rr CRACK Trigonometry I G. Tewani I Cengage App available on I |S| ■■ □ SilS Supplemented with Chapterwise/Topicwise □ Daily Practice Problems (DPP) See inside cover for details MATHEMATICS for Joint Entrance Examination JEE (Advanced) 2nd edition Trigonometry G. Tewani Z** CENGAGE Australia • Brazil • India • Mexico • Singapore • United Kingdom • United States I > CENGAGE F Mathematics: © 2009,2018 Cengage Learning India Pvt. Ltd. Trigonometry, 2e G.Tewani ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, without the prior written permission of the publisher. For permission to use material from this text or product submit all requests online at www.cengage.com/permissions Further permission questions can be emailed to [email protected] ISBN-13: 978-93-87511-24-8 ISBN-10:93-87511-24-3 Cengage Learning India Pvt. Ltd. 418, F.I.E., Patparganj I Delhi 110092 Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Australia, Brazil, India, Mexico, Singapore, United Kingdom, and United States. Locate your local office at: www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd. For product information, visit www.cengage.co.in Printed in India Brief Contents Preface 1. Logarithm and Its Applications 2. Trigonometric Functions 3. Trigonometric Ratios for Compound, Multiple, Sub-multiple Angles, and Transformation Formulas 4. Trigonometric Equations 5. Inverse Trigonometric Functions 6. Solutions and Properties of Triangle Appendix 1: Logarithm and Antilogarithm Tables Appendix 2: Solutions Appendix 3: Chaptenvise Solved 2014 and 2015 JEE Advanced Questions Chapterwise Solved 2016 JEE Advanced Questions Chaptenvise Solved 2017 JEE Advanced Questions I f r. ' 'I L i . : H.-' i ■ > c i • j Contents Preface ix 1. Logarithm and Its Applications 1.1-1.24 Exponential Function: 1.1 Graph of Exponential Function and Its Properties 1.1 Graph of Exponential Function for Different Bases 1.2 Logarithmic Function 1.3 Graphs of the Logarithmic Function 1.3 Graphs of the Logarithmic Function for Different Bases 1.3 Fundamental Laws of Logarithms 1.5 Logarithmic Equations 1.9 Logarithmic Inequalities 1.11 Standard Logarithmic Inequalities 1.11 Frequently Used Inequalities 1.11 Finding logarithm 1.14 Characteristic and Mantissa of a Logarithm 1.14 Mantissa of the Logarithm of a Given Number 1.15 Significant Digits 1.15 Negative Characteristics 1.15 Antilogarithm 1.16 To Find Antilog of a Number 1.16 Exercises 1.18 Subjective Type 1.18 Single Correct Answer Type 1.19 Multiple Correct Answers Type 1.21 Matrix-Match Type 1.22 Integer Answer Type 1.22 Archives 1.23 Answer Key 1.24 2. Trigonometric Functions 2.1-2.28 Trigonometric Ratios/Functions of Acute Angles 2.1 Trigonometric Ratios of Standard Angles 2.1 Trigonometric Identities 2.2 Problems Based on Identities 2.2 Use of Trigonometry in Geometry 2.3 Measurement of an Angle 2.6 Degree Measurement 2.6 Radian Measurement 2.6 Conversion between Radians and Degrees 2.6 Relation between Radians and Real Numbers 2.6 Conversion of Some Common Angles 2.6 Angles in Trigonometry 2.8 Angles Exceeding 360° 2.8 Sign of Angles 2.8 Reference Angle 2.9 Trigonometric Functions of Any Angle 2.10 Trigonometric Functions Defined as Circular Function 2.11 Sign of Trigonometric Functions in Different Quadrants 2.11 Variation in the Values of Trigonometric Functions in Different Quadrants 2.11 vi Contents 2.12 Graph of Sine Function with Its Varying Values on Unit Circle 2.12 Graph and Other Useful Data of Trigonometric Functions 2.13 Transformation of the Graphs of Trigonometric Functions 2.18 Trigonometric Ratios of Allied Angles 2.21 Exercises 2.21 Subjective Type 2.21 Single Correct Answer Type 2.24 Multiple Correct Answers Type • 2.25 Reasoning Type 2.25 Linked Comprehension Type 2.26 Matrix-Match Type . ■ z 2.26 Integer Answer Type 2.27 Archives 2.28 Answer Key 3.1-3.45 3.. Trigonometric Ratios for Compound, Multiple, Sub-multiple Angles, and Transformation Formulas 3.1 Trigonometric Ratios for Compound Angles 3.1 Cosine of the Difference and Sum of Two Angles 3.2 Sine of the Difference and Sum of Two Angles I... 3.2 Tangent of the Difference and Sum of Two Angles 3.2 Some More Results 3.2 Some Important Conditional Identities 3.7 Range of/(0) = acos 0 + bsin 0 3.8 Transformation Formula 3.8 Formula to Transform the Product into Sum or Difference 3.8 Formulas to Transform the Sum or Difference into Product . 