Download Trig Identities Notes and Practice

Trigonometric Identities
I. Pythagorean Identities
A. sin 2   cos2   1
B. tan 2   1  sec 2 
C. cot 2   1  csc 2 
II. Sum and Difference of Angles Identities
A. sin      sin  cos   cos  sin 
B. sin      sin  cos   cos  sin 
C. cos     cos  cos   sin  sin 
D. cos     cos  cos   sin  sin 
tan   tan 
E. tan     
1  tan  tan 
tan   tan 
F. tan     
1  tan  tan 
III. Double Angle Identities
A. sin 2   2 sin  cos 
B. cos2   cos 2   sin 2 
= 2 cos2  1
= 1  2 sin 2 
2 tan 
C. tan 2  
1  tan 2 
IV. Half Angle Identities

1  cos
A. sin  
2
2

1  cos
B. cos  
2
2

1  cos
C. tan  
2
1  cos
Trigonometric Identities
I. Reciprocal Identities
1
1
A) sin  
D) csc  
csc 
sin 
1
1
B) cos 
E) sec  
sec 
cos
1
1
C) tan  
F) cot  
cot 
tan 
II. Quotient Identities
sin 
A) tan  
cos
cos
B) cot  
sin 
III. Pythagorean Identities
A) sin 2   cos2   1
B) tan 2   1  sec 2 
C) cot 2   1  csc 2 
Simplify each expression.
1. csc 2   1
2. 1  sin x1  sin x
4. cos tan csc
5.
7. sin 2 x  sin 2 x cot 2 x
csc 
1  cot 2 


3. sin 2  csc 2   1
6.
1
1

2
sin  tan 2 
6-3 Trig Identities
I. Guidelines for Verifying Trig Identities
A. Begin with the most complicated side
B. Work on one side only
C. Rewrite sums or differences of quotients as one single quotient
D. Rewrite in terms of sine and cosine only
E. Factor ( GCF or Difference of Squares)
F. Multiply ( Foil or Distributive Property)
Verify each identity.
1. csc  sin   sin 2   cos2 
2. sin  cot   tan    sec 
3. csc   cot  csc   cot    1
4. csc 4   csc 2   cot 4   cot 2 
5. csc x  cot x 
7.
sin x
1  cos x
6. 8 csc 2   3 cot 2   3  5 csc 2 
1
1

 2 cot 2 
1  sec  1  sec 
6-4 Sum and Difference Formulas
p. 491: 1-23 odd, 29, 31
I. Sum and Difference Identities
A. cos    
B. cos    
C. sin     
D. sin     
E. tan     
F. tan     
Find the exact value of each expression.

1. cos
2. tan105
12
5. sin 40 cos 20 cos 40 sin 20
Find each of the following.
a) sin    
b) tan    
1

, 0 
2
2
7 
cos  
,
 0
3
2
7. sin  
8. Verify:
cos      cos 
3. sin
13
12
4. csc15
6. cos100 cos 80 sin 100 sin 80
6-5 Double-Angle and Half-Angle Identities
I. Double Angle Identities
A) sin 2  2 sin  cos
B) cos 2  cos2   sin 2 
 2 cos2   1
 1  2 sin 2 
2 tan 
C) tan 2 
1  tan 2 
II. Half Angle Identities

1  cos
A) sin  
2
2
B) cos
C) tan

2

2

1  cos
2

1  cos
1  cos
where + or – is determined by

2
Find the exact value of each expression.
a) sin 2 
1. sin  
b) cos2 
12

, 0 
13
2
 
c) sin  
2
2. tan  
15
3
,   
8
2
Use the half angle identity to find each exact value.
5
3. sin
4. tan 22.5
8
Find the exact value.

3

5. sin  2 cos 1
2 

5

6. cos sin 1 
13 

6-7 Trig Equations (I)
Solve each equation for 0    2 .
3
3
1. sin  
2. cot  
2
3
4. sin 2   
7. csc
2
2
3 2 3

2
3
5. 2 tan  1  1
  
8. cot     1
2 6
3. cos 
1
2
6. 4 sin 2   3  0
6-8 Trig Equations (II)
I. Solving Trig Equations
A) Set trig function equal to a numerical value
B) Apply trig identities to rename expression in terms of one trig function
C) Factor:
1. GCF
2. Product of binomials
D) Divide cos to produce tan
Solve for 0    2 .
1. cos2   cos  0
2. 2 sin 2   3 sin   1  0
4. 2 cos2   sin   1
5.
7. cos 2  1  sin
3 sin  cos
3. cot  1csc   1  0
6. sin 2  sin  0
8. sin tan  3 sin