Bettinger A.K., Englund J.A. - Algebra and Trigonometry (1960)

The authorsbelieve thatin this bookthebasic materialof college algebra and trigonometry has been presented with suflicient rigor to provide a firm and coherent groundwork for subsequent

TEXT FLY WITHIN

DO 1620025m

Algebra and Trigonometry

Scranton, Pennsylvania

INTERNATIONAL TEXTBOOKS IN MATHEMATICS

L R. W//COX

Illinois Institute of Technology

The authorsbelieve thatin this bookthebasic materialof college

algebra and trigonometry has been presented with suflicient rigor

to provide a firm and coherent groundwork for subsequent courses

The first chapter consists of a number of introductory topics

which are intended to serve as a review of elementary algebra

Actually, something more than a mere review is available in this

chapter Not only are the review topics considered from a more mature point of view than is usual, but the treatment is inter-

woven with concepts that are basic for an understanding of more

algebra are derived and logically connected with the bcisic

assump-tions. We are led naturally to an ordering of the real-number

one customarily finds in textbooks

The second chapter introduces the student to the function

the book Certain aspects of the discussion become somewhat

of the true nature of a function is important for virtually all later

courses in mathematics

In line with modern demands, the trigonometric functions are

initially introduced in the third chapter as functions of real

angles is relatively simple

The rest of the volume contains all the usual topics from college

algebra and trigonometry In certain instances a particular

devel-opment may differ somewhat from that usually found In such

We are indebted to our colleague, Professor Morris Dansky, for

his valuable suggestions while the manuscript was in preparation

We wish particularly to express our deep appreciation to fessor L R Wilcox for his thorough criticism of the manuscriptand hisinvaluable suggestions for improvementofthetext. Finally,

Com-pany for its cooperation and patience.

A K BETTINGER

Omaha, Nebraska

August, 1960

1-7. Fundamental Operations on Fractions 9

1-8. Order Relations forReal Numbers 12

1-10. Inequalities Involving Absolute Values 14 1-11 Positive Integral Exponents 16

1-12. Algebraic Expressions 17 1-13. Equations and Identities 18

1-15. Order of Fundamental Operations 20 1-16. Addition and Subtraction ofAlgebraic Expressions 21 1-17. Multiplication of Algebraic Expressions 22

1-19 Division of Algebraic Expressions 23

1-21. Important Type Forms forFactoring 27

1-22 GreatestCommon Divisor 30

1-23. Least Common Multiple 32 1-24. Reduction of Fractions 33 1-25. Signs Associated With Fractions 34 1-26. Addition and Subtraction of Fractions 36 1-27. Multiplication and Division of Fractions 39

1-30. Linear EquationsinOne Unknown 43

3. THE TRIGONOMETRIC FUNCTIONS G3

3-1. The Point Function P(t) 63

3-2 Definitions of the Trigonometric Functions (54

3-4 Tables ofTrigonometric Functions 71

3-5 Positiveand Negative Angles and Standard Position 75

3-7. The RelationBetweenRadians and Degrees 77

3-10 Tables ofNatural Trigonometric FunctionsofAngles 82

4-1 Positive IntegralExponents 86

4-4 Scientific Notation 1)2

4-6. The Factorial Symbol J)7

6-7 ScalarandVectorQuantities 125

7-1 Derivation of the Addition Formulas 135

8. GRAPHS OF TRIGONOMETRIC FUNCTIONS; INVERSE FUNCTIONS

8-1 Variation of the Trigonometric Functions 146

8-2. The Graph of the Sine Function 117

8-3. The Graphsof the Cosineand TangentFunctions 148

8-4. Periodicity, Amplitude, and Phase 14i)

