Table of Contents

CONTENTS1 FRACTIONS AND DECIMALS 21.1 Integers 21.2 Order of operations 31.3 Fractions 31.4 Decimal fractions 71.5 Rounding 101.6 Significant figures 111.7 Using a scientific calculator 141.8 Evaluating mathematical functions using acalculator 15Self-test 152 RATIO, PROPORTION ANDPERCENTAGE 172.1 Ratio 172.2 Direct variation: the unitary method 192.3 Direct variation: the algebraic method 212.4 Inverse variation 242.5 Joint variation 262.6 Percentages 27Self-test 293 MEASUREMENT AND MENSURATION 303.1 SI units (système internationale d’unités) 303.2 Estimations 323.3 Approximations 333.4 Accuracy of measurement 333.5 Stated accuracy 353.6 Implied accuracy in a stated measurement 353.7 Additional calculator exercises involvingsquares and square roots 363.8 Pythagoras’ theorem 373.9 ‘Ideal’ figures 383.10 Perimeters and areas of right-angled triangles,quadrilaterals and polygons 403.11 The area of a rectangle 406 INTRODUCTION TO GEOMETRY 946.1 Points, lines, rays and angles 946.2 Angles 956.3 Complements and supplements 966.4 Vertically opposite angles 986.5 Parallel lines and a transversal 983.12 The area of a triangle 413.13 The circle: circumference and area 443.14 Volumes of prisms 453.15 Surface areas of prisms 47Self-test 494 INTRODUCTION TO ALGEBRA 504.1 Substitutions 504.2 Addition of like terms 524.3 Removal of brackets 534.4 Multiplication and division of terms 534.5 The distributive law 544.6 HCF and LCM 564.7 Algebraic fractions 574.8 Algebraic fractions of the forma 6 bc 594.9 Two important points 604.10 Solving linear equations 624.11 Solving more difficult linear equations 654.12 Using linear equations to solve practicalproblems 664.13 Simultaneous linear equations 694.14 The substitution method 704.15 The elimination method 714.16 Practical problems 73Self-test 755 FORMULAE: EVALUATION ANDTRANSPOSITION 775.1 Evaluation of the subject of a formula 775.2 More SI units 795.3 Evaluation of a formula 805.4 Transposition 835.5 Transpositions in which grouping is required 865.6 More transposition and evaluation 88Self-test 916.6 The angles of a triangle 996.7 The use of the rule, protractor and setsquare 1006.8 Geometrical constructions using acompass 103Self-test 105PART 1 NUMERACY AND ALGEBRAPART 2 GEOMETRY AND TRIGONOMETRYAbout the authors viiiAcknowledgments viiiPreface ixText at a glance xE-student/E-instructor xiiVI Contents7 GEOMETRY OF TRIANGLES ANDQUADRILATERALS 1077.1 Classification of triangles 1077.2 Detecting equal angles 1097.3 The incentre, the circumcentre, the centroidand the orthocentre 1107.4 Congruent triangles 1127.5 Similar triangles 1147.6 Special quadrilaterals and their properties 1177.7 Hero’s formula 1207.8 Areas of the parallelogram and the rhombus 121Self-test 1248 GEOMETRY OF THE CIRCLE 1268.1 The circle: some terms used 1268.2 The circle: proportional relationships 1288.3 Angles associated with the circle 1298.4 Chords of a circle 1338.5 Tangents to circles 1348.6 The angle in the alternate segment 1368.7 Intersecting circles 1378.8 Practical situations 138Self-test 1399 STRAIGHT LINE COORDINATEGEOMETRY 1429.1 The number plane 1429.2 The gradient of a line 1449.3 Intercepts 147PART 3 APPLIED MATHEMATICS9.4 The straight line equation y 5 mx 1 b 1489.5 Finding the equation for a particular straightline 1519.6 Dependent and independent variables 1529.7 Plotting and interpolation 1549.8 Plotting graphs from equations 1589.9 Graphical solution of linear equations andsystems of two (simultaneous) linearequations 160Self-test 16310 INTRODUCTION TO TRIGONOMETRY 16610.1 Conversion of angles between sexagesimalmeasure (degrees, minutes and seconds)and decimal degrees 16610.2 The tangent ratio: construction anddefinition 16710.3 The tangent ratio: finding the length of a sideof a right-angled triangle 16910.4 The tangent ratio: evaluating an angle 17010.5 The sine and cosine ratios 17110.6 Applications 17410.7 How to find a trigonometrical ratio from onealready given 17710.8 Compass directions; angles of elevation anddepression 17810.9 Area of a triangle 180Self-test 18111 INDICES AND RADICALS 18411.1 Radicals 18411.2 Positive integral indices 18611.3 Substitutions and simplifications 18711.4 The four laws for indices 18811.5 Zero, negative and fractional indices 19111.6 Scientific notation 19411.7 Engineering notation 19611.8 Transposition in formulae involving exponentsand radicals 198Self-test 19912 POLYNOMIALS 20112.1 Definition 20112.2 Multiplication of polynomials 20212.3 Factorising 20312.4 Common factors 20412.5 The difference of two squares 20512.6 Trinomials 20612.7 Composite types 21012.8 Algebraic fractions (simplification byfactorising) 21012.9 Quadratic equations: