Chapter 6 - Trig Identities and Conditional Equations_1_

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```					        Chapter 7
Lesson 3
Sum and Difference Identities
Lesson Objectives
• Use the sum and
difference identities
for sine, cosine, and
tangent.
Introduction
• The basic identities discussed so far have
involved only 1 variable such as x. In this
lesson we consider identities that involve
two variables.
Trig Values of Uncommon Angles

• The identities we will learn about today
allow us to find the trig values of some
uncommon angles.
• The “uncommon” angle must be the sum or
difference of 2 common angles.
Example 1
• Write each angle as the sum of 2 common angles:
75  30o  45o
165  45o  120o
4 3  
7/12  12  12  3  4
• Write each as the difference of 2 common angles:
15  45o  30o
105  135o  30o
4 3  
/12    
12 12 3 4
Sum & Difference Identities
Example 2
cos (x + y) = cos x  cos y  sin x  sin y
• Find the exact value of cos 75 using the sum
identity above.
cos(30 o 45 o )  cos30o  cos 45o  sin 30o  sin 45o
3 2 1 2
     
2 2 2 2
6    2
    
4    4
6 2

4
Example 3
cos (x  y) = cos x  cos y + sin x  sin y
• Find the exact value of cos (195) using the
difference identity above.
cos(240 o 45 o )  cos240  cos45  sin 240  sin 45
o       o         o        o

1 2  3 2
       
2 2   2   2
 2    6
     
4    4
 2 6

4
Example 4
sin( x  y)  sin x  cos y  cos x  sin y
• Find the exact value of sin(/12) using the
difference identity for sine.
                       
sin     sin  cos  cos  sin
3 4      3     4     3      4
3 2 1 2
        
2 2 2 2
6    2
    
4    4
6 2

4
Example 5

• Find the exact value of tan(285) using a sum.
tan 240 o  tan 45 o                  3
tan(240 45 ) 
o     o
2   3 2  3
1  tan 240 o  tan 45 o   tan 240 
o
1         2  1
2
3 1

1  3 1
1 3 1 3
      
1 3 1 3

1 3  3  3         42 3
                               2  3
1 3               2
Example 6
• Verify         cos(180 o  x)   cos x
cos180o cos x  sin 180o sin x   cos x
(1) cos x  (0) sin x   cos x
 cos x   cos x
Example 7
Find the exact value of sin(x + y), given that sin x = 2/3,
cos y = -1/4, x is an angle in Quad. II, and y is an angle
in Quad. III.                                  sin y  cos2
sin 2 2 x  cos2yx 1
1
sin( x  y)  sin x  cos y  cos x  sin y               2
 2   1  2
2

    x
sin 2 y   cos  1 1
 2   1   5
 2    1   cos  sin y             3  4 
      
3  4                x
  sin y                               15 5

 3   4   3   
sin 2 y2  
cos x
16 9
 2    1    5    15                                   5
               
 3  4 
                       cosyx 15
sin
3  4                                                    43
 2  5 3 
         
 12 
 12       
25 3

12
Assignment
#7-3
Pg. 442
14-18, 20, 26-28, 34, 36, 38

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