Module 5: Lesson 5

 

Self-Check 1

 

 “Your Turn” from “Example 1” on page 311 of the textbook

  1. The left side of the equation  can be rewritten as This means that both cos x and sin x cannot be equal to 0. The non-permissible values when cos x = 0 are , and the non-permissible values when sin x = 0 are . Thus, the non-permissible values for the identity are .
  2. The graphical solution may look like the one that follows.

     
    This is a graph of two sinusoidal functions. One function is y equals tangent x times cosine x divided by cosecant x. The other function is y equals one minus cosine squared x. The two functions appear as one curve on the graph because the two functions overlap.

    Because the two graphs are superimposed on one another, it appears as though the two graphs are identical, and therefore an identity. While this is a verification over the domain 0 ≤ x ≤ 2π, it is not a proof.

    Answers will vary depending on the value for x chosen to substitute. Verification when x = 60° is shown.

     
    LS RS



    The left side is equal to the right side, so the identity is verified for x = 60°.

  3. One possible proof of the identity is as follows.

     
    LS RS



    The left side is equal to the right side, so the identity is proven.


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