Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sum and Difference Identities KEY
Classify each statement as true or false, then explain your reasoning.
TRUE
1) sin(90) = sin(90) + sin(0)
FALSE
3) sin(30) + sin(60) = sin(90)
FALSE
No, ½ + 3 /2  1
4) tan(45) + tan(45) = tan(90)
No, Left side = 2, right side is undefined
Yes, 1 = 1 + 0
FALSE
2) cos(90) = cos(90) + cos(0)
No, 0  0 + 1
From these examples, recognize that the distributive property does NOT apply
when using the trigonometric functions.
…So what rules do apply?
Sum and Difference Identities
sin( A  B )  sin A cos B  cos A sin B
sin( A  B )  sin A cos B  cos A sin B
tan( A  B) 
tan A  tan B
1  tan A tan B
tan( A  B) 
tan A  tan B
1  tan A tan B
cos( A  B )  cos A cos B  sin A sin B
cos( A  B )  cos A cos B  sin A sin B
Demonstrate that each identity works by substituting the given values for A and B.
A. Demonstrate
Sample: cos( A  B )  cos A cos B  sin A sin B , when A  3 and B  6
cos( A  B )  cos A cos B  sin A sin B
Write the identity:
cos( 3  6 )  cos 3 cos 6  sin 3 sin 6
Substitute:
cos2   21  
Evaluate:
0  43 
0=0
Simplify:
5)
     
3
2
3
2

3
4
3
2

3
2
©2012, TESCCC

3
2

3
2
1
2
3
4
cos( A  B )  cos A cos B  sin A sin B , when A 
cos( A  B)  cos A cos B  sin A sin B
cos( 3  6 )  cos 3 cos 6  sin 3 sin 6
cos 6  21 
3
2

3
and B 

6
 21
3
4
04/25/13
page 1 of 5
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sum and Difference Identities KEY
6)
sin( A  B )  sin A cos B  cos A sin B , when A 
sin( A  B)  sin A cos B  cos A sin B
sin( 3  6 )  sin 3 cos 6  cos 3 sin 6
sin 2 
1  34 
3
2

3
2

3
and B 

6

3
and B 

6
 21  21
1
4
1 1
7)
sin( A  B )  sin A cos B  cos A sin B , when A 
sin( A  B)  sin A cos B  cos A sin B
sin( 3  6 )  sin 3 cos 6  cos 3 sin 6
sin 6 
8)
1
2
 34 
1
2

3
2
3
2
 21  21
1
4
1
2
tan A  tan B
, when A 
1  tan A tan B
tan 3  tan 6


tan( 3  6 ) 
1  tan 3 tan 6
tan( A  B) 
tan( 6 ) 
3
3
9)


3
3
3
1 3 
3
3


3
and B 

6
and B 

6
 33 2 33 2 3


1 1
2
6
3 3
3
3
3
tan A  tan B
, when A 
1  tan A tan B
tan 3  tan 6
tan( 3  6 ) 
1  tan 3 tan 6
tan( A  B) 
tan( 2 ) 
3
3
3


3
 33 4 3 3

1 1
0
3 3
3
1  3  33
(Both sides of the equation are undefined.)
©2012, TESCCC
04/25/13
page 2 of 5
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sum and Difference Identities KEY
B. Find
Find the exact values for the trigonometric functions at 105 and 15 by writing the
angles as a sum or difference of other special values (such as 60 and 45).
Sample: cos(105)  cos(60  45)  cos 60 cos 45  sin 60 sin 45
 21 

10)
11)

3
2


6
4

3
2

6
4


2
2
2 6
4
2
2
 21 
2
2

6 2
4
2
4
tan( 105)  tan( 60  45)
tan60  tan 45
tan(60  45) 
1  tan60 tan 45

3 1
1 3

( 3  1) (1  3)

(1  3) (1  3)
42 3
 2  3
2
cos(15)  cos(60  45)
cos(60  45)  cos 60 cos 45  sin60 sin45
 21 

