Comprehensive Trigonometry 13 February 2017 02:12:47 PM ABOUT THE AUTHOR REJAUL MAKSHUD (R M) Post graduated from Calcutta University in PURE MATHEMATICS having teaching experience of 15+ years in many prestigious institution of India. Presently, he trains IIT Aspirants at RACE IIT ACADEMY, Jamshedpur, playing a role of DIRECTOR cum HOD OF MATHEMATICS. Rejaul Makshud McGraw Hill Education (India) Private Limited CHENNAI Chennai New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto Published by McGraw Hill Education (India) Private Limited, 444/1, Sri Ekambara Naicker Industrial Estate, Alapakkam, Porur, Chennai-600116 Comprehensive Trigonometry with Challenging Problems & Solutions for Jee Main and Advanced Copyright © 2017, McGraw Hill Education (India) Private Limited. No part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication. This edition can be exported from India only by the publishers, McGraw Hill Education (India) Private Limited. ISBN (13): 978-93-5260-510-1 ISBN (10): 93-5260-510-1 Information contained in this work has been obtained by McGraw Hill Education (India), from sources believed to be reliable. However, neither McGraw Hill Education (India) nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill Education (India) nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw Hill Education (India) and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at Script Makers, 19, A1-B, DDA Market, Paschim Vihar, New Delhi, India, and text and cover printed at Cover Designer: Creative Designer visit us at: www.mheducation.co.in PREFACE This text book on TRIGONOMETRY with Problems & Solutions for JEE Main and Advanced is meant for aspirants preparing for the entrance examination of different technical institutions, especially NIT/IIT/BITSAT/IISc. In writing this book I have drawn heavily from my long teaching experience at National Level Institutes. After many years of teaching I have realised the need of designing a book that will help the readers to build their base, improve their level of mathematical concepts and enjoy the subject. This book is designed keeping in view the new pattern of questions asked in JEE Main and Advanced Exams. It has eight chapters. Each chapter has a large number of worked out problems and exercise based problems as given below: Level – I: Questions based on Fundamentals Level – II: Mixed Problems (Objective Type Questions) Level – III: Problems for JEE Advanced Exam (0.......9): Integer type Questions Passages: Comprehensive link passages Matching: Match Matrix Reasoning: Assertion and Reasoning Previous years papers: Questions asked in past IIT-JEE Exams become easy. So please don’t jump to exercise problems before you go through the Concept Booster and the objectives. Once you are arranged in a manner that they gradually require advanced thinking. tackle any type of problem easily and skilfully. My special thanks goes to Mr. M.P. Singh (IISc. Bangalore), Mr. Manoj Kumar (IIT, Delhi), Mr. Nazre Hussain (B. Tech.), Dr. Syed Kashan Ali (MBBS) and Mr. Shahid Iqbal, who have helped, inspired and motivated me to accomplish this task. As a matter of fact, teaching being the best learning process, I must thank all my students who inspired me most for writing this book. I would like to convey my affectionate thanks to my wife, who helped me immensely and my children who bore with patience my neglect during the period I remained devoted to this book. I also convey my sincere thanks to Mr Biswajit of McGraw Hill Education for publishing this book in such a beautiful format. and to all my learned teachers— Mr. Swapan Halder, Mr. Jadunandan Mishra, Mr. Mahadev Roy and Mr. Dilip Bhattacharya, who instilled the value of quality teaching in me. I have tried my best to keep this book error-free. I shall be grateful to the readers for their constructive suggestions toward the improvement of the book. REJAUL MAKSHUD M. Sc. (Calcutta University, Kolkata) Dedicated to My Beloved Mom and Dad Contents Preface v 1. The Ratios and Identities 1-99 1.1 Introduction 1 1.2 Application of Trigonometry 1 1.3 Trigonometrical Functions 1 1.4 Measurement of Angles 2 1.5 Some Solved Examples 2 Exercise 1 4 1.6 Trigonometrical Ratios 5 1.7 Limits of the Values of Trigonometrical Functions 5 1.8 Some Solved Examples 5 Exercise 2 10 1.9 Measurement of the Angles of Different T-ratios 11 1.10 Some Solved Examples 13 Exercise 3 15 Exercise 4 17 1.11 T-ratios of Compound Angles 17 1.12 Some Important Deductions 19 1.13 Some Solved Examples 20 