Trigonometric Functions,
Identities and Equations
Angle
When a ray OA starting from its initial B
position OA rotates about its end point O
e
d
si
and takes the final position OB, we say
al
that angle AOB (written as ∠ AOB) has
in
rm
Te
been formed.
θ°
The amount of rotation from the initial side A
to the terminal side is called the measure O Initial side
(Vertex)
of the angle.
Positive and Negative Angles
An angle formed by a rotating ray is said to be positive or negative
depending on whether it moves in an anti-clockwise or a clockwise
direction, respectively.
B O Initial side
A
−θ°
ed
si
al
in
Te
rm
rm
Te
in
al
θ°
si
d
A
e
O Initial side B
(Positive angle) (Negative angle)
Measurement of Angles
There are three system for measuring the angles, which are given
below
1. Sexagesimal System (Degree Measure)
In this system, a right angle is divided into 90 equal parts, called the
degrees. The symbol 1° is used to denote one degree. Each degree is
divided into 60 equal parts, called the minutes and one minute is
,divided into 60 equal parts, called the seconds. Symbols 1′ and 1′ ′ are
used to denote one minute and one second, respectively.
i.e. 1 right angle = 90°, 1° = 60′, 1′ = 60′ ′
2. Circular System (Radian Measure)
In this system, angle is measured in radian. A radian is the angle
subtended at the centre of a circle by an arc, whose length is equal to
the radius of the circle. The number of radians in an angle subtended
arc
by an arc of circle at the centre is equal to .
radius
3. Centesimal System (French System)
In this system, a right angle is divided into 100 equal parts, called the
grades. Each grade is subdivided into 100 min and each minute is
divided into 100 s.
i.e. 1 right angle = 100 grades = 100 g , 1 g = 100′, 1′ = 100′ ′
Relation between Degree and Radian
(i) π radian = 180°
180° 22
or 1 radian = = 57°16′ 22′ ′ where, π = = 3.14159
π 7
π
(ii) 1° = rad = 0.01746 rad
180
(iii) If D is the number of degrees, R is the number of radians and G
is the number of grades in an angle θ, then
D G 2R
= =
90 100 π
Length of an Arc of a Circle
If in a circle of radius r, an arc of length l subtend an angle θ radian at
the centre, then
l
Q
θ r
1c
r P
l Length of arc
θ= = or l = r θ
r Radius
, Trigonometric Ratios For acute Angle
Relation between different sides and angles of a right angled triangle
are called trigonometric ratios or T-ratios.
Trigonometric ratios can be represented as
C
Perpendicular BC
sin θ = = ,
Perpendicular
Hypotenuse AC
e
n us
Base AB
te
cos θ = =
o
,
yp
Hypotenuse AC
H
Perpendicular BC
tan θ = = , θ
Base AB A B
Base
1
cosec θ =
sin θ
1 cos θ 1
sec θ = , cot θ = =
cos θ sin θ tan θ
Trigonometric (or Circular) Functions
Let X ′OX and YOY ′ be the coordinate axes. Taking O as the centre
and a unit radius, draw a circle, cutting the coordinate axes at
A, B, A′ and B′, as shown in the figure.
Y
B
P(x, y)
1 l
y
θ
X' A' O x M A X
B'
Y'
arc AP θ l
Q∠AOP = radius OP = 1 = θ °, using θ = r
Now, six circular functions may be defined as
(i) cosθ = x (ii) sinθ = y
1 1
(iii) sec θ = , x ≠ 0 (iv) cosec θ = , y ≠ 0
x y
y x
(v) tanθ = , x ≠ 0 (vi) cotθ = , y ≠ 0
x y
Identities and Equations
Angle
When a ray OA starting from its initial B
position OA rotates about its end point O
e
d
si
and takes the final position OB, we say
al
that angle AOB (written as ∠ AOB) has
in
rm
Te
been formed.
θ°
The amount of rotation from the initial side A
to the terminal side is called the measure O Initial side
(Vertex)
of the angle.
Positive and Negative Angles
An angle formed by a rotating ray is said to be positive or negative
depending on whether it moves in an anti-clockwise or a clockwise
direction, respectively.
B O Initial side
A
−θ°
ed
si
al
in
Te
rm
rm
Te
in
al
θ°
si
d
A
e
O Initial side B
(Positive angle) (Negative angle)
Measurement of Angles
There are three system for measuring the angles, which are given
below
1. Sexagesimal System (Degree Measure)
In this system, a right angle is divided into 90 equal parts, called the
degrees. The symbol 1° is used to denote one degree. Each degree is
divided into 60 equal parts, called the minutes and one minute is
,divided into 60 equal parts, called the seconds. Symbols 1′ and 1′ ′ are
used to denote one minute and one second, respectively.
i.e. 1 right angle = 90°, 1° = 60′, 1′ = 60′ ′
2. Circular System (Radian Measure)
In this system, angle is measured in radian. A radian is the angle
subtended at the centre of a circle by an arc, whose length is equal to
the radius of the circle. The number of radians in an angle subtended
arc
by an arc of circle at the centre is equal to .
radius
3. Centesimal System (French System)
In this system, a right angle is divided into 100 equal parts, called the
grades. Each grade is subdivided into 100 min and each minute is
divided into 100 s.
i.e. 1 right angle = 100 grades = 100 g , 1 g = 100′, 1′ = 100′ ′
Relation between Degree and Radian
(i) π radian = 180°
180° 22
or 1 radian = = 57°16′ 22′ ′ where, π = = 3.14159
π 7
π
(ii) 1° = rad = 0.01746 rad
180
(iii) If D is the number of degrees, R is the number of radians and G
is the number of grades in an angle θ, then
D G 2R
= =
90 100 π
Length of an Arc of a Circle
If in a circle of radius r, an arc of length l subtend an angle θ radian at
the centre, then
l
Q
θ r
1c
r P
l Length of arc
θ= = or l = r θ
r Radius
, Trigonometric Ratios For acute Angle
Relation between different sides and angles of a right angled triangle
are called trigonometric ratios or T-ratios.
Trigonometric ratios can be represented as
C
Perpendicular BC
sin θ = = ,
Perpendicular
Hypotenuse AC
e
n us
Base AB
te
cos θ = =
o
,
yp
Hypotenuse AC
H
Perpendicular BC
tan θ = = , θ
Base AB A B
Base
1
cosec θ =
sin θ
1 cos θ 1
sec θ = , cot θ = =
cos θ sin θ tan θ
Trigonometric (or Circular) Functions
Let X ′OX and YOY ′ be the coordinate axes. Taking O as the centre
and a unit radius, draw a circle, cutting the coordinate axes at
A, B, A′ and B′, as shown in the figure.
Y
B
P(x, y)
1 l
y
θ
X' A' O x M A X
B'
Y'
arc AP θ l
Q∠AOP = radius OP = 1 = θ °, using θ = r
Now, six circular functions may be defined as
(i) cosθ = x (ii) sinθ = y
1 1
(iii) sec θ = , x ≠ 0 (iv) cosec θ = , y ≠ 0
x y
y x
(v) tanθ = , x ≠ 0 (vi) cotθ = , y ≠ 0
x y
