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Pre-Calculus 12A
Section 6.2, 6.3, 6.4
Sum, Difference, and Double-Angle Identities
The sum and difference identities are used to simplify expressions and to determine the exact trigonometric
values of some angles.
SUM IDENTITIES
EXAMPLES
sin(A  B )  sin A cos B  cos A sin B
sin(12  23)  sin12 cos 23  cos 12 sin 23
cos(A  B )  cos A cos B  sin A sin B




  
cos     cos cos  sin sin
6
4
6
4
6 4
tan 40  tan 25
tan(40  25) 
1  tan 40 tan 25
tan(A  B ) 
tan A  tan B
1  tan A tan B
DIFFERENCE IDENTITIES
EXAMPLES
sin(A  B )  sin A cos B  cos A sin B
sin(52  33)  sin 52 cos 33  cos 52 sin33
cos(A  B )  cos A cos B  sin A sin B




  
cos     cos cos  sin sin
3
4
3
4
3
4


tan70  tan35
tan(70  35) 
1  tan70 tan35
tan A  tan B
tan(A  B ) 
1  tan A tan B
Example 1: Simplify Expressions Using Sum and Difference Identities
Write each expression as a single trigonometric function and determine the exact value for each expression.
cos 290 cos80  sin290 sin80
 5 
 
 5    
b. sin 
 cos    cos 
 sin  
 12 
3
 12   3 
a.
Solution:
cos 290 cos80  sin290 sin80
 5 
 
 5    
sin 
 cos    cos 
 sin  
 12 
3
 12   3 
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Pre-Calculus 12A
Section 6.2, 6.3, 6.4
Example 2: Determine Exact Trigonometric Values for Angles
Find the exact value for each of the following:
a.
sin(15)
 5 
b. cos  
 12 
Solution:
sin(15)
 5 
cos 

 12 
Example 3: Prove an Identity Using the Compound Angle Identities
Prove each of the following identities for all permissible values of the variable(s).


sin   x   cos x
2





b. sin  x    cos  x    sin x
3
6


sin(a  b )
 tan a  tan b
c.
cos a cos b
a.
Solution:
a.


sin   x   cos x
2

LHS


sin   x 
2

RHS
cos x
2
Pre-Calculus 12A
Section 6.2, 6.3, 6.4
b.




sin  x    cos  x    sin x
3
6


LHS




sin  x    cos  x  
3
6



RHS
sin x
c.
LHS
sin(a  b )
cos a cos b
sin(a  b )
 tan a  tan b
cos a cos b
RHS
tan a  tan b
Example 4: Solve an Equation Involving the Compound Angle Identities
Solve sin x cos

6
 cos x sin

6

1
,0  x  2
2
Solution:
sin x cos

6
 cos x sin

6

1
,0  x  2
2
3
Pre-Calculus 12A
Section 6.2, 6.3, 6.4
DOUBLE ANGLE IDENTITIES
EXAMPLES
sin 2A  2sin A cos A
sin

4
 2sin

8
cos

8
cos 2A  cos A  sin A
2
2
cos 2A  2cos 2 A  1
cos 140  cos2 70  sin2 70
cos 2A  1  2sin2 A
cos 140  2cos 2 70  1
cos 140  1  2sin2 70
tan 2A 
2 tan A
1  tan2 A
tan

6
2 tan

1  tan

12
2

12
Example 5: Simplify Expressions Using Double-Angle Identities
Write each expression as a single trigonometric function and determine the exact value for each expression.
a.
cos2
b. 2sin

3

12
 sin2
cos


3
12
Solution:
a.
cos
2

3
 sin
2

3
b.
2sin

12
cos

12
4
Pre-Calculus 12A
Section 6.2, 6.3, 6.4
Example 6: Simplify Expressions Using Identities & Recognizing Non-Permissible Values
Consider the expression
1  cos 2x
sin 2x
a. Determine the non-permissible values for the expression.
b. Simplify the expression.
Solution:
a.
sin2x  0
This means that 2sin x cos x  0
so sin x  0
cos x  0
sin x  0 when x   n , n  I
cos x  0 when x 
b.

2
  n,n  I
Therefore combining these values means x 
n
2
,n  I
1  cos 2x
sin 2x
Example 7: Prove an Identity Using the Double Angle Identity
Prove the following identities for all permissible values of x.
a.
sin 2x
 tan x
1  cos 2x
b.
cos 2x
 cos x  sin x
cos x  sin x
c.
tan2x  2tan2x sin2 x  sin2x
d.
1  cos 2x
 sec x
sin x sin 2x
5
Pre-Calculus 12A
Section 6.2, 6.3, 6.4
Solution:
a.
sin 2x
 tan x
1  cos 2x
LHS
sin 2x
1  cos 2x
b.
sin 2x
 tan x
1  cos 2x
RHS
tan x
cos 2x
 cos x  sin x
cos x  sin x
LHS
cos 2x
cos x  sin x
cos 2x
 cos x  sin x
cos x  sin x
RHS
cos x  sin x
6
Pre-Calculus 12A
c.
Section 6.2, 6.3, 6.4
tan2x  2tan2x sin2 x  sin2x
LHS
tan2x  2tan2x sin2 x
d.
tan2x  2tan2x sin2 x  sin2x
RHS
sin2x
1  cos 2x
 sec x
sin x sin 2x
LHS
1  cos 2x
sin x sin 2x
1  cos 2x
 sec x
sin x sin 2x
RHS
sec x
7
Pre-Calculus 12A
Section 6.2, 6.3, 6.4
Example 8: Solving Equations Using the Double Angle Identity
Solve for  in general form and then in the specified domain. Use exact solutions when possible.
a.
cos 2  1  cos   0, 0    2
b. 1  cos2   3sin  2,      
c. sin2  sin  6cos   3;  180    180
Solution:
a.
cos 2  1  cos   0, 0    2
b. 1  cos2   3sin  2,      
c. sin2  sin  6cos   3;  180    180
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