Download Lesson 10 Sum Difference And Double Angle Identities

6.2 Sum, Difference, and Double Angle Identities
The expressions sin (A + B) and cos (A + B) occur frequently enough in
math that it is necessary to find expressions equivalent to them that
involve sines and cosines of single angles. So….
Does
sin (A + B) = Sin A + Sin B
Let A = 30 and B = 60
sin(30  60 )  sin 30  sin 60
1
3
sin(90 )  
2 2
1 3
1
2
Math 30-1
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Sum and Difference Identities Formula Sheet
sin (A + B) = sin A cos B + cos A sin B
sin (A - B) = sin A cos B - cos A sin B
cos (A + B) = cos A cos B - sin A sin B
cos (A - B) = cos A cos B + sin A sin B
tanA  tanB
tan(A  B) 
1  tan Atan B
tanA  tanB
tan(A  B) 
1  tan Atan B
Math 30-1
2
Simplifying Trigonometric Expressions
1.
Express cos 1000 cos 800 + sin 800 sin 1000 as a
trigonometric function of a single angle.
This expression has the same pattern as cos (A - B),
with A = 1000 and B = 800.
cos 100 cos 80 + sin 80 sin 100 = cos(1000 - 800)
= cos 200
2.




Express sin cos  cos sin as a single trig function.
3
6
3
6
This expression has the same pattern as sin(A - B),


with A  and B  .
3
6
 


sin cos  cos sin  sin    
3 6
3
6
3
6




Math 30-1
 sin

6
3
Determine Exact Values using Sum or Difference Identities
1. Determine the exact value for sin 750.
Think of the angle measures that produce exact values:
300, 450, and 600.
Use the sum and difference identities - which angles,
used in combination of addition or subtraction, would
give a result of 750?
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sin 750 = sin(300 + 450)
= sin 300 cos 450 + cos 300 sin 450
2    3
1
2


     

2 2
2
 2


2 6
4
Math 30-1
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Finding Exact Values
2. Determine the exact value for cos 150.
cos 150 = cos(450 - 300)
= cos 450 cos 300 + sin 450 sin300
 2
1 
3   2
  
    
2 
2   2
 2
6 2
4

5
3. Find the exact value for sin .
12
5
 
sin
 sin(  )
12
4 6




 sin cos  cos sin
4
6
4
6
 2
1 
3   2
  
    
2 
2   2
 2

6 2
4
Math 30-1

4

3 12

3
4 12
 2

6 12

5
Determine the exact value of tan105
tan105  tan 135  30

tan135  tan 30

1  tan135 tan 30


1
 1 

3 


 1 

 1   1 
3



1


1


3

 1 1

3







Determine a common
denominator
Combine terms in numerator
Rationalize the denominator
or…….
 3


 3
Math 30-1
 3 1 
 

 1 3 
6
Using the Sum and Difference Identities




Prove cos      sin.
2


cos    
2

cos

cos   sin


 sin 
sin 
sin
2
2
(0) (cos )  (1)(sin)
(1)(sin  )
 sin 
L.S. = R.S.
Math 30-1
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Using the Sum and Difference Identities
2
4
Given sin A= and cos B = , where A and B are
3
5
acute angles, determine the exact value of sin(A + B).
sin(A + B)=sin A cos B  cos A sin B
 2  4   5   3 
     
  

 3  5   3   5 
A
x
y
r
5
2
3
B
4
3
5
8+3 5

15
8+3 5
Therefore, sin(A + B)=
.
15
Math 30-1
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Double Angle Identities
The identities for the sine and cosine of the sum of two
numbers can be used, when the two numbers A and B
are equal, to develop the identities for sin 2A and cos 2A.
cos 2A = cos (A + A)
sin 2A = sin (A + A)
= sin A cos A + cos A sin A
= cos A cos A - sin A sin A
= 2 sin A cos A
= cos2 A - sin2A
Identities for sin 2A and cos 2A:
cos 2A = cos2A - sin2A
cos 2A = 2cos2A - 1
cos 2A = 1 - 2sin2A
sin 2A = 2sin A cos A
Math 30-1
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Double Angle Identities
Express each in terms of a single trig function.
a) 2 sin 45° cos 45 °
sin 2x = 2sin x cos x
sin 2(45 ° ) = 2sin 45 ° cos 45 °
= sin 90 °
Math 30-1
b) cos2 5 - sin2 5
cos 2x = cos2 x - sin2 x
cos 2(5) = cos2 5 - sin2 5
= cos 10
10
Double Angle Identities
Verify the identity
tan A 
tan A
L.S = R.S.
1  cos 2 A
.
sin 2A
1  (cos 2 A  sin 2 A)
2sin A cos A
1  cos 2 A  sin 2 A
2sin A cos A
sin 2 A  sin 2 A
2sin A cos A
2sin 2 A
2sin A cos A
sin A
cos A
Math 30-1
tan A
11
Double Angle Identities
sin 2x
.
Verify the identity tan x 
1  cos 2x
tan x
2sin x cos x
1  2 cos 2 x  1
2sin x cos x
2 cos 2 x
sin x
cos x
tan x
L.S = R.S.
Math 30-1
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Identities
Prove
2tanx
1  tan2 x

sin2x.
2sin x cos x
2sin x
cos x
sec 2 x
2sin x
cos x
1
cos 2 x
2sin x cos 2 x

cos x
1
2sin x cos x
L.S. = R.S.
Math 30-1
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Textbook p. 306 – 308
Low: (Basic Drill and Practice)
1 – 8, 10, 11, 12
Medium: (Problem Solving and Word Problems)
9, 13, 15, 16, 19, 20
High: (Extension and Higher Level)
14, 17, 18, 21, 22, 24
Math 30-1
14