Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.5
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Double & Half Angle Formulas
Double Angle Formulas for Sine, Cosine
Fill in these formulas
1)
sin 2    
2)
cos2    
(Fill out all three versions of this)
Using the Double-Angle Formulas
Find the exact value of sin(2θ), and cos(2θ), each given expression, assuming that 0 ≤θ≤2π
3) sin    
7 
, 
25 2
(You may need to draw out a triangle & fill in the missing side)
4) csc      5 , cos   0
3

5) cos   ,0   
5
2
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.5
Page 2 / 4
4
3
6) tan   ,    
3
2
1
3
7) tan   ,    
2
2
Establishing the Double-Angle Formulas
8) Establish/prove the formula for sin 2 , starting with sin      cos  sin   sin  cos 
9) Establish/prove the formula for the three versions of a formula for cos  2 , starting with the sum
formula for cosine.
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.5
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10) Find a formula for tan 2 in terms of the tangent function (i.e., a formula that only has tangent
in it – no sine, cosine, cotangent, etc)
Half Angle Formulas for Sine, Cosine
Fill in these formulas
 
11) Start with the identity that cos  2  1  2 sin2  to find a formula for sin   .
2


Hint: Notice that θ is ½ of 2θ – let's say that θ = , and then try to isolate the term w/
2
2
12) Find the exact value of sin(15 )

 
13) Find the exact value of sin  
8

 
14) Start with the identity that cos  2  2 cos2   1 to find a formula for cos   .
2
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.5
15) Find the exact value of cos(165 )

16) Find the exact value of cos  
 12 
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