(x) is deﬁned there, f(2) < f(3), and f(5) < f(10). In the language of coordinate geometry, this means that every chord slopes upwards, because the ratio f(b) − f(a)

b − a must

be positive, for all pairs of distinct numbers a and b. Decreasing functions are deﬁned similarly.

2

INCREASING AND DECREASING FUNCTIONS: Suppose thatf(x) is a function.

• f(x) is called an increasing function if every chord slopes upwards, that is, f(a) < f(b), whenever a < b.

• f(x) is called a decreasing function if every chord slopes downwards, that is, f(a) > f(b), whenever a < b. x y x y x y

An increasing function A decreasing function Neither of these Note: _{These are global deﬁnitions, looking at the graph of the function as a} whole. They should be contrasted with the pointwise deﬁnitions introduced in Chapter Ten of the Year 11 volume, where a functionf(x) was called increasing

atx = a if f(a) > 0, that is, if the tangent slopes upwards at the point.

Throughout our course, a tangent describes the behaviour of a function at a particular point, whereas a chord relates the values of the function at two diﬀerent points.

The exact relationship between the global and pointwise definitions of increasing are surprisingly difficult to state, as the examples in the following paragraphs demonstrate, but in this course it will be sufficient to rely on the graph and common sense.

The Inverse Relation of an Increasing or Decreasing Function: When a horizontal line crosses a graph twice, it generates a horizontal chord. But every chord of an in-creasing function slopes upwards, and so an inin-creasing function cannot possibly fail the horizontal line test. This means that the inverse relation of every increas-ing function is a function. The same argument applies to decreasincreas-ing functions.

3

INCREASING OR DECREASING FUNCTIONS AND THE INVERSE RELATION:

• The inverse of an increasing or decreasing function is a function.

• The inverse of an increasing function is increasing, and the inverse of a

de-creasing function is dede-creasing.

To justify the second remark, notice that reﬂection in y = x maps lines sloping upwards to lines sloping upwards, and maps lines sloping downwards to lines sloping downwards.

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Example — The Cube and Cube Root Functions: The functionf(x) = x3

and its inverse functionf−1(x) = √3 x are graphed to the right. • f(x) = x3 _{is an increasing function, because every chord}

slopes upwards. Hence it passes the horizontal line test, and its inverse is a function, which is also increasing.

• f(x) is not, however, increasing at every point, because

the tangent at the origin is horizontal. Correspondingly, the tangent to y = √3 x at the origin is vertical.

y=x3 x y= x y 1 1 −1 −1 y=3 x • For all x, √3 x3 ₌_{x and (}√3 x )3 ₌x.

Example — The Logarithmic and Exponential Functions:

The two functionsf(x) = ex and f−1(x) = log x provide a particularly clear example of a function and its inverse.

• f(x) = ex_{is an increasing function, because every chord}

slopes upwards. Hence it passes the horizontal line test, and its inverse is a function, which is also increasing.

• f(x) = ex _{is also increasing at every point, because its}

derivative is f(x) = ex which is always positive.

y=ex x y= log x y= x y 1 2 1 2 e e

• For all x, log ex ₌_{x, and for x > 0, e}logx₌_x.

Example — The Reciprocal Function: The function f(x) = 1/x is its own inverse, because the reciprocal of the reciprocal of any nonzero number is always the original number. Correspondingly, its graph is symmetric iny = x.

• f(x) = 1/x is neither increasing nor decreasing, because

chords joining points on the same branch slope down-wards, and chords joining points on diﬀerent branches slope upwards. Nevertheless, it passes the horizontal line test, and its inverse (which is itself) is a function.

x y= x y 1 x y= 1 1

• f(x) = 1/x is decreasing at every point, because its

derivative is f(x) = −1/x2, which is always negative.

Restricting the Domain — The Square and Square Root Functions: The two functions

y = x2 _and _{y =}√_{x give our ﬁrst example of restricting the domain so that the}

inverse of the restricted function is a function.

