Download TRIGONOMETRIC IDENTITIES

Table of Values of Trigonometric Functions
 (Radians)  (Degrees)
0
0
sin 
0
cos 
1
tan 
0
csc 
Undefined
sec 
1
2
2 3
3

6
30
1
2
3
2
3
3

4
45
2
2
2
2
1
2

3
60
3
2
1
2
3
2 3
3
2

2
90
1
0
Undefined
1
Undefined

180
0
-1
0
Undefined
-1
3
2
270
-1
0
Undefined
-1
Undefined
cot 
Undefined
3
2
1
3
3
0
Undefined
TRIGONOMETRIC IDENTITIES
In terms of a right triangle with angle :
1) sin  =
opp
hyp
2) cos  =
adj
hyp
opp is the side opposite angle 
adj is the side adjacent to angle 
hyp is the hypotenuse
3) tan  =
opp sin 

adj cos 
0
Reciprocal Identities
1) csc  =
1
sin 
2) sec  =
1
cos 
3) cot  =
1
tan 
4) sin  =
1
csc 
5) cos  =
1
sec 
6) tan  =
1
cot 
Pythagorean Identities
1) sin2  + cos2  = 1
2) 1 + tan2  = sec2 
3) 1 + cot2  = csc2 
Even-Odd Identities
1) sin (-) = - sin 
2) cos (-) = cos 
3) tan (-) = - tan 
4) csc (-) = - csc 
5) sec (-) = sec 
6) cot (-) = -cot 
Sum and Difference Formulas
1) sin (α + ) = sin α cos  + cos α sin 
2) sin (α - ) = sin α cos  - cos α sin 
3) cos (α + ) = cos α cos  - sin α sin 
4) cos (α - ) = cos α cos  + sin α sin 
5) tan (α + ) =
tan   tan 
1  tan  tan 
6) tan (α - ) =
tan   tan 
1  tan  tan 
Double Angle Formulas
1) sin (2) = 2 sin  cos 
2) cos (2) = cos2  - sin2 
4) cos (2) = 2 cos2  - 1
5) tan (2) =
7) cos2 θ =
3) cos (2) = 1 – 2 sin2 
2 tan 
1  tan 2 
6) sin2 θ =
1  cos 2
2
1  cos 2
2
Half Angle Formulas
1) sin

2

1  cos 
2
2) cos

2

1  cos 
2
3) tan
Where the + or – sign is determined by the quadrant of the angle

2

1  cos 
1  cos 

2
Product-to-Sum Formulas
sin  sin  
1
cos     cos   
2
sin  cos  
cos  cos  
1
cos     cos   
2
1
sin      sin    
2
Sum-to-Product Formulas
sin   sin   2 sin
 
cos   cos   2 cos
2
cos
 
2
 
cos
2
 
2
sin   sin   2 sin
 
cos   cos   2 sin
2
cos
 
Law of Sines
Law of Cosines
sin A sin B sin C


a
b
c
c2 = a2 + b2 – 2ab cos C
b2 = a2 + c2 – 2ac cosB
a2 = b2 + c2 – 2bc cosA
Complex Roots Formula
 
2k 
2k 

z k  n r cos 0 
  i sin  0 
 where k = 0, 1, 2, …, n – 1
n 
n 
n
 n
2
 
sin
2
 
2