introduction 21112.10 Solution of the equation p 3 q 5 0 21212.11 Solution of the equation x2 5 c 21212.12 Solution of the equation ax2 1 bx 5 0 21412.13 Solution of the equation ax2 1 bx 1 c 5 0 21512.14 ‘Roots’ and ‘zeros’ 21712.15 The quadratic formula 21712.16 Graphical solution of a quadratic equation 223Self-test 22413 FUNCTIONS AND THEIR GRAPHS 22513.1 Function notation 22513.2 The parabola: graphing the curve from itsequation 22713.3 The parabola: finding the equation knowingthe coordinates of three points 23113.4 Practical applications of the parabola 23213.5 The circle 23413.6 The rectangular hyperbola 23613.7 Algebraic solution of simultaneous equationsthat involve quadratic functions 23713.8 Graphical solution of simultaneous equationsthat involve quadratic functions 23813.9 Verbally formulated problems involvingfinding the maximum or minimum value of aquadratic function by graphical means 240Self-test 24214 LOGARITHMS AND EXPONENTIALEQUATIONS 24414.1 Definition of a logarithm: translation betweenexponential and logarithmic languages 24414.2 Evaluations using the definition (logsand antilogs) 24514.3 Powers and logarithms with base 10 or e 24614.4 Decibels 247Contents VII14.5 The three laws of logarithms 25014.6 Change of base 25314.7 Evaluations using the laws of logarithms 25414.8 Solution of logarithmic equations 25514.9 Exponential equations 25514.10 Change of subject involving logarithms 25814.11 Applications 25814.12 The curve y 5 k logb Cx 25914.13 A quick but important revision of exponentsand logarithms 26314.14 Limits 264Self-test 26515 NON-LINEAR EMPIRICAL EQUATIONS 26715.1 Introduction 26715.2 Conversion to linear form by algebraicmethods 26715.3 The use of logarithms to produce a linearform 269Self-test 27216 COMPOUND INTEREST: EXPONENTIALGROWTH AND DECAY 27316.1 Compound interest 27316.2 Exponential growth 27416.3 Graphs of exponential functions 27816.4 Exponential relationships 281Self-test 28517 CIRCULAR FUNCTIONS 28717.1 Angles of any magnitude 28717.2 The reciprocal ratios 29617.3 The co-ratios 29817.4 Circular measure 29817.5 Angular velocity (rotational speed) 30017.6 The circle: length of arc, area of sector, area ofsegment 301Self-test 30218 TRIGONOMETRIC FUNCTIONS AND PHASEANGLES 30318.1 Graphs of a sin bu, a cos bu 30318.2 The tangent curves 30818.3 Sketch-graphs of a sin (u 6 a), a cos (u 6 a) 30818.4 The functions a sin (bu 1 a), a cos (bu 1 a) 31018.5 Phase angles 31418.6 Adding sine and cosine functions 316Self-test 31819 TRIGONOMETRY OF OBLIQUETRIANGLES 31919.1 The sine rule 31919.2 The ambiguous case 32219.3 The cosine rule 32319.4 The use of the sine and cosine rules 326Self-test 32820 TRIGONOMETRIC IDENTITIES 33020.1 Definition 33020.2 Trigonometric identities 33020.3 Proving an identity 33220.4 Summary: trigonometric identities 33320.5 Other trigonometric identities 336Self-test 33821 INTRODUCTION TO VECTORS 34021.1 A mathematical vector 34021.2 Scalar and vector quantities 34021.3 Addition of vectors and vector quantities 34121.4 Analytical addition of two vectors at rightangles to each other 34321.5 Resolution of a vector into two components atright angles 34421.6 Resultant of a number of vectors analytically 34821.7 Addition of vectors graphically 35021.8 Equilibrium 353Self-test 35622 ROTATIONAL EQUILIBRIUM AND FRAMEANALYSIS 35722.1 A few reminders 35722.2 The moment of a force about a point 35822.3 An introduction to the analysis of frames 36123 DETERMINANTS AND MATRICES 36723.1 Definition and evaluation of a 2 3 2determinant 36723.2 Solution of simultaneous equations using 2 3 2determinants 36823.3 Matrices: introduction 37023.4 Some definitions and laws 37223.5 Multiplication of matrices 37323.6 Compatibility 37623.7 The identity matrix, I 37723.8 The inverse matrix, A21 37923.9 The algebra of matrices 38123.10 Expressing simultaneous equations in matrixform 38423.11 Solving simultaneous linear equations usingmatrices 38523.12 3 3 3 Determinants: definition andevaluation 39423.13 Solutions of simultaneous linear equationsusing 3 3 3 determinants 39624 STATISTICS AND PROBABILITY 40124.1 Statistics 40124.2 Median, mode and mean 40124.3 Range, variance and standard deviation 40324.4 Exploring the location of the data 40524.5 Graphs 40924.6 Probability 41024.7 Mean and standard deviation of binomialevents 41324.8 Normal curve 415Self-test 421Appendix A 423Appendix B 425Answers 426Index 457

About the Author

Blair Alldis BA, Bsc (Sydney) has taught mathematics, physics and chemistry in schools and TAFE for more than 35 years, and was formerly head teacher of mathematics at Randwick College of TAFE. He is now retired and living in Queensland but continues to publish books for TAFE.