• y = x2 _{is neither increasing nor decreasing, because}

some of its chords slope upwards, some slope down-wards, and some are horizontal. Its inverse x = y2 is not a function — for example, the number 1 has two square roots, 1 and−1.

• Deﬁne the restricted function f(x) by f(x) = x2_{, where}

x ≥ 0. This is the part of y = x2 _{shown undotted}

in the diagram on the right. Then f(x) is an increas-ing function, and so has an inverse which is written as

f−1₍_{x) =}√_{x, and which is also increasing.} y=x

x y y=f−1( )x y= ( )f x 1 1 −1 −1

• For all x > 0, √x2 ₌x and (√x )2 ₌_x.

Further Examples of Restricting the Domain: These two worked exercises show the process of restricting the domain applied to more general functions. Sincey = x is the mirror exchanging the graphs of a function and its inverse, and since points on a mirror are reﬂected to themselves, it follows that if the graph of the function intersects the liney = x, then it intersects the inverse there too.

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WORKEDEXERCISE: _{Explain why the inverse relation of}f(x) = (x−1)2_{+2 is not a} function. Deﬁneg(x) to be the restriction of f(x) to the largest possible domain containing x = 0 so that g(x) has an inverse function. Write down the equation ofg−1(x), then sketch g(x) and g−1(x) on one set of axes.

SOLUTION: The graph of y = f(x) is a parabola with vertex (1, 2).

This fails the horizontal line test, so the inverse is not a function. (Alternatively,f(0) = f(2) = 3, so y = 3 meets the curve twice.) Restrictingf(x) to the domain x ≤ 1 gives the function

g(x) = (x − 1)2_{+ 2}_{, where x ≤ 1,}

which is sketched opposite, and includes the value at x = 0. Sinceg(x) is a decreasing function, it has an inverse with equation

x = (y − 1)2_{+ 2}_{, where y ≤ 1.} x y= x y y=g x−1( ) y= ( )g x 1 1 2 2 3 3

Solving for y, (y − 1)2=x − 2, where y ≤ 1,

y = 1 +√x − 2 or 1 −√x − 2, where y ≤ 1.

Hence g(x) = 1 −√x − 2, since y ≤ 1.

WORKEDEXERCISE: _{Use calculus to find the turning points and points of inflexion} of y = (x − 2)2(x + 1), then sketch the curve. Explain why the restriction f(x) of this function to the part of the curve between the two turning points has an inverse function. Sketchy = f(x), y = f−1(x) and y = x on one set of axes, and write down an equation satisfied by the x-coordinate of the point M where the function and its inverse intersect.

SOLUTION: For y = (x − 2)2(x + 1) = x3− 3x2+ 4,

y _{= 3}_x2_{− 6x = 3x(x − 2),}

and y= 6x − 6 = 6(x − 1).

So there are zeroes atx = 2 and x = −1, and (after testing) turning points at (0, 4) (a maximum) and (2, 0) (a minimum), and a point of inﬂexion at (1, 2).

The part of the curve between the turning points is decreasing, so the functionf(x) = (x − 2)2(x + 1), where 0 ≤ x ≤ 2, has an inverse functionf−1(x), which is also decreasing.

x y= x y y=f−1( )x y= ( )f x 2 4 2 4 M

The curves y = f(x) and y = f−1(x) intersect on y = x, and substituting y = x into the function,

x = x3_{− 3x}2_{+ 4}_,

so thex-coordinate of M satisﬁes the cubic x3− 3x2− x + 4 = 0.

Exercise 1A

1. Consider the functionsf = {(0, 2), (1, 3), (2, 4)} and g = {(0, 2), (1, 2), (2, 2)}.

(a) Write down the inverse relation of each function.

(b) Graph each function and its inverse relation on a number plane, using separate dia-grams forf and g.

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2. The functionf(x) = x + 3 is deﬁned over the domain 0 ≤ x ≤ 2.

(a) State the range of f(x).

(b) State the domain and range of f−1(x).

(c) Write down the rule for f−1(x).

