Detailed Notes: Trigonometry | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Trigonometry 
The word 'trigonometry' is derived from the Greek words 'trigon' and ' metron' and it 
means 'measuring the sides and angles of a triangle'. 
Angle: 
Angle is a measure of rotation of a given ray about its initial point. The original ray is 
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. 
 
(i) Positive angle 
 
 
(anticlockwise measurement) 
(ii) Negative angle (clockwise measurement) 
= 
Page 2


Trigonometry 
The word 'trigonometry' is derived from the Greek words 'trigon' and ' metron' and it 
means 'measuring the sides and angles of a triangle'. 
Angle: 
Angle is a measure of rotation of a given ray about its initial point. The original ray is 
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. 
 
(i) Positive angle 
 
 
(anticlockwise measurement) 
(ii) Negative angle (clockwise measurement) 
= 
Trigonometric Ratios for Acute Angles: 
Let a revolving ray OP starts from OA and revolves into the position OP, thus tracing out 
the angle AOP. 
In the revolving ray take any point ?? and draw ???? perpendicular to the initial ray ???? . 
In the right angle triangle MOP, OP is the hypotenuse, ???? is the perpendicular, and ???? is 
the base. 
The trigonometrical ratios, or functions, of the angle AOP are defined as follows: 
?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ?? ?? ? ( ? ?????? ) 
 ?? ?? ???? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ???? 
 ?? ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ???? 
=
????
????
 
?? ?? ?? ?? ?? ?? ?? 
=
????
????
 
?? ?? ?? ?? ???? ?? =
????
????
 
 
It can be noted that the trigonometrical ratios are all real numbers. 
Trigonometric ratios for angle ? ?? : 
We will now extend the definition of trigonometric ratios to any angle in terms of radian 
measure and study them as trigonometric functions. (also called circular functions) 
Consider a unit circle (radius 1 unit) with centre at origin of the coordinate axes. Let at 
origin of the coordinate axes. Let ?? ( ?? , ?? ) be any point on the circle with angle ?????? = ?? 
radian, i.e., length of arc ???? = ?? We define ?? ?? ?? ? ?? = ?? and ?? ?? ?? ? ?? = ?? Since ?? OMP is a right 
triangle, we have ?? ?? 2
+ ?? ?? 2
= ?? ?? 2
 or ?? 2
+ ?? 2
= 1 Thus, for every point on the unit circle, 
we have ?? 2
+ ?? 2
= 1 or ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 
Since one complete revolution subtends an angle of 2 ?? radian at the centre of the circle, 
? ?????? =
?? 2
, 
Page 3


Trigonometry 
The word 'trigonometry' is derived from the Greek words 'trigon' and ' metron' and it 
means 'measuring the sides and angles of a triangle'. 
Angle: 
Angle is a measure of rotation of a given ray about its initial point. The original ray is 
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. 
 
(i) Positive angle 
 
 
(anticlockwise measurement) 
(ii) Negative angle (clockwise measurement) 
= 
Trigonometric Ratios for Acute Angles: 
Let a revolving ray OP starts from OA and revolves into the position OP, thus tracing out 
the angle AOP. 
In the revolving ray take any point ?? and draw ???? perpendicular to the initial ray ???? . 
In the right angle triangle MOP, OP is the hypotenuse, ???? is the perpendicular, and ???? is 
the base. 
The trigonometrical ratios, or functions, of the angle AOP are defined as follows: 
?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ?? ?? ? ( ? ?????? ) 
 ?? ?? ???? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ???? 
 ?? ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ???? 
=
????
????
 
?? ?? ?? ?? ?? ?? ?? 
=
????
????
 
?? ?? ?? ?? ???? ?? =
????
????
 
