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Sum and Difference
Identities
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 )
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.
and
cos   
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
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