3. The functionF is deﬁned by F (x) =√x over the domain 0 ≤ x ≤ 4.

(a) State the range of F (x).

(b) State the domain and range of F−1(x).

(c) Write down the rule forF−1(x).

(d) GraphF and F−1.

4. Sketch the graph of each function. Then use reﬂection in the line y = x to sketch the inverse relation. State whether or not the inverse is a function, and ﬁnd its equation if it is. Also, state whetherf(x) and f−1(x) (if it exists) are increasing, decreasing or neither.

(a) f(x) = 2x (b) f(x) = x3+ 1 (c) f(x) =1− x2 (d) f(x) = x2− 4 (e) f(x) = 2x (f) f(x) =√x − 3 5. Consider the functionsf(x) = 3x + 2 and g(x) = 1₃(x − 2).

(a) Find f (g(x)) and g (f(x)). (b) What is the relationship betweenf(x) and g(x)?

6. Each functiong(x) is deﬁned over a restricted domain so that g−1(x) exists. Find g−1(x) and write down its domain and range. (Sketches ofg and g−1 will prove helpful.)

(a) g(x) = x2, x ≥ 0 (b) g(x) = x2+ 2, x ≤ 0 (c) g(x) = −4− x2_{, 0}_{≤ x ≤ 2}

7. (a) Write down dy

dx for the functiony = x3− 1. (b) Make x the subject and hence ﬁnd dx

dy. (c) Hence show that dy

dx× dx dy = 1. 8. Repeat the previous question fory =√x .

D E V E L O P M E N T

9. The functionF (x) = x2+ 2x + 4 is deﬁned over the domain x ≥ −1. (a) Sketch the graphs ofF (x) and F−1(x) on the same diagram. (b) Find the equation of F−1(x) and state its domain and range.

10. (a) Solve the equation 1− ln x = 0.

(b) Sketch the graph of f(x) = 1 − ln x by suitably transforming the graph of y = ln x. (c) Hence sketch the graph of f−1(x) on the same diagram.

(d) Find the equation of f−1(x) and state its domain and range. (e) Classify f(x) and f−1(x) as increasing, decreasing or neither.

11. (a) Carefully sketch the function deﬁned byg(x) = x + 2

x + 1, forx > −1.

(b) Find g−1(x) and sketch it on the same diagram. Is g−1(x) increasing or decreasing? (c) Find any values of x for which g(x) = g−1(x). [Hint: The easiest way is to solve

g(x) = x. Why does this work?]

12. The previous question seems to imply that the graphs of a function and its inverse can only intersect on the liney = x. This is not always the case.

(a) Find the equation of the inverse of y = −x3.

(b) At what points do the graphs of the function and its inverse meet? (c) Sketch the situation.

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13. (a) Explain how the graph of f(x) = x2 must be transformed to obtain the graph of

g(x) = (x + 2)2_{− 4.}

(b) Hence sketch the graph ofg(x), showing the x and y intercepts and the vertex. (c) What is the largest domain containingx = 0 for which g(x) has an inverse function? (d) Let g−1(x) be the inverse function corresponding to the domain of g(x) in part (c).

What is the domain ofg−1(x)? Is g−1(x) increasing or decreasing? (e) Find the equation ofg−1(x), and sketch it on your diagram in part (b). (f) Classifyg(x) and g−1(x) as either increasing, decreasing or neither.

14. (a) Show thatF (x) = x3− 3x is an odd function.

(b) Sketch the graph of F (x), showing the x-intercepts and the coordinates of the two stationary points. IsF (x) increasing or decreasing?

(c) What is the largest domain containingx = 0 for which F (x) has an inverse function? (d) State the domain ofF−1(x), and sketch it on the same diagram as part (b).

15. (a) State the domain off(x) = e

x

1 +ex. (b) Show thatf

₍_{x) =} ex

(1 +ex)2 .

(c) Hence explain whyf(x) is increasing for all x.

(d) Explain whyf(x) has an inverse function, and ﬁnd its equation.

16. (a) Sketchy = 1 + x2 and hence sketchf(x) = 1

1 +x2 . Is f(x) increasing or decreasing?