 
It can be noted that the trigonometrical ratios are all real numbers. 
Trigonometric ratios for angle ? ?? : 
We will now extend the definition of trigonometric ratios to any angle in terms of radian 
measure and study them as trigonometric functions. (also called circular functions) 
Consider a unit circle (radius 1 unit) with centre at origin of the coordinate axes. Let at 
origin of the coordinate axes. Let ?? ( ?? , ?? ) be any point on the circle with angle ?????? = ?? 
radian, i.e., length of arc ???? = ?? We define ?? ?? ?? ? ?? = ?? and ?? ?? ?? ? ?? = ?? Since ?? OMP is a right 
triangle, we have ?? ?? 2
+ ?? ?? 2
= ?? ?? 2
 or ?? 2
+ ?? 2
= 1 Thus, for every point on the unit circle, 
we have ?? 2
+ ?? 2
= 1 or ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 
Since one complete revolution subtends an angle of 2 ?? radian at the centre of the circle, 
? ?????? =
?? 2
, 
? ?????? = ?? and ? ?????? =
3 ?? 2
. All angles which are integral multiples of 
?? 2
 are called 
quadrantal angles. 
The coordinates of the points A, B, C and D are, respectively, ( 1 , 0 ) , ( 0 , 1 ) , ( - 1 , 0 ) and 
( 0 , - 1 ). Therefore, for quadrantal angles, we have 
?? ?? ?? ? 0 = 1 ? ?? ?? ?? ? 0 = 0 ? ?? ?? ?? ?
?? 2
= 0 ? ?? ?? ?? ?
?? 2
= 1 ? ?? ?? ?? ? ?? = - 1 ? ?? ?? ?? ? ?? = 0 ? ?? ?? ?? ?
3 ?? 2
= 0 ? ?? ?? ?? ?
3 ?? 2
= - 1 ? ?? ?? ?? ? 2 ?? = 1 ? ?? ?? ?? ? 2 ?? = 0 ? 
Now if we take one complete revolution from the position OP, we again come back to same 
position OP. Thus, we also observe that if ?? increases (or decreases) by any integral 
multiple of 2 ?? , the values of sine and cosine functions do not change. Thus, ?? ?? ?? ? ( 2 ???? +
?? ) = ?? ?? ?? ? ?? , ?? ? ?? , ?? ?? ?? ? ( 2 ???? + ?? ) = ?? ?? ?? ? ?? , ?? ? ?? . Further, ?? ?? ?? ? ?? = 0, if ?? =
0 , ± ?? , ± 2 ?? , ± 3 ?? …., i.e., when ?? is an integral multiple of ?? and ?? ?? ?? ? ?? = 0, if ?? =
±
?? 2
, ±
3 ?? 2
, ±
5 ?? 2
, …. i.e., ?? ?? ?? ? ?? vanishes when ?? is an odd multiple of 
?? 2
. Thus ?? ?? ?? ? ?? = 0 
implies ?? = ???? , where ?? is any integer ?? ?? ?? ? ?? = 0 implies ?? = ( 2 ?? + 1 )
?? 2
, where ?? is any 
integer. 
We now define other trigonometric functions in terms of sine and cosine functions: 
?? ?? ?? ?? ?? ? ?? =
1
???? ?? ? ?? , ?? ? ???? , ? where ?? is any integer. 
?? ?? ?? ? ?? =
1
?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
???? ?? ? ?? ?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
?? ?? ?? ? ?? ???? ?? ? ?? , ?? ? ???? , where ?? is any integer. 
We have shown that for all real ?? , ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 ???? ? ?? ?? ?? ?? ?? ???? ? ?? h ???? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? 2
? ?? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? ?? ?? 2
? ?? ? 
(Think!) 
(Think !) 
? { ?? ? ( 2 ?? + 1 )
?? 2
; ?? ? ?? } ? ? ? { ?? ? ???? ; ?? ? ?? } ? 
Sign of The Trigonometric Functions 
(i) If ?? is in the first quadrant then ?? ( ?? , ?? ) lies in the first quadrant. Therefore ?? > 0 , ?? > 0 
and hence the values of all the trigonometric functions are positive. 
(ii) If ?? is in the II quadrant then ?? ( ?? , ?? ) lies in the II quadrant. Therefore ?? < 0 , ?? > 0 and 
hence the values sin, cosec are positive and the remaining are negative. 
(iii) If ?? is in the III quadrant then ?? ( ?? , ?? ) lies in the III quadrant. Therefore ?? < 0 , ?? < 0 
and hence the values of tan, cot are positive and the remaining are negative. 
Page 4


Trigonometry 
The word 'trigonometry' is derived from the Greek words 'trigon' and ' metron' and it 
means 'measuring the sides and angles of a triangle'. 
Angle: 
Angle is a measure of rotation of a given ray about its initial point. The original ray is 
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. 
 
