Section 5.4 Notes Page 1
5.4 Product-to-Sum and Sum-to-Product Formulas
Product-to-Sum Formulas
1
cos( x  y)  cos( x  y)
2
1
cos x cos 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
sin x sin y 
EXAMPLE: Simplify: sin(6 ) sin(4 ) using a product-to-sum formula.
We will use the first formula sin x sin y 
1
cos( x  y)  cos( x  y) with x  6 and y  4
2
1
cos(6  4 )  cos(6  4 )
2
Now simplify.
1
cos(2 )  cos(10 )
2
This is as far as we can go.
EXAMPLE: Simplify: cos(3 ) cos( ) using a product-to-sum formula.
We will use the formula cos x cos y 
1
cos( x  y)  cos( x  y) with x  3 and y   .
2
1
cos(3   )  cos(3   )
2
Now simplify.
1
cos(2 )  cos(4 )
2
This is as far as we can go.
EXAMPLE: Simplify: sin(3 ) cos(5 ) using a product-to-sum formula.
We will use the formula sin x cos y 
1
sin( x  y)  sin( x  y) with x  3 and y  5 .
2
1
Now simplify.
sin(3  5 )  sin(3  5 )
2
1
sin(2 )  sin(8 )
We will use the identity sin(2 )  sin 2 .
2
1
 sin(2 )  sin(8 ) or 1 sin(8 )  sin(2 ) is as far as we can go.
2
2
EXAMPLE: Find the exact value of cos
We will use the formula cos x sin y 
1
sin( x  y)  sin( x  y) with x  5 and y   .
2
12
12
1   5  
 5  
   sin 
 
sin 

2   12 12 
 12 12 
1   
  
sin    sin  

2 2
 3 

5
using a product-to-sum formula.
sin
12
12
Section 5.4 Notes Page 2
Now simplify.
We can use our table to get the values of these trig functions.
3 1
3 2 3
1
.

1 
 
2  2 4
4
2
Sum-to-Product Formulas
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 
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 
EXAMPLE: Simplify: sin 5   sin 3  using a sum-to-product formula.
x y x y
We will use the formula sin x  sin y  2 sin 
 cos
 with x  5 and y  3 .
 2   2 
 5  3   5  3 
2 sin 
 cos

 2   2 
Now simplify.
 2   8 
2 sin   cos 
 2   2 
2 sin   cos4 
We can’t simplify this anymore, so we are done.
EXAMPLE: Simplify: cos3   cos2  using a sum-to-product formula.
Section 5.4 Notes Page 3
x y x y
We will use the formula cos x  cos y  2 cos
 cos
 with x  3 and y  2 .
 2   2 
 3  2   3  2 
2 cos
 cos

 2   2 
Now simplify.
 5    
2 cos  cos 
 2  2
We can’t simplify anymore, so we are done.
EXAMPLE: Simplify: cos4   cos7  using a sum-to-product formula.
x y x y
We will use the formula cos x  cos y  2 sin 
 cos
 with x  4 and y  7 .
 2   2 
 4  7   4  7 
 2 sin 
 sin

 2   2 
Now simplify.
 11    3 
 2 sin 
 sin 

 2   2 
  3 
 3 
We can use the identity sin 
   sin   .
 2 
 2 
 11 
 3 
 2 sin 
   sin  
 2 
 2 
 11   3 
2 sin 
 sin  
 2   2 
EXAMPLE: Find the exact value of sin 15   sin 75  using a sum-to-product formula.
x y x y


We will use the formula sin x  sin y  2 sin 
 cos
 with x  15 and y  75 .
 2   2 
 15   75 
2 sin 
2

  15   75 
 cos
2
 
2 sin 45  cos 30  
2
2 3
6
.


2 2
2



Simplify.
From here we can use our table to get the exact values.