(b) What is the largest domain containingx = −1 for which f(x) has an inverse function? (c) State the domain off−1(x), and sketch it on the same diagram as part (a).

(d) Find the rule forf−1(x).

(e) Isf−1(x) increasing or decreasing?

17. (a) Show that any linear functionf(x) = mx + b has an inverse function if m = 0. (b) Does the constant functionF (x) = b have an inverse function?

18. The functionf(x) is deﬁned by f(x) = x − 1

x, forx > 0.

(a) By considering the graphs ofy = x and y = 1_x forx > 0, sketch y = f(x). (b) Sketchy = f−1(x) on the same diagram.

(c) By completing the square or using the quadratic formula, show that

f−1₍_{x) =} 1 2

x +4 +x2 _.

19. The diagram shows the functiong(x) = 2x

1 +x2 , whose domain is all realx.

g x( ) x 1 −1 1 −1

(a) Show thatg(1_a) =g(a), for all a = 0.

(b) Hence explain why the inverse ofg(x) is not a function. (c) (i) What is the largest domain ofg(x) containing x = 0

for whichg−1(x) exists?

(ii) Sketchg−1(x) for this domain of g(x).

(iii) Find the equation ofg−1(x) for this domain of g(x).

(d) Repeat part (c) for the largest domain ofg(x) that does not contain x = 0.

(e) Show that the two expressions for g−1(x) in parts (c) and (d) are reciprocals of each other. Why could we have anticipated this?

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20. Consider the functionf(x) = 1₆(x2− 4x + 24).

(a) Sketch the parabola y = f(x), showing the vertex and any x- or y-intercepts.

(b) State the largest domain containing only positive numbers for which f(x) has an inverse function f−1(x).

(c) Sketch f−1(x) on your diagram from part (a), and state its domain. (d) Find any points of intersection of the graphs of y = f(x) and y = f−1(x).

(e) Let N be a negative real number. Find f−1(f(N)).

21. (a) Prove, both geometrically and algebraically, that if an odd function has an inverse function, then that inverse function is also odd.

(b) What sort of even functions have inverse functions?

22. [The hyperbolic sine function] The function sinhx is deﬁned by sinh x = 1₂(ex− e−x). (a) State the domain of sinhx.

(b) Find the value of sinh 0.

(c) Show that y = sinh x is an odd function. (d) Find d

dx(sinhx) and hence show that sinh x is increasing for all x.

(e) To which curve isy = sinh x asymptotic for large values of x?

(f) Sketch y = sinh x, and explain why the function has an inverse function sinh−1x. (g) Sketch the graph of sinh−1x on the same diagram as part (f).

(h) Show that sinh−1x = log

x +x2_{+ 1} _{, by treating the equation} x = 1

2(ey − e−y)

as a quadratic equation in ey. (i) Find d

dx(sinh−1x), and hence ﬁnd

dx √

1 +x2 . E X T E N S I O N

23. Suppose that f is a one-to-one function with domain D and range R. Then the function

g with domain R and range D is the inverse of f if

f (g(x)) = x for every x in R and g (f(x)) = x for every x in D. Use this characterisation to prove that the functions

f(x) = −2 3 9− x2_{, where 0 ≤ x ≤ 3,} _and _{g(x) =} 3 2 4− x2_{, where − 2 ≤ x ≤ 0,}

are inverse functions.

24. Theorem: If f is a diﬀerentiable function for all real x and has an inverse function g, theng(x) = 1

f₍_g(x)), provided that f(g(x)) = 0.

(a) It is known that d

dx(lnx) = 1x and that y = ex is the inverse function of y = ln x.

Use this information and the above theorem to prove that d

dx(ex) =ex.

(b) (i) Show that the function f(x) = x3+ 3x is increasing for all real x, and hence that it has an inverse function, f−1(x). (ii) Use the theorem to ﬁnd the gradient of the tangent to the curve y = f−1(x) at the point (4, 1).

