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Sum and Difference
Identities
Section 5.2
Objectives
• Apply a sum or difference identity to
evaluate the sine or cosine of an angle.
Sum and Difference Identities
sin(a  b )  sin(a ) cos( b )  sin(b ) cos( a )
The identity above is a short hand method for writing two
identities as one. When these identities are broken up, they look
like
sin(a  b )  sin(a ) cos( b )  sin(b ) cos( a )
sin(a  b )  sin(a ) cos( b )  sin(b ) cos( a )
cos( a  b )  cos( a ) cos( b )  sin(a ) sin(b )
The identity above is a short hand method for writing two
identities as one. When these identities are broken up, they
look like
cos( a  b )  cos( a ) cos( b )  sin(a ) sin(b )
cos( a  b )  cos( a ) cos( b )  sin(a ) sin(b )
Use a sum or difference
identity to find the exact
value of
 

sin 
 12 
In order to answer this question, we need to find two of the angles
that we know to either add together or subtract from each other that
will get us the angle π/12. Let’s start by looking at the angles that we
know:

2

6 12

4

3 12
2 8

3
12
5 10

6
12
3

4 12

6

2 12
3 9

4 12
12

12

continued on next slide
Use a sum or difference
identity to find the exact
value of
 

sin 
 12 
We have several choices of angles that we can subtract from each
other to get π/12. We will pick the smallest two such angles:
2

6 12

3

4 12

Now we will use the difference formula for the sine function to
calculate the exact value.
sin(a  b )  sin(a ) cos( b )  sin(b ) cos( a )
continued on next slide
Use a sum or difference
identity to find the exact
value of
 

sin 
 12 
For the formula a will be
This will give us
3

4 12

and b will be
2

6 12


 3 2 
sin   sin


12
12
12
 


 




sin    sin  cos    sin  cos  
4 6
4
6
6
4
   2  3   1  2 

sin   
  




12
2
2
  

  2  2 
continued on next slide
Use a sum or difference
identity to find the exact
value of
 

sin 
 12 
For the formula a will be
This will give us
3

4 12

and b will be
 
2 3
2

sin  

4
4
 12 
 
2 3 2

sin  
4
 12 

 
2 3 1

sin  
4
 12 

2

6 12

Simplify


sin x  
4

using a sum or difference
identity
In order to answer this question, we need to use the sine formula for
the sum of two angles.
sin(a  b )  sin(a ) cos( b )  sin(b ) cos( a )
For the formula a will be
x
and b will be

4



sin x    sin( x ) cos   sin  cos( x )
4

4
4
continued on next slide
Simplify


sin x  
4

using a sum or difference
identity



sin x    sin( x ) cos    sin  cos( x )
4

4
4
 2  2


 cos( x )
sin x    sin( x )



4

 2   2 
   2 

sin( x )  cos( x )
sin x    

4  2 

Simplify


cos x  
2

using a sum or difference
identity
In order to answer this question, we need to use the cosine formula for
the difference of two angles.
cos( a  b )  cos( a ) cos( b )  sin(a )cin (b )
For the formula a will be
x
and b will be

2



cos x    cos( x ) cos   sin( x ) sin 
2

2
2
continued on next slide
Simplify


cos x  
2

using a sum or difference
identity



cos  x    cos( x ) cos    sin( x ) sin 
2

2
2

cos  x    cos( x )(0)  sin( x )(1)
2


cos  x    sin( x )
2

Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
1.
cos   
2.
sin   
and
For this problem, we have two angles.
We do not actually know the value of
either angle, but we can draw a right
triangle for each angle that will allow us
to answer the questions.
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
and
32  b 2  7 2
Triangle for α
9  b 2  49
7
b
b 2  40
b   40
length is positive
α
3
b  40
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
Triangle for β
and
a 2  4 2  52
a 2  16  25
5
4
a2  9
a  9
length is positive
β
a
a 3
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
1.
cos   
Now that we have our triangles, we
can use the cosine identity for the
sum of two angles to complete the
problem.
and
7
40
α
3
5
4
cos(    )  cos( a ) cos(  )  sin( a ) sin(  )
40  4 
 3  3  
 
cos(    )        
7  5 
 7  5  
β
3
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
1.
cos   
Now that we have our triangles, we
can use the cosine identity for the
sum of two angles to complete the
problem.
7
40
Note: Since α is in quadrant
Iv, the sine value will be
negative
α
3
5
cos(    )  cos( a ) cos(  )  sin( a ) sin(  )
40  4 
 3  3  
 
cos(    )        
7  5 
 7  5  
and
β
4
Note: Since β is in quadrant
II, the cosine value will be
negative
3
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
1.
cos   
40  4 
 3  3  
 
cos(    )        
7  5 
 7  5  
9  4 40 

cos(    )  
  
35 
35 
cos(    ) 
 9  4 40
35
and
7
40
α
3
5
4
β
3
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
1. cos   
and
 9  4 40
cos(    ) 
35
While we are here, what are the possible quadrants in which the angle α+β can
fall?
In order to answer this question, we need to know if cos(α+β) is positive or
negative. We can type the value into the calculator to determine this. When
we do this, we find that cos(α+β) is positive. The cosine if positive in
quadrants I and IV. Thus α+β must be in either quadrant I or IV. We cannot
narrow our answer down any further without knowing the sign of sin(α+β).
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
2.
sin   
Now that we have our triangles, we
can use the cosine identity for the
sum of two angles to complete the
problem.
7
40
Note: Since α is in quadrant
Iv, the sine value will be
negative
α
3
5
sin(   )  sin( a ) cos(  )  cos( a ) sin(  )

40  3   3  4 
      
sin(   )   
 5   7  5 
7


and
β
4
Note: Since β is in quadrant
II, the cosine value will be
negative
3
continued on next slide
Find the exact value of the following
trigonometric functions below given
3
cos    and  is in quadrant IV
7
4
sin   and  is in quadrant II
5
2.
sin   
7
40
α

40  3   3  4 
      
sin(   )   
 5   7  5 
7


3 40 12
sin(   ) 

35
35
3 40  12
sin(   ) 
35
and
3
5
4
β
3