Page 1
vA mathematician knows how to solve a problem,
he can not solve it. – MILNE v
3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engineers and others. Currently, trigonometry is
used in many areas such as the science of seismology,
designing electric circuits, describing the state of an atom,
predicting the heights of tides in the ocean, analysing a
musical tone and in many other areas.
In earlier classes, we have studied the trigonometric
ratios of acute angles as the ratio of the sides of a right
angled triangle. We have also studied the trigonometric identities and application of
trigonometric ratios in solving the problems related to heights and distances. In this
Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions
and study their properties.
3.2 Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is
Chapter 3
TRIGONOMETRIC FUNCTIONS
Arya Bhatt
(476-550)
Fig 3.1
V er tex
2024-25
Page 2
vA mathematician knows how to solve a problem,
he can not solve it. – MILNE v
3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engineers and others. Currently, trigonometry is
used in many areas such as the science of seismology,
designing electric circuits, describing the state of an atom,
predicting the heights of tides in the ocean, analysing a
musical tone and in many other areas.
In earlier classes, we have studied the trigonometric
ratios of acute angles as the ratio of the sides of a right
angled triangle. We have also studied the trigonometric identities and application of
trigonometric ratios in solving the problems related to heights and distances. In this
Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions
and study their properties.
3.2 Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is
Chapter 3
TRIGONOMETRIC FUNCTIONS
Arya Bhatt
(476-550)
Fig 3.1
V er tex
2024-25
44 MATHEMATICS
called the initial side and the final position of the ray after rotation is called the
terminal side of the angle. The point of rotation is called the vertex. If the direction of
rotation is anticlockwise, the angle is said to be positive and if the direction of rotation
is clockwise, then the angle is negative (Fig 3.1).
The measure of an angle is the amount of
rotation performed to get the terminal side from
the initial side. There are several units for
measuring angles. The definition of an angle
suggests a unit, viz. one complete revolution from the position of the initial side as
indicated in Fig 3.2.
This is often convenient for large angles. For example, we can say that a rapidly
spinning wheel is making an angle of say 15 revolution per second. We shall describe
two other units of measurement of an angle which are most commonly used, viz.
degree measure and radian measure.
3.2.1 Degree measure If a rotation from the initial side to terminal side is
th
1
360
? ?
? ?
? ?
of
a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is
divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is
called a minute, written as 1', and one sixtieth of a minute is called a second, written as 1?.
Thus, 1° = 60', 1' = 60?
Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are
shown in Fig 3.3.
Fig 3.2
Fig 3.3
2024-25
Page 3
vA mathematician knows how to solve a problem,
he can not solve it. – MILNE v
3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engineers and others. Currently, trigonometry is
used in many areas such as the science of seismology,
designing electric circuits, describing the state of an atom,
predicting the heights of tides in the ocean, analysing a
musical tone and in many other areas.
In earlier classes, we have studied the trigonometric
ratios of acute angles as the ratio of the sides of a right
angled triangle. We have also studied the trigonometric identities and application of
trigonometric ratios in solving the problems related to heights and distances. In this
Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions
and study their properties.
3.2 Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is
Chapter 3
TRIGONOMETRIC FUNCTIONS
Arya Bhatt
(476-550)
Fig 3.1
V er tex
2024-25
44 MATHEMATICS
called the initial side and the final position of the ray after rotation is called the
terminal side of the angle. The point of rotation is called the vertex. If the direction of
rotation is anticlockwise, the angle is said to be positive and if the direction of rotation
is clockwise, then the angle is negative (Fig 3.1).
The measure of an angle is the amount of
rotation performed to get the terminal side from
the initial side. There are several units for
measuring angles. The definition of an angle
suggests a unit, viz. one complete revolution from the position of the initial side as
indicated in Fig 3.2.
This is often convenient for large angles. For example, we can say that a rapidly
spinning wheel is making an angle of say 15 revolution per second. We shall describe
two other units of measurement of an angle which are most commonly used, viz.
degree measure and radian measure.
