Module 5: Lesson 3

 

Self-Check 3
  1. Questions 4.c., 7, 11, 13, and 14 on pages 296 to 298 of the textbook
    1.  
      1. To solve this question, it will be necessary to make use of the Pythagroean theroem identities that were just discussed.

         
    1.  
      1. By using the Pythagorean theorem identity sin2 θ + cos2 θ = 1 and rearranging it, the expression 1−sin2 θ becomes cos2 θ.
    1.  
      1. The graphs of and y = csc x are shown.

        The graph shows the two equations y is equal to the cosecant of x , squared minus the cotangent of x, squared, all divided by the cosine of x and y is equal to the cosecant of x. The two graphs do not overlap. Thus, the two expressions do not form an identity.

        The graphs of and y = sec x are shown.

        The graph shows two equations: y is equal to the cosecant of x, squared, minus the cotangent of x, squared, all divided by the cosine of x and y is equal to the secant of x. The two graphs overlap. It appears from the graph that y is equal to the cosecant of x, squared, minus the cotangent of x, all divided by the cosine of x is equivalent to y is equal to the secant of x.

        Based on the graphs, it appears as though is equivalent to y = sec x.
      2. For the equation the non-permissible values will occur when either csc2 x, cot2 x, or sec x is undefined. It will also occur when cos x = 0.

         
        cos x = 0 at

         
        csc2 x is undefined at {x = nπ, n ∈ I, x ∈ R}.

         
        cot2 x is undefined at {x = nπ, n ∈ I, x ∈ R}.

         
        sec x is undefined at

        When the two sets of restrictions are put together, the resulting set of restrictions is
      3. While there is more than one way to rearrange only one method is shown.

         
    1.  
      1. After Stan’s substitution, the following answer is obtained:

         
      2. After Lina’s substitution, the following answer is obtained:

         


        The left side of the equation is equal to 1, but the right side of the equation is undefined.
      3. Lina’s substitution gives an undefined solution for tan x. Because one of the terms is undefined, this is a non-permissible value.
      4. After Giselle’s substitution, the following answer is obtained:

         


        She discovers that the two answers are not equivalent.

      5. The three students have enough information to determine that the equation given is not an identity. Once Giselle identified a permissible value for which the equation was not true, they could safely conclude that the equation was not an identity.
    2. While there are multiple ways in which this expression can be simplified, only one method is shown.

       


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