Download Double-Angle Formulas

Section 5.3
Double-Angle,
Power-Reducing, and
Half-Angle Formulas
Objectives:
• Use the double-angle formulas.
• Use the power-reducing formulas.
• Use the half-angle formulas.
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Double-Angle Formulas
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Example1 Part A: Using Double-Angle Formulas
to Find Exact Values
4
If sin   and
5
 lies in quadrant II,
find the exact value of
sin 2 .
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Example 1A Continued
If sin   4 and  lies in quadrant II, find the exact value
5
of sin 2 .
sin 2  2sin  cos
x  3
y4
r 5
r 5
4

4  3 

 2    
 5  5 
24

25
x
y 4
sin   
r 5
x
3
cos   
r
5
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Example1 Part B
4
If sin   and
5
 lies in quadrant II,
find the exact value of
cos 2 .
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Example1B Continued
If sin   4 and  lies in quadrant II, find the exact value
5
of cos 2 .
cos 2  cos 2   sin 2 
2
x  3
y4
r 5
y 4
sin   
r 5
x
3
cos   
r
5
3  4

    
 5  5
9 16
 
25 25
7

25
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6
Example1 Part C: Using Double-Angle Formulas
to Find Exact Values
4
If sin   and
5
 lies in quadrant II,
find the exact value of
tan 2 .
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Example 1 C Continued
If sin   4 and  lies in quadrant II, find the exact value
5
of tan 2 .
2 tan 
tan 2 
2
1

tan

x  3
4

y4
2  
3


r 5

2
4

1   
y 4
 3
sin   
y
4
r 5
tan    
Continued
x
3
x
3
cos   
r
5
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Example 1 C Continued
Continued
2 tan 
tan 2 
1  tan 2 
4

8
8
2  


3


3  3

2 
7
4
16


1   
1
9
9
 3
8  9 

     
 3  7 
24

7
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Example 2
Find the exact value of the trigonometric
expression 2sin165 cos165
2sin165 cos165  sin[2(165)]
 sin 330
1

2
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Three Forms of the Double-Angle Formula for cos 2
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Example3: Verifying an Identity
3
sin
3


3sin


4sin

Verify the identity:
now expand using sin(A+B)
sin(2   ) 
sin 2 cos  cos 2 sin  
sin 2  2sin  cos
2sin  cos cos  (1  2sin  ) sin  
2
cos 2  1  2sin 2 
2sin  cos 2   sin   2sin 3  
2sin  (1  sin 2  )  sin   2sin 3  
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Example 3 Continued
Cont.
2sin  (1  sin 2  )  sin   2sin 3  
2sin   2sin 3   sin   2sin 3  
 3sin   4sin 
3
distribute
Combine like terms
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Half-Angle Formulas
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Example 4: Using the Half-Angle Formula
to Find an Exact Value
3
Use cos 210  
to find the exact value of cos105°.
2
 3
210
1   
cos105  cos
2 

2

2

1  cos 
cos  
2
2
210
1  cos 210
cos

2
2
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Example 4 Continued
3
Use cos 210  
to find the exact value of cos105°.
2
 3
210
1  cos 210
1   
cos

2  2
2
2


2
2
 3
1   
2 
2 3



4
2
2 3

2
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Example 5
Find sin α/2, cos α/2, and tan α/2 if
sec α = -5/4 and α lies in Quadrant II.
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Half-Angle Formulas for Tangent
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Power-Reducing Formulas - Look at but won’t assign
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Example: Reducing the Power of a Trigonometric
Function
Write an equivalent expression for sin4 x that does not
contain powers of trigonometric functions greater than 1.
2
2
1  cos 2 x  1  2cos 2 x  cos 2 x
2

4
2
sin x   sin x   
 
4
2


1 1
1  1  cos 2(2 x) 
1 1
1 2
  cos 2 x  cos 2 x   cos 2 x  

4 2
4
2

4 2
4
1 1
1
1 1
1 1
  cos 2 x  1  cos 4 x    cos 2 x   cos 4 x
4 2
8
4 2
8 8
3 1
1
  cos 2 x  cos 4 x
8 2
8
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