Re: Secant/Cosecant Graphs-
The graphs of co-secant and secant graphs are both similar and different.
1. A co-secant graph is a shift of 90 degrees or pi/2 to the right of a secant graph.
2. The asymptotes for a secant graph cut through the center of the parabolic-structure of co-secant graphs.
3. The asymptotes for co-secant graphs cut through the center of a secant graph.
4. When a sine graph's x-value is zero, the x-value for a secant graph is also zero.
5. When the x-value for a cosine graph is zero, the x-value of a co-secant graph is also zero.
6. All odd intervals of pi have the secant graphs opening downward.
7. All odd intervals of 3pi/2 have co-secant graphs opening downward.
8. All even intervals of pi have secant graphs opening upwards.
9. All even intervals of 3pi/2 have co-secant graphs opening upwards
Question: Is there an algebraic way of modeling "all odd intervals of"
"all even intervals of"
Hint: It may involve k's and Z's
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I find it interesting that a sine graph's x-value is zero at the same time that a secant x-value is also at zero because generally you associate secant with 1/cos(x). Likewise, I find it interesting that we generally associate co-secant with sine, but they do not have matching x-values. Maybe this has something to do with the fact that secant does not experience a horizontal shift, meaning that its point at (0,1) matches the sine graph's point at (0,0), while cosine starts at its maximum point, (0,1).
ReplyDeleteThen again, I might be completely wrong, so don't hesitate to correct my reasoning if it's illogical.
ReplyDelete"All odd intervals of": x= ∈ℝ∣x≠(2k+1)(π/2) k∈z
ReplyDelete"All even intervals of": x= ∈ℝ∣x≠(2k)(π/2) k∈z
This pertains to graphs in which the intervals are(π/2). The last portion of the above expressions would change if the period of the cosecant (or secant) functions was different. These expressions (of domain) also apply to tangent and cotangent functions.
Some of the symbols might look unclear;ℝ= all real numbers (it's an R) and π= pi.
All of this is covered in our notes from Wednesday
Good observations Gunnar. I think by stepping back and looking at when you see the 0's and 1's it helps reinforce your understanding of all these relationships - reciprocal functions/phase shifts - and to relate to our most recent lesson, the difference between reciprocals and inverses. I also like how you "associate" Secant with Cosine, etc. These types of associations help out with proofs too. Everything all relates back to the three main functions sine cosine tangent - soh cah toa.
ReplyDeleteMolly, those even/odd representations using k's and referring to the set of integers (Z) was what was asked. Although it was after class, you were the first to answer that question so you get the E.C. Plus, I am very impressed by the symbol usage and curious how you got those characters. Just pasted? or did you find some way to type them. In any case, they show up clearly. Thanks for posting.
ReplyDelete