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Trigonometric Functions Class 11 Notes Maths Chapter 3

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 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
? ?
? ? ? ?
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FAQs on Trigonometric Functions Class 11 Notes Maths Chapter 3

1. What are the main trigonometric functions?
Ans. The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a triangle to the ratios of its sides.
2. How are trigonometric functions used in real-life applications?
Ans. Trigonometric functions are used in various real-life applications such as navigation, engineering, physics, and astronomy. They help in calculating distances, angles, heights, and trajectories in these fields.
3. What is the unit circle and its relationship with trigonometric functions?
Ans. The unit circle is a circle with a radius of 1 unit. It is centered at the origin of a coordinate plane. The trigonometric functions can be defined using the coordinates of points on the unit circle, where the angle is measured from the positive x-axis.
4. How are trigonometric functions related to right-angled triangles?
Ans. Trigonometric functions are primarily used to calculate the ratios of sides in a right-angled triangle. For example, sine (sin) is the ratio of the length of the side opposite an angle to the hypotenuse, cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the length of the opposite side to the adjacent side.
5. How can trigonometric functions be used to solve problems involving angles and sides in triangles?
Ans. Trigonometric functions can be used to solve problems involving angles and sides in triangles using trigonometric identities and formulas. By knowing the values of certain trigonometric functions and one side or angle of a triangle, we can calculate the other unknown sides or angles. This is commonly done using the sine, cosine, and tangent functions, along with their inverses.
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