Module 5: Lesson 3

 

Self-Check 1
  1. Questions 1, 2, and 5 on page 296 of the textbook
    1.  
      1. For the expression the non-permissible values will occur when sin x = 0.
         
        sin x = 0 at {x = nπ, n ∈ I, x ∈ R}

        Thus, the non-permissible values are {x ≠ nπ, n ∈ I, x ∈ R}.
      2. For the expression the non-permissible values will occur when tan x = 0 and when tan x

        is undefined.

         

        tan x = 0 at {x = nπ, n ∈ I, x ∈ R}

         

        tan x is undefined at .

        When the two sets of restrictions are put together the resulting set of restrictions is

      3. For the expression the non-permissible values will occur when 1 − sin x = 0 and when cot x is undefined.

         
        1 − sin x = 0 at

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

        The two sets of restrictions can be combined and written as .
      4. For the expression the non-permissible values will occur when cos x + 1 = 0 and when tan x is undefined.

         
        cos x + 1 = 0 at {x = π + 2nπ, n ∈ I, x ∈ R}

         
        tan x is undefined at .

        The two sets of restrictions can be combined and written as .

    2. Some identities have non-permissible values because the identity contains trigonometric functions that have non-permissible values, or a trigonometric function is in the denominator of the identity.  For example, tan θ has non-permissible values of θ = 90° ± 180°n, where n ∈ I. Since  the non-permissible values occur when cos θ = 0; this happens when θ = 90° ± 180°n, where n ∈ I.

    1.  
      1. Substituting x = 30° in the equation gives the following:

         
        LS RS




        Using x = 30° in the equation yields the same answer on both sides of the equation. Based on this, the equation could be an identity.

        Note that this question could also be solved by using exact values. Substituting into the equation gives the following:

         
        LS RS




        Using in the equation yields the same answer on both sides of the equation. Based on this, the equation could be an identity.
      2. Based on the complexity of the denominator, it is likely that you would use graphing to determine the non-permissible values. The non-permissible values occur when tan x + cot x = 0 or when it is undefined, and when sec x is undefined.

         
        Over the domain 0° ≤ x < 360°, sec x is undefined at 90° and 270°.

         
        Over the domain 0° ≤ x < 360°, tan x + cot x is undefined at 0°, 90°, 180°, and 270°.


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