8-5 Inverse Functions ir>r>

8-6 Inverses of the Trigonometric Functions 156

!)-! Solutions of Simultaneous Equations 1(5*5

10-1 I )eti'rminants of the Second Order 173

10-2. Determinants of theThird Order 175

11-2. The StandardNotation forComplex Numbers 191

11-3. Operations on Complex Numbers in Standard Form 192

11-4. Graphical Representation 105

11-5. Trigonometric Representation 106

11-6 Multiplication and Division in TrigonometricForm 198

11-7. DC Moivrc's Theorem 199

11-8. Roots of Complex Numbers 200

12-1. Quadratic Equations in OneUnknown 204

1 2-2 Solution ofQuadratic Equations by Factoring 20 1

12-3. Completing the Square 206

12-4 Solution of Quadratic Equations bytheQuadratic Formula 209

12-5. Equations Involving Radicals 212

12-6. Equations in Quadratic Form 214

12-7. The Discriminant 215

12-8. Sum and Product of the Roots 217

12-9. GraphsofQuadratic Functions 218

12-10. Quadratic Equations in Two Unknowns 221

12-11. GraphicalSolutions of Systems ofEquationsInvolving

13-3. The Remainder Theorem 240

13-4. The Fundamental Theoremof Algebra 241 13-5 Pairs of Complex Rootsofan Equation 243

13-6. The Graph ofa Polynomial for Large Valuesof a- 244

13-7. Roots Between aand b If /(a) and f(b) Have Opposite Signs 245

15-3. The General Term ofan Arithmetic Progression 260

15-4. Sum of the First 71 Terms ofan Arithmetic Progression 261

15-5. Arithmetic Means 262

15-6. Harmonic Progressions 264

15-8. TheGeneralTermof a GeometricProgression 265

15-9. Sum of the Firstn Termsofa GeometricProgression 266

16-1. Method ofMathematicalInduction 273

16-2. Proofof the Binomial Theoremfor Positive IntegralExponents 275

17-7. Most Probable Numberand Mathematical Expectation 284

17-8 Statistical, or Empirical, Probability 284

17-9. MutuallyExclusive Events 285 17-10. Dependent and Independent Events 286

18-1 Classes of Problems 289

18-2. The Lawof Sines 289

18-3 Solution ofCase I bythe Lawof Sines: Given One Sideand

18-4 Solution of Case II by the Lawof Sines: Given Two Sidesand

18-5. The Law of Cosines 296

18-6 Solution ofCaseIIIand CaseIV by the Lawof Cosines 297

18-7. The Law of Tangents 298

18-8. The Half-Angle Formulas 300

1 Introductory Topics

1-1. THE REAL-NUMBER SYSTEM

course is a development from the original counting numbers, or

the invention of positive integers, practical problems of

Much later, in comparativelymoderntimes,the concepts ofnegative

numbers and of other types of numbers were gradually developed

one number from a smaller one presented itself. Thus, the number

defined to be any number that can be expressed as the quotient, or

ratio, of two integers. For example, 2/3, 5 (which may be sidered as 5/1), and 7 are rational numbers

con-The number system was then extended to include also numbers which cannot be expressed as the quotient of two integers, namely,

numbers are so called in contrast to the imaginary or complex

numbers considered in Chapter 11.

operations in arithmetic and algebra without being conscious that

certain basic laws were being obeyed. We shall introduce the four

assumptions governing them

i

Introductory Topics

Addition It is assumed that there is a mode of combining any two real numbers a and 6 so as to produce a definite real number

terms

any two real numbers a and b to produce a definite real number

multiplica-tion. Theproductofaand bis denotedbya b orbyab.The

individ-ual numbers a and b are called factors of the product

Commutative LawforAddition If aand b areanyreal numbers,then

(1-1) a+ b = b + a.

Thus1

, the sum of two mumbers is the same regardless of the order

inwhichthey are added For example,

Associative Law for Addition If a, b, c are any real numbers,then

(1-2) (a + 6) + c = a + (6 + c).

Thatis, we obtain thesame resultwhether weaddthe sum ofa and

order in which they are multiplied. For example,

2-3=3-2.

Associative Law for Multiplication.,If a, 6, c are any real

(1-4) (o6)c = a(6c).