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CHAPTER1: The Inverse Trigonometric Functions 1B Deﬁning the Inverse Trigonometric Functions 9

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1 B

Deﬁning the Inverse Trigonometric Functions

Each of the six trigonometric functions fails the horizontal line test completely, in that there are horizontal lines which cross each of their graphs inﬁnitely many times. For example,y = sin x is graphed below, and clearly every horizontal line betweeny = 1 and y = −1 crosses it inﬁnitely many times.

x y π 2π −π −2π π 2 π 2 − 3π 2 − 3π 2 A B C D 1 −1

To create an inverse function fromy = sin x, we need to restrict the domain to a piece of the curve between two turning points. For example, the piecesAB, BC andCD all satisfy the horizontal line test. Since acute angles should be included, the obvious choice is the arc BC from x = −π₂ tox = π₂.

The Deﬁnition of sin−1x: The functiony = sin−1x (which is read as ‘inverse sine ex’) is accordingly deﬁned to be the inverse function of the restricted function

y = sin x, where −π

2 ≤ x ≤ π2.

The two curves are sketched below. Notice, when sketching the graphs, that

y = x is a tangent to y = sin x at the origin. Thus when the graph is reﬂected in y = x, the line y = x does not move, and so it is also the tangent to y = sin−1_x

at the origin. Notice also that y = sin x is horizontal at its turning points, and hencey = sin−1x is vertical at its endpoints.

x y= x y π 2 π 2 − 1 −1 x y= x y π 2 π 2 − 1 −1 y = sin x, −π 2 ≤ x ≤ π2 y = sin−1x 4 THE DEFINITION OFy = sin−1x:

• y = sin−1_{x is not the inverse relation of y = sin x, it is the inverse function of}

the restriction ofy = sin x to −π₂ ≤ x ≤ π₂.

• y = sin−1_{x has domain −1 ≤ x ≤ 1 and range −}π

2 ≤ y ≤ π2.

• y = sin−1_{x is an increasing function.}

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Note: _{In this course, radian measure is used exclusively when dealing with the} inverse trigonometric functions. Calculations using degrees should be avoided, or at least not included in the formal working of problems.

5 RADIAN MEASURE: Use radians when dealing with inverse trigonometric functions.

The Deﬁnition of cos−1x: The function y = cos x is graphed below. To create a satisfactory inverse function from y = cos x, we need to restrict the domain to a piece of the curve between two turning points. Since acute angles should be included, the obvious choice is the arcBC from x = 0 to x = π.

x y π 2π −π −2π π 2 π 2 − 3π 2 − 3π 2 A B C D 1 −1

Thus the function y = cos−1x (read as ‘inverse cos ex’) is deﬁned to be the inverse function of the restricted function

y = cos x, where 0 ≤ x ≤ π,

and the two curves are sketched below. Notice that the tangent toy = cos x at its x-intercept (π₂, 0) is the line t: x + y = π₂ with gradient −1. Reﬂection in

y = x reﬂects this line onto itself, so t is also the tangent to y = cos−1_{x at its}

y-intercept (0,π

2). Like y = sin−1x, the graph is vertical at its endpoints.

x y= + =y x π₂ x y π π 2 1 −1 x y= + =y x π₂ x y π π 2 1 −1 y = cos x, 0 ≤ x ≤ π y = cos−1_x 6 THE DEFINITION OFy = cos−1x:

• y = cos−1x is not the inverse relation of y = cos x, it is the inverse function

of the restriction of y = cos x to 0 ≤ x ≤ π.

• y = cos−1_{x has domain −1 ≤ x ≤ 1 and range 0 ≤ y ≤ π.}

• y = cos−1_{x is a decreasing function.}

• y = cos−1_{x has gradient −1 at its y-intercept, and is vertical at its endpoints.}

The Deﬁnition of tan−1x: The graph of y = tan x on the next page consists of a collection of disconnected branches. The most satisfactory inverse function is formed by choosing the branch in the interval−π₂ < x < π₂.