3.11 Trigonometric Ratios of Multiple and Sub-Multiple Angles 3.11 Formulas for Multiple Angles 3.18 Values of Trigonometric Ratios of Typical Angles 3.20 Sum of Sines or Cosines of n Angles in A.P. 3.22 Conditional Identities 3.22 Some Standard Identities in Triangle 3.25 Some Important Results and Their Applications 3.27 Finding Range of Expressions Using Trigonometric Substitution 3.28 Important Inequalities 3.30 Exercises 3.30 Subjective Type 3.31 Single Correct Answer Type 3.36 Multiple Correct Answers Type 3.38 Reasoning Type 3.38 Linked Comprehension Type 3.40 Matrix-Match Type 3.42 Integer Answer Type 3.43 Archives 3.45 Answer Key 4.1-4.29 4. Trigonometric Equations 4.1 Trigonometric Equations 4.1 Solution or Root of a Trigonometric Equation 4.1 Principal Solution of a Trigonometric Equation 4.1 General Solution of a Trigonometric Equation 4.1 General Solution of Some Elementary Equations 4.2 Solving Elementary Equations 4.4 General Solution of Some Standard Equations 4.4 General Solution of Equation sin 0 = sin a 4.7 General Solutions of Equation tan 6 = tan a Contents vii General Solutions of Equation sin20 = sin2aor cos20= cos2a 4.9 General Solutions of Equation tan20= tan2 a 4.9 Solutions of Equations of Form a cos 6 + b sin 6 = c 4.10 Problems Based on Extreme Values of Functions 4.11 Solving Equations Using Graphs 4.13 Inequalities 4.14 Trigonometric Inequations 4.14 Exercises 4.15 Subjective Type 4.15 Single Correct Answer Type 4.16 Multiple Correct Answers Type 4.20 Reasoning Type 4.23 Linked Comprehension Type 4.23 Matrix-Match Type 4.24 Integer Answer Type 4.25 Archives 4.25 Answer Key 4.28 5. Inverse Trigonometric Functions 5.1-5.42 Introduction 5.1 Inverse Functions 5.1 Inverse Trigonometric Functions 5.2 Domain, Range, and Graphs of Inverse Trigonometric Functions 5.2 Functions of the Form j{fi(x')), whereis Trigonometric Function 5.6 Principal Value of the Function/"’(/(x)) 5.6 Relating Different Inverse Trigonometric Functions 5.11 Simplifying Expression Using Trigonometric Substitution 5.12 Relatingy"*(x) with f\-x) and/’(1/x) 5.14 Relating/"1^) with/’1(-x) 5.14 Relatingy"’(x) with/"’(l/x) 5.14 Complementary Angles 5.15 Sum and Difference of Angles in Terms of tan”1 5.18 Multiple Angles in Terms of tan"’x 5.20 Sum and Difference of Angles in Terms of sin"1 and cos"1 5.22 Multiple Angles in Terms of sin" ’x and cos" ’x 5.24 Results 5.24 Results 5.25 Some More Multiple Angle Formulas 5.26 Exercises 5.28 Subjective Type 5.28 Single Correct Answer Type 5.29 Multiple Correct Answers Type 5.35 Reasoning Type 5.36 Linked Comprehension Type 5.37 Matrix-Match Type 5.38 Integer Answer Type 5.39 Archives 5.39 Answer Key 5.42 6. Solutions and Properties of Triangle 6.1-6.51 Standard Symbols 6.1 Sine Rule 6.1 Napier’s Formula 6.2 Cosine Rule . 6.5 Projection Formula 6.8 Half-Angle Formulas 6.9 viii Contents Area of Triangle 6.11 Different Circles and Centers Connected with Triangle 6.13 Circumcircle and Circumcenter 6.13 Distance of the Circumcenter from the Sides 6.13 F Incircle and Incenter 6.15 Orthocenter 6.19 Centroid of Triangle 6.22 Escribed Circles of a Triangle and Their Radii 6.24 Geometry Relating to Excenters 6.26 Distance between Incenter and Excenter 6.26 Distance between Excenters 6.26 Miscellaneous Topics 6.27 m-n Theorem 6.27 Inequality 6.28 Area of Quadrilateral 6.29 Cyclic Quadrilateral 6.29 Regular Polygon 6.30 Solution of Triangles (Ambiguous Cases) 6.31 Exercises 6.34 Subjective Type 6.34 Single Correct Answer Type 6.35 Multiple Correct Answers Type 6.39 Reasoning Type 6.41 Linked Comprehension Type 6.42 Matrix-Match Type 6.45 Integer Answer Type 6.45 Archives 6.46 Answer Key ' .’ S' 6.50 Appendix 1: Logarithm and Antilogarithm Tables A.1-A.6 Appendix 2: Solutions S.1-S.198 Chapter 1 S.l Concept Application Exercises S.l Exercises S.6 Chapter 2 S.19 Concept Application Exercises S.19 Exercises S.23 Chapter 3 S.36 Concept Application Exercises S.36 Exercises S.45 Chapter 4 S.80 Concept Application Exercises S.80 Exercises S.86 Chapter 5 S.112 Concept Application Exercises S.112 Exercises S.l 20 Chapter 6 S.148 Concept Application Exercises S.148 Exercises S.155 Appendix 3: Chapterwise Solved 2014 and 2015 JEE Advanced Questions Q.1-Q.4 Chapterwise Solved 2016 JEE Advanced Questions Q.5-Q.6 Chapterwise Solved 2017 JEE Advanced Questions Q.7 Preface ■< 'T Thile the paper-setting pattern and assessment methodology have been revised many times over and newer criteria devised \/\ / to help develop more aspirant-friendly engineering entrance tests, the need to standardize the selection processes and V V their outcomes at the national level has always been felt. The Joint Entrance Examination (JEE) to India’s prestigious engineering institutions (IITs, NITs, IllTs, ISM, IlSERs, and other engineering colleges) aims to serve as a common national-level engineering entrance test, thereby eliminating the need for aspiring engineers to sit through multiple entrance tests. While the methodology and scope of an engineering entrance test are prone to change, there are two basic objectives that any test needs to serve: 1. The objective to test an aspirant’s caliber, aptitude, and attitude for the engineering field and profession. 2. The need to test an aspirant’s grasp and understanding of the concepts of the subjects of study and their applicability at the grassroots level. Students appearing for various engineering entrance examinations cannot bank solely on conventional shortcut measures to crack the entrance examination. Conventional techniques alone are not enough as most of the questions asked in the examination are based on concepts rather than on just formulae. Hence, it is necessary for students appearing for joint entrance examination to not only gain a thorough knowledge and understanding of the concepts but also develop problem-solving skills to be able to relate their understanding of the subject to real-life applications based on these concepts. This series of books is designed to help students to get an all-round grasp of the subject so as to be able to make its useful appli­ cation in all its contexts. It uses a right mix of fundamental principles and concepts, illustrations which highlight the application of these concepts, and exercises for practice. The objective of each book in this series is to help students develop their problem-solving skills/accuracy, the ability to reach the crux of the matter, and the speed to get answers in limited time. These books feature all types of problems asked in the examination—be it MCQs (one or more than one correct), assertion-reason type, matching column type, comprehension type, or integer type questions. These problems have skillfully been set to help students develop a sound problem­ solving methodology. Not discounting the need for skilled and guided practice, the material in the books has been enriched with a number of fully solved concept application exercises so that every step in learning is ensured for the understanding and application of the subject. The books in this series present a latest topicwise/chapterwise collection of questions asked in JEE Advanced, so that students who have studied and practised concepts thoroughly can have a feel of the type and difficulty of questions asked in the examination. This whole series of books adopts a multi-faceted approach to mastering concepts by including a variety of exercises asked in the examination. A mix of questions helps stimulate and strengthen multi-dimensional problem-solving skills in an aspirant. It is imperative to note that this book would be as profound and useful as you want it to be. Therefore, in order to get maximum benefit from this book, we recommend the following study plan for each chapter. Step 1: Go through the entire opening discussion about the fundamentals and concepts. Step 2: After learning the theory/concept, follow the illustrative examples to get an understanding of the theory/concept. Overall the whole content of the book is an amalgamation of the theme of mathematics with ahead-of-time problems, which equips the students with the knowledge of the field and paves a confident path for them to accomplish success in the JEE. With best wishes! G. Tewani