Download Section 3.5 - Canton Local

Chapter 6
Trigonometric
Identities and
Equations
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All rights reserved
© 2010
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SECTION 6.4
Product-to-Sum and Sum-to-Product Identities
OBJECTIVES
1
2
3
Derive product-to-sum identities.
Derive sum-to-product identities.
Verify trigonometric identities involving
multiple angles.
PRODUCT-TO-SUM IDENTITIES
1
cos x cos y   cos x  y   cos x  y 
2
1
sin x sin y   cos x  y   cos x  y 
2
1
sin x cos y  sin x  y   sin x  y 
2
1
cos x sin y  sin x  y   sin x  y 
2
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EXAMPLE 2
Using a Product-to-Sum Identity
Find the exact value of sin 75º sin 15º.
Solution
Use the identity
1
sin x sin y  cos  x  y   cos  x  y   .
2
1
sin 75ºsin15º  cos  75º 15º   cos  75º 15º  
2
1
  cos60º  cos90º 
2
1 1
 1
   0 
2 2
 4
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SUM-TO-PRODUCT IDENTITIES
 x  y
 x  y
cos x  cos y  2 cos 
cos 

 2 
 2 
 x  y  x  y
cos x  cos y  2 sin 
sin 

 2   2 
 x  y
 x  y
sin x  sin y  2 sin 
cos 


 2 
 2 
 x  y
 x  y
sin x  sin y  2 sin 
cos 

 2 
 2 
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EXAMPLE 3
Using Sum-to-Product Identities
Write each expression as a product of two
trigonometric functions and simplify where
possible.
a. sin 4  sin 6
b. cos65º  cos55º
Solution
a. Use the identity
 x y  x y
sin x  sin y  2sin 
 cos 
.
 2   2 
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EXAMPLE 3
Using Sum-to-Product Identities
Solution continued
 4  6
sin 4  sin 6  2sin 
 2

 4  6 
 cos 


 2 
 2 
 10 
 2 sin 
cos 


 2 
 2 
 2sin    cos  5 
 2sin  cos 5
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EXAMPLE 3
Using Sum-to-Product Identities
Solution continued
b. Use the identity
 x  y
 x  y
cos x  cos y  2 cos 
cos 
.


 2 
 2 
 65º 55º 
 65º 55º 
cos 65º  cos 55º  2 cos 
cos 





2
2
 2 cos60º cos5º
 1
 2   cos 5º
 2
 cos5º
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EXAMPLE 4
Expressing a Sum of a Sine and a Cosine
as a Product
Write sin 5θ + cos 3θ as a product of two
trigonometric functions.
Solution
Use the identity
 x  y
 x  y
cos x  cos y  2 cos 
cos 
.


 2 
 2 
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EXAMPLE 4
Expressing a Sum of a Sine and a Cosine
as a Product
Solution continued
sin 5  cos 3


 cos   5   cos 3
2





5


3


5


3

2

2
 2cos 
 cos 
2
2










 2cos     cos   4 
4

4

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




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EXAMPLE 5
Verifying an Identity
sin 5  sin 9
Verify the identity
 cot 2 .
cos5  cos9
Solution
5  9
5  9
2sin
cos
sin 5  sin 9
2
2

cos5  cos9 2sin 5  9 sin 5  9
2
2
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EXAMPLE 5
Verifying an Identity
Solution continued

2 sin 7 cos  2 
2 sin 7 sin  2 
cos 2 

 sin 2 
cos 2

sin 2
 cot 2
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