(i) Positive angle 
 
 
(anticlockwise measurement) 
(ii) Negative angle (clockwise measurement) 
= 
Trigonometric Ratios for Acute Angles: 
Let a revolving ray OP starts from OA and revolves into the position OP, thus tracing out 
the angle AOP. 
In the revolving ray take any point ?? and draw ???? perpendicular to the initial ray ???? . 
In the right angle triangle MOP, OP is the hypotenuse, ???? is the perpendicular, and ???? is 
the base. 
The trigonometrical ratios, or functions, of the angle AOP are defined as follows: 
?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ?? ?? ? ( ? ?????? ) 
 ?? ?? ???? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ???? 
 ?? ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ???? 
=
????
????
 
?? ?? ?? ?? ?? ?? ?? 
=
????
????
 
?? ?? ?? ?? ???? ?? =
????
????
 
 
It can be noted that the trigonometrical ratios are all real numbers. 
Trigonometric ratios for angle ? ?? : 
We will now extend the definition of trigonometric ratios to any angle in terms of radian 
measure and study them as trigonometric functions. (also called circular functions) 
Consider a unit circle (radius 1 unit) with centre at origin of the coordinate axes. Let at 
origin of the coordinate axes. Let ?? ( ?? , ?? ) be any point on the circle with angle ?????? = ?? 
radian, i.e., length of arc ???? = ?? We define ?? ?? ?? ? ?? = ?? and ?? ?? ?? ? ?? = ?? Since ?? OMP is a right 
triangle, we have ?? ?? 2
+ ?? ?? 2
= ?? ?? 2
 or ?? 2
+ ?? 2
= 1 Thus, for every point on the unit circle, 
we have ?? 2
+ ?? 2
= 1 or ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 
Since one complete revolution subtends an angle of 2 ?? radian at the centre of the circle, 
? ?????? =
?? 2
, 
? ?????? = ?? and ? ?????? =
3 ?? 2
. All angles which are integral multiples of 
?? 2
 are called 
quadrantal angles. 
The coordinates of the points A, B, C and D are, respectively, ( 1 , 0 ) , ( 0 , 1 ) , ( - 1 , 0 ) and 
( 0 , - 1 ). Therefore, for quadrantal angles, we have 
?? ?? ?? ? 0 = 1 ? ?? ?? ?? ? 0 = 0 ? ?? ?? ?? ?
?? 2
= 0 ? ?? ?? ?? ?
?? 2
= 1 ? ?? ?? ?? ? ?? = - 1 ? ?? ?? ?? ? ?? = 0 ? ?? ?? ?? ?
3 ?? 2
= 0 ? ?? ?? ?? ?
3 ?? 2
= - 1 ? ?? ?? ?? ? 2 ?? = 1 ? ?? ?? ?? ? 2 ?? = 0 ? 
Now if we take one complete revolution from the position OP, we again come back to same 
position OP. Thus, we also observe that if ?? increases (or decreases) by any integral 
multiple of 2 ?? , the values of sine and cosine functions do not change. Thus, ?? ?? ?? ? ( 2 ???? +
?? ) = ?? ?? ?? ? ?? , ?? ? ?? , ?? ?? ?? ? ( 2 ???? + ?? ) = ?? ?? ?? ? ?? , ?? ? ?? . Further, ?? ?? ?? ? ?? = 0, if ?? =
0 , ± ?? , ± 2 ?? , ± 3 ?? …., i.e., when ?? is an integral multiple of ?? and ?? ?? ?? ? ?? = 0, if ?? =
±
?? 2
, ±
3 ?? 2
, ±
5 ?? 2
, …. i.e., ?? ?? ?? ? ?? vanishes when ?? is an odd multiple of 
?? 2
. Thus ?? ?? ?? ? ?? = 0 
implies ?? = ???? , where ?? is any integer ?? ?? ?? ? ?? = 0 implies ?? = ( 2 ?? + 1 )
?? 2
, where ?? is any 
integer. 
We now define other trigonometric functions in terms of sine and cosine functions: 
?? ?? ?? ?? ?? ? ?? =
1
???? ?? ? ?? , ?? ? ???? , ? where ?? is any integer. 
?? ?? ?? ? ?? =
1
?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
???? ?? ? ?? ?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
?? ?? ?? ? ?? ???? ?? ? ?? , ?? ? ???? , where ?? is any integer. 
We have shown that for all real ?? , ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 ???? ? ?? ?? ?? ?? ?? ???? ? ?? h ???? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? 2
? ?? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? ?? ?? 2
? ?? ? 
(Think!) 
(Think !) 
? { ?? ? ( 2 ?? + 1 )
?? 2
; ?? ? ?? } ? ? ? { ?? ? ???? ; ?? ? ?? } ? 
Sign of The Trigonometric Functions 
(i) If ?? is in the first quadrant then ?? ( ?? , ?? ) lies in the first quadrant. Therefore ?? > 0 , ?? > 0 
and hence the values of all the trigonometric functions are positive. 
(ii) If ?? is in the II quadrant then ?? ( ?? , ?? ) lies in the II quadrant. Therefore ?? < 0 , ?? > 0 and 
hence the values sin, cosec are positive and the remaining are negative. 
(iii) If ?? is in the III quadrant then ?? ( ?? , ?? ) lies in the III quadrant. Therefore ?? < 0 , ?? < 0 
and hence the values of tan, cot are positive and the remaining are negative. 
(iv) If ?? is in the IV quadrant then ?? ( ?? , ?? ) lies in the IV quadrant. Therefore ?? > 0 , ?? < 0 
and hence the values of cos, sec are positive and the remaining are negative. 
 ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? 
?? ???? 
 Quadrant + + + + + + 
II ?
???? 
 Quadrant + - - - - + 
III ?
???? 
 Quadrant - - + + - - 
IV ?
?? h 
 Quadrant - + - - + - 
 