1 Illustrations of the laws are given here only for the mostfamiliar

num-bers, the positive integers It is understood, however, that the laws apply to

Sec. 12 Introductory Topics 3

Thatis, we obtain the sameresult whether we multiply theproduct

of a and b by c, or we multiply a by the product of b and c. Sincethe way in which we associate or group these numbers is imma-

terial, we may write the result as abc without fear of ambiguity

This law, which is usually known as the distributive lawfor

addition and multiplication. The distributive law forms the basisfor the factoring process in algebra, as will be seen

A simple example of the distributive law is

three or more terms, as in the following illustration:

repeated addition Thus, by the distributive law,

Zero It is assumed that there is a special number called zero

(1 (i) a + = a.

i] + o=;j, o + i = j, o + o = o.

of can exist. For let 0' be another such number Then, since

a+ = a and b + 0'= b for any numbers r/, b, it follows, by taking

()' + = 0', and + 0' = 0.

From the commutative law, 0'.

2The right side of (1-5) should read (ah) -f (ar). However, by

conven-tion, we agree to omit the parentheses when all multiplications are to be

4 Introductory Topics Sec. 12

a there exists a corresponding number, called the negative of aand

designated by a, such that

(1 8) a 1 = a.

i r =

i, r- 1 - r,whence 1 = 1'.

Reciprocal of a Number. It is assumed that for every number a

rccip-a

'7 = L

The reader may verify the fact that there is only one reciprocal

a x 1, then x ~

-a

is needed

Subtraction The difference a 6, of any real numbers a and

6, is defined by

(1 10) o - b =a + (- 6).

Sec. 14 /nfroc/ucfory Topics 5

The operation indicated by the si#n minus which produces for any two real numbers a and b the real number a b is calledsubtraction

t)and &, where b ^0, is defined by'

The operation associating with real numbers a and b (b =/-0) their

It should be noted that subtraction and division are subordinate

to addition and multiplication, in that they are defined in terms of

these latter. The difference a b is that number .r for which

b y ~a. It should be noted that + (-) = for every

number a, and that a 'a a (I/ a) 1 for every number a~f 0.

1-3. OPERATIONS WITH ZERO

It has already been notedthat the special number hasthe erty (t + - a for every real number a. In particular, we may let

prop-a= to obtain

+ = 0.

It has already been noted that = 0, so that = a4- = a

for every real numbera.

Next, we prove that for every real number a,

It was noted that every non-zero number has a reciprocal. We

then x= 1. Sinceit hasbeenshown that x= 0,we would have

6 Introductory Topics

to conclude that = 1. However, if = 1 isallowed, then forevery

number awe have

expressed in another way:

In the precedingproof, free use has been made of the commutative

1-5. THE REAL-NUMBER SCALE

Real numbers may be represented by points on a straight line.

On such a line select an arbitrary point as origin and lay off

equal unit distances in both directions, asshown in Fig 1-1 (The

Sec. 16 Introductory Topics 7

the left, and so on. Rational numbers that are not integers

and 1; and 7/3 represents the point one-third of the

dis-tance from the point 2 to the point 3 It is a basic assumption

exactly one point The full significance of this assumption cannot

be developed in an elementary text.

One observation of importance can be made at this time The

non-zero real numbers are divided into two classes. One class

con-sist of numbers representing points to the "right" of 0, and theother consists of numbers representing points to the "left" of 0.

The first class consists of positive numbers, and the second of

negative numbers The number maybe considered as constituting

a third class It is understoodthatno two ofthe three classes zero,

common Thus a number cannot be both positive and zero, both

desig-nation ofany negativenumberwillincludean explicitsign ,which

possi-bility of allowing a general symbol, such as x, to stand for a

It is to be assumed that the sum of two positive numbers is

To operate effectively with real numbers, a knowledge of the

In each of the following relationships, a and b are any two real

8 Introductory Topics Sec.

It has already been observed that non-zero numbers are divided

into two classes, namely, positive and negative It is assumed that

a is positive. All calculations involving negative numbers can be

is equal to the negative of the sum of the negatives of the given

negative numbers is positive, and is equal to the product of thenegatives of the given numbers. By (1-19) the product of a posi-tive number and a negative number is negative

Introductory Topics 9

A further study of the algebra of real numbers leads us to theconsideration of thefundamentaloperations as appliedto fractions

By definition, a fraction is the quotient obtained by dividing

one number a by another number 6, where b is not zero We

call a the numerator and b the denominator; and we generally

write the fraction a/6, read "a over b" or "a divided by b."

four operations to fractions In them a, b, c, d are any real

c

'

which statesthat the reciprocal of a fraction is found by inverting

to multiplyingby its reciprocal

each of the followingequations:

d) Theassociativeand commutativelaws for addition.

