Trigonometry : Course in Mathematics for the IIT-JEE and Other Engineering Entrance Examinations

272 Pages·2011·13.644 MB·English

by  K. R Choubey, Ravikant Choubey,  Chandrakant Choubey,

COURSE IN MATHEMATICS
( )
FOR IIT JEE AND OTHER ENGINEERING ENTRANCE EXAMINATIONS
T
RIGONOMETRY
K.R. C
HOUBEY
R C
AVIKANT HOUBEY
C C
HANDRAKANT HOUBEY
Chandigarh • Delhi • Chennai
The aim of this publication is to supply information taken from sources believed to be valid and reliable. This
is not an attempt to render any type of professional advice or analysis, nor is it to be treated as such. While
much care has been taken to ensure the veracity and currency of the information presented within, neither
the publisher nor its authors bear any responsibility for any damage arising from inadvertent omissions,
negligence or inaccuracies (typographical or factual) that may have found their way into this book.
Copyright © 2011 Dorling Kindersley (India) Pvt. Ltd
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C
ONTENTS
Preface v
PART A TRIGONOMETRIC FUNCTIONS AND IDENTITIES
Lecture 1 Measurement of Angles A.3–A.16
Lecture 2 Trigonometric Functions A.17–A.33
Lecture 3 Quadrants A.35–A.48
Lecture 4 Trigonometric Functions of Compound Angles A.49–A.59
Lecture 5 Sum and Difference of Two Angles A.61–A.75
Lecture 6 Trigonometric Ratios of Multiple and Sub-Multiple Angles A.77–A.92
Lecture 7 Identities of Trigonometric Functions A.93–A.103
Lecture 8 Graphs of Trigonometric Functions A.105–A.111
PART B TRIGONOMETRIC EQUATIONS AND INEQUALITIES
Lecture 1 Trigonometric Equations B.3–B.23
PART C PROPERTIES AND SOLUTIONS OF TRIANGLES
Lecture 1 Properties of Triangles-I C.3–C.20
Lecture 2 Properties of Triangles-II C.21–C.46
PART D INVERSE TRIGONOMETRIC FUNCTIONS
Lecture 1 Inverse Trigonometric Function D.3–D.22
PART E HEIGHTS AND DISTANCES
Lecture 1 Heights and Distances E.3–E.29
PART F TEST YOUR SKILLS
Lecture 1 Test Your Skills F.3–F.31
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P
REFACE
When a new book is written on a well known subject like Trigonometry for class XI/XII Academics/
AIEEE/IIT/State engineering entrance exams and NDA, several questions arise like—why, what, how and
for whom? What is new in it? How is it different from other books? For whom is it meant? The answers
to these questions are often not mutually exclusive. Neither are they entirely satisfactory except perhaps
to the authors. We are certainly not under the illusion that there are no good books. There are many good
books available in the market.
However, none of them caters specifically to the needs of students. Students find it difficult to solve
most of the problems of any of the books in the absence of proper planning. This inspired us to write
this book Trigonometry, to address the requirements of students of class XI/XII CBSE and State Board
Academics. In this book, we have tried to give a connected and simple account of the subject. It gives a
detailed, lecture wise description of basic concepts with many numerical problems and innovative tricks
and tips. Theory and problems have been designed in such a way that the students can themselves pursue
the subject. We have also tried to keep this book self contained. In each lecture all relevant concepts,
prerequisites and definitions have been discussed in a lucid manner and also explained with suitable illus-
trated examples including tests.
Due care has been taken regarding the Board (CBSE/ State) examination need of students and
nearly 100 per cent articles and problems set in various examinations including the IIT-JEE have been
included.
The presentation of the subject matter is lecturewise, intelligent and systematic, the style is lucid and
rational, and the approach is comprehensible with emphasis on improving speed and accuracy. The basic
motive is to attract students towards the study of mathematics by making it simple, easy and interesting
and on a day-to-day basis. The instructions and method for grasping the lectures are clearly outlined topic
wise. The presentation of each lecture is planned for better experiential learning of mathematics which is
as follows:
1. Basic Concepts: Lecture Wise
2. Solved Subjective Problems (XII Board (C.B.S.E./State): For Better Understanding and Concept
Building of the Topic.
3. Unsolved Subjective Problems (XII Board (C.B.S.E./State): To Grasp the Lecture Solve These
Problems.
4. Solved Objective Problems: Helping Hand.
5. Objective Problem: Important Questions with Solutions.
6. Unsolved Objective Problems (Identical Problems for Practice)
For Improving Speed with Accuracy.
vi Preface
7. Worksheet: To Check Preparation Level
8. Assertion-Reason Problems : Topic Wise Important Questions and Solutions with Reasoning
9. Mental Preparation Test: 01
10. Mental Preparation Test: 02
11. Topic Wise Warm Up Test: 01: Objective Test
12. Topic Wise Warm Up Test: 02: Objective Test
13. Objective Question Bank Topic Wise: Solve These to Master.
This book will serve the need of the students of class XI/XII board, NDA, AIEEE and SLEEE (state level
engineering entrance exam) and IIT-JEE. We suggest each student to attempt as many exercises as possible
without looking up the solutions. However, one should not feel discouraged if one needs frequent help of the
solutions as there are many questions that are either tough or lengthy. Students should not get frustrated if
they fail to understand some of the solutions in the first attempt. Instead they should go back to the beginning
of the solution and try to figure out what is being done At the end of every topic, some harder problems with
100 per cent solutions and Question Bank are also given for better understanding of the subject.
There is no end and limit to the improvement of the book. So, suggestions for improving the book are
always welcome.
We thank our publisher, Pearson Education for their support and guidance in completing the project
in record time.
K.R. CHOUBEY
RAVIKANT CHOUBEY
CHANDRAKANT CHOUBEY
P A
A R T
Trigonometric
Functions and
Identities
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LECTURE
11
Measurement
of Angles
2.1 Types of Angles
1. MEASUREMENT OF AN ANGLE
An angle  is defi ned to be
The measurement of an angle is the amount of rota-
tion from the initial side to the terminal side. (i) an acute angel, if 0°   < 90°
(ii) an obtuse angle, if 90° <  < 180°
B
(iii) a refl ex angle, if 180 <  < 360
e
nal sid ((ivv)) aa rsitgrahitg ahnt galneg, lief , i f=  9 =0 °180°
mi
Ter
2.2 Coterminal Angles
Vertex O A Two angles having different measures but same
Initial side
initial and terminal sides are said to be coterminal
angles. e.g., (i) Angles with measure –30 and 330
are coterminal. (ii) Angles with measure 45 and
2. SENSE OF AN ANGLE
–315 are coterminal.
The sense of angle is positive or negative according
as the initial side rotates in anti-clockwise or clock-
3. SYSTEMS OF MEASURING ANGLES
wise direction to get the terminal side.
There are three systems of measuring angles:
O A
–(cid:1) (i) Sexagesimal system
(ii) Centesimal system
(iii) Circular system
3.1 Sexagesimal System
Negative angle
In sexagesimal system, we have 1 right angle =
B
90 degrees (90), 1 = 60 minutes = (60′) 1′ = 60
seconds (= 60″)
Some of the angles whose measures are 360,
180°, 270°, 420° and –30° are shown in fi gures
below:
(cid:1) 360° A
Vertex O O
Positive angle B