3.2.1 Degree measure If a rotation from the initial side to terminal side is
th
1
360
? ?
? ?
? ?
of
a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is
divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is
called a minute, written as 1', and one sixtieth of a minute is called a second, written as 1?.
Thus, 1° = 60', 1' = 60?
Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are
shown in Fig 3.3.
Fig 3.2
Fig 3.3
2024-25
TRIGONOMETRIC FUNCTIONS 45
3.2.2 Radian measure There is another unit for measurement of an angle, called
the radian measure. Angle subtended at the centre by an arc of length 1 unit in a
unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the Fig
3.4(i) to (iv), OA is the initial side and OB is the terminal side. The figures show the
angles whose measures are 1 radian, –1 radian, 1
1
2
radian and –1
1
2
radian.
(i)
(ii)
(iii)
Fig 3.4 (i) to (iv)
(iv)
We know that the circumference of a circle of radius 1 unit is 2p. Thus, one
complete revolution of the initial side subtends an angle of 2p radian.
More generally, in a circle of radius r, an arc of length r will subtend an angle of
1 radian. It is well-known that equal arcs of a circle subtend equal angle at the centre.
Since in a circle of radius r, an arc of length r subtends an angle whose measure is 1
radian, an arc of length l will subtend an angle whose measure is
l
r
radian. Thus, if in
a circle of radius r, an arc of length l subtends an angle ? radian at the centre, we have
? =
l
r
or l = r ?.
2024-25
Page 4
vA mathematician knows how to solve a problem,
he can not solve it. – MILNE v
3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engineers and others. Currently, trigonometry is
used in many areas such as the science of seismology,
designing electric circuits, describing the state of an atom,
predicting the heights of tides in the ocean, analysing a
musical tone and in many other areas.
In earlier classes, we have studied the trigonometric
ratios of acute angles as the ratio of the sides of a right
angled triangle. We have also studied the trigonometric identities and application of
trigonometric ratios in solving the problems related to heights and distances. In this
Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions
and study their properties.
3.2 Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is
Chapter 3
TRIGONOMETRIC FUNCTIONS
Arya Bhatt
(476-550)
Fig 3.1
V er tex
2024-25
44 MATHEMATICS
called the initial side and the final position of the ray after rotation is called the
terminal side of the angle. The point of rotation is called the vertex. If the direction of
rotation is anticlockwise, the angle is said to be positive and if the direction of rotation
is clockwise, then the angle is negative (Fig 3.1).
The measure of an angle is the amount of
rotation performed to get the terminal side from
the initial side. There are several units for
measuring angles. The definition of an angle
suggests a unit, viz. one complete revolution from the position of the initial side as
indicated in Fig 3.2.
This is often convenient for large angles. For example, we can say that a rapidly
spinning wheel is making an angle of say 15 revolution per second. We shall describe
two other units of measurement of an angle which are most commonly used, viz.
degree measure and radian measure.
3.2.1 Degree measure If a rotation from the initial side to terminal side is
th
1
360
? ?
? ?
? ?
of
a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is
divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is
called a minute, written as 1', and one sixtieth of a minute is called a second, written as 1?.
Thus, 1° = 60', 1' = 60?
Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are
shown in Fig 3.3.
Fig 3.2
Fig 3.3
2024-25
TRIGONOMETRIC FUNCTIONS 45
3.2.2 Radian measure There is another unit for measurement of an angle, called
the radian measure. Angle subtended at the centre by an arc of length 1 unit in a
unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the Fig
3.4(i) to (iv), OA is the initial side and OB is the terminal side. The figures show the
angles whose measures are 1 radian, –1 radian, 1
1
2
radian and –1
1
2
radian.