Exercise 5 22 1.14 Transformation Formulae 23 Exercise 6 25 1.15 Multiple Angles 26 1.16 Some Important Deductions 27 Exercise 7 33 1.17 T he Maximum and Minimum Values of f (x) = a cos x + b sin x + c 34 Exercise 8 36 1.18 Sub–Multiple Angles 37 1.19 Some Solved Examples 40 Exercise 9 43 1.20 Conditional Trigonometrical Identities 43 1.21 Some Solved Examples 44 Exercise 10 47 1.22 Trigonometrical Series 47 1.23 Different Types of the Summation of a Trigonometrical Series 48 Exercise 11 48 Exercise 12 49 Exercise 13 50 Problems for JEE Advanced Exam 51 Level I (Problems Based on Fundamentals) 65 Level II (Mixed Problems) 67 Level III (Tougher Problems for JEE Advanced) 69 Integer Type Questions 70 Link Comprehension Type (For JEE Advanced Exam Only) 71 Match Matrix 72 Assertion & Reason 74 (Questions Asked in Past IIT-JEE Exams) 74 Answers 77 Hints and Solutions 79 Contents viii Integer Type Questions 89 Questions asked in IIT-JEE Exams 91 2. Graphs of Trigonometric Functions 100-118 2.1 Introduction 100 2.2 Characteristics of co-sine Function 109 2.3 Characteristics of Tangent Function 111 2.4 Characteristics of co-tangent Function 112 2.5 Characteristics of co-secant Function 112 2.6 Characteristics of Secant Function: 113 Level I (Questions Based on Fundamentals) 115 Level II (For JEE Main Exam Only) 116 Level III (For JEE Advanced Exam Only ) 117 Answers 118 3. The Trigonometric Equation 119-183 3.2 Solution of a Trigonometric Equation 119 3.3 General solution of Trigonometric Equations 119 3.4 Ranges of Trigonometric Functions 119 3.5 Some Solved Examples 120 Exercise 1 121 3.6 A Trigonometric Equation is of the Form 122 Exercise 2 123 3.7 Principal Value 123 3.8 Method to Find Out the Principal Value 123 Exercise 3 124 3.9 Solutions in Case of Two Equations are Given: 124 Exercise 4 127 3.10 Some Important Remarks to Keep in Mind While Solving a Trigonometric Equation 128 3.11 Types of Trigonometric Equations 128 Exercise 5 128 Exercise 6 129 Exercise 7 129 Exercise 8 130 Exercise 9 130 Exercise 10 131 Exercise 11 131 Exercise 12 132 Exercise 13 132 Exercise 14 133 Level I (Questions Based on Fundamentals) 140 Level II (Mixed Problesms) 141 Level III (Problems for JEE Advanced Exam) 144 Integer Type Questions 145 Linked Comprehension Type (For JEE Advanced Exams Only) 146 Match Matrix 147 Assertion and Reason 149 Questions Asked in Past IIT-JEE exams 150 Answers 151 Hints and Solutions 156 Level III 170 Integer Type Questions 175 Past IIT-JEE Questions 177 4. Trigonometric In-Equation 184-195 4.1 Trigonometric Inequalities 184 Type - I: An inequation is of the form sin x > k. 184 Contents ix Type - II: An in-equation is of the form sin x < k. 185 Type - III: An in-equation is of the form cos x > k. 185 Type - IV: An in-equation is of the form cos x < k. 186 Type - V: An in-equation is of the form tan x > k. 186 Type -VI: An inequation is of the form tan x < k. 187 4.2 Some Solved Examples 187 Comprehensive Link Passage 192 Answers 194 5. Logarithm 196-231 5.1 Introduction 196 5.2 Some Solved Examples 198 5.3 Logarithmic Equation 201 5.4 Logarithmic Inequation 204 Problems for JEE Advanced Exam 207 Comprehensive Link Passages 218 Match Matrix 219 Integer Type Questions 219 Questions Asked in Past IIT-JEE Exams 220 Answers 220 Hints and Solutions 222 Integer Type Questions 226 Questions Asked In Past IIT-JEE Exams 228 6. Inverse Trigonometric Function 232-329 6.1 Introduction to Inverse Function 232 6.2 Some Solved Examples 233 Exercise 1 235 6.3 Inverse Trigonometric Functions 235 6.4 Graphs of Inverse Trigonometric Functions 236 Characteristics of arc sine function 236 Characteristics of arc cosine function: 236 Characteristics of arc tangent function 236 Characteristics of arc co-tangent function: 237 Characteristics of arc co-secant function: 237 Characteristics of arc secant function: 237 Exercise 2 240 6.5 Constant Property 241 Exercise 3 242 6.5 Conversion of Inverse Trigonometric Functions 243 Exercise 4 245 6.6 Composition of Trigonometric functions and its Inverse 245 6.7 Composition of Inverse Trigono Metric Functions and Trigonometric Functions 247 Exercise 5 251 6.9 Sum of Angles 252 Exercise 6 260 6.10 Multiple Angles 261 Exercise 7 263 6.11 More Multiple Angles 263 Exercise 8 265 Problems for JEE Main Exam 266 Questions with Solutions of Past JEE Main Exams 273 Problems for JEE Advanced Exam 275 Level II (Mixed Problems) 289 Level III (Problems for JEE Advanced) 292 Integer Type Questions 294 Comprehensive link passage 295 Match-Matrix 296