Unit 5.3

1. 5.3
Sum and
Difference
Identities
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Cosine of a Difference
 Cosine of a Sum
 Sine of a Difference or Sum
 Tangent of a Difference or Sum
 Verifying a Sinusoid Algebraically
… and why
These identities provide clear examples of how different
the algebra of functions can be from the algebra of real
numbers.
Copyright © 2011 Pearson, Inc. Slide 5.3 - 2

3. Cosine of a Sum or Difference
cosu  v cosucos v msinusin v
(Note the sign switch in either case.)
Copyright © 2011 Pearson, Inc. Slide 5.3 - 3

4. Example Using the Cosine-of-a-
Difference Identity
Find the exact value of cos 75º without using a calculator.
Copyright © 2011 Pearson, Inc. Slide 5.3 - 4

5. Example Using the Cosine-of-a-
Difference Identity
Find the exact value of cos 75º without using a calculator.
cos 75º  cos 45º 30º
 cos45º cos30º  sin 45º sin 30º

2
2






3
2






2
2






1
2

 

 

6  2
4
Copyright © 2011 Pearson, Inc. Slide 5.3 - 5

6. Sine of a Sum or Difference
sinu  v sinu cosv  sinv cosu
(Note that the sign does not switch in either case.)
Copyright © 2011 Pearson, Inc. Slide 5.3 - 6

7. Example Using the Sum and
Difference Formulas
Write the following expression as the sine or
cosine of an angle: sin

3
cos

4
 sin

4
cos

3
Copyright © 2011 Pearson, Inc. Slide 5.3 - 7

8. Example Using the Sum and
Difference Formulas
Write the following expression as the sine or
cosine of an angle: sin
Recognize sin

3
cos

4

3
cos
 sin

4

4
 sin
cos

3

4
cos

3
as the sin(u  v).
sin

3
cos

4
 sin

4
cos

3
 sin

3


4

 

 
 sin
7
12
Copyright © 2011 Pearson, Inc. Slide 5.3 - 8

9. Tangent of a Difference of Sum
tanu  v
sinu  v
cosu  v

sinu cosv  sinv cosu
cosu cosvmsinusinv
or
tanu  v
tanu  tanv
1mtanu tanv
Copyright © 2011 Pearson, Inc. Slide 5.3 - 9

10. Example Expressing a Sum of
Sinusoids as a Sinusoid
Find values for a, b, and h so that for all x,
4 cos2x  7sin2x  asin b x  h    .
Copyright © 2011 Pearson, Inc. Slide 5.3 - 10

11. Example Expressing a Sum of
Sinusoids as a Sinusoid
4 cos2x  7sin2x  asin b x  h   

is a sinusoid with period
2
2
  , so b  2
4 cos2x  7sin2x  asin 2 x  h   

 asin2x  2h
 asin2x cos2h  acos2x sin2h
4 cos2x  7sin2x  asin2hcos2x  acos2hsin2x,
so 4  asin2h and  7  acos2h
Copyright © 2011 Pearson, Inc. Slide 5.3 - 11

12. Example Expressing a Sum of
Sinusoids as a Sinusoid
so 4  asin2h and  7  acos2h
sin2h  
4
a
and cos2h  
7
a
so tan2h 
4 a
7 a

4
7
so possible values of 2h are tan1 4
7

 

 
 n
Copyright © 2011 Pearson, Inc. Slide 5.3 - 12

13. Example Expressing a Sum of
Sinusoids as a Sinusoid
To find a, we use sin2h  
4
a
and cos2h  
7
a
Because a is the hypotenuse,
a  72
 42
 65
Thus,


4 cos2x  7sin2x  65 sin 2 x 
1
2
tan1 4
7

 

 


2
 

 

Copyright © 2011 Pearson, Inc. Slide 5.3 - 13

14. Example Expressing a Sum of
Sinusoids as a Sinusoid
Support Graphically
The graphs of
y1  4 cos2x  7sin2x
and


y2  65 sin 2 x 
1
2
tan1 4
7

 

 


2
 

 

are identical.
Copyright © 2011 Pearson, Inc. Slide 5.3 - 14

15. Example Expressing a Sum of
Sinusoids as a Sinusoid
Interpret
The values are a  65, b  2, and h 
1
2
tan1 4
7

 

 

2
(other values are possible).
The amplitude is 65, the peridod is  , and
the phase shift is
1
2
tan1 4
7

 

 

2
radians.
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16. Quick Review
Express the angle as a sum or difference of special angles
(multiples of 30o, 45o,  /6, or  /4). Answers are not unique.
1. 15o
2.  /12
3. 75o
Tell whether or not the identity f (x  y)  f (x)  f (y)
holds for the function f .
4. f (x)  16x
5. f (x)  2ex
Copyright © 2011 Pearson, Inc. Slide 5.3 - 16

17. Quick Review Solutions
Express the angle as a sum or difference of special angles
(multiples of 30o, 45o,  /6, or  /4). Answers are not unique.
1. 15o 45o  30o
2.  /12

3


4
3. 75o 30o  45o
Tell whether or not the identity f (x  y)  f (x)  f (y)
holds for the function f .
4. f (x)  16x yes
5. f (x)  2ex no
Copyright © 2011 Pearson, Inc. Slide 5.3 - 17