Trigonometric Functions Class 11 Notes Maths Chapter 3

 Page 1


 
KEY POINTS
? A radian is an angle subtended at the centre of a circle by an arc whose 
length is equal to the radius of the circle. We denote 1 radian by 1
c
. 
? ? radian = 180 degree, 1° = 60’
1 radian = 
180
?
degree and 1’ = 60’ ’ 
1 degree = 
180
?
radian
? If an arc of length l makes an angle qradian at the centre of a circle of 
radius r, we have 
l
r
? ?
Page 2


 
KEY POINTS
? A radian is an angle subtended at the centre of a circle by an arc whose 
length is equal to the radius of the circle. We denote 1 radian by 1
c
. 
? ? radian = 180 degree, 1° = 60’
1 radian = 
180
?
degree and 1’ = 60’ ’ 
1 degree = 
180
?
radian
? If an arc of length l makes an angle qradian at the centre of a circle of 
radius r, we have 
l
r
? ?
 
Some Standard Results 
?
 
ta n ta n
ta n(x + y)
1 ta n .ta n
x y
x y
?
?
?
Page 3


 
KEY POINTS
? A radian is an angle subtended at the centre of a circle by an arc whose 
length is equal to the radius of the circle. We denote 1 radian by 1
c
. 
? ? radian = 180 degree, 1° = 60’
1 radian = 
180
?
degree and 1’ = 60’ ’ 
1 degree = 
180
?
radian
? If an arc of length l makes an angle qradian at the centre of a circle of 
radius r, we have 
l
r
? ?
 
Some Standard Results 
?
 
ta n ta n
ta n(x + y)
1 ta n .ta n
x y
x y
?
?
?
 
cot(x +y) = 
cot x.cot y-1
coty + cotx
? sin (x - y) = sinx.cosy - cosx siny 
cos (x - y) = cosx cosy + sinxsiny 
ta n ta n
ta n(x-y)
1 ta n .ta n
x y
x y
?
?
?
cot .cot 1
cot(x-y) 
cot cot
x y
y x
?
?
?
? tan(x +y + z) = 
ta n ta n ta n ta n ta n ta n
1 ta n ta n ta n .ta n ta n ta n
x y z x y z
x y y z z x
? ? ?
? ? ?
? 2sinx cosy = sin(x + y) + sin(x - y) 
2cosx siny = sin(x + y) - sin(x - y) 
2cosx cosy = cos (x + y) + cos (x -y) 
2sinx siny = cos (x + y) - cos (x + y) 
? sin x + sin y = 2 sin 
x y											 ?
 cos 
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
?
?
?
?
Page 4


 
KEY POINTS
? A radian is an angle subtended at the centre of a circle by an arc whose 
length is equal to the radius of the circle. We denote 1 radian by 1
c
. 
? ? radian = 180 degree, 1° = 60’
1 radian = 
180
?
degree and 1’ = 60’ ’ 
1 degree = 
180
?
radian
? If an arc of length l makes an angle qradian at the centre of a circle of 
radius r, we have 
l
r
? ?
 
Some Standard Results 
?
 
ta n ta n
ta n(x + y)
1 ta n .ta n
x y
x y
?
?
?
 
cot(x +y) = 
cot x.cot y-1
coty + cotx
? sin (x - y) = sinx.cosy - cosx siny 
cos (x - y) = cosx cosy + sinxsiny 
ta n ta n
ta n(x-y)
1 ta n .ta n
x y
x y
?
?
?
cot .cot 1
cot(x-y) 
cot cot
x y
y x
?
?
?
? tan(x +y + z) = 
ta n ta n ta n ta n ta n ta n
1 ta n ta n ta n .ta n ta n ta n
x y z x y z
x y y z z x
? ? ?
? ? ?
? 2sinx cosy = sin(x + y) + sin(x - y) 
2cosx siny = sin(x + y) - sin(x - y) 
2cosx cosy = cos (x + y) + cos (x -y) 
2sinx siny = cos (x + y) - cos (x + y) 
? sin x + sin y = 2 sin 
x y											 ?
 cos 
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
?
?
?
?
 