6) Theassociativelaw for addition.

c) Theassociativelawfor multiplication.

d) Thedistributive law.

Example 1-2. Ineach of thefollowing,performthe indicated operation:

Example 1-3. Byusing the fundamental assumptions andrules for operations^

andtransforming the left side into therightside, justifythe equation

(a +6)

-(c - d) = (6

-c) + (a +d),

We shall use the notation a > to express the fact that a is a

nega-tive. The symbol > means is greater than, and < means is less

Assume that a and b are any two given numbers If a- b >

0,

to the rightof 6 on thereal-number line. When a> b, that is,when

b -a <0.

The student is familiar with the symbol - (for equality), which

means that the two symbols a and b represent the same

mathe-matical object For example, 6-3*2.

If a and 6 are two distinct numbers on the scale, we say "a is

Sometimes we shall find it convenient to combine the symbols

< and = or > and =. We write ^ tomean is less than or equal to,

and wewrite ^ tomeanisgreater than or equal to.

We thus have order relations on pairs of real numbers, defined

a > b (or b < a) ifand only ifa b is positive;

a > b (or6 < a) if andonly if b ais negative.

Introductory Topics 13

only one of the following relationships holds:

0, so that

ifa < band b < c, then a < c.

Proof of Property 2: If a< b and 6 < c, then both 6 a and

have assumed that the sum of two positive numbers is positive

Since c b and 6 a are positive by assumption, their sum, which

is c a, is alsopositive. Hence,c > a,ora < c.

if a > b andc > 0, then ac > be.

Proof of Property 4: We have assumed that the product of two

follows that their product is also positive But (a 6)c =ac be.

Therefore, ac be is positive, and ac> be.

ifa > b and c < 0, then ac < be.

Introductory Topics

numbers with unlike signs is negative Here a -b is

-6c, and (a

follows that ac be < 0, or that ac< be.

sides It therefore follows that a term on one side of an inequality

reversed if both sides are multiplied or divided by the same tive number.

nega-1-9. ABSOLUTE VALUE

non-negative number called its absolute value For any real number a,

Thus, |3 | = 3, since 3 > 0; also |- 3 | = - (- 3) = 3, since - 3 < 0.

1-10. INEQUALITIES INVOLVING ABSOLUTE VALUES

number scale, then \x\ is thenumerical distance between P and the

origin. If we let a be a positive number, then \x\ <a means thatthe point P is less than a units from the origin; that is, x lies

\x\ < aand a <x < a mean exactly the same thing

A more general inequality which often occurs is |# 6| <a,

where a> 0. This is equivalent to a < x 6 < a. If 6 is added

to each term, we may write 6 a< x< b +a. Hence, the

state-ments \x b\<a and 6 a<x<b +a mean exactly the same

thing

means that the distance between x and 3 is less than 2. To solvethis inequality for xt we add 3 to each term of the inequality,obtaining 1<x < 5.

The following illustrative examples may help to give a better

Example 1-7. Findintegers aand6suchthat a <\/2 < b.

Solution: Since \/2 may be represented approximately by 1.414, the values

a = 1 and6=2 satisfy theinequalities. Thus, 1 < \/2 <2. Any otherpair of

integers a and 6 such that a ^ 1 and 6^2 would also satisfy the inequalities,

Example 1-8. Expressthe inequality |x \ <3 without using the absolute-value symbol.

Solution: Weknowthat the statements

| x

\ < aand a < x <ameanexactly

thesame thing. Here a is the positive number 3, and

|x

\ < 3 means that the point representedbyx is lessthan3 unitsfromtheorigin;thatis,x isbetween 3

and3. Theinequalitymaybe written 3 <x < 3.

Example 1-9 Explain themeaningof the inequality |x 2 | < 1and write it

without using the absolute- value symbol.

