prelims 9/2/2005 10: 51 page i Basic Engineering Mathematics prelims 9/2/2005 10: 51 page ii In memory of Elizabeth prelims 9/2/2005 10: 51 page iii Basic Engineering Mathematics Fourth Edition John Bird, BSc(Hons), CMath, CEng, FIMA, MIEE, FIIE(Elec), FCollP prelims 9/2/2005 10: 51 page iv Newnes An imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published 1999 Second edition 2000 Third edition 2002 Reprinted 2000 (twice), 2001, 2003 Fourth edition 2005 Copyright © 2005, John Bird. All rights reserved. The right of John Bird to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP.Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK: phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 6575 0 For information on all Newnes publications visit our website at www.newnespress.com Typeset by Charon Tec Pvt. Ltd, Chennai, India www.charontec.com Printed and bound in Great Britain prelims 9/2/2005 10: 51 page v Contents Preface xi 1. Basic arithmetic 1 1.1 Arithmetic operations 1 1.2 Highest common factors and lowest common multiples 3 1.3 Order of precedence and brackets 4 2. Fractions, decimals and percentages 6 2.1 Fractions 6 2.2 Ratio and proportion 8 2.3 Decimals 9 2.4 Percentages 11 Assignment 1 13 3. Indices, standard form and engineering notation 14 3.1 Indices 14 3.2 Worked problems on indices 14 3.3 Further worked problems on indices 16 3.4 Standard form 17 3.5 Worked problems on standard form 18 3.6 Further worked problems on standard form 19 3.7 Engineering notation and common prefixes 19 4. Calculations and evaluation of formulae 21 4.1 Errors and approximations 21 4.2 Use of calculator 22 4.3 Conversion tables and charts 25 4.4 Evaluation of formulae 27 Assignment 2 29 5. Computer numbering systems 30 5.1 Binary numbers 30 5.2 Conversion of binary to denary 30 5.3 Conversion of denary to binary 31 5.4 Conversion of denary to binary via octal 32 5.5 Hexadecimal numbers 33 6. Algebra 37 6.1 Basic operations 37 6.2 Laws of indices 39 6.3 Brackets and factorization 41 prelims 9/2/2005 10: 51 page vi vi Contents 6.4 Fundamental laws and precedence 43 6.5 Direct and inverse proportionality 45 Assignment 3 46 7. Simple equations 47 7.1 Expressions, equations and identities 47 7.2 Worked problems on simple equations 47 7.3 Further worked problems on simple equations 49 7.4 Practical problems involving simple equations 50 7.5 Further practical problems involving simple equations 52 8. Transposition of formulae 54 8.1 Introduction to transposition of formulae 54 8.2 Worked problems on transposition of formulae 54 8.3 Further worked problems on transposition of formulae 55 8.4 Harder worked problems on transposition of formulae 57 Assignment 4 59 9. Simultaneous equations 60 9.1 Introduction to simultaneous equations 60 9.2 Worked problems on simultaneous equations in two unknowns 60 9.3 Further worked problems on simultaneous equations 62 9.4 More difficult worked problems on simultaneous equations 63 9.5 Practical problems involving simultaneous equations 65 10. Quadratic equations 69 10.1 Introduction to quadratic equations 69 10.2 Solution of quadratic equations by factorization 69 10.3 Solution of quadratic equations by ‘completing the square’ 71 10.4 Solution of quadratic equations by formula 72 10.5 Practical problems involving quadratic equations 73 10.6 The solution of linear and quadratic equations simultaneously 75 11. Inequalities 77 11.1 Introduction to inequalities 77 11.2 Simple inequalities 77 11.3 Inequalities involving a modulus 78 11.4 Inequalities involving quotients 79 11.5 Inequalities involving square functions 79 11.6 Quadratic inequalities 80 Assignment 5 82 12. Straight line graphs 83 12.1 Introduction to graphs 83 12.2 The straight line graph 83 12.3 Practical problems involving straight line graphs 88 13. Graphical solution of equations 94 13.1 Graphical solution of simultaneous equations 94 13.2 Graphical solutions of quadratic equations 95 prelims 9/2/2005 10: 51 page vii Contents vii 13.3 Graphical solution