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x y π 2π −π −2π π 2 π 2 − 3π 2 − 3π 2

Thus the functiony = tan−1x is deﬁned to be the inverse function of

y = tan x, where −π

2 < x < π2.

The line of reﬂectiony = x is the tangent to both curves at the origin. Notice also that the vertical asymptotesx = π₂ andx = −π₂ are reﬂected into the horizontal asymptotesy = π₂ and y = −π₂. x y π 2 π 4 π 2 − π 4 − −1 1 x y π 2 π 4 π 2 − π 4 − −1 1 y = tan x, −π 2 < x < π2 y = tan−1x 7 THE DEFINITION OFy = tan−1x:

• y = tan−1_{x is not the inverse relation of y = tan x, it is the inverse function}

of the restriction of y = tan x to −π₂ < x < π₂.

• y = tan−1_{x has domain the real line and range −}π

2 < y < π2.

• y = tan−1_{x is an increasing function.}

• y = tan−1_{x has gradient 1 at its y-intercept.}

• The lines y = π

2 and y = −π2 are horizontal asymptotes.

Inverse Functions of cosecx, sec x and cot x: It is not convenient in this course to deﬁne the functions cosec−1x, sec−1x and cot−1x because of diﬃculties associ-ated with discontinuities. Extension questions in Exercises 1C and 1D investigate these situations.

Calculations with the Inverse Trigonometric Functions: The key to calculations is to in-clude the restriction every time an expression involving the inverse trigonometric functions is rewritten using trigonometric functions.

8

INTERPRETING THE RESTRICTIONS:

• y = sin−1_{x means x = sin y where −}π

2 ≤ y ≤ π2.

• y = cos−1_{x means x = cos y where 0 ≤ y ≤ π.}

• y = tan−1_{x means x = tan y where −}π

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WORKEDEXERCISE: _Find: _(a) _cos−1₍−1

2) (b) tan−1(−1)

SOLUTION:

(a) Let α = cos−1(−1₂).

Then cosα = −1₂, where 0 ≤ α ≤ π. Hence α is in the second quadrant, and the related angle is π₃,

so α = 2₃π.

(b) Let α = tan−1(−1).

Then tanα = −1, where − π₂ < α < π₂. Henceα is in the fourth quadrant, and the related angle is π₄,

so α = −π₄.

WORKEDEXERCISE: _Find: _(a) _{tan sin}−1₍−1

5) (b) sin(2 cos−1 45)

SOLUTION:

(a) Let α = sin−1(−1 5).

Then sinα = −1₅, where −π₂ ≤ α ≤ π₂. Hence α is in the fourth quadrant, and tanα = √−1 24 5 1 α 24 =−₁₂1√6. (b) Let α = cos−1 4₅. Then cosα = 4 5, where 0 ≤ α ≤ π.

Hence α is in the ﬁrst quadrant, and sin 2α = 2 sin α cos α

α 5 4 3 = 2×3₅ × 4₅ = 24₂₅.

Exercise 1B

x y −1 1 π 3 2 1 π 2

1. Read oﬀ the graph the values of the following correct to two decimal places:

(a) cos−10·4 (b) cos−10·8 (c) cos−10·25 (d) cos−1(−0·1) (e) cos−1(−0·4) (f) cos−1(−0·75)

2. Find the exact value of each of the following:

(a) sin−10 (b) sin−1 1₂ (c) cos−11 (d) tan−11 (e) sin−1(−1) (f) cos−10 (g) tan−10 (h) tan−1(−1) (i) sin−1(−√₂3) (j) cos−1(−√1 2) (k) tan−1(−√1 3) (l) cos−1(−1)

3. Use your calculator to ﬁnd, correct to three decimal places, the value of:

(a) cos−10·123 (b) cos−1(−0·123) (c) sin−1 2₃ (d) sin−1(−2₃) (e) tan−15 (f) tan−1(−5)

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4. Find the exact value of:

(a) sin−1(−1₂) + cos−1(−1₂)