Solution: Theinequality

6 In each of the following, find a pair of integers, a and 6, such that the given

inequalities aresatisfied:

a a <5 <b b. a < - 3 < b e. a < <b.

d, a < TT <6 e. a < \/3 <& f a < 1 1 - 2

| <6

7 Ifa ^ 3, place the proper ordersymbol between a +7and 10.

1-11. POSITIVE INTEGRAL EXPONENTS

If two or more equal quantities are multiplied by one another,

, read "5 cubed/'

means 5-5-5. In general, anmeans the product of n factors each

equal to a. We call a the base and nthe exponent of the power It

follows from the associativelaw that

laws are reserved for a later chapter.

.

power of the same base, subtract the exponents:

(1-32)

'

^ = a"*-*, ifa ^ 0,m > n.

Introductory Topics 17

Law fora Powerofa Power.Toraiseapowerofa given basetoa

Law for a Power of a Product To obtain a power of a product,

Law for a Power of a Quotient. To obtain a powerof aquotient,

of the expression connected by plus and minus signs are called

terms The terms of the expression 3x2

-5xy2+7z are 3#2

, -5xy2

,

and 7z. Here the numbers 3, 5, and 7 are called numerical

coffi-cients, or just coefficients; x2

An expression containing one or more terms is called a nomial A multinomial consisting of one term is a monomial A

whose terms are of the form axmyn

z p

,where m, n, p, are

more of the factors xm, y", zp, may be absent Thus, 7, 5#4

, and 3xy + 2 are polynomials, while x+ - is not.

y

is 2, the

term Thus, in the trinomial 3x2

-5xy* +7z, the third-degreeterm, -5xy2

, is its highest-degreeterm? Therefore, 3#2

-5xy*+7z

18 IntroductoryTopics Sec. 112

form

where the coefficients a , &i, * , an are numerical coefficients,

1-13. EQUATIONS AND IDENTITIES

An equation is a statement of equality between two numbers oralgebraic expressions The two expressions are called members, or

sides, of the equation. Equations are of two kinds, namely,

conditional equations and identities. A conditional equation, or

none at all) of the literal quantities appearing An identity istrue

same meaningas parentheses, namely, brackets [ ],braces { }, and

to be combined to form a single quantity. The word "parentheses"

Introductory Topics 19

is often used to indicate any or all of these symbols of grouping

Removal of the symbols of grouping is accomplished by applying

law The following examples illustrate the procedure

Example 1-10. Removeparenthesesfrom

To write a given expression in parentheses preceded by a plus

and write + in front of the parentheses Thus,

To write a given expression in parentheses preceded by aminus

parentheses, and write in front of the parentheses Thus,

a) Since the sign before the parentheses is to be +, we enclose 3x -y in its

givenform within the parentheses precededby a plus sign. In this case the ex*

pressionbecomes2 -f (3x

-y).

Introductory Topics

6) Since the sign before the parentheses is tobe ,we change thesign ofeachtermof3x -yandenclose 3x +ywithin the parentheses precededbyaminus

sign. Thenthe expressionbecomes2 ( 3# +y).

1-15. ORDER OF FUNDAMENTAL OPERATIONS

indicat-ing which operation is to be performed first. We have used them

unneces-sarily, as has been already pointed out, the convention is adopted

to perform all multiplications first and then the additions (or

subtractions) If two or more of these symbols of grouping are

remove the innermost pair of symbols first.

Sec. Introductory Topics

45. a*2 - 2bxy +[(6y -2cx*)

-(axy +i/

2 )].

46 3n6 {4ac -f [ab 2ac +ab] 3a6

useaplussign beforetheparentheses, andthen use aminussign.

y3and5x2yB, whichhavethesameliteralparts,

is obtained as (5 -2)xy* or Say2

.

algebraic expressions In practice, we usually arrange like terms

prefixing the sum of the numerical coefficients in thecolumn The

1-17. MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

term of one by each term of the other and combining like terms

Example 1-16. Multiply 3x2 - 2xy +y2by2x - 3y.

They should be learned thoroughly