Values of trigonometric functions of certain popular angles are shown in the following 
table: 
 0 
?? 6
 
?? 4
 
?? 3
 
?? 2
 
?? ?? ?? v
0
4
= 0 
v
1
4
=
1
2
 
v
2
4
=
1
v 2
 
v
3
4
=
v 3
2
 
v
4
4
= 1 
?? ?? ?? 1 
v 3
2
 
1
v 2
 
1
2
 0 
?? ?? ?? 0 
1
v 3
 1 v 3 N.D. 
 
N.D. implies not defined 
The values of ?? ?? ?? ?? ?? ? ?? , ?? ?? ?? ? ?? and ?? ?? ?? ? ?? are the reciprocal of the values of ?? ?? ?? ? ?? , ?? ?? ?? ? ?? and 
?? ?? ?? ? ?? , respectively. 
Trigonometric Ratios of allied angles 
If ?? is any angle, then - ?? ,
?? 2
± ?? , ?? ± ?? ,
3 ?? 2
± ?? , 2 ?? ± ?? etc. are called allied angles. 
 
 - ?? 
?? 2
- ?? 
?? 2
+ ?? ?? - ?? ?? + ?? 
3 ?? 2
- ?? 
3 ?? 2
+ ?? 
2 ?? - ?? 
2 ?? + ?? 
Page 5


Trigonometry 
The word 'trigonometry' is derived from the Greek words 'trigon' and ' metron' and it 
means 'measuring the sides and angles of a triangle'. 
Angle: 
Angle is a measure of rotation of a given ray about its initial point. The original ray is 
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. 
 