? sin 2x = 2 sin x cos x =
2
2 tan
1 tan
x
x ?
?
2
2
1 ta n
1 ta n
x
x
?
?
? tan 2x =
2
2 tan
1 tan
x
x ?
? tan 3x  = 
3
2
3ta n ta n
1 3ta n
x x
x
?
?
? sin(x + y) sin(x — y) = sin x — sin y= cos y — cos x 
?
? General	solution	—	A	solution	of	a	trigonometric	equation,	generalised	by	means	of	
periodicity,	is	known	as	the	general	solution.	Principal	solutions	—	The	solutions	
of	a	trigonometric	equation	which 
lie in [0,2 ?) are called its principal solutions. 
? sin ? = 0 ? ? ?? = n ? ? ? where ? n ? ? z 
? cos ? = 0 ? ? ? ? = (2n ? ? ?
2
?
, where n ? ? z
? tan ? = 0 ? ? ?? = n ? ? ? where ? n ? ? z 
? sin ? = sin ? ? ? ? ? = n ? ? ?(— 1)
n
? ? where n ? z 
? cos ? = cos ? ? ? ? ?? = 2n ? ? ? ? ? ? where n ? ? z 
? tan ? = tan ? ? ? ? ?? = n ? ? ? ? ? ? where n ? ? z 
?
?
Page 5


 
KEY POINTS
? A radian is an angle subtended at the centre of a circle by an arc whose 
length is equal to the radius of the circle. We denote 1 radian by 1
c
. 
? ? radian = 180 degree, 1° = 60’
1 radian = 
180
?
degree and 1’ = 60’ ’ 
1 degree = 
180
?
radian
? If an arc of length l makes an angle qradian at the centre of a circle of 
radius r, we have 
l
r
? ?
 
Some Standard Results 
?
 
ta n ta n
ta n(x + y)
1 ta n .ta n
x y
x y
?
?
?
 
cot(x +y) = 
cot x.cot y-1
coty + cotx
? sin (x - y) = sinx.cosy - cosx siny 
cos (x - y) = cosx cosy + sinxsiny 
ta n ta n
ta n(x-y)
1 ta n .ta n
x y
x y
?
?
?
cot .cot 1
cot(x-y) 
cot cot
x y
y x
?
?
?
? tan(x +y + z) = 
ta n ta n ta n ta n ta n ta n
1 ta n ta n ta n .ta n ta n ta n
x y z x y z
x y y z z x
? ? ?
? ? ?
? 2sinx cosy = sin(x + y) + sin(x - y) 
2cosx siny = sin(x + y) - sin(x - y) 
2cosx cosy = cos (x + y) + cos (x -y) 
2sinx siny = cos (x + y) - cos (x + y) 
? sin x + sin y = 2 sin 
x y											 ?
 cos 
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
?
?
?
?
 
? sin 2x = 2 sin x cos x =
2
2 tan
1 tan
x
x ?
?
2
2
1 ta n
1 ta n
x
x
?
?
? tan 2x =
2
2 tan
1 tan
x
x ?
? tan 3x  = 
3
2
3ta n ta n
1 3ta n
x x
x
?
?
? sin(x + y) sin(x — y) = sin x — sin y= cos y — cos x 
?
? General	solution	—	A	solution	of	a	trigonometric	equation,	generalised	by	means	of	
periodicity,	is	known	as	the	general	solution.	Principal	solutions	—	The	solutions	
of	a	trigonometric	equation	which 
lie in [0,2 ?) are called its principal solutions. 
? sin ? = 0 ? ? ?? = n ? ? ? where ? n ? ? z 
? cos ? = 0 ? ? ? ? = (2n ? ? ?
2
?
, where n ? ? z
? tan ? = 0 ? ? ?? = n ? ? ? where ? n ? ? z 
? sin ? = sin ? ? ? ? ? = n ? ? ?(— 1)
n
? ? where n ? z 
? cos ? = cos ? ? ? ? ?? = 2n ? ? ? ? ? ? where n ? ? z 
? tan ? = tan ? ? ? ? ?? = n ? ? ? ? ? ? where n ? ? z 
?
?
 
5 1
sin18
4
?
? ? ;  
10 2 5
cos18
4
?
? ?
10 2 5 5 1
sin36 ; cos36
4 4
? ?
? ? ? ?