Comprehensive Trigonometry with Challenging Problems and Solutions JEE Main and Advanced by Rejaul Makshud

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Comprehensive Trigonometry with Challenging Problems and Solutions JEE Main and Advanced 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 The 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

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

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

Assertion and Reason 297

Questions Asked in Past IIT-JEE Exams 298

Answers 300

Hints and Solution 304

Integer Type Questions 316

Question Asked in Part IIT-JEE Exams 319

Questions with Solutions of Past IIT-JEE Exams from 1981 to 2015 323

7. Properties of Triangles 330-440

7.1 Introduction 330

Exercise 1 333

7.2 Cosine Rule 333

Exercise 2 337

7.3 Projection Formulae 337

Exercise 3 338

7.4 Napier’s Analogy (Law of Tangents) 338

Exercise 4 339

7.5 Half-Angled Formulae 340

Exercise 5 344

7.6 Area of triangle 344

Exercise 6 349

7.6 Radii of Circle Connected with a Triangle 349

Exercise 7 351

7.8 Inscribed Circle and its Radius 352

Exercise 8 355

7.9 Escribed Circle of a Triangle and their Radii 356

Exercise 9 360

7.10 Regular Polygon 361

Exercise 12 365

7.11 Orthocentre and Pedal Triangle of any Triangle 366

7.12 Distance between the Circumcentre and Orthocentre 367

7.13 Distance between the circumcentre and the Incentre 368

7.14 Distance between the Circumcentre and Centroid 368

7.15 Distance between the Incentre and Orthocentre 368

7.16 Excentral Triangle 369

7.17 Quadrilateral 372

Problems for JEE Advanced Exam 375

Level I (Questions Based on Fundamentals) 396

Level II Mixed Problems 398

Level III (For JEE Advanced Exam Only) 401

Integer Type Questions 403

(Questions Asked in IIT-JEE Exams with their Solutions) 404

Comprehensive Link Passage (For JEE Advanced Exam Only) 408

Match Matrix (For JEE Advanced Exam Only) 410

Assertion and Reason 412

Answers 413

Level III (Problems for JEE Advanced) 414

Integer Type Questions 424

Hints & Solutions of Past IIT-JEE Questions 426

8. The Heights and Distances 441-448

8.1 Introduction 441

8.2 Angle of Elevation, Angle of Depression and the Line of Sight 441

8.3 Bearing of a Line 441

8.4 Some Solved Examples 442

Level I (Problems Based on Fundamentals) 446

Level II (Mixed Problems) 447

Level III (Problems for JEE Main) 447

Answers 448