7-3:
Sum and Difference
Identities
Objectives
Use the sum and difference identities for
the sine, cosine and tangent functions.
Sum and Difference Identities for Cosine
If α and β represent the measures of two angles,
then the following identities hold for all values
of α and β
cos(   )  cos  cos 
Notice the difference!
sin  sin 
Example
Show by producing a counterexample that
cos(x-y)≠cosx-cosy.
Let x=π/4 and y=π/4.
?
cos(π/4 – π/4) = cos π/4 – cos π/4
?
cos 0 = √2/2 – √2/2
1≠0
Example
Use the sum or difference identity to find the exact
value of cos 75°.
You could have also
used 135° - 60°.
cos(75°)
= cos(30°+ 45°)
= cos30°cos45° – sin30°sin45°
 
2  1  2 
  3 

2 
2 
2 
 2 
  6    2 
4 
4

If you use the calculator, you
6 2

4
will get a decimal
approximation!!!
Sum and Difference Identities for Sine
If α and β represent the measures of two angles,
then the following identities hold for all values
of α and β
sin(   )  sin  cos   cos  sin 
Now the signs match!
Example
Find the value of sin(x+y) if 0<x<π/2,
0<y<π/2, sinx=4/5 and siny=5/13.
sin(x+y) = sinxcosy+cosxsiny
= (4/5)(12/13) + (5/13)(3/5)
= 48/65 + 15/65
= 63/65
Sum and Difference Identities for Tangent
If α and β represent the measures of two angles,
then the following identities hold for all values
of α and β
tan   tan 
tan(   ) 
1 tan  tan 
Notice the signs now!!!
Example
Use the sum or difference identity to find the exact
value of tan 255°.
You could have also
used 210° + 45°.
tan(255°)
= tan(225 + 30°)
3 3
3
3  3


1  (1)  3  1  3 3 3  3
 3
3
1 3

tan 225  tan 30
1  tan 225 tan 30

3 3 3 3 9 6 3 3


93
3 3 3 3

12  6 3
 2 3
6
1 3
If you use the calculator, you
will get a decimal
approximation!!!
Example
Verify that sec(π+A) = - secA is an identity.
Homework
7-3: p. 442
#15-24 multiples of 3
#26-30
#34-38