(i)
(ii)
(iii)
Fig 3.4 (i) to (iv)
(iv)
We know that the circumference of a circle of radius 1 unit is 2p. Thus, one
complete revolution of the initial side subtends an angle of 2p radian.
More generally, in a circle of radius r, an arc of length r will subtend an angle of
1 radian. It is well-known that equal arcs of a circle subtend equal angle at the centre.
Since in a circle of radius r, an arc of length r subtends an angle whose measure is 1
radian, an arc of length l will subtend an angle whose measure is
l
r
radian. Thus, if in
a circle of radius r, an arc of length l subtends an angle ? radian at the centre, we have
? =
l
r
or l = r ?.
2024-25
46 MATHEMATICS
3.2.3 Relation between radian and real numbers
Consider the unit circle with centre O. Let A be any point
on the circle. Consider OA as initial side of an angle.
Then the length of an arc of the circle will give the radian
measure of the angle which the arc will subtend at the
centre of the circle. Consider the line PAQ which is
tangent to the circle at A. Let the point A represent the
real number zero, AP represents positive real number and
AQ represents negative real numbers (Fig 3.5). If we
rope the line AP in the anticlockwise direction along the
circle, and AQ in the clockwise direction, then every real
number will correspond to a radian measure and
conversely. Thus, radian measures and real numbers can
be considered as one and the same.
3.2.4 Relation between degree and radian Since a circle subtends at the centre
an angle whose radian measure is 2p and its degree measure is 360°, it follows that
2p radian = 360° or p radian = 180°
The above relation enables us to express a radian measure in terms of degree
measure and a degree measure in terms of radian measure. Using approximate value
of p as
22
7
, we have
1 radian =
180
p
°
= 57° 16' approximately.
Also 1° =
p
180
radian = 0.01746 radian approximately.
The relation between degree measures and radian measure of some common angles
are given in the following table:
A
O
1
P
1
2
-1
-2
Q
0
Fig 3.5
Degree 30° 45° 60° 90° 180° 270° 360°
Radian
p
6
p
4
p
3
p
2
p
3p
2
2p
2024-25
Page 5
vA mathematician knows how to solve a problem,
he can not solve it. – MILNE v
3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engineers and others. Currently, trigonometry is
used in many areas such as the science of seismology,
designing electric circuits, describing the state of an atom,
predicting the heights of tides in the ocean, analysing a
musical tone and in many other areas.
In earlier classes, we have studied the trigonometric
ratios of acute angles as the ratio of the sides of a right
angled triangle. We have also studied the trigonometric identities and application of
trigonometric ratios in solving the problems related to heights and distances. In this
Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions
and study their properties.
3.2 Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is
Chapter 3
TRIGONOMETRIC FUNCTIONS
Arya Bhatt
(476-550)
Fig 3.1
V er tex
2024-25
44 MATHEMATICS
called the initial side and the final position of the ray after rotation is called the
terminal side of the angle. The point of rotation is called the vertex. If the direction of
rotation is anticlockwise, the angle is said to be positive and if the direction of rotation
is clockwise, then the angle is negative (Fig 3.1).
The measure of an angle is the amount of
rotation performed to get the terminal side from
the initial side. There are several units for
measuring angles. The definition of an angle
suggests a unit, viz. one complete revolution from the position of the initial side as
indicated in Fig 3.2.
This is often convenient for large angles. For example, we can say that a rapidly
spinning wheel is making an angle of say 15 revolution per second. We shall describe
two other units of measurement of an angle which are most commonly used, viz.
degree measure and radian measure.
3.2.1 Degree measure If a rotation from the initial side to terminal side is
th
1
360
? ?
? ?
? ?
of
a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is
divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is
called a minute, written as 1', and one sixtieth of a minute is called a second, written as 1?.
Thus, 1° = 60', 1' = 60?
Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are
shown in Fig 3.3.