(i) Positive angle 
 
 
(anticlockwise measurement) 
(ii) Negative angle (clockwise measurement) 
= 
Trigonometric Ratios for Acute Angles: 
Let a revolving ray OP starts from OA and revolves into the position OP, thus tracing out 
the angle AOP. 
In the revolving ray take any point ?? and draw ???? perpendicular to the initial ray ???? . 
In the right angle triangle MOP, OP is the hypotenuse, ???? is the perpendicular, and ???? is 
the base. 
The trigonometrical ratios, or functions, of the angle AOP are defined as follows: 
?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ? ( ? ?????? ) ?? ?? ?? ?? ?? ? ( ? ?????? ) 
 ?? ?? ???? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ?? 
=
????
????
 
 ?? ?? ???? 
 ?? ?? ?? ?? 
=
????
????
 
 ?? ?? ?? ?? 
 ?? ?? ???? 
=
????
????
 
?? ?? ?? ?? ?? ?? ?? 
=
????
????
 
?? ?? ?? ?? ???? ?? =
????
????
 
 
It can be noted that the trigonometrical ratios are all real numbers. 
Trigonometric ratios for angle ? ?? : 
We will now extend the definition of trigonometric ratios to any angle in terms of radian 
measure and study them as trigonometric functions. (also called circular functions) 
Consider a unit circle (radius 1 unit) with centre at origin of the coordinate axes. Let at 
origin of the coordinate axes. Let ?? ( ?? , ?? ) be any point on the circle with angle ?????? = ?? 
radian, i.e., length of arc ???? = ?? We define ?? ?? ?? ? ?? = ?? and ?? ?? ?? ? ?? = ?? Since ?? OMP is a right 
triangle, we have ?? ?? 2
+ ?? ?? 2
= ?? ?? 2
 or ?? 2
+ ?? 2
= 1 Thus, for every point on the unit circle, 
we have ?? 2
+ ?? 2
= 1 or ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 
Since one complete revolution subtends an angle of 2 ?? radian at the centre of the circle, 
? ?????? =
?? 2
, 
? ?????? = ?? and ? ?????? =
3 ?? 2
. All angles which are integral multiples of 
?? 2
 are called 
quadrantal angles. 
The coordinates of the points A, B, C and D are, respectively, ( 1 , 0 ) , ( 0 , 1 ) , ( - 1 , 0 ) and 
( 0 , - 1 ). Therefore, for quadrantal angles, we have 
?? ?? ?? ? 0 = 1 ? ?? ?? ?? ? 0 = 0 ? ?? ?? ?? ?
?? 2
= 0 ? ?? ?? ?? ?
?? 2
= 1 ? ?? ?? ?? ? ?? = - 1 ? ?? ?? ?? ? ?? = 0 ? ?? ?? ?? ?
3 ?? 2
= 0 ? ?? ?? ?? ?
3 ?? 2
= - 1 ? ?? ?? ?? ? 2 ?? = 1 ? ?? ?? ?? ? 2 ?? = 0 ? 
Now if we take one complete revolution from the position OP, we again come back to same 
position OP. Thus, we also observe that if ?? increases (or decreases) by any integral 
multiple of 2 ?? , the values of sine and cosine functions do not change. Thus, ?? ?? ?? ? ( 2 ???? +
?? ) = ?? ?? ?? ? ?? , ?? ? ?? , ?? ?? ?? ? ( 2 ???? + ?? ) = ?? ?? ?? ? ?? , ?? ? ?? . Further, ?? ?? ?? ? ?? = 0, if ?? =
0 , ± ?? , ± 2 ?? , ± 3 ?? …., i.e., when ?? is an integral multiple of ?? and ?? ?? ?? ? ?? = 0, if ?? =
±
?? 2
, ±
3 ?? 2
, ±
5 ?? 2
, …. i.e., ?? ?? ?? ? ?? vanishes when ?? is an odd multiple of 
?? 2
. Thus ?? ?? ?? ? ?? = 0 
implies ?? = ???? , where ?? is any integer ?? ?? ?? ? ?? = 0 implies ?? = ( 2 ?? + 1 )
?? 2
, where ?? is any 
integer. 
We now define other trigonometric functions in terms of sine and cosine functions: 
?? ?? ?? ?? ?? ? ?? =
1
???? ?? ? ?? , ?? ? ???? , ? where ?? is any integer. 
?? ?? ?? ? ?? =
1
?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
???? ?? ? ?? ?? ?? ?? ? ?? , ?? ? ( 2 ?? + 1 )
?? 2
, where ?? is any integer. 
?? ?? ?? ? ?? =
?? ?? ?? ? ?? ???? ?? ? ?? , ?? ? ???? , where ?? is any integer. 
We have shown that for all real ?? , ?? ?? ?? 2
? ?? + ?? ?? ?? 2
? ?? = 1 
 ???? ? ?? ?? ?? ?? ?? ???? ? ?? h ???? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? 2
? ?? ? ? 1 + ?? ?? ?? 2
? ?? = ?? ?? ?? ?? ?? 2
? ?? ? 
(Think!) 
(Think !) 
? { ?? ? ( 2 ?? + 1 )
?? 2
; ?? ? ?? } ? ? ? { ?? ? ???? ; ?? ? ?? } ? 
Sign of The Trigonometric Functions 
(i) If ?? is in the first quadrant then ?? ( ?? , ?? ) lies in the first quadrant. Therefore ?? > 0 , ?? > 0 
and hence the values of all the trigonometric functions are positive. 
(ii) If ?? is in the II quadrant then ?? ( ?? , ?? ) lies in the II quadrant. Therefore ?? < 0 , ?? > 0 and 
hence the values sin, cosec are positive and the remaining are negative. 
(iii) If ?? is in the III quadrant then ?? ( ?? , ?? ) lies in the III quadrant. Therefore ?? < 0 , ?? < 0 
and hence the values of tan, cot are positive and the remaining are negative. 
(iv) If ?? is in the IV quadrant then ?? ( ?? , ?? ) lies in the IV quadrant. Therefore ?? > 0 , ?? < 0 
and hence the values of cos, sec are positive and the remaining are negative. 
 ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? 
?? ???? 
 Quadrant + + + + + + 
II ?
???? 
 Quadrant + - - - - + 
III ?
???? 
 Quadrant - - + + - - 
IV ?
?? h 
 Quadrant - + - - + - 
 
