Download Trigonometric Identities

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
6-1 Inverse Trig Functions
p. 468: 1-31 odd
I. Inverse Trig Functions
-

y
2
2
1
B. y  cos x if and only if x  cos y where 0  y  
-

y
C. y  tan 1 x if and only if x  tan y where
2
2
A. y  sin 1 x if and only if x  sin y where
Find the exact value of each expression
 1
1. cos1 0
2. sin 1   
 2
 3

4. cos 1 

2


5. tan 1 1
3. tan 1 3
6. sin 1 1
Use a calculator to find each value.
 3
7. sin 1   
8. tan 1 5.1
 7
9. cos1
Find the exact value of each expression.
3 

10. sin sin 1 0.24
11. tan 1  tan

7 



12. cos 1  cos 
3



3
5
6-2 Inverse Trig Functions Continued
p. 474:1-41 odd
I. Inverse Trig Functions


y
2
2
1
B. y  sec x if and only if sec y  x and 0  y  
A. y  csc 1 x if and only if csc y  x and
C. y  cot 1 x if and only if cot y  x and 0  y  
Find the exact value of each expression.
 1 

3
2 



1. cos sin 1
2.
tan
sin


 2 
2






4. csc cos 1 0


5. sin tan 1 4

2 

6. cot 1  cot

3 

Find the exact value of each.
2 3
8. sec 1
9. cot 1  1
3
Use a calculator to find each value.
11. cot 1 6.1

 1 
3. sec cos 1  
 2 

10. csc 1 2
12. sec 1  4
4 

7. cos 1  cos

3 

Trigonometric Identities
Trig Identities Worksheet: 1-6 all, 9, 13, 15, 19
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
p. 480: 1-29 odd, omit 3
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
1
1

 2 cot 2 
1  sec  1  sec 
6. 8 csc2   3 cot 2   3  5 csc2 
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
p. 501: 1-5 odd, 13-17 odd, 21, 51, 53
I. Double Angle Identities
A) sin 2  2 sin  cos
B) cos 2  cos2   sin 2 
 2 cos2   1
 1  2 sin 2 
C) tan 2 
2 tan 
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)
p. 511: 1-29 odd
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)
p. 518: 1-19 odd, 25
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