ALL CHAPTERS COVERED
, Contents
Preface vii
Chapter 1 The Ratios and Identities 1.1–1.87
Introduction 1.1
Measurement of Angles 1.1
Trigonometrical Ratios 1.2
Limits of the values of Trigonometrical Functions 1.2
Sign of Trigonometric Ratios 1.2
T-ratios of the Angle (–q ), in Terms of q, for All Values of q 1.3
T-ratios of the Different Angles in Terms of q, for All Values of q 1.3
Graph of Trigonometric Functions 1.3
T-ratios of Compound Angles 1.4
Some Important Deductions 1.4
Transformation Formulae 1.5
Multiple Angles 1.6
Some Important Deductions 1.6
The Maximum and Minimum Values of 1.7
Sub–Multiple Angles 1.8
Conditional Trigonometrical Identities 1.11
Trigonometrical Series 1.12
Exercises 1.12
Answers 1.30
Hints and Solutions 1.30
Chapter 2 Trigonometric Equations 2.1–2.53
Definition 2.1
Solution of a Trigonometric Equation 2.1
General solution of Trigonometric Equations 2.1
A Trigonometric Equation is of the Form 2.1
Principal Value 2.1
Method to Find Out the Principal Value 2.2
Solutions in Case of Two Equations are Given 2.2
Some Important Remarks to Keep in Mind while Solving a Trigonometric Equation 2.2
Types of Trigonometric Equations: 2.2
Exercises 2.3
Answers 2.17
Hints and Solutions 2.18
,x Contents
Chapter 3 Trigonometric In-Equation 3.1–3.15
Trigonometric Inequalities 3.1
Exercises 3.2
Answers 3.4
Hints and Solutions 3.5
Chapter 4 Inverse Trigonometric Functions 4.1–4.84
Inverse Function 4.1
Introduction to Inverse Function 4.1
Inverse Trigonometric Functions 4.1
Graphs of Inverse Trigonometric Functions 4.2
Constant Property 4.4
Conversion of Inverse Trigonometric Functions 4.4
Composition of Trigonometric Functions and its Inverse 4.5
Composition of Inverse Trigonometric Functions and Trigonometric Functions 4.5
Sum of Angles 4.6
Multiple Angles 4.7
More Multiple Angles 4.7
Exercises 4.8
Answers 4.24
Hints and Solutions 4.25
Chapter 5 Properties of Triangles 5.1–5.92
Properties of Triangles 5.1
Introduction 5.1
Sine Rule 5.1
Cosine Rule 5.1
Projection Formulae 5.1
Napier’s Analogy (Law of Tangents) 5.1
Half-Angled Formulae 5.1
Area of Triangle 5.2
m-n Theorem 5.2
Radii of Circle Connected with a Triangle 5.3
Inscribed Circle and its Radius 5.3
Escribed Circle of a Triangle and Their Radii 5.3
Regular Polygon 5.4
Orthocentre and Pedal Triangle of Any Triangle 5.4
Distance between the Circumcentre and Orthocentre 5.5
Distance Between the Circumcentre and the Incentre 5.6
Distance between the Circumcentre and Centroid 5.6
Distance Between the Incentre and Orthocentre 5.6
Excentral Triangle 5.7
Quadrilateral 5.9
Exercises 5.12
Answers 5.30
Hints and Solutions 5.30
, The Ratios and Identities 1.1
C H A P T E R
1 The Ratios and Identities
–ve angle
CONCEPT BOOSTER O C
q
1.1 INTRODUCTION
Trigonometry (from Greek trigonon “triangle” + metron
D
“measure”) is a branch of mathematics that studies triangles
and the relationships of the lengths of their sides and the an- 3. System of measuring angles
gles between those sides. There are three systems of measuring angles such as
Trigonometry defines the trigonometric functions, which (i) Sexagesimal system
describe those relationships and have applicability to cyclical (ii) Centisimal system
phenomena, such as waves. This field, evolved during the (iii) Circular system
third century BC as a branch of geometry, was used exten- 4. In sexagesimal system, we have
sively for astronomical studies. It is also the foundation of the 1 right angle = 90o
practical art of surveying. 1o = 60¢
Trigonometry basics are often taught in school either as a 1¢ = 60≤
separate course or as part of a pre-calculus course. The trigo- 5. In centasimal system, we have
nometric functions are pervasive in parts of pure mathemat- 1 right angle = 100g
ics and applied mathematics such as Fourier analysis and the 1g = 100¢
wave equation, which are in turn essential to many branches 1¢ = 100≤
of science and technology. 6. In circular system, the unit of measurement is radian
Radian: One radian is the measure of an angle sub-
1.2 MEASUREMENT OF ANGLES tended at the centre of a circle by an arc of length equal
1. Angle: The measurement of an angle is the amount of to the radius of the circle.
rotation from the initial side to the terminal side. Here, –AOB = 1 radian = 1e.
2. Sense of an Angle: The sense of an angle is +ve or B
–ve based on whether the initial side rotates in the anti-
clock-wise or clockwise direction to get the terminal 1°
O A
side.
B
Notes
O A
(i) When an angle is expressed in radians, the word ra-
dian is omitted.