(b) sin(cos−10) (c) tan(tan−11) (d) cos−1(sinπ₃) (e) sin(cos−1 1₂√3 ) (f) cos−1(cos3₄π) (g) tan−1(− tanπ₆) (h) cos2 tan−1(−1)

(i) tan−1(√6 sinπ₄)

D E V E L O P M E N T

5. Find the exact value of: (a) sin−1(sin4₃π) (b) cos−1cos(−π₄)

(c) tan−1(tan5₆π) (d) cos−1(cos5₄π)

(e) sin−12 sin(−π₆) (f) tan−1(3 tan7₆π)

6. (a) In each part use a right-angled triangle within a quadrants diagram to help ﬁnd the exact value of:

(i) sin(cos−1 3₅) (ii) tan(sin−1 5₁₃)

(iii) cos(sin−1 2₃) (iv) sincos−1(−15

17)

(v) cos

tan−1(−1₃) (vi) tancos−1(−3

4)

(b) Use a right-angled triangle in each part to show that:

(i) sin(cos−1x) =1− x2 _{(ii) sin}−1_{x = tan}−1

x √

1− x2

7. Use an appropriate compound-angle formula and the techniques of the previous question where necessary to ﬁnd the exact value of:

(a) sin(sin−1 4₅ + sin−1 12₁₃) (b) cos(tan−1 1₂ + sin−1 1₄)

(c) tan(tan−1 1₄ + tan−1 3₅) (d) tan(sin−1 3₅ + cos−1 12₁₃)

8. Use an appropriate double-angle formula to ﬁnd the exact value of:

(a) cos(2 cos−1 1₃) (b) sin(2 cos−1 6₇) (c) tan2 tan−1(−2)

9. (a) Ifα = tan−1 1₂ and β = tan−1 1₃, show that tan(α + β) = 1. (b) Hence ﬁnd the exact value of tan−1 1₂ + tan−1 1₃.

10. Use a technique similar to that in the previous question to show that: (a) sin−1 1√ 5 + sin −1 1_√ 10 = π 4

(b) tan−1 1₂ − tan−1 1₄ = tan−1 2₉

(c) cos−1 3₁₁ − sin−1 3₄ = sin−1 19₄₄ (d) sin−1 1₃ + cos−1 1₃ = π₂

11. (a) Ifθ = sin−1 3₅, show that cos 2θ = ₂₅7 . (b) Hence show that cos−1 7₂₅ = 2 sin−1 3₅.

12. Use techniques similar to that in the previous question to prove that:

(a) tan−1 3₄ = 2 tan−1 1₃ (b) 2 cos−1x = cos−1(2x2− 1), for 0 ≤ x ≤ 1 (c) 2 tan−12 =π − cos−1 3₅ [Hint: Use the fact that tan(π − x) = − tan x.]

13. (a) Explain why sin−1(sin 2)= 2. (b) Sketch the curvey = sin x for 0 ≤ x ≤ π, and use symmetry to explain why sin 2 = sin(π − 2).

(c) What is the exact value of sin−1(sin 2)?

14. Letx be a positive number and let θ = tan−1x.

(a) Simplify tan(π₂ − θ). (b) Show that tan−1 1_x = π₂ − θ. (c) Hence show that tan−1x + tan−1 1_x = π₂, forx > 0.

(d) Use the fact that tan−1x is odd to ﬁnd tan−1x + tan−1 1_x, forx < 0.

15. (a) Ifα = tan−1x and β = tan−12x, write down an expression for tan(α+β) in terms of x. (b) Hence solve the equation tan−1x + tan−12x = tan−13.

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16. Using an approach similar to that in the previous question, solve forx:

(a) tan−1x + tan−12 = tan−17 (b) tan−13x − tan−1x = tan−1 1₂

17. (a) If α = sin−1x, β = tan−1x and α + β =π₂, show that cos(α + β) =

√

1− x2_{− x}2

√

1 +x2 .

(b) Hence show that x2 =

√

5− 1 2 .

18. (a) If u = tan−1 1₃ and v = tan−1 1₅, show that tan(u + v) =4 7.