13)

sin(105)  sin( 60  45)
sin(60  45)  sin60 cos 45  cos 60 sin 45

12)
2
4
2
2
2
2

2
4

3
2
6
4


2
2
2 6
4
sin(15)  sin( 60  45)
sin(60  45)  sin60 cos 45  cos 60 sin 45
©2012, TESCCC

3
2


6
4

2
2
 21 
2
2

6 2
4
2
4
04/25/13
page 3 of 5
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sum and Difference Identities KEY
14)
tan( 15)  tan( 60  45)
tan60  tan45
tan(60  45) 
1  tan60 tan45


C. Find
3 1
1 3

( 3  1) (1  3)

(1  3) (1  3)
4  2 3
 2 3
2
Use the sum and difference identities to evaluate functions of other rotation angles.

R  257 , 24
25
R+S
S 35 , 54 
R and S are rotation angles (between 0 and 90) whose terminal sides
) and ( 35 , 54 ) , respectively.
intersect the unit circle at ( 257 , 24
25
15)
RS
(1, 0)
16)
17)
sin R = 24/25
cos R = 7/25
tan R = 24/7
sin S = 4/5
cos S = 3/5
tan S = 4/3
With a calculator in degree mode, solve for the measures of R and S. Then use these values
to estimate R+S and RS.
R = 73.74
R+S = 126.87
S = 53.13
RS = 20.61
Use the sum and difference identities to find the exact values for the trigonometric functions for
R+S and RS.
sin( R  S )  sinR cosS  cosR sinS
sin( R  S )  sinR cosS + cosR sinS
24 3
7
4
24 3
7
4



25  5  25  5
25 5
25 5
72
125
cos(R  S ) 
28
 125

72
125
44
125
cos(R  S ) 
cosR cosS + sinR sinS
3
7
24 4
25  5  25  5
21
125
18)
Write down the sine, cosine and tangent values for each angle.
96
 125

117
125
28
 125

100
125

4
5
cosR cosS  sinR sinS
3
7
24 4
25  5  25  5
21
125
96
75
 125
  125
  53
How can the answers to #16 be used to check the answers to #17 (using a calculator)?
sin(126.87)  0.8 = 4/5. cos(126.87)  -0.6 = -3/5.
sin(20.61)  0.352 = 44/125. cos(20.61)  0.936 = 117/125.
More exact values could be found using the store key for angles R and S, i.e. sin -1(24/25)
ENTER 73.73979529 STO ALPHA R and do the same for S, then use sin(R + S).
©2012, TESCCC
04/25/13
page 4 of 5
Precalculus
HS Mathematics
Unit: 05 Lesson: 02
Sum and Difference Identities KEY
D. Prove
Use the sum and difference identities to verify (or prove) other identities.
Sample: Verify that sin( 2  x )  cos x
(Sum/Difference Identity)
sin( 2  x )  sin 2 cos x  cos 2 sin x
 (1) cos x  (0) sin x
 cos x
19)
(Evaluate/Substitute)
(Simplify)
Prove cos( 2  x )  sin x
cos( 2  x )  cos 2 cos x  sin 2 sin x
 (0)cos x  (1)sin x
 sin x
20)
Prove sec( 2  x )  csc x
1
sec( 2  x ) 
cos( 2  x )

1
1
1


 csc x

cos 2 cos x  sin 2 sin x (0)cos x  (1)sin x sin x

21)
Prove cos(  x )  cos x
cos( x )  cos(0  x )
 cos0cos x  sin0sin x
 (1)cos x  (0)sin x
 cos x
Hint: Rewrite x as “0 – x”
22)
Prove sin( x )   sin x
sin( x )  sin(0  x )
 sin0cos x  cos0sin x
 (0)cos x  (1)sin x
  sin x
Hint: Rewrite x as “0 – x”
23)
Prove tan(  x )   tan x
tan(  x )  tan(0  x )
Hint: Rewrite x as “0 – x”

©2012, TESCCC
tan0  tan x
0  tan x
 tan x


  tan x
1  tan0 tan x 1  (0) tan x
1
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