of linear and quadratic equations simultaneously 99 13.4 Graphical solution of cubic equations 100 Assignment 6 102 14. Logarithms 103 14.1 Introduction to logarithms 103 14.2 Laws of logarithms 103 14.3 Indicial equations 105 14.4 Graphs of logarithmic functions 106 15. Exponential functions 107 15.1 The exponential function 107 15.2 Evaluating exponential functions 107 15.3 The power series for ex 108 15.4 Graphs of exponential functions 110 15.5 Napierian logarithms 111 15.6 Evaluating Napierian logarithms 111 15.7 Laws of growth and decay 113 Assignment 7 116 16. Reduction of non-linear laws to linear-form 117 16.1 Determination of law 117 16.2 Determination of law involving logarithms 119 17. Graphs with logarithmic scales 124 17.1 Logarithmic scales 124 17.2 Graphs of the form y = axn 124 17.3 Graphs of the form y = abx 127 17.4 Graphs of the form y = aekx 128 18. Geometry and triangles 131 18.1 Angular measurement 131 18.2 Types and properties of angles 132 18.3 Properties of triangles 134 18.4 Congruent triangles 136 18.5 Similar triangles 137 18.6 Construction of triangles 139 Assignment 8 141 19. Introduction to trigonometry 142 19.1 Trigonometry 142 19.2 The theorem of Pythagoras 142 19.3 Trigonometric ratios of acute angles 143 19.4 Solution of right-angled triangles 145 19.5 Angles of elevation and depression 147 19.6 Evaluating trigonometric ratios of any angles 148 20. Trigonometric waveforms 151 20.1 Graphs of trigonometric functions 151 20.2 Angles of any magnitude 152 20.3 The production of a sine and cosine wave 154 prelims 9/2/2005 10: 51 page viii viii Contents 20.4 Sine and cosine curves 155 20.5 Sinusoidal form A sin(ωt ± α) 158 Assignment 9 161 21. Cartesian and polar co-ordinates 162 21.1 Introduction 162 21.2 Changing from Cartesian into polar co-ordinates 162 21.3 Changing from polar into Cartesian co-ordinates 163 21.4 Use of R → P and P → R functions on calculators 164 22. Areas of plane figures 166 22.1 Mensuration 166 22.2 Properties of quadrilaterals 166 22.3 Worked problems on areas of plane figures 167 22.4 Further worked problems on areas of plane figures 171 22.5 Areas of similar shapes 172 Assignment 10 173 23. The circle 174 23.1 Introduction 174 23.2 Properties of circles 174 23.3 Arc length and area of a sector 175 23.4 The equation of a circle 178 24. Volumes of common solids 180 24.1 Volumes and surface areas of regular solids 180 24.2 Worked problems on volumes and surface areas of regular solids 180 24.3 Further worked problems on volumes and surface areas of regular solids 182 24.4 Volumes and surface areas of frusta of pyramids and cones 186 24.5 Volumes of similar shapes 189 Assignment 11 190 25. Irregular areas and volumes and mean values of waveforms 191 25.1 Areas of irregular figures 191 25.2 Volumes of irregular solids 193 25.3 The mean or average value of a waveform 194 26. Triangles and some practical applications 198 26.1 Sine and cosine rules 198 26.2 Area of any triangle 198 26.3 Worked problems on the solution of triangles and their areas 198 26.4 Further worked problems on the solution of triangles and their areas 200 26.5 Practical situations involving trigonometry 201 26.6 Further practical situations involving trigonometry 204 Assignment 12 206 27. Vectors 207 27.1 Introduction 207 27.2 Vector addition 207 27.3 Resolution of vectors 209 prelims 9/2/2005 10: 51 page ix Contents ix 27.4 Vector subtraction 210 27.5 Relative velocity 212 28. Adding of waveforms 214 28.1 Combination of two periodic functions 214 28.2 Plotting periodic functions 214 28.3 Determining resultant phasors by calculation 215 29. Number sequences 218 29.1 Simple sequences 218 29.2 The n’th term of a series 218 29.3 Arithmetic progressions 219 29.4 Worked problems on arithmetic progression 220 29.5 Further worked problems on arithmetic progressions 221 29.6 Geometric progressions 222 29.7 Worked problems on geometric progressions 223 29.8 Further worked problems on geometric progressions 224 Assignment 13 225 30.