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Chapter 6 - Trig Identities and Conditional Equations_1_

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Chapter 6 - Trig Identities and Conditional Equations_1_ Powered By Docstoc
					        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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