Formulae For
Trigonometric Functions
By OP Gupta [Indira Award Winner, +91-9650 350 480]
Relation between trigonometric ratios
a) tan θ 
sin θ
cos θ
b) tan θ 
1
cot θ
d) cot θ 
cos θ
sin θ
e) cosec θ 
c) tan θ.cot θ  1
1
sin θ
f) sec θ 
Trigonometric identities
Multiple angle formulae involving 2A and 3A
a) sin 2 θ  cos2 θ  1
2
1
.
cos θ
a) sin 2A  2sin AcosA
2
b) 1  tan θ  sec θ
2
b) sin A  2sin
2
c) 1  cot θ  cosec θ
A
2
cos
A
2
c) cos 2A  cos2 A  sin 2 A
Addition / subtraction formulae & some related results
a) sin(A  B)  sin Acos B  cos A sin B
d) cos A  cos2
b) cos(A  B)  cos A cos B  sin Asin B
A
A
 sin 2
2
2
e) cos2A  2 cos 2 A  1
c) cos(A  B) cos(A  B)  cos2 A  sin 2 B  cos2 B  sin 2 A
f) 2 cos 2 A  1  cos2A
d) sin(A  B) sin(A  B)  sin 2 A  sin 2 B  cos2 B  cos2 A
e) tan(A  B) 
tan A  tan B
1  tan A tan B
g) cos 2A  1  2sin 2 A
f) cot (A  B) 
cot Bcot A  1
cot B  cot A
h) 2sin 2 A  1  cos2A
Transformation of sums / differences into products & vice-versa
C D
CD
cos
2
2
CD
C

D
b) sin C  sin D  2cos
sin
2
2
C
D
CD
c) cosC  cos D  2cos
cos
2
2
C D C D
d) cosC  cos D  2sin
sin
2
2
e) 2sin Acos B  sin ( A  B)  sin (A  B)
f) 2 cosAsin B  sin (A  B)  sin (A  B)
g) 2cosAcosB  cos (A  B)  cos(A  B)
h) 2sinAsinB  cos (A  B)  cos (A  B)
a) sinC  sin D  2sin
i) sin 2A 
2 tan A
1  tan 2 A
j) cos2A 
1  tan 2 A
1  tan 2 A
k) tan 2A 
2 tan A
1  tan 2 A
l) sin 3A  3sin A  4sin 3 A
m) cos3A  4 cos3 A  3cos A
n) tan 3A 
3tan A  tan 3 A
1  3tan 2 A
Relations in Different Measures of Angle
 Angle in Radian Measure =  Angle in Degree Measure ×
 Angle in Degree Measure =  Angle in Radian Measure ×

180
180

l
  (in radian measure)  , where  is the angle made by an arc of length l, and r is radius of the circle.
r
 Also followings are of importance as well:
 1Right angle  90 o
o
1 =

180
= 0.01745 radians  approximately 
List Of Formulae By OP Gupta
 1 o = 60 , 1 = 60 
o
 1 radian = 57 17 45 or 206265 seconds .
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List Of Formulae for Class XI By OP Gupta (Electronics & Communications Engineering)
General Solutions
n
a) sin x  sin y  x  n  ( 1 ) y , where n  Z .
b) cos x  cos y  x  2n  y , where n Z .
c) tan x  tan y  x  n  y , where n  Z .
Relation in Degree & Radian Measures
Angles in Degree
0
30
Angles in Radian
c
 
 
6
0
45
c
 
 
4
60
c
 
 
 3
90
c
 
 
2
 In actual practice, we omit the exponent ‘c’ and instead of writing  c
180
270
c
 3 
 
 2 
c
 
we simply write

360
c
 2 
c
and similarly for others.
Trigonometric Ratio of Standard Angles
Degree /Radian   
T – Ratios

0
0
0
sin
cos
45
60
90
π
6
1
2
π
4
1
π
3
3
2
1
2
π
2
2
1
3
2
1
1
tan
30
0
2
1

sec
2
3

cot
3

2
2
2
1
0
3
3
cosec
1
1
3
2
2

1
1
0
3
Trigonometric Ratios of Allied Angles
Angles

T- Ratios

sin
cos
tan
cot
sec
cosec
π
θ
2
π
θ
2
πθ
cos θ
sin θ
cot θ
tan θ
cosecθ
secθ
cos θ
– sin θ
– cot θ
– tan θ
– cosecθ
sin θ
– cos θ
 tan θ
– cot θ
– secθ
secθ
cosecθ
3π
θ
2
3π
θ
2
2π  θ
θ
2π  θ
OR
– sin θ
– cos θ
– cos θ
– sin θ
– cos θ
– sin θ
tan θ
cot θ
– secθ
– cosecθ
cot θ
tan θ
– cosecθ
– secθ
sin θ
– cot θ
– tan θ
cos θ
– tan θ
– cot θ
cosecθ
secθ
– cosecθ
sin θ
cos θ
tan θ
cot θ
secθ
πθ
– secθ
cosecθ
Table for the Domain and Range of trigonometric functions
T- Functions   
Domain
Range
sin x
R
[1, 1]
cos x
R
[1, 1]
tan x
cot x
cosec x
{ x  R : x  (2n  1) π 2, n  Z}
{ x  R : x  n π, n  Z}
{ x  R : x  n π, n  Z}
R
R  ( 1, 1)
sec x
{ x  R : x  (2n  1) π 2, n  Z}
R  ( 1, 1)
List Of Formulae By OP Gupta
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MATHEMATICS – List Of Formulae for Class XI By OP Gupta (+91-9650350480)
Remember The Followings For Quick Use

π
sin θ  ( 1) n if θ  (2n  1) , n  Z
2

cos θ  ( 1) n if θ  n π, n  Z

sin θ  0 if θ  n π, n  Z

π
cosθ  0 if θ  (2n  1) , n  Z .
2
 Law of sines: The sine rule states that the lengths of the sides of a triangle are proportional to the
a
b
c
.
sines of angles opposite to them i.e., in ABC , we have


sin A sin B sin C
a
b
c



 k ( say )  a  k sin A, b  k sin B, k  c sin C .
sin A sin B sin C
sin A sin B sin C



  ( say )  sin A  a , sin B  b , sin C  c .
a
b
c
 Law of cosines: In any ABC , we have
b2  c 2  a 2
or a 2  b2  c 2  2bc cos A
2bc
c 2  a 2  b2
b) cos B 
or b2  c 2  a 2  2ca cos B
2ca
a) cos A 
c) cosC 
a 2  b2  c 2
or c 2  a 2  b2  2ab cos C .
2ab
Hii, All!
I hope this texture may have proved beneficial for you.
While going through this material, if you noticed any error(s) or, something which doesn’t make
sense to you, please bring it in my notice through SMS or Call at +91-9650 350 480 or Email at
[email protected].
With lots of Love & Blessings!
- OP Gupta
Electronics & Communications Engineering, Indira Award Winner
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List Of Formulae By OP Gupta
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