Download 5.5 – Multiple Angle and Product to Sum Identities Solving a Multiple

5.5 – Multiple Angle and Product to Sum Identities
Solving a Multiple Angle Identity
cos 2 x  cos x  0
2 cos 2 x  1  cos x  0
2 cos 2 x  cos x  1  0
(2 cos x  1)(cos x  1)  0
1
or  1
2

5
x   2 n,
 2 n,   2 n
3
3
cos x 
Using Double-Angle Formulas to Analyze Graphs
Use a double-angle formula to rewrite the equation. Then sketch the graph of the equation over
the interval [0, 2 ]
g ( x)  3  6sin 2 x
g ( x)  3(1  2sin 2 x)
g ( x)  3cos(2 x)
Evaluating Functions Involving Double Angles
Find sin(2x), cos(2x), tan(2x)
3

sin x  , 0  x 
5
2
4
5
 3   4  24
sin(2 x)  2sin x cos x  2     
 5   5  25
Draw a picture to find : cos x 
2
4
cos(2 x)  2 cos 2 x  1  2    1
5
7
 16 
 2   1 
25
 25 
24
sin(2 x) 25 24
tan(2 x) 


7
cos(2 x)
7
25
Deriving a Triple-Angle Formula
Derive the triple angle formula for cos(3x)
cos 3x  cos(2 x  x)
 cos 2 x cos x  sin 2 x sin x
 (2 cos 2 x  1) cos x  (2sin x cos x)sin x
 2 cos3 x  cos x  2sin 2 x cos x
 2 cos3 x  cos x  2(1  cos 2 x) cos x
 2 cos3 x  cos x  2 cos x  2 cos3 x
 4 cos3 x  3cos x
Rewrite tan 4 x as a quotient of first powers of the cosines of multiple angles.

tan 4 x  tan 2 x

2
 1  cos 2 x 


 1  cos 2 x 
1  2 cos 2 x  cos 2 2 x

1  2 cos 2 x  cos 2 2 x
1  cos 4 x
1  2 cos 2 x 
2

1  cos 4 x
1  2 cos 2 x 
2
2  4 cos 2 x  1  cos 4 x

2  4 cos 2 x  1  cos 4 x
3  4 cos 2 x  cos 4 x

3  4 cos 2 x  cos 4 x
2
Using Half-Angle Formulas
Find the exact value of cos 105 degrees.
cos105  

3
2
1
2

2 3
4

2 3
2
1  cos 210
2
Solving a Trigonometric Equation
Find all the solutions of cos 2 x  sin 2
x
2
1  cos x
cos 2 x 
2
2
2 cos x  1  cos x
cos 2 x  sin 2
2 cos 2 x  cos x  1  0
(2 cos x  1)(cos x  1)  0
1
cos x  or  1
2
 5
x  , ,
3 3
Common Mistake:
x
 cos 2 x
2
x
 1  cos x 
2 cos 2  2 

2
2


2 cos 2
Rewrite Products as Sums
Rewrite as a sum or a difference.
sin 5 x cos 3 x
1
 [sin(5 x  3 x)  sin(5 x  3 x)]
2
1
 [sin(8 x)  sin(2 x)]
2
x
on the interval from [0, 2 ]
2
Find the exact value of sin195+sin105
 195  105 
 195  105 
sin195  sin105  2sin 
 cos 

2
2




 300 
 90 
 2sin 
 cos  
 2 
 2 
 2sin150 cos 45
 1  2 
 2   

 2   2 

2
2
Solve: sin4x-sin2x=0 List all the values between 0 and Pi inclusive.
sin 4 x  sin 2 x  0
 4x  2x   4x  2x 
2 cos 
 sin 

 2   2 
 6x   2x 
2 cos   sin  
 2   2 
2 cos 3 x sin x  0
cos 3 x  0
3x 
x


2
6
x  0,
sin x  0
n


3
x  0n
n
  5
,
6 2
,
6
, ,
7 3 11
, ,
, 2
6 2 6
Example 11: Verify the Identity
sin 6 x  sin 4 x
 tan 5 x
cos 6 x  cos 4 x
 6x  4x 
 6x  4x 
2sin 
 cos 

 2 
 2 

 6x  4x 
 6x  4x 
2 cos 
 cos 

 2 
 2 
2sin 5 x cos x

2 cos 5 x cos x
sin 5 x

 tan 5 x
cos 5 x
Homework #36
pp 415-416 1-6 all, 9-37 odd, 41-77 EOO, 95-100 all