Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.6
Page 1 / 2
§7.6 – Sum-To-Product ; Product-To-Sum
Product-To-Sum Formulas:
sin  A  sin B  
1
cos A  B   cos A  B 
2
1
cos A  cosB   cos A  B   cos A  B 
2
1
sin  A  cosB   sin  A  B   sin  A  B 
2
Practice w/ Product-To-Sum Formulas
Basically, we want to make sure that we can use these to translate expressions that contain no
multiplication. We don't need exact (or even approximate) answers
Ex:
1
1
sin 4   sin 3   cos4  3   cos4  3   cos   cos 7 
2
2
Establishing the Product-To-Sum Formulas
cos A  cosB, start with
cos     cos  cos   sin  sin 
cos     cos  cos   sin  sin 
For
Add together (in order to get rid of the sinsin's)
cos    cos    =
cos  cos   sin  sin   cos  cos   sin  sin 
(the sinsin's will cancel, leaving:)
cos  cos   cos  cos  , meaning that we've established:
cos    cos     2 cos  cos  , which we can now treat as an equation
cos    cos   
cos  cos  
2
sin  A  sin B , start with
cos     cos  cos   sin  sin 
cos     cos  cos   sin  sin 
For
Subtract from each other (in order to get rid of the coscos's)
cos    cos    , etc
sin  A  cosB , start with
sin      sin  cos   sin  cos 
sin      sin  cos   sin  cos  (since we need the mixed sincos terms)
For
Add to each other (in order to get rid of the sinBcosA's)
sin      sin    
= sin  cos   sin  cos  sin  cos   sin  cos 
Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.6
Page 2 / 2
= sin  cos  sin  cos 
= 2 sin  cos 
Therefore
sin      sin      2 sin  cos 
sin      sin    
sin  cos  
2
Sum-To-Product Formulas:
 A B 
 A B 
sin A  sin B  2 sin 
  cos

 2 
 2 
 A B 
 A B 
sin A  sin B  2 sin 
  cos

 2 
 2 
 A B 
 A B 
cos A  cos B  2 cos
  cos

 2 
 2 
 A B 
 A B 
cos A  cos B  2 sin 
  sin 

 2 
 2 
Practice w/ Sum-To-Product Formulas
Basically, we want to make sure that we can use these to translate expressions into new
expressions, that contain no addition/subtraction.
We don't need exact (or even approximate) answers, we just need to do the rewrite
 2  5 
 2  5 
 7 
  3 
Ex: sin 2   sin 5   2 sin 
  cos
 = 2 sin    cos

 2 
 2 
 2 
 2 
(In this case, we could also have done:
 2  5 
 5  2 
 7 
 3 
sin 2   sin 5  = sin 5   sin 2  = 2 sin 
  cos
 = 2 sin    cos  )
 2 
 2 
 2 
 2 
Establishing Sum-To-Product Identities
The keys to these are
1. Realizing that we're starting with an identity we want to establish, so we can choose either
side
2. Using the product-to-sum identities to split the multiplication on the left-hand-side up (into
addition/subtraction)
Establishing Identities Using Sum-To-Product and Product-To-Sum
Basically, look for places where you can substitute one of the identities for an addition,
subtraction, or multiplication