Values of trigonometric functions of certain popular angles are shown in the following 
table: 
 0 
?? 6
 
?? 4
 
?? 3
 
?? 2
 
?? ?? ?? v
0
4
= 0 
v
1
4
=
1
2
 
v
2
4
=
1
v 2
 
v
3
4
=
v 3
2
 
v
4
4
= 1 
?? ?? ?? 1 
v 3
2
 
1
v 2
 
1
2
 0 
?? ?? ?? 0 
1
v 3
 1 v 3 N.D. 
 
N.D. implies not defined 
The values of ?? ?? ?? ?? ?? ? ?? , ?? ?? ?? ? ?? and ?? ?? ?? ? ?? are the reciprocal of the values of ?? ?? ?? ? ?? , ?? ?? ?? ? ?? and 
?? ?? ?? ? ?? , respectively. 
Trigonometric Ratios of allied angles 
If ?? is any angle, then - ?? ,
?? 2
± ?? , ?? ± ?? ,
3 ?? 2
± ?? , 2 ?? ± ?? etc. are called allied angles. 
 
 - ?? 
?? 2
- ?? 
?? 2
+ ?? ?? - ?? ?? + ?? 
3 ?? 2
- ?? 
3 ?? 2
+ ?? 
2 ?? - ?? 
2 ?? + ?? 
?? ?? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? 
?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? 
?? ?? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? 
?? ?? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? ?? ?? ?? ? ?? 
?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ? ?? - ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? 
 
Think, and fill up the blank blocks in following table. 
 0 
?? 6
 
?? 4
 
?? 3
 
?? 2
 
2 ?? 3
 
5 ?? 6
 ?? 
7 ?? 6
 
4 ?? 3
 
3 ?? 2
 
5 ?? 3
 
11 ?? 6
 2 ?? 
?? ?? ?? 0 
1
2
 
1
v 2
 
v 3
2
 
1          
?? ?? ?? 1 
v 3
2
 
1
v 2
 
1
2
 0          
?? ?? ?? 0 
1
v 3
 1 
v 3 
N.D
. 
         
 
Trigonometric functions: