5 4 Trig Identities

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					Chapter 5 Trigonometric Equations
5.4




                               MATHPOWERTM 12, WESTERN EDITION   5.4.1
               Trigonometric Identities
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
                                            sin
                          Prove that tan       .
Recall the basic                            cos
trig identities:                        y     y x
       y                                    
sin                                   x     r r
       r                                      y r
       x                                     
cos                                         r x
       r                                      y
       y                                    
tan                                         x
       x
                                          L.S. = R.S.
                                                             5.4.2
Trigonometric Identities
     Quotient Identities
                  sin                        cos
           tan                      cot  
                  cos                        sin

     Reciprocal Identities
                    1                    1               1
           sin                cos            tan 
                  csc                 se c            cot

     Pythagorean Identities
      sin2 + cos2 = 1      tan2 + 1 = sec2       cot2 + 1 = csc2
      sin2 = 1 - cos2      tan2 = sec2 - 1       cot2 = csc2 - 1
      cos2 = 1 - sin2



                                                                    5.4.3
Trigonometric Identities [cont’d]
   sinx x sinx = sin2x

            1   cos2    1   cos2   1
   cos                 
          cos cos cos       cos


   sinA  cos A2  sin2 A  2sinAcos A  cos 2 A
                     1 2sinAcosA
    cos A
    sin A  cos A  sin A
      1     sin A     1
    sin A

           = cosA


                                                     5.4.4
               Simplifying Trigonometric Expressions
     Identities can be used to simplify trigonometric expressions.

Simplify.
                                                 cot2 
a)       cos  sin tan            b)
                                               1  sin2 
                       sin                  cos 2 
        cos  sin
                       cos                  sin2 
                                           
                  sin2                      cos 
                                                 2

        cos 
                  cos                         1
                                             cos2    1
         cos   sin 
             2        2
                                                   
                                            sin  cos2 
                                                2

               cos 
           1                                     1
                                          
         cos                                  sin2 

        sec                               csc2 

                                                                     5.4.5
Simplifing Trigonometric Expressions
                                                cscx
 c)     (1 + tan x)2 - 2 sin x sec x   d) tanx  cotx
                               1                   1
       (1  tanx)  2 sinx
                  2
                                         
                            cosx                 sinx
                                             sinx cosx
                               sinx                
       1  2 tanx  tan x  2
                        2
                                            cos x sin x
                               cosx
                                                    1
       1  tan2 x  2tanx  2 tanx              sinx
       sec2 x                              sin2 x  cos 2 x
                                               sin xcos x
                                                  1
                                         
                                                sinx
                                                  1
                                             sinx cos x
                                              1      sinx cos x
                                                 
                                            sinx         1
                                           cos x                 5.4.6
                        Proving an Identity
   Steps in Proving Identities
1. Start with the more complex side of the identity and work
   with it exclusively to transform the expression into the
   simpler side of the identity.
2. Look for algebraic simplifications:
         • Do any multiplying , factoring, or squaring which is
           obvious in the expression.
          • Reduce two terms to one, either add two terms or
            factor so that you may reduce.
3. Look for trigonometric simplifications:
          • Look for familiar trig relationships.
          • If the expression contains squared terms, think
            of the Pythagorean Identities.
          • Transform each term to sine or cosine, if the
           expression cannot be simplified easily using other ratios.
4. Keep the simpler side of the identity in mind.
                                                                   5.4.7
Proving an Identity
   Prove the following:

  a) sec x(1 + cos x)      = 1 + sec x

    = sec x + sec x cos x    1 + sec x
    = sec x + 1

                      L.S. = R.S.

   b)    sec x   = tan x csc x           c)     tan x sin x + cos x = sec x
                    sinx    1                   sinx sinx               secx
         secx                                               cosx
                    cos x sinx                  cosx      1
                      1                         sin2 x  cos 2 x
                                             
                    cosx                             cos x
                   secx                          1
                                              
                                                cosx
           L.S. = R.S.                         secx             L.S. = R.S.
                                                                        5.4.8
Proving an Identity
        d)              sin4x - cos4x                = 1 - 2cos2 x

                  = (sin2x - cos2x)(sin2x + cos2x) 1 - 2cos2x
                  = (1 - cos2x - cos2x)
                  = 1 - 2cos2x
                                                 L.S. = R.S.
                   1           1
         e)                          2 csc x
                                            2

               1  cosx 1  cosx
              (1  cosx)  (1  cosx)
                                      2 csc2 x
               (1  cosx)(1  cosx)
                    2
            
              (1  cos2 x)
                 2
             
               sin2 x
              2 csc2 x
                                   L.S. = R.S.                       5.4.9
Proving an Identity
          cos A    1  sinA
 f)                                        2 secA
        1  sinA     cos A
        cos2 A  (1  sinA)(1  sin A)
                                            2secA
              (1  sin A)(cos A)
        cos2 A  (1  2sin A  sin2 A)
      
              (1  sinA)(cos A)
        cos2 A  sin2 A  1  2sin A
      
              (1  sinA)(cos A)
            2  2sin A
      
        (1  sin A)(cos A)
           2(1  sin A)
      
        (1  sin A)(cos A)
           2
      
        (cos A)
       2secA                          L.S. = R.S.
                                                      5.4.10
            Using Exact Values to Prove an Identity
Consider sinx  1  cosx .
          1  cos x   sinx
a) Use a graph to verify that the equation is an identity.
                                                
b) Verify that this statement is true for x =       .
                                                6
c) Use an algebraic approach to prove that the identity is true
   in general. State any restrictions.
a)                                                         1  cosx
                                                             sinx
                                                        y
                                                           1  cosx
                                                             sinx




                                                                5.4.11
Using Exact Values to Prove an Identity [cont’d]
                                             
b) Verify that this statement is true for x = .
                                                6
           sinx         1  cosx
                    
         1  cosx         sinx                      Rationalize the
                                                  denominator:
           sin          1  cos
              6                 6                       1
                     
         1  cos                                     2 3
                  6       sin
                               6                         1       2 3
             1                                               
                              3                      2  3 2 3
             2         1
                3             2                       2 3
         1                                         
               2           1                           43
         1       2         2
                                                  2 3
         2 2 3  2  3  2
                            2      1
            1                                Therefore, the identity is
       
         2 3                                true for the particular
                      2 3                               
                                             case of  x .
       2 3                     L.S. = R.S.              6
                                                                      5.4.12
Using Exact Values to Prove an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
   in general. State any restrictions.
                                          Restrictions:
       sinx                1  cosx
                                        Note the left side of the
     1  cosx                sinx
                                         equation has the restriction
       sinx     1  cosx   1  cosx      1 - cos x ≠ 0 or cos x ≠ 1.
             
     1  cosx 1  cosx       sinx         Therefore, x ≠ 0 + 2 n,
     sinx(1  cosx)
                                          where n is any integer.
  
      1  cos2 x                         The right side of the
    sinx(1  cosx)                       equation has the restriction
                                        sin x ≠ 0. x = 0 and 
        sin2 x
                                         Therefore, x ≠ 0 + 2n
    1  cosx                             and x ≠  + 2n, where
  
      sinx                               n is any integer.

                     L.S. = R.S.
                                                                5.4.13
                  Proving an Equation is an Identity
     Consider the equation   sin2 A  1         1
                                         1         .
                           sin A  sinA
                              2
                                              sinA
 a) Use a graph to verify that the equation is an identity.
 b) Verify that this statement is true for x = 2.4 rad.
     c) Use an algebraic approach to prove that the identity is true
        in general. State any restrictions.
a)
                                                            sin2 A  1
                                                                1
                                                      y  1 2
                                                               A
                                                          sin sinAsinA




                                                                   5.4.14
Proving an Equation is an Identity [cont’d]
  b) Verify that this statement is true for x = 2.4 rad.
             sin2 A 1                  1
                                  1
           sin2 A  sinA              sinA
               (s in 2.4)2  1            1
                                  1
           (s in 2.4)2  sin2.4        sin 2.4

         = 2.480 466              = 2.480 466

                           L.S. = R.S.

       Therefore, the equation is true for x = 2.4 rad.




                                                           5.4.15
Proving an Equation is an Identity [cont’d]

 c) Use an algebraic approach to prove that the identity is true
    in general. State any restrictions.      Note the left side of the
     sin2 A 1                  1                equation has the restriction:
                          1                      sin2A - sin A ≠ 0
   sin2 A  sinA              sinA              sin A(sin A - 1) ≠ 0
                                                sin A ≠ 0 or sin A ≠ 1
  (s inA  1)(sinA  1)    1
                                1                               
                                               A  0,  orA 
      sin A(s inA  1)        sinA                              2
                                                Therefore,  0  2  n or
                                                            A
  (sinA 1)                                     A   + 2n, or

     sinA                                            
                                                A   2  n, wher e isn
  sinA     1                                         2
                                              any integer.
  sinA sinA
                                                The right side of the
        1                                       equation has the restriction
 1                                            sin A ≠ 0, or A ≠ 0.
      sinA
                                                Therefore, A ≠ 0,  + 2 n,
                                                where n is any integer.
                   L.S. = R.S.                                         5.4.16
Suggested Questions:
Pages 264 and 265
A 1-10, 21-25, 37,
   11, 13, 16
B 12, 20, 26-34




                       5.4.16

				
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