Download and half-angle formulas.

Double- And Half-Angle Formulas
Double-Angle Formulas
cos2u  cos u  sin u
2
 2 cos u  1
2
 1 2 sin u
sin 2u  2 sin u cosu
2 tan u
tan 2u 
2
1  tan u
2
2
Double- And Half-Angle Formulas
Half-Angle Formulas
u
1  cosu
sin  
2
2
u
1  cosu
cos  
2
2
u 1  cosu
sin u
tan 

2
sin u
1  cosu
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.
1
1. sin 15  sin 30
2
3
1 cos 30
1

2

2
2
2 3 1


2
2
2 3

2
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.
1
2. cos 22.5 cos 45
2
2
1 cos 45
1

2

2
2

2 2 1

2
2
2 2

2
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.
1
3. tan 15  tan  30
2
3
1
1  cos 30
2


1
sin  30

2
2
2 3
   2  3

1
2
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.
1  
  
4. sin     sin   
2 6 
 12 


1  cos  

3
1
6
2

2
2
2 3 1
2

3

 
2
2
2
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.
1  5 
5
5. cos
 cos  
8
2 4 
5
2
1  cos
1
4
2


2
2
2 2
2 2 1
 

2
2
2
Evaluating Trigonometric Expressions
Evaluate expressions using double- and half-angle formulas.

1



6. tan
 tan  
8
2 4 
1  cos 
4

sin 
4
 
 
2
1
2

2
2
2 2 2
2 2 2 2




 2 1
2
2
2
2
Evaluating Trigonometric Expressions
Find the exact value of the function.
u
4

7. cos if cos u  , 0  u 
2
5
2
4
1
9
1
1  cos u
9
5
 



10
5 2
2
2
3 10

10
Evaluating Trigonometric Expressions
Find the exact value of the function.
4 
8. tan 2u if sin u  ,  u  
5 2
2 tan u

2
1  tan u
4
 4
8
2  

3  3


2
 4
1  
 3
9  16
9
5
4
tan u  
3
-3
8 9
  
3 7
3
8

 3
7

9
24

7
Evaluating Trigonometric Expressions
Verify the identity.
9. 2sin 2 x  cos 4 x  1
2
2 sin 2 x  cos 22 x   1
2
2
2 sin 2 x  1  2 sin 2 x  1
2
11
Evaluating Trigonometric Expressions
Verify the identity.
10. sin x  cos x   1  sin 2 x
2
sin x  2 sin x cos x  cos x
2
2
sin x  cos x  2 sin x cos x
1 2 sin xcos x
2
2
1 sin 2 x
Evaluating Trigonometric Expressions
Solve the equation for 0 < x < 2.
11. sin x  cos 2x
sin x  1  2 sin x
2
2 sin x  sin x  1  0
2 sin x  1  sin x  1   0
2
2 sin x  1 sin x  1
1
 5 3
sin x 
x , ,
2
6 6 2
Evaluating Trigonometric Expressions
Solve the equation for 0 < x < 2.
12. cos 2x  cos x  2  0
2 cos x  1  cos x  2  0
2
2 cos x  cos x  3  0
2 cos x  3 cos x  1   0
2
2 cos x  3 cos x  1
3
x


cos x 
2
Evaluating Trigonometric Expressions