Fig 3.2
Fig 3.3
2024-25
TRIGONOMETRIC FUNCTIONS 45
3.2.2 Radian measure There is another unit for measurement of an angle, called
the radian measure. Angle subtended at the centre by an arc of length 1 unit in a
unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the Fig
3.4(i) to (iv), OA is the initial side and OB is the terminal side. The figures show the
angles whose measures are 1 radian, –1 radian, 1
1
2
radian and –1
1
2
radian.
(i)
(ii)
(iii)
Fig 3.4 (i) to (iv)
(iv)
We know that the circumference of a circle of radius 1 unit is 2p. Thus, one
complete revolution of the initial side subtends an angle of 2p radian.
More generally, in a circle of radius r, an arc of length r will subtend an angle of
1 radian. It is well-known that equal arcs of a circle subtend equal angle at the centre.
Since in a circle of radius r, an arc of length r subtends an angle whose measure is 1
radian, an arc of length l will subtend an angle whose measure is
l
r
radian. Thus, if in
a circle of radius r, an arc of length l subtends an angle ? radian at the centre, we have
? =
l
r
or l = r ?.
2024-25
46 MATHEMATICS
3.2.3 Relation between radian and real numbers
Consider the unit circle with centre O. Let A be any point
on the circle. Consider OA as initial side of an angle.
Then the length of an arc of the circle will give the radian
measure of the angle which the arc will subtend at the
centre of the circle. Consider the line PAQ which is
tangent to the circle at A. Let the point A represent the
real number zero, AP represents positive real number and
AQ represents negative real numbers (Fig 3.5). If we
rope the line AP in the anticlockwise direction along the
circle, and AQ in the clockwise direction, then every real
number will correspond to a radian measure and
conversely. Thus, radian measures and real numbers can
be considered as one and the same.
3.2.4 Relation between degree and radian Since a circle subtends at the centre
an angle whose radian measure is 2p and its degree measure is 360°, it follows that
2p radian = 360° or p radian = 180°
The above relation enables us to express a radian measure in terms of degree
measure and a degree measure in terms of radian measure. Using approximate value
of p as
22
7
, we have
1 radian =
180
p
°
= 57° 16' approximately.
Also 1° =
p
180
radian = 0.01746 radian approximately.
The relation between degree measures and radian measure of some common angles
are given in the following table:
A
O
1
P
1
2
-1
-2
Q
0
Fig 3.5
Degree 30° 45° 60° 90° 180° 270° 360°
Radian
p
6
p
4
p
3
p
2
p
3p
2
2p
2024-25
TRIGONOMETRIC FUNCTIONS 47
Notational Convention
Since angles are measured either in degrees or in radians, we adopt the convention
that whenever we write angle ?°, we mean the angle whose degree measure is ? and
whenever we write angle ß, we mean the angle whose radian measure is ß.
Note that when an angle is expressed in radians, the word ‘radian’ is frequently
omitted. Thus,
p
p 180 and 45
4
= ° = °
are written with the understanding that p
and
p
4
are radian measures. Thus, we can say that
Radian measure =
p
180
×
Degree measure
Degree measure =
180
p
×Radian measure
Example 1 Convert 40° 20' into radian measure.
Solution We know that 180° = p radian.
Hence 40° 20' = 40
1
3
degree =
p
180
×
121
3
radian =
121p
540
radian.
Therefore 40° 20' =
121p
540
radian.
Example 2 Convert 6 radians into degree measure.
Solution We know that p radian = 180°.
Hence 6 radians =
180
p
× 6 degree =
1080 7
22
×
degree
= 343
7
11
degree = 343° +
7 60
11
×
minute [as 1° = 60']
= 343° + 38' +
2
11
minute [as 1' = 60?]
= 343° + 38' + 10.9? = 343°38' 11? approximately.
Hence 6 radians = 343° 38' 11? approximately.
Example 3 Find the radius of the circle in which a central angle of 60° intercepts an
arc of length 37.4 cm (use
22
p
7
=
).
2024-25
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