Positive angle
, Contents
Preface vii
Chapter 1 The Ratios and Identities 1.1–1.87
Introduction 1.1
Measurement of Angles 1.1
Trigonometrical Ratios 1.2
Limits of the values of Trigonometrical Functions 1.2
Sign of Trigonometric Ratios 1.2
T-ratios of the Angle (–q ), in Terms of q, for All Values of q 1.3
T-ratios of the Different Angles in Terms of q, for All Values of q 1.3
Graph of Trigonometric Functions 1.3
T-ratios of Compound Angles 1.4
Some Important Deductions 1.4
Transformation Formulae 1.5
Multiple Angles 1.6
Some Important Deductions 1.6
The Maximum and Minimum Values of 1.7
Sub–Multiple Angles 1.8
Conditional Trigonometrical Identities 1.11
Trigonometrical Series 1.12
Exercises 1.12
Answers 1.30
Hints and Solutions 1.30
Chapter 2 Trigonometric Equations 2.1–2.53
Definition 2.1
Solution of a Trigonometric Equation 2.1
General solution of Trigonometric Equations 2.1
A Trigonometric Equation is of the Form 2.1
Principal Value 2.1
Method to Find Out the Principal Value 2.2
Solutions in Case of Two Equations are Given 2.2
Some Important Remarks to Keep in Mind while Solving a Trigonometric Equation 2.2
Types of Trigonometric Equations: 2.2
Exercises 2.3
Answers 2.17
Hints and Solutions 2.18
,x Contents
Chapter 3 Trigonometric In-Equation 3.1–3.15
Trigonometric Inequalities 3.1
Exercises 3.2
Answers 3.4
Hints and Solutions 3.5
Chapter 4 Inverse Trigonometric Functions 4.1–4.84
Inverse Function 4.1
Introduction to Inverse Function 4.1
Inverse Trigonometric Functions 4.1
Graphs of Inverse Trigonometric Functions 4.2
Constant Property 4.4
Conversion of Inverse Trigonometric Functions 4.4
Composition of Trigonometric Functions and its Inverse 4.5
Composition of Inverse Trigonometric Functions and Trigonometric Functions 4.5
Sum of Angles 4.6
Multiple Angles 4.7
More Multiple Angles 4.7
Exercises 4.8
Answers 4.24
Hints and Solutions 4.25
Chapter 5 Properties of Triangles 5.1–5.92
Properties of Triangles 5.1
Introduction 5.1
Sine Rule 5.1
Cosine Rule 5.1
Projection Formulae 5.1
Napier’s Analogy (Law of Tangents) 5.1
Half-Angled Formulae 5.1
Area of Triangle 5.2
m-n Theorem 5.2
Radii of Circle Connected with a Triangle 5.3
Inscribed Circle and its Radius 5.3
Escribed Circle of a Triangle and Their Radii 5.3
Regular Polygon 5.4
Orthocentre and Pedal Triangle of Any Triangle 5.4
Distance between the Circumcentre and Orthocentre 5.5
Distance Between the Circumcentre and the Incentre 5.6
Distance between the Circumcentre and Centroid 5.6
Distance Between the Incentre and Orthocentre 5.6
Excentral Triangle 5.7
Quadrilateral 5.9
Exercises 5.12
Answers 5.30
Hints and Solutions 5.30
, The Ratios and Identities 1.1
C H A P T E R
1 The Ratios and Identities
–ve angle
CONCEPT BOOSTER O C
q
1.1 INTRODUCTION
Trigonometry (from Greek trigonon “triangle” + metron
D
“measure”) is a branch of mathematics that studies triangles
and the relationships of the lengths of their sides and the an- 3. System of measuring angles
gles between those sides. There are three systems of measuring angles such as
Trigonometry defines the trigonometric functions, which (i) Sexagesimal system
describe those relationships and have applicability to cyclical (ii) Centisimal system
phenomena, such as waves. This field, evolved during the (iii) Circular system
third century BC as a branch of geometry, was used exten- 4. In sexagesimal system, we have
sively for astronomical studies. It is also the foundation of the 1 right angle = 90o
practical art of surveying. 1o = 60¢
Trigonometry basics are often taught in school either as a 1¢ = 60≤
separate course or as part of a pre-calculus course. The trigo- 5. In centasimal system, we have
nometric functions are pervasive in parts of pure mathemat- 1 right angle = 100g
ics and applied mathematics such as Fourier analysis and the 1g = 100¢
wave equation, which are in turn essential to many branches 1¢ = 100≤
of science and technology. 6. In circular system, the unit of measurement is radian
Radian: One radian is the measure of an angle sub-
1.2 MEASUREMENT OF ANGLES tended at the centre of a circle by an arc of length equal
1. Angle: The measurement of an angle is the amount of to the radius of the circle.
rotation from the initial side to the terminal side. Here, –AOB = 1 radian = 1e.
2. Sense of an Angle: The sense of an angle is +ve or B
–ve based on whether the initial side rotates in the anti-
clock-wise or clockwise direction to get the terminal 1°
O A
side.
B
Notes
O A
(i) When an angle is expressed in radians, the word ra-
dian is omitted.
Positive angle