(b) Show that tan−1 1₃ + tan−1 1₅ + tan−1 1₇ + tan−1 1₈ = π₄.

19. Show that tan−1 1₂ + tan−1 2₅ + tan−1 8₉ = π₂.

20. Solve tan−1 x x + 1+ tan−1 x 1− x = tan −1 6 7. E X T E N S I O N

21. Prove by mathematical induction that for all positive integer values ofn, tan−1 1 2× 12 + tan −1 1 2× 22 +· · · + tan −1 1 2n2 = π 4 − tan −1 1 2n + 1.

22. Given that a2+b2 = 1, prove that the expression tan−1 ax 1− bx − tan−1x − b a is independent ofx.

23. (a) Show that x

2

x4₊x2_{+ 1} ≤

1

3, for all real x. (b) Determine the range of y = tan−1

1 1 +x2

and the range ofy = tan−1

x2

1 +x2

.

(c) Show that tan−1 1 1 +x2 + tan−1 x2 1 +x2 = tan−1 1 + x 2 1 +x2₊x4 .

(d) Hence determine the range of y = tan−1 1 1 +x2 + tan−1 x2 1 +x2 .

1 C

Graphs Involving Inverse Trigonometric Functions

This section deals mostly with graphs that can be obtained using transformations of the graphs of the three inverse trigonometric functions. Graphs requiring calculus will be covered in the next section.

Graphs Involving Shifting, Reflecting and Stretching: The usual transformation pro-cesses can be applied, but substitution of key values should be used to confirm the graph. In the case of tan−1x, it is wise to take limits so as to confirm the horizontal asymptotes.

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CHAPTER1: The Inverse Trigonometric Functions 1C Graphs Involving Inverse Trigonometric Functions 15

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WORKEDEXERCISE: _{Sketch, stating the domain and range:} (a) y = 2 sin−1(x − 1) (b) y = π − tan−13x SOLUTION:

(a) y = 2 sin−1(x − 1) is y = sin−1x shifted right 1 unit, then stretched vertically by a factor of 2. This should be conﬁrmed by making the following three substitutions:

x 0 1 2 y −π 0 π x y 1 2 π −π

The domain is 0≤ x ≤ 2, and the range is −π ≤ y ≤ π. (b) y = π − tan−13x is y = tan−1x stretched horizontally

by a factor of1

3, then reﬂected in they-axis, then shifted

upwards byπ. This should be conﬁrmed by the follow-ing table of values and limits:

x → −∞ −1 3 0 1 3 → ∞ y → 3π 2 5π 4 π 3π 4 → π2 x y π π 2 1 3 3π 2 3π 4

The domain is all real numbers, and the range is π₂ < y < 3₂π.

More Complicated Transformations: A curve like y = −1₂cos−1(1− 2x) could be ob-tained by transformations. But the situation is so complicated that the best approach is to construct an appropriate table of values, combined with knowl-edge of the general shape of the curve.

WORKEDEXERCISE: _Sketchy = −1

2 cos−1(1−2x), and state its domain and range.

SOLUTION: Using a table of values:

x 0 1₂ 1 y 0 −π₄ −π₂ x y 1 π 2 − π 4 − 1 2

The domain is 0≤ x ≤ 1 and the range is −π₂ ≤ y ≤ 0.

Symmetries of the Inverse Trigonometric Functions: The two functions y = sin−1x and y = tan−1x are both odd, but

y = cos−1_{x has odd symmetry about its y-intercept (0,}π

2).

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SYMMETRIES OF THE INVERSE TRIGONOMETRIC FUNCTIONS:

• y = sin−1x is odd, that is, sin−1(−x) = − sin−1x.

• y = tan−1_{x is odd, that is, tan}−1₍_{−x) = − tan}−1_x.

• y = cos−1_{x has odd symmetry about its y-intercept (0,}π

2), that is,

cos−1(−x) = π − cos−1x Only the last identity needs proof.

Proof: _Let α = cos−1₍−x).

Then −x = cos α, where 0 ≤ α ≤ π,