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 
? sin (x + y) = sinx cosy + cosx siny 
cos (x + y) = cosx cosy — sinx siny 
 
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 
? sin (x + y) = sinx cosy + cosx siny 
cos (x + y) = cosx cosy — sinx siny 
 
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 + sinx siny 
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 
2
x y ?
 cos 
2
x y ?
sin x — sin y = 2 cos 
2
x y ?
 sin
2
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
cos x — cos y = — 2 sin 
2
x y ?
 sin
2
x y ?
2sin s in
2 2
x y y x ? ? ? ? ? ?
?
? ? ? ?
? ? ? ?
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 
? sin (x + y) = sinx cosy + cosx siny 
cos (x + y) = cosx cosy — sinx siny 
 
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 + sinx siny 
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 
2
x y ?
 cos 
2
x y ?
sin x — sin y = 2 cos 
2
x y ?
 sin
2
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
cos x — cos y = — 2 sin 
2
x y ?
 sin
2
x y ?
2sin s in
2 2
x y y x ? ? ? ? ? ?
?
? ? ? ?
? ? ? ?
 
? sin 2x = 2 sin x cos x =
2
2 tan
1 tan
x
x ?
? cos 2x = cos
2
x — sin
2
x = 2 cos
2
x — 1 = 1 — 2sin
2
x = 
2
2
1 ta n
1 ta n
x
x
?
?
? tan 2x =
2
2 tan
1 tan
x
x ?
? sin 3x = 3 sinx — 4 sin
3
x 
? cos 3x = 4 cos
3
x — 3 cos x 
? tan 3x  = 
3
2
3ta n ta n
1 3ta n
x x
x
?
?
? sin(x + y) sin(x — y) = sin
2
x — sin
2
y= cos
2
y — cos
2
x 
? cos(x + y) cos(x — y) = cos
2
x — sin
2
y= cos
2
y — sin
2
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 
? sin (x + y) = sinx cosy + cosx siny 
cos (x + y) = cosx cosy — sinx siny 
 
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 + sinx siny 
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 
2
x y ?
 cos 
2
x y ?
sin x — sin y = 2 cos 
2
x y ?
 sin
2
x y ?
cos x +cos y = 2 cos
2
x y ?
 cos 
2
x y ?
cos x — cos y = — 2 sin 
2
x y ?
 sin
2
x y ?
2sin s in
2 2
x y y x ? ? ? ? ? ?
?
? ? ? ?
? ? ? ?
 
? sin 2x = 2 sin x cos x =
2
2 tan
1 tan
x
x ?
? cos 2x = cos
2
x — sin
2
x = 2 cos
2
x — 1 = 1 — 2sin
2
x = 
2
2
1 ta n
1 ta n
x
x
?
?
? tan 2x =
2
2 tan
1 tan
x
x ?
? sin 3x = 3 sinx — 4 sin
3
x 
? cos 3x = 4 cos
3
x — 3 cos x 
? tan 3x  = 
3
2
3ta n ta n
1 3ta n
x x
x
?
?
? sin(x + y) sin(x — y) = sin
2
x — sin
2
y= cos
2
y — cos
2
x 
? cos(x + y) cos(x — y) = cos
2
x — sin
2
y= cos
2
y — sin
2